Advances in Quantum Chemical Topology Beyond QTAIM 0323908918, 9780323908917

Citation preview

ADVANCES IN QUANTUM CHEMICAL TOPOLOGY BEYOND QTAIM

ADVANCES IN QUANTUM CHEMICAL TOPOLOGY BEYOND QTAIM Edited By

JUAN I. RODRI´GUEZ Instituto Politecnico Nacional, Mexico

FERNANDO CORTE´S-GUZMA´N Universidad Nacional Auto´noma de Mexico, Mexico

JAMES S.M. ANDERSON Universidad Nacional Auto´noma de Mexico, Mexico

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-90891-7 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Susan Dennis Acquisitions Editor: Charles Bath Editorial Project Manager: Kathrine Esten Production Project Manager: Kumar Anbazhagan Cover Designer: Greg Harris Typeset by STRAIVE, India

Contributors Ricardo Almada-Monter Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico ´ lvarez-Gonzaga Escuela Superior de Omar A. A Fı´sica y Matema´ticas, Instituto Politecnico Nacional, Mexico City, Mexico

Mark E. Eberhart Department of Chemistry, Colorado School of Mines, Golden, CO, United States

James S.M. Anderson Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Evelio Francisco Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, Universidad de Oviedo, Oviedo, Spain

Paul W. Ayers Department of Chemistry, McMaster University, Hamilton, Canada

Marco Antonio Garcia-Revilla Department of Chemistry, Natural and Exact Sciences Division, University of Guanajuato, Guanajuato, Mexico

Alberto Ferna´ndez-Alarco´n Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Jose E. Barquera-Lozada Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Carlo Gatti CNR-SCITEC, Istituto di Scienze e Tecnologie Chimiche “Giulio Natta”; Istituto Lombardo, Accademia di Scienze e Lettere, Milano, Italy

Yoshio Barrera Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Rosa M. Go´mez-Espinosa Centro Conjunto de Investigacio´n en Quı´mica Sustentable UAEM-UNAM, Toluca, Estado de Mexico, Mexico

Giovanna Bruno Dipartimento di Chimica, Universita` degli Studi di Milano, Milano, Italy Pablo Carpio-Martı´nez Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Jos e M. Guevara-Vela Departamento de Quı´mica Fı´sica Aplicada, Universidad Auto´noma de Madrid, Madrid, Spain Jesu´s Herna´ndez-Trujillo Departamento de Fı´sica y Quı´mica Teo´rica, Facultad de Quı´mica, UNAM, Mexico City, Mexico

Julia Contreras-Garcı´a Sorbonne Universite, CNRS, Laboratoire de Chimie Theorique, Paris, France Fernando Cortes-Guzma´n Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico

Jesu´s Jara-Cort es Unidad Academica de Ciencias Ba´sicas e Ingenierı´as, Universidad Auto´noma de Nayarit, Tepic, Mexico

Aurora Costales Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, Universidad de Oviedo, Oviedo, Spain

Samantha Jenkins Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine

Emiliano Dorantes-Herna´ndez Escuela Superior de Fı´sica y Matema´ticas, Instituto Politecnico Nacional, Mexico City, Mexico

ix

x

Contributors

Utilization of Resources, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China Laurent Joubert Normandy University, COBRA UMR, Universite de Rouen, INSA Rouen, CNRS, Mont St Aignan Cedex, France Airi Kawasaki Department of Chemistry, Graduate School of Science, Tokyo Metropolitan University, Tokyo, Japan Steven R. Kirk Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine Utilization of Resources, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China Bruno Landeros-Rivera Departamento de Fı´sica y Quı´mica Teo´rica, Facultad de Quı´mica, UNAM, Mexico City, Mexico Cherif F. Matta Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, Nova Scotia, Canada Hector D. Morales-Rodrı´guez Escuela Superior de Fı´sica y Matema´ticas, Instituto Politecnico Nacional, Mexico City, Mexico Amanda Morgenstern Chemistry and Biochemistry Department, University of Colorado, Colorado Springs, CO, United States Aldo de Jesu´s Mortera-Carbonell Departamento de Fı´sica y Quı´mica Teo´rica, Facultad de Quı´mica, UNAM, Mexico City, Mexico Xing Nie Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine Utilization of Resources, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China

Eduardo Orozco-Valdespino Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico ´ Angel Martı´n Penda´s Departamento de Quı´mica Fı´sica y Analı´tica, Facultad de Quı´mica, Universidad de Oviedo, Oviedo, Spain David I. Ramı´rez-Palma Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico Toma´s Rocha-Rinza Instituto de Quı´mica, Universidad Nacional Auto´noma de Mexico, Mexico City, Mexico Juan I. Rodrı´guez CICATA-Queretaro, Instituto Politecnico Nacional, Queretaro, Mexico; Escuela Superior de Fı´sica y Matema´ticas, Instituto Politecnico Nacional, Mexico City, Mexico Shant Shahbazian Department of Physics, Shahid Beheshti University, Tehran, Iran David C. Thompson Chemical Computing Group, Montreal, QC, Canada Vincent Tognetti Normandy University, COBRA UMR, Universite de Rouen, INSA Rouen, CNRS, Mont St Aignan Cedex, France Ismael Vargas-Rodrı´guez Department of Chemistry, Natural and Exact Sciences Division, University of Guanajuato, Guanajuato, Mexico Timothy R. Wilson Department of Chemistry, Colorado School of Mines, Golden, CO, United States Tianlv Xu Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine Utilization of Resources, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China Yong Yang Key Laboratory of Chemical Biology and Traditional Chinese Medicine

Contributors

Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine Utilization of Resources, College of Chemistry

xi

and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China

C H A P T E R

20 Photochemistry: A topological perspective Marco Antonio Garcia-Revilla and Ismael Vargas-Rodrı´guez Department of Chemistry, Natural and Exact Sciences Division, University of Guanajuato, Guanajuato, M
exico

1. Introduction The study subjects of Photochemistry are the phenomena associated with the absorption and emission of electromagnetic radiation by chemical systems. Among such processes are the following [1]: • Spectroscopic phenomenon as fluorescence and phosphorescence. • Luminescent chemical reactions, as the combustion and bioluminescence, for instance the plankton luminescence and the flashlights of fireflies. • Photo promoted chemical reactions, as the photosynthesis [1] and the retinal reaction of the human vision [1]. • Nonlinear optical properties, related to the nonlinear behavior of the induced polarization of the charge distribution by the interaction of a substance with the electromagnetic radiation. This phenomenon has become of great importance in telecommunications, computing, optical devices, microscopy, among others. Photochemical reactions (PR) are chemical transformations in which an excited state is reached because of the incidence of electromagnetic radiation, in these reactions the excitation degree of reactants can be easily controlled using a monochromatic beam of the required wavelength. Hence, PR are preferred over the thermal reactions, also known as dark reactions. Besides, thermal reactions deal with a Maxwell-Boltzman distribution of ground state energies of translational, vibrational, rotational, electronic, and nuclear contributions, for this reason, the probability of causing a chemical reaction is lower than in the photochemical case [2]. In addition, there are many new synthetic routes to obtain organic compounds which are not possible to be synthesized by thermal reactions [3].

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00015-3

515

Copyright # 2023 Elsevier Inc. All rights reserved.

516

20. Photochemistry: A topological perspective

The Stark-Einstein law of photochemical equivalence stablishes that one light-quantum is absorbed by one molecule promoting a chemical reaction [4], and that is all, there is not a continuous molecular absorption of light during a photochemical transformation. The efficient absorption of a light-quantum causes an electronic excitation, for this reason the characterization of the ground and excited states is fundamental for the understanding of the effects of the electromagnetic radiation in a substance. An electronic excited state is a solution of the electronic Schr€ odinger equation with a superior energy than the ground electronic state [5], the quality of the description of the excited state is related to the chosen electronic structure method. The following are examples of processes in which excited states are reached from ground states: • Elemental particle bombing. • Alternate current or fire heating exposure. • Electromagnetic radiation exposure. The processes experienced by substances induced by electromagnetic radiation can be viewed as the succession of two phenomena: the excitation of the ground state reaching an excited state, and the deactivation of the excited state. Some photochemical processes return to the starting point, the ground state of the substance. Nevertheless, there are some chemical transformations caused by deactivation of excited states.

1.1 Excitation of the ground state An excited state is reached after the absorption of electromagnetic radiation, fulfilling the Frank-Condon principle [4] and the selection rules [4]. Fig. 20.1 shows the Jablonski diagram, which is a general scheme of the electronic structure related to photochemical processes [4]. The molecule initially relies in the ground state, let’s say a singlet state S0, above the S0 there are successive singlet excited states S1, S2, …, the consecutive numbering is related to the increasing order in energy. Nevertheless, some molecules hold ground states with different multiplicity, that is the case of the oxygen molecule in the gas phase at 1 atm and 25°C displaying a triplet ground state, 3Σ. The singlet excited states are extremely relevant for photochemical processes, several molecules of chemical interest display singlet ground states. Besides, selection rules stablished that the permitted excitations are those with the same Vibrational relaxation

Internal conversion

Vibrational relaxation

Cross between systems

Absorption Fluorescence External conversion Phosphorescence

FIG. 20.1

Jablonski diagram.

1. Introduction

517

multiplicity [4]. Meanwhile, excited states with multiplicities different than singlets are located above the ground state, among such excited states are the triplet states, T1, T2, …, which are of special interest because of the crucial role of triplet states in phosphorescence. For this reason, a commonly used Jablonski diagram display S0, S1, S2, …, and T1, T2, … electronic states. Besides, accordingly to Hund rules, the triplet states hold lower energy than the excited singlet states. In addition, the quantized vibrational structure is embedded in each electronic state, the vibrational normal modes are represented as sublevels of each electronic state.

1.2 Deactivation mechanisms of excited states Once an excited state is reached, deactivation of such excited state takes place. Deactivation processes can be classified as external (induced by collisions), nonradiative (isoenergetic) and radiative (luminescence) displayed in Scheme 20.1. Regarding external processes, the deactivation process could happen through: I. Vibrational relaxation (VR), in which the molecules in the S1 excited state vibrate and collide each other dispersing the energy of the system, arriving to the lower vibrational state of S1. The time to perform this process is about 10
12 s. II. An external conversion (EC), in which the molecule travels from the lower vibrational mode of S1 to the fundamental singlet state. III. Chemical Reaction (CR), in which the potential energy surface profile of the singlet and excited state changes, causing breaking and forming chemical bonds. CR deactivation processes, represented in Fig. 20.2, show a vertical excitation reaching an excited state with a vibrational energy large enough to overcome the activation energy to reach a secondary minimum in the excited state profile. This could drive changes in the connectivity (molecular graph), modifications in binding properties and consequently chemical

SCHEME 20.1

Deactivation processes classification.

518

FIG. 20.2

20. Photochemistry: A topological perspective

Energetic profiles of photochemical processes causing chemical reactions.

transformation. Fig. 20.2B shows the case in which a vertical excitation hits an excited state with no energy barrier, in such case the chemical transformation occurs straightforward. A nonradiative process takes place through an isoenergetic energy transference, such processes are classified as: I. Internal Conversion (IC), which happens through an isoenergetic transference from the S1 excited state to a high energetic vibrational mode of S0, followed by a full vibrational relaxation of the ground state. II. Intersystem Crossing (ISC), which is performed though an isoenergetic transference from a vibrational mode of S1 excited state to a vibrational mode of a tripled state, let’s say to the T1 triplet state. Radiative processes are classified as: I. Normal Fluorescence, which is observed when a full vibrational relaxation along the S1 vibrational modes is followed by a S1 ➔ S0 emissive transition. II. Resonant Fluorescence, which is observed through a S1➔ S0 emissive transition from the initial vibrational mode of S1 to the S0 ground state. Lifetime of Fluorescence processes is about 10
7 s. III. Phosphorescence, which is observed when a ISC is followed by a transition between electronic states of different multiplicity, T1 ➔ S0. Lifetime of T1 is about 10
5 s and the transition probability to S0 is low. This is the reason why this process is slower than the fluorescence. The radiative processes discussed above are the basis of the fluorescence spectroscopy in instrumental analysis [6], green fluorescent protein as in vivo marker in genetics [7], tumor markers in anticancer treatment [8], among others.

519

2. Theory

Another relevant phenomenon is the Chemiluminescence, which is the light emission caused by a chemical reaction forming a excited state followed by the deactivation of this excited state to the ground state [9]. The bioluminescence phenomena are those in which biochemical reactions are involved in the formation of the referred excited states, such reactions are commonly catalyzed by enzymes [10]. There are several applications of chemiluminescence, for instance, the luminol reaction in forensic chemistry, the understanding of luciferase activity in fireflies, the use of oxidation reactions of tetracene in light bars, among others.

2. Theory A semiclassical approach is useful to study the photophysical phenomena causing chemical transformations in a substance by the interaction with electromagnetic radiation. The mentioned approach deals with the substance in question using the quantum mechanics, meanwhile, the electromagnetic radiation is treated through the classic electromagnetism theory. From the wave-behavior approach, the electromagnetic radiation is defined as an electromagnetic oscillating field which propagates as a wave. The classic picture states that an electromagnetic radiation is produced when a charge q experiences acceleration, for instance, an alternate current circuit and the collision of a proton beam. The electric and magnetic fields generated by any charge and current distribution are determined by the Maxwell equations [4], ρ r  FðEÞ ¼ (20.1) E0 r  B ¼ 0, rF¼


(20.2)

∂B , ∂t

r  B ¼ μ0 J + μ0 0

(20.3) ∂F , ∂t

(20.4)

where F and B represent the electric and magnetic vectors, respectively; E0 and μ0 are the permittivity (E0¼ 8.85 * 10
12 C2N
1 m
2) and permeability (μ0 ¼ 1.25 * 10
6 NS2C
2) vacuum constants, respectively. The Gauss laws for electric and magnetic fields are presented in Eqs. (20.1) and (20.2). Eq. (20.1) shows that an electric field is produced by a charge distribution, meanwhile, Eq. (20.2) shows that the net magnetic flux of a space enclosed by a surface is zero. The consequence of this law is that there are not magnetic monopoles, at variance whit electric counterpart in which the electric monopole is possible. The Faraday electromagnetic induction law is placed in Eq. (20.3), which states that a temporary variation of a magnetic field generates an electric field. Finally, Eq. (20.4) is the Ampere Law, which indicates that an electric current and the variation of an electric field generates a magnetic field. The solutions of the four Maxwell equations are electromagnetic waves, in which the energy is transported through the space. The electric and magnetic field are perpendicular each other and oscillate during propagation forming a right angle with the propagation direction.

520

FIG. 20.3

20. Photochemistry: A topological perspective

Plane electromagnetic wave.

Electromagnetic plane waves (Fig. 20.3) are of special interest because its simplicity, they propagate in a single direction having a plane wavefront, the points of the wave packet are in the same phase forming a plane. In addition, the electric and magnetic fields oscillate each one in a plane as well, forming a linear polarized electromagnetic wave. The expressions for the electric and magnetic field of such a wave are FðEÞ ¼ iFx ¼ iF0x cos ðwt
kzÞ,

(20.5)

B ¼ jBy ¼ jB0y cos ðwt
kzÞ,

(20.6)

where F0x and B0y are the amplitudes of the electric and magnetic field, respectively, and w and k are the wave frequency and wave number, respectively. An extra simplification can be made, the spatial contribution of the electric field and the entire magnetic component can be neglected, FðEÞ ¼ iFx ¼ iF0x cos ðwtÞ,

(20.7)

for the case of interaction of electromagnetic waves with organic compounds. Such approximation is based on (a) the fact that the magnitude of the electric field is larger than the magnitude of the magnetic field. The correlation between electric and magnetic field magnitudes is given by F(E) ¼ cB, where c is the speed of light; and (b) the wave lengths of the electromag˚ , larger than the size of a molecular system. netic radiation are 3 orders of magnitude, 103 A Nevertheless, such approximation is not valid for metallic systems, in which the electric bands, and therefore the respective wavelengths, are close to the size of the metal. There are two correlated phenomena experienced by a molecule during the interaction with an electromagnetic wave: the vertical excitation, and the charge density polarization. QCT is a useful tool to study such phenomena. The quality of the wavefunction describing the ground and excited states is related to the choice of the electronic structure method to solve the electronic Schr€ odinger equation. There is a large variety of methods to address such issue. As a matter of fact, the multiconfigurational (MC) methods display the best agreement with experimental measurements.

2. Theory

521

Nevertheless, MC implementations scaling, and the resulting computational cost becomes prohibitive for the characterization of systems of chemical interest, however, MC results are used as a reference. For large systems with chemical interest, a way to afford this problem is using the Time Depending Density Functional Theory [11]. Two issues must be studied during a photochemical process: 1. Polarization of the charge density. 2. Transition probabilities between possible excited states.

2.1 Polarization The electric field, F, of the photonic incident beam interact with the substance charge distribution and produce the forces fi ¼ qiE [12], where q is the charge. The induced polarization during the electric field interaction is observed in Fig. 20.4A. The polarization can be defined as the change in the charge distribution in a substance due to the interaction with the electromagnetic radiation. Any modification in the charge configuration between nuclei and the electron density in a molecular system induces polarization (Fig. 20.4B). There are two types of polarization that can be observed: linear (Fig. 20.4), and nonlinear (Fig. 20.5). The linear polarization happens when the polarization behaves as a linear function of the electric field (Fig. 20.4C). Besides, induced polarization in macroscopic scale can be expressed as [12,13]: P ¼ χ ð1Þ F,

(20.8)

where P is the induced polarization, F is the electric field and χ (1) is the coefficient of linear susceptibility. Linear polarization is observed when a material interacts with a moderate electric field, nevertheless, when the intensity of the electric field is relatively large, the polarization could behave in a nonlinearly manner. Observe the nonlinear behavior in induced polarization as a function of time (Fig. 20.5A), modifications in the charge distribution (Fig. 20.5B) and in the induced polarization as a function of the applied electric field (Fig. 20.5C), in comparison with the linear counterpart.

FIG. 20.4

Linear polarization, (A) light beam electric field (solid curve) as a function of time, induced polarization (solid curve) as a function of time, (B) polarization of a material as a function of time, (C) linear induced polarization as a function of the applied electric field. Reprinted with permission from M. Pinnow, Materials for nonlinear optics: Chemical perspectives. ACS symposium series no. 455. Edited by S. R. Marder, J. E. Sohn and G. D. Stucky. ISBN 0-8412-1939-7. American Chemical Society, Washington, DC, 1991. US $ 129.95. Acta Polym. 44 (1993) 112–112. https://doi.org/10.1002/actp.1993. 010440216. Copyright 2022 American Chemical Society.

522

–

– CHARGE DISTRIBUTION +

+

– +

t0

t1

t2

INDUCED • POLARIZATION t0

t1

INDUCED POLARIZATION

ELECTRIC FIELD

20. Photochemistry: A topological perspective

t2

LINEAR

NONLINEAR

APPLIED FIELD

TIME

(a)

(b)

(c)

FIG. 20.5 Nonlinear polarization, (A) light beam electric field (solid curve) as a function of time, nonlinear-induced polarization (solid curve) as a function of time, (B) polarization of a material with nonlinear optical activity as a function of time, (C) nonlinear-linear behavior of second order induced polarization as a function of the applied electric field. Reprinted with permission from M. Pinnow, Materials for nonlinear optics: Chemical perspectives. ACS symposium series no. 455. Edited by S. R. Marder, J. E. Sohn and G. D. Stucky. ISBN 0-8412-1939-7. American Chemical Society, Washington, DC, 1991. US $ 129.95. Acta Polym. 44 (1993) 112–112. https://doi.org/10.1002/actp.1993.010440216. Copyright 2022 American Chemical Society.

The mathematical expression of the macroscopic nonlinear polarization is unknown. Nevertheless, the following Taylor series expansion can be used because the induced polarization is a continuous function of the electric field [13], P ¼ P0 + χ ð1Þ F + χ ð2Þ FF + χ ð3Þ FFF + …,

(20.9)

where χ (1), χ (2) and χ (3) are the susceptibility coefficients of first, second and third order, respectively. F is the electric field, P is the induced polarization and P0 is the charge distribution polarization in the absence of the electric field. Linear optical properties (refraction, reflection, and absorption, among others) are related to the magnitude of χ (1). Second order nonlinear optical properties (second harmonic generation, frequency mixing, amplification, electrooptic effect) are related to the magnitude of χ (2). Third-order nonlinear optical properties (third harmonic generation, frequency mixing, optic bio-stability) are related to the magnitude of χ (3), and so on. Some examples in which the nonlinear optical properties are applied in technology follow: lithium niobate third order properties in telecommunications [14], processing of optical signaling [15], optical devices [16]. Furthermore, in a biological system the phase changes are proportional to the radiative intensity changes, getting a clear cell imaging without the use of staining substances [17], this phenomenon is used to visualize the heat waves produced by materials. From the microscopic perspective, the quantum analog to the induced polarization is the induced dipolar moment μ [12,13] given by a Taylor series expansion as μ ¼ μ0 + αF +

1 1 βFF + γFFF…:, 2 6

(20.10)

where μ is the induced dipolar moment, μ0 is the dipolar moment in absence of the electric field F, α is defined as the polarizability, β is the first hyperpolarizability and γ is the second hyperpolarizability. Comparing expressions of macroscopic and microscopic frameworks, α, β and γ are the microscopic analogs to the first χ (1), second χ (2)and third χ (3) order susceptibility coefficients, respectively.

2. Theory

523

The application of the Quantum Chemical Topology to the characterization of the linear and nonlinear optical properties needs a deep understanding of the microscopic polarizabilities and hyperpolarizabilities. The energy of the interaction of a charge distribution ρ(r) with an electric potential Φ(r) can be calculated as follows [18]: ð E ¼ ρðrÞΦðrÞdr, (20.11) The electric field, defined as F ¼
∂Φ ∂r , is uniform at the molecular framework, for this reason the energy can be expressed by the multipolar expression 1 E ¼ qΦ
μF
QF0 …, 2

(20.12)

where q is the net charge or monopole, μ is the dipolar moment and Q is the quadrupole moment. Using the last equation, the dipolar moment can be expressed as μ¼


∂E , ∂F

(20.13)

In the absence of an external electric field, the dipolar and quadrupole moments are calculated from the wavefunction |ψi, μ0 ¼
hψjrjψ i,   Q0 ¼ ψjrrt jψ ,

(20.14) (20.15)

where r is the position vector and rt is its transpose. Moreover, the presence of an electric field conduces to an induced dipolar moment, which is expressed with Eq. (20.10). Besides, for a homogeneous electric field the total energy is expressed as a Taylor series, where the derivatives are evaluated in F ¼ 0.     ∂E 1 ∂2 E 1 ∂3 E 1 ∂4 E 2 3 EðFÞ ¼ Eð0Þ +  F + F + F + F4 …:, (20.16) ∂F F¼0 2 ∂F2 F¼0 6 ∂F3 F¼0 24 ∂F4 F¼0 this expression can be used to evaluate the induced dipolar moment as     ∂E ∂E ∂2 E 1 ∂3 E 1 ∂4 E 2 ¼

F
F
F3 …, μ¼
∂F ∂F F¼0 ∂F2 F¼0 2 ∂F3 F¼0 6 ∂F4 F¼0 Comparing with Eq. (20.10), the following expression are deduced:  ∂E μ0 ¼
 , ∂F F¼0  ∂ 2 E α ¼
2  , ∂F F¼0  ∂3 E β ¼
3 , ∂F F¼0  ∂4 E γ ¼
4 , ∂F F¼0

(20.17)

(20.18) (20.19) (20.20) (20.21)

524

20. Photochemistry: A topological perspective

2.2 Theory of transition probabilities The interaction of the electric field of an electromagnetic radiation can be studied as a perturbation of the time-independent Hamiltonian, such perturbation is a function of the position and time. This enables us to use the time-independent Schr€ odinger equation in the study b ð0Þ be the electronic of the mater-radiation interaction, a time-dependent phenomenon. Let H Hamiltonian of a molecule and ψ (0) n the wavefunction of the nth stationary state of such system, the result eigen value set of equations are b ð0Þ ðrÞψ ð0Þ ðrÞ ¼ Eð0Þ ψ ð0Þ ðrÞ, H n n n

n ¼ 1,2,3…,

(20.22)

E(0) n

is the electronic energy of the nth state. The zero super index stablishes the equivwhere alence of the electronic Hamiltonian with the unperturbed Hamiltonian of this approach, for (0) this reason ψ (0) n is the wavefunction of the nth unperturbed system and En the energy of the unperturbed nth state. The inclusion of the perturbation gets the Hamiltonian b 0 ðr, tÞ, b ðr, tÞ ¼ H b ð0Þ ðrÞ + H H

(20.23)

If the system initially is in the state ψ (0) n (r) and, over the effect of electromagnetic radiation represented by the perturbation, evolves to a different electronic state ψ (0) m (r) the transition probability Pn!m(t) is a relevant quantity to calculate, because the most probable transitions are those observed in a spectroscopic experiment. In first place, the solution of the time-dependent Schr€ odinger equation for this perturbed system is needed, using a wavefunction Ψ(r, t) the equation to solve is h ð0Þ i ∂Ψðr, tÞ b b ðrÞ + H b 0 ðr, tÞ Ψðr, tÞ, ¼ Hðr, tÞΨðr, tÞ ¼ H iħ (20.24) ∂t Expanding the wavefunction Ψ(r, t) as a linear combination of the unperturbed systems ψ (0) n , X ð 0Þ
iE t Ψðr, tÞ ¼ cn ðtÞψ ðn0Þ e n =ħ , (20.25) n

And solving for a particular m state with a small perturbation the transition probability is 2 ð 0 1  t b 0
iωmn t0 0  Pn!m ðtÞ ¼ 2  Hmn e dt  , (20.26)  ħ  0 D E ð 0Þ ð 0Þ b 0 ¼ ψ ðm0Þ jH b 0 ðr, tÞjψ ðn0Þ and ωmn ¼ Em
En . The perturbation considered in the interwhere H mn ħ action of electromagnetic radiation with mater is just the temporal contribution of the electric field, and the interaction of such electric field is given by: V int ¼
μ  F, where the μ is the electric dipolar moment of the molecule (μ ¼

P i

(20.27) qi ri) and F is the temporal

contribution to the electric field (F ¼ Fox cos (wt)). For this reason, in the framework of the dipolar approximation, the perturbative Hamiltonian is written as b 0 ðtÞ ¼
μx Fox cos ðwtÞ, H

(20.28)

3. QTAIM-photochemistry

where μx ¼

P

i q i xi .

525

Substituting this expression in the transition probability, we get D E2 2  ð0Þ ð0Þ    jFox j2  ψ m jμx jψ n  ð t0 0 0
iωmn t 0 cos ð wt Þe dt (20.29) Pn!m ðtÞ ¼  ,    0 ħ2

It is of special interest for the QCT community to compute the functional group contributions of such probabilities.

3. QTAIM-photochemistry 3.1 Polarizabilities and transition probabilities Bader and co-workers [19,20] address the partition of the polarizability, transition probabilities and dipolar moment of diatomic and polyatomic molecules through the Quantum theory of Atoms in Molecules. Such work is based on the perturbation theory approach, demonstrating that the polarizability, transition probabilities and dipolar moment can be expressed as a function of charge transference and polarization components. In addition, the shape and volume of the atomic basin is modified for the electric field because of the charge transference. The main conclusion of this work is that the total polarizability, transition probabilities and dipolar moment are determined by the atomic polarizability and the charge transference between topological atoms. Furthermore, is demonstrated that a functional group contribution to the total optical properties studied can be defined. A more detailed description of this contribution follows. Within the dipole approximation and considering a weak field, the transition Pn!k probability between n and k states is defined by Eq. (20.29), a simple rearrangement of the equation is 3 2 ð 0 2   t  0
iωmn t0 0  F cos ð wt Þe dt j j 7  ox D E2 6 7  0 6  ð0Þ ð0Þ  6 7 , Pn!k ðtÞ ¼  ψ m jμx jψ n  ∙6 (20.30) 7 ħ 4 5 where the time-independent contribution to the transition probability is the first term of the product, Pn!k. Following with the time-independent term, it can be written in the integral form "ð ! #2 N X ∗ Pn!k ¼ dτψ n
er i ψ k , (20.31) i

where (
eri) is the dipole operator of the ith electron and the sum is over all the N electrons of the system. Due to the multiplicative properties of the dipolar moment operator, the probability can be written as ð 2 Pn!k ¼ N dτψ ∗n ð
er Þψ k , (20.32)

526

20. Photochemistry: A topological perspective

where the integration is over all N electronic coordinates. Defining a transition density ð ∗ ∗ N dτ0ðψ n ψ k +ψ k ψ n Þ ρnk ðrÞ ¼ , (20.33) 2 where the primed integrand means an integration over all spin and spatial coordinates except for the spatial coordinates of electron 1. The probability is written as ð 2 Pn!k ¼ N drð
er Þρnk ðrÞ ¼ jMnk j2 , (20.34) where Mnk is the transition moment. Considering the unperturbed and first perturbative correction to the wave function, ψ (0) n and ψ (1) n , respectively, the expressions to the first and second energy corrections to an atomic basin, Ω, energy are ð ð  D E  ð 0 ð1Þ ð0Þ b 0 ð0Þ ð0Þ∗ b 0 ð0Þ En ðΩÞ ¼ ψ n jH jψ n ¼ N dr dτ ψ n H ψ n ¼ drHðn1Þ ðrÞ, (20.35) ð D E b 0 jψ ð1Þ Eðn2Þ ðΩÞ ¼ ψ ðn0Þ jH ¼ n

Ω

Ω

ð

Ω

Ω

  b 0 ψ ð1Þ + ψ ð1Þ∗ H b 0 ψ ð0Þ dr dτ0 ψ ðn0Þ∗ H n n n 2

Ω

ð ¼

Ω

drHðn2Þ ðrÞ,

(20.36)

(2) where H(1) n (r) and Hn (r) are the first- and second-order perturbation densities, respectively. This perturbation modifies the interatomic surface of an atom in a molecule, for this reason it is important to consider this shift in the surface, δS, in the evaluation of the second order energy. Whit this regard, the energy contribution to the second order energy due to the firstorder change density should be evaluated, þ Eðns2Þ ðΩÞ ¼ dSHðn1Þ ðrÞ δS, (20.37)

The first-order change in the density is evaluated by D E ð0Þ b 0 ð0Þ jψ n X ψ n jH ρðn1Þ ðrÞ ¼ ρnk ðrÞ, ½ En
Ek  k6¼n

(20.38)

Hence, the first polarizability, αn(r), using the dipolar approximation, considering the perb 0 ¼
eF  r, is written in terms of the first order change in the density as turbation H ð   r ψ ðn1Þ + ψ ðn1Þ∗b r ψ ðn0Þ , αn ðrÞ ¼
erρðn1Þ ðrÞ ¼
eN dτ0 ψ ðn0Þ∗ b (20.39) αn ðrÞ ¼
e

X

Mnk rρ ðrÞ, ½En
Ek  nk k6¼n

(20.40)

527

3. QTAIM-photochemistry

Integration of such polarizability in an atomic basin gives the atomic polarizability ð αbn ðΩÞ ¼ αn ðrÞdr, (20.41) Ω

The sum of all atomic first polarizabilities is the molecular polarizability X α n ðrÞ ¼ αbn ðΩÞ,

(20.42)

Ω

One of the main problems of the atomic polarizabilities, αn(r), is the origin dependence because of the position vector r. Nevertheless, to solve this origin dependence a substitution of r 5 rΩ +XΩ is used to build the following expression of the total first polarizability 2 3 ð h i X6 ð 7 X αðrÞ ¼ αp ðΩÞ + qð1Þ ðΩÞXΩ , (20.43) 4
e dr rΩ ρð1Þ ðrÞ
e dr ρð1Þ XΩ 5 ¼ Ω

Ω

Ω

Ω

where the atomic polarization is defined as ð

ð

αp ðΩÞ ¼
e dr rΩ ρð1Þ ðrÞ ¼
e dr rΩ Ω

k6¼n

Ω

ð

αp ðΩÞ ¼
e dr rΩ Ω

D E ð0Þ b 0 ð0Þ jψ n X ψ n jH ½En
Ek 

ρnk ðrÞ,

X

Mnk ρ ðrÞ, ½En
Ek  nk k6¼n

(20.44)

(20.45)

The integration of over the atomic basin gives α p ð ΩÞ ¼

X Mkn Mnk ðΩÞ ½ En
Ek 

k6¼n

,

where the atomic moment Mnk(Ω) is obtained with the following integration: ð Mnk ðΩÞ ¼
e dr rΩ ρnk ðrÞ,

(20.46)

(20.47)

Ω

Besides, it is possible to regroup the nuclear coordinate terms into contributions proportional to the charge transferred from an atomic basing (Ω) to a neighboring atomic basin (Ω0 ). With this regard, the atomic polarizability gets two contributions one that comes from the polarization, αp(Ω), and the second one by the charge transference between neighboring atomic basins, αc(Ω). X ½Xc ðΩjΩ0 Þ
XΩ qð1Þ ðΩ0 Þ, (20.48) αðΩÞ ¼ αp ðΩÞ + αc ðΩÞ ¼ αp ðΩÞ + Ω0

The transition probability density can be constructed in analogy with Eqs. (20.38) and (20.44), Pn!k ðrÞ ¼ Mnk Mkn ðrÞ ¼ Mkn ∙ð
erÞρnk ðrÞ,

(20.49)

528

20. Photochemistry: A topological perspective

where Mkn(r) is determined by the respective transition density. Once again, the substitution of r 5 rΩ +XΩ and subsequent integration gives the expression ð X

Pn!k ¼ dr Pn!k ðrÞ ¼ Mnk ðΩÞ + qnk ðΩÞXΩ , (20.50) Ω

Ð where Mnk(Ω) is the atomic transition dipole, and qnk ðΩÞ ¼
e dr ρnk ðrÞ is the induced atomic Ω

charge produced by charge density polarization. As in the case of the polarization, it is possible to solve the origin-dependent problem replacing the sum of nuclear coordinates by a set of terms describing the dipolar contribution to the charge, qnk(Ω), which is induced by the polarization between a neighboring atomic basin, Ω0 . " # X c 0 0 Pn!k ¼ Mnk ∙ Mnk ðΩÞ + ½X ðΩjΩ Þ
XΩ  qnk ðΩ Þ , (20.51) Ω0

At first order, the electronic polarizability is written in the form μ ¼ μ0 + αF,

(20.52)

μ
μ0 ¼ Δμ ¼ αF,

(20.53)

Which can be rearranged as

Considering a system of a system with nuclear charges Za and position vectors Xa the dipole moment is written as ð X μ¼ Za Xa
rρðrÞdτ, (20.54) a

Imposing the QTAIM partitioning the dipolar moment is written as X μ¼ ½qðΩÞXΩ + MðΩÞ ¼ μc + μp ,

(20.55)

Ω

where the XΩ is the position vector of the nuclear attractor of the topological atom Ω, μc and μp are the charge transfer and polarization terms, respectively. In addition, the atomic charge is defined as ð qðΩÞ ¼ ZΩ
ρ0 ðrÞdτ ¼ ZΩ
N ðΩÞ, (20.56) Ω

and the first moments M(Ω) are calculated with the following equation: ð MðΩÞ ¼
r Ω ρ0 ðrÞdτ, Ω

(20.57)

where the electronic position vector with origin at the nuclear attractor of the topological atom Ω is rΩ 5r 2 XΩ. Hence, the change in the dipolar moment of such system is calculated by ð X Δμ ¼ Za ΔXa
rρðn1Þ ðrÞdτF, (20.58) a

4. QTAIM applications in photochemistry

529

where the topological expression of μ and 4μ are independent of the choice of the origin for a neutral molecule. With the corresponding form using the QTAIM partitioning X Δμ ¼ (20.59) f½ΔqðΩÞX Ω + ZΩ ΔXΩ  + ΔMðΩÞg ¼ Δμc + Δμp , Ω

As in the case of Polarizabilities and transition probabilities it is possible to solve the origin dependence considering regrouping terms involving nuclear position vectors, as a result the dipolar moment origin independent is calculated as the sum of atomic dipolar moment contributions in the following manner: ( ) X X X c 0 0 μnk ¼ μnk ðΩÞ ¼ Mnk ðΩÞ + ½X ðΩjΩ Þ
XΩ qn ðΩ Þ , (20.60) Ω

Ω

Ω0

4. QTAIM applications in photochemistry A relevant correlation between the ellipticity and the photophysical properties was reported by Presselt and co-workers [21], where the relation between π-conjugation and excitation efficiency is used to study optical efficiency of Phenyl-Terpyridine Ru(II) and Zn(II) complexes (Fig. 20.6). Presselt and co-workers analyzed the correlation between π-conjugation, ellipticity and electronic transition energies of pyridyl-phenyl ligands of such complexes, the lowest transitions are related to a Metal-to-ligand charge transfer dominated by HOMO-LUMO excitations. The changes in the ellipticity are performed changing the substituent (R) at the para position of phenyl group. The conclusion was that the HOMO-LUMO energy gap decreases with increasing pyridyl-phenyl ellipticity, therefore it is possible to get a desirable photophysical properties tunning pyridyl-phenyl ellipticity. In 2012, J.N. Latosinska and co-workers [22] analyzed the photodegradation of nifedipine (NIF), an antihypertensive drug, such molecule is highly sensitive to the electromagnetic radiation and degrades to nitrosoniphelipine (NO-NIF), which is not an antihypertensive molecule but an antioxidative agent. In a combined experimental-computational study,

FIG. 20.6

Schematic representation of the ligands (tpy-ph-R, left), the homoleptic Zn(II) complexes ([Zn(tpy-phR)2]2+, middle), and the heteroleptic Ru(II) complexes ([(tert-butyl-tpy)Ru(tpy-ph-R)]2+, right) as well as the various substituents R in the 4-ph position. Reprinted with permission from M. Presselt, B. Dietzek, M. Schmitt, S. Rau, A. Winter, M. J€ ager, et al., A concept to tailor electron delocalization: applying QTAIM analysis to phenylterpyridine compounds. J. Phys. Chem. A 114 (50) (2010) 15163–74. Copyright 2021 American Chemical Society.

530

20. Photochemistry: A topological perspective

Latosinska and co-workers detected nifedipine photodegradation products by means of 1H-14N Nuclear Quadrupole Double Resonance (NQDR). Besides, a DFT/QTAIM study was performed to validate the computational capability to predict such photodegradation products. The computational reactivity criterion was based on the topological QTAIM Charges and the Laplacian of the electron density at the bond critical pointr2ρ(r)bcp, and the electron density at the bond critical point ρ(r)bcp. A relevant observation is that r2ρ(r)bcp, ρ(r)bcp display the largest values at the bonds that break because of irradiation. Hence, r2ρ(r)bcp, ρ(r)bcp are proper descriptors of the bond susceptibility to be broken by irradiation. In a recent publication, Chavez-Calvillo and Herna´ndez-Trujillo studied the temporal evolution of the electron density topological properties in the QTAIM framework [23]. The molecular weave function is represented as a combination of electronic states, in agreement with the superposition of sates principle, ΨðX, x, tÞ ¼

n X

χ i ðX, tÞψ i ðx; XÞ,

(20.61)

i

where χ i(X, t) is the time-dependent nuclear wave packet, and χ i(X, t) is the ith electronic state. Ψ(X, x, t) is calculated through the solution of the time-dependent Schr€ odinger equation, at the same time the set of adiabatic states {ψ ai } is obtained together with their respective eigenvalues, {Eai }, and nuclear wave packets {Xai }. The time evolution of the average of some properties of the electron density is characterized by means of the QTAIM partitioning. The time evolution of the electron density itself can be expressed as ð n X ij χ i ðX, tÞ∗ χ j ðX, tÞρX ðrÞ, (20.62) hρðrÞi ¼ dX ij ij ρX(r)

where is the transition density between ith and jth states. In the same way, the time evolution of the atomic charge is calculated with the expression ð ð hqðΩÞiðtÞ ¼ ZΩ
N dr dτ0 jΨj2 , (20.63) ð hqðΩÞiðtÞ ¼ ZΩ
dX

Ω

n X

χ i ðX, tÞ∗ χ j ðX, tÞqX ðΩÞ, ij

(20.64)

ij

The reactive scattering of H+3 and the photodissociation of LiF is studied based on the time evolution of the atomic charges. An important conclusion is that the time evolution of the atomic charges is consistent with the adiabatic and diabatic nature of the studied processes.

5. Alternative topological approaches A molecular graph MG is defined as the set of critical points and flux lines connecting these critical points [24], among such set is the bond path which is the conjunction of bond critical points (BCP) and flux lines connecting such BCP with nonboring nuclei. The characterization of the evolution of the molecular graph along a chemical transformation is a commonly used tool to get insight of a chemical reaction. Besides, the quantity and type of critical points and

5. Alternative topological approaches

531

paths linking such critical points serve as a qualitative characterization of the connectivity in a molecule. Changes in molecular graphs along a chemical reaction display qualitative information about breaking and forming chemical interactions. In addition, in QTAIM it is possible to quantitatively measure the bond-path curvature (BPC) separating two bonded nuclei using the ratio BPC ¼

BPL
GBL , GBL

(20.65)

where BPL is the bond path length and GBL is the geometric bond length defined as the internuclear separation. Nevertheless, molecular graphs and BPC did not show significant changes during electronic state excitations, MG and BPC are qualitatively the same in the ground and excited states at a particular molecular configuration. For this reason, molecular graphs cannot be used to track the evolution of a molecule that experiences photochemical transformations, a similar situation happens for some chemical descriptors of QTAIM. For instance, the Laplacian of the electron density at the bond critical point (r2ρ(r)bcp) get an average of the total curvature of the electron density, collapsing in a scalar the information of the charge accumulation of each spatial direction. So, 2-D(MG) and 1-D (r2ρ(r)bcp) topological descriptors are unsuccess in the study of photochemical transformations, there are some approaches to solve this problem: the New Generation QTAIM and the stress tensor approaches. The New Generation QTAIM methods (NG-QTAIM) is presented as a re-interpretation of topological aspects of the chemical bonding, a “bond-path framework set” (B) is defined and it can be used to differentiate the topology of the electron density of excited states from the ground state. Such set is built by three linkages p, q and r, which are related to the eigenvalues of the hessian matrix e1, e2 and e3, respectively. The path r is the orthodox QTAIM bond path where e3 is the eigenvector with the positive eigenvalue, the direction of depletion of charge density. Furthermore, e1, e2 are the second and first preferred directions in which the charge is accumulated, respectively. The p and q paths are constructed as the collection of tip path points defined by the expressions pi ¼ ri + εi e1,i qi ¼ ri + εi e2:i

(20.66)

where ri is the ith point at the r path, εi is the ellipticity evaluated in ri, e1, i and e2, i are the eigenvectors of the accumulative charge density at ri. Such eigenvectors are weighted by the ellipticity εi, this permit to track the changes in the charge accumulation pattern not only at the BCP, but in all the bond path. There is no sense to define the path r in terms of the ellipticity because the eigenvector e3 is independent of the eigenvalues defining the ellipticity. This approach is an alternative to the characterization of the Laplacian of the electron density, in which the information of each curvature collapses at the moment of the evaluation of the trace of the hessian matrix at the BCP or at any point along the bond path. An important observation is that p, q and r paths are orthogonal each other because are constructed using orthogonal eigenvectors e1,i, e2,i and e3,i. The p and q are used to define the path lengths Xn
1   p
p , H∗ ¼ (20.67) i+1 i i¼1 H¼

Xn
1  q i¼1

i+1


qi ,

(20.68)

532

20. Photochemistry: A topological perspective

The path lengths are related to the behavior of the charge accumulation along the bond path. In the limit of zero ellipticity, H path length is equal to the bond path length of r. The simplicity of this approach is useful to characterize bonding in excited states of photochemical processes. An useful quantity to characterize the covalency of a chemical bond is the local energy density defined as [24]: Hðrb Þ ¼ Gðrb Þ + V ðrb Þ,

(20.69)

where G(rb) and V(rb) are the local kinetic and potential energy densities at the bond critical point, respectively. H(rb) < 0 of closed shell interactions (r2ρ(rb) > 0) show a covalent interaction, for the same closed shell interaction H(rb) > 0 means a lack of covalent character. A strong shared shell interaction displays r2ρ(rb) < 0 and H(rb) < 0, for example, such situation corresponds to CdC and CdH bonds.

5.1 Stress tensor Stress tensor connects, through Virial theorem, the quantum information and the classical picture of mechanical forces equilibria in a molecular system. Such forces are the Ehrenfest forces experienced by a system, this approach gives a justification of the small frequencies of the normal modes related to intramolecular rearrangements [24]. The characterization of the eigenvectors of the stress tensor is a useful tool to characterize the changes in the electron distribution during photochemical processes, such changes are not observed in traditional QTAIM approaches (molecular graph, Laplacian of electron density at the bond critical point, etc.). In the QTAIM framework, the stress tensor is defined as " # ! 1 ∂2 ∂2 ∂2 ∂2 0 σ ðrÞ ¼
+ 0

, (20.70)  γ ðr, r Þ 2 ∂ri ∂r0j ∂ri ∂rj ∂ri ∂rj ∂r0i ∂r0j 0 r¼r

0

where γ(r, r ) is the first order density matrix. The principal electronic stresses, Πxx, Πyy and Πzz, are obtained by the diagonalization of σ(r), which are equivalent to the stress tensor eigenvalues λ1σ , λ2σ and λ3σ related to the corresponding eigenvectors e1σ , e2σ and e3σ . A negative stress tensor eigenvalue λ1σ means a compression in the direction of the e1σ eigenvector. Besides, the stress tensor eigenvector trajectories Tσ(s) are constructed as the set of shifts dr(s) with step s at a BCP [25]. The ordered set of shifts in the eigenvector projection space Uσ are direction vectors defined by dr0 (s) ¼ {dr  e1σ , dr  e2σ, dr  e3σ }, related to the direction vectors in the real space dr(s) ¼ {dr  e1, dr  e2, dr  e3} which are related to the follow of the Intrinsic Reaction Coordinate (IRC) from the transition state to the related minima. The respective trajectory length in the stress tensor projection space Uσ is defined as X (20.71) Lσ ¼ jdr0 ðs + 1Þ
dr0 ðsÞj, s¼0

533

5. Alternative topological approaches

At the real space, the longitude of Tσ(s) is given by X l¼ jdrðsÞj,

(20.72)

s

Lσ ¼ 0 when the magnitude of BCPs shift and the direction of dr0 (s) are constant. Using NG-QTAIM, Huang and co-workers [26] studied the photoisomerization of the double bond of fulvene at the ground and excited states, with the objective of finding a topological representation of the chemical bond that differentiates the ground and excited states binding properties. Besides, an exhaustive exploration of the Potential Energy Surface (PES) was performed, connecting the Frank-Condon point with the conical intersections occurring through the torsional coordinate (TC) and the Bond Length Alternation (BLA). Those findings reveal that traditional QTAIM bond paths cannot differentiate between the binding properties of ground and excited states. Nevertheless, the bond-framework set B¼¼ [26] was successful to reveal binding differences, the path longitude H displays clear differences for both states in the deactivation of excited state through the BLA and TC see Fig. 20.7. Tian and co-workers [27] have analyzed the photochemical ring aperture of 1,3cyclohexadiene (CGD) at 1,3,5-hezatriene (HT), an archetypical ring rupture mediated by electromagnetic radiation relevant in biochemistry and in organic chemistry. One of the most relevant experimental fact is an observed ratio between CHD and HT of (70:30), Tian and co-workers proposed a rationalization of such ratio based on NG-QTAIM results. The calculated mechanism at the SSR/ωPBE/6-31G* show a ratio CHD/HT of (60:40), elemental steps of such mechanism are: The photoexcitation is a π-bonding π-antibonding electronic promotion of a S0 ground state (11A) to a S1 excited state (11B in the C2 symmetry);, where upon the 3.20 3.10

c2v C2-C6(S0) c2v C2-C6(S1)

3.00

H

2.90 2.80 2.70 2.60 2.50 –0.25 –0.20 –0.15 –0.10 –0.05 0.00 BLA

0.05 0.10

0.15

0.20

FIG. 20.7 The corresponding variations of length H of the eigenvector-following path. The variation of the BPL with the BLA coordinate is consistent with the variation of the H(rb) values in that the BPL decreases increase in the BLA coordinate showing the weakening of the C2–C6 BCP. Reprinted with permission from W.J. Huang, R. Momen, A. Azizi, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited states of fulvene, Int. J. Quantum Chem. 118 (2018) e25768. https://doi.org/10.1002/qua.25768.

534

20. Photochemistry: A topological perspective

nuclear wave-packet slides down the steep slope on the S1 PES and the character of the state changes from 11B to 21A (a π to σ * transition). After that, the nuclear wave-packet continues to move on the S1 PES and reaches an S1/S0 conical intersection (CI), where it switches to the S0 state. Having switched to the S0 state, the nuclear wave-packet splits into two branches, one traveling back to CHD and another propagating forward to HT. The latest gas phase experimental measurements yield 70:30 CHD:HT branching ratio, which is generally consistent with 60:40 ratio for the reaction in solution, see Fig. 20.8. The authors demonstrate that the dynamic correlation is extremely important to optimize the conical intersections. The analysis of the NG-QTAIM of C5dC6 bonds along the minimum energetic profile shows that, before reaching the conical intersection, H(rb), λ3σ and ε display values indicating a large covalent stability of the C5dC6 bond at the S0 state compared with the S1. The latter is a plausible explanation of the (70:30) CHD/HT experimental ratio. A similar example of the rationalization of photochemical process by NG-QTAIM is the reported by Bin and co-workers [28], in which the oxirane photochemical ring-opening is modeled and characterized by 3-D topological paths. Light-driven molecular motors are systems suitable to perform unidirectional rotary motion, in which a double bond photoisomerization is the driving force of such movement [29–32]. The photoexcitation breaks a π-bond, resulting in a rotary motion of a molecular moiety in the direction dictated by the properties of the potential energy surface of the return to the ground state. It is known that such photoisomerization occurs through a conical intersection S1/S0 [33–35]. Wang and co-workers have studied the topological properties of 3-[(2S)-2fluoro-2-methyl-1-indanylidene]-1-methyl-2-methyl-indole (F-NAIBP) along non adiabatic molecular dynamic trajectories [36,37]. This study deals with the characterization of two types of dynamic trajectories, Fast (F) and Slow (s). Topological and NG-QTAIM properties were calculated at conformations along F and S trajectories with the goal of get chemical insight relative to the speed of such dynamic trajectories. Wang conclude that the bond-path framework set, together with H and H* trajectories, are better descriptors than MG and BPC, providing a better explanation of the preference of S1 over S0 to activate the rotary molecular S2 (21A) S1 (11B)

C.I. S1 (11B)

S0 (11A)

C.I.

HT CHD

FIG. 20.8 Scheme of photochemical ring-opening reaction of CHD. Reprinted with permission from T. Tian, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited state of the ringopening of cyclohexadiene, Int. J. Quantum Chem. 119 (2019) e25862. Copyright 2021.

5. Alternative topological approaches

535

motor. In addition, there is a fluorine-hydrogen bond path (F13-H36) that is present only in the S dynamic trajectory, such interaction displays a covalent character that stuck the molecular rotor. Furthermore, the set of one of the main conclusions is that traditional 2-D molecular graphs and local descriptors are not sufficient to make clear correlations between topological properties and trajectory speed. Nevertheless, Wang present the stress tensor trajectory for dynamic trajectories of the reaction pathway, such stress tensor trajectory can be used to explain the different lifetimes of dynamic trajectories. Along the F trajectory, up the conical intersection, the components of the stress tensor in the real space are isotropic, this means that a spherical trajectory is expected with an efficient rotation. Meanwhile, at the S trajectory the t3 component (t3 ¼ e3σ  dr) is significantly larger than the rest. Besides, t3 show large positive values, which indicates important translational movements along the bond path additional to the torsion, this situation is interpreted as an inefficient rotatory movement. The Interacting Quantum Atoms Approach (IQA) is a modern tool to obtain a partition of the total energy of a molecular system, the contributions to the energy of a system in IQA hold great physical significance and are used to interpret chemical information of reactivity of molecules. Penda´s and co-coworkers have defined a partitioning of energy using the QTAIM partitioning of the real space through the density matrix formulation of the average of the total energy of a Quantum System [38]. In IQA, the total energy of a system can be expressed ΩΩ0 as a sum of intra-basing energy components, EΩ net, and an inter-basin term, Eint . X X 0 E¼ EΩ EΩΩ (20.73) int , net + Ω>Ω0

Ω

where the EΩ net includes the electronic kinetic energy, electron-nucleus attraction, and 0 electron-electron repulsion at the Ω basin. Meanwhile, EΩΩ int takes account of electron-nucleus, electron-electron, and nucleus-nucleus interactions between basins Ω and Ω0 [38]. In addition, the inter-basin terms can be separated in two, the classical and the exchange-correlation contributions. 0

0

0

ΩΩ ΩΩ EΩΩ int ¼ Eclass + Exc ,

(20.74)

0

EΩΩ classtakes account of Coulomb-like nucleus-nucleus and electron-electron potential, for 0 this reason it is used as the ionic contribution to a chemical interaction. In addition, EΩΩ xc is the quantum counterpart of an interaction, commonly related to a covalent-like contribution to a chemical bond. In addition, the deformation energy is defined as Ω Ω EΩ def ¼ Enet
Enet,0 ,

EΩ net, 0

(20.75)

where is the reference state, usually the isolated moiety of a molecular system. EΩ def becomes a useful quantity to evaluate the change in energy from a reference state to the studied one. In a recent contribution, Ferna´ndez-Alarco´n and co-workers [39] show the capability of IQA to reveal the effect of intermolecular interactions in excitation energies of molecular clusters, for the case of the solvatochormic effect in the water dimer. In the first place, authors reveal the changes in the interatomic surfaces and the dipolar moment of water monomer because of the S0 ➔ S1 excitation. Besides, such changes affect the energetic contributions,

536

20. Photochemistry: A topological perspective

EO
H reduce in magnitude, nevertheless, delocalization index OdH, DI(O,H), increases in xc magnitude. This is a not so common situation and is related to the strengthen of the zwitterionic character of the OdH interaction, which is a not normal covalent interaction. Besides, H…H EH…H reduces in magnitude because of the reduction of EO
H and EO
H int xc class , Eclass decreases H…H with a marginal stabilization of Exc . These results are a symptom of the weakening of the binding of a water molecule because of the excitation process. In other hand, the excited states of water dimer 21A0 and 11A00 were analyzed, which are respectively blue- and redshifted with respect to S0 ➔ S1 vertical excitation of the water monomer. In first place, the observed charge transfer from the hydrogen bond donor acceptor to the hydrogen bond donor increases during the successive excitation S0 ➔ 21A0 ➔ 11A00 . Besides, the analysis of the IQA components show that the blue- and red-shifts are related to the wakening and strengthen of 0 EΩΩ int , respectively, with respect to the ground state (Fig. 20.9). Finally, based on those IQA results, an analysis of the changes in dipolar moments during the excitation process, S0 ➔ 21A0 ➔ 11A00 is presented, which show the capability of IQA to describe the traditional picture of intermolecular interactions from a QCT perspective. Montilla and co-workers [40] developed a method for the partitioning of static polarization which independent or the origin. The polarizability is originated through an external perturbation, an electric field, therefore a real-space analysis displays some difficulties because some polarizability contributions are origin dependent. Bader and co-workers have purposed a solution of this problem based on the substitution of the position vector by r 5 rΩ +XΩ. An alternative to this procedure is to use the LoProp method purposed by Karlstrom [41], where the total nonrelativist energy is obtained by: ...H2O

H2O DEclass Eint (IQA)

–0.6 –0.8

...H2O

H2O DExc

21A¢

...H2O

H2O DEint

DEint (IQA) 0.36 Blue shift 0.44

–8 · 10–2

0.19

–0.96 –0.72 –0.24

–1.8

–0.85 –1.17 –1.0 Energy (eV)

–0.45 –0.4

11A²

–0.64 –2.0

0 0 0

S0

0.2

–1.0

Red shift 0.5

Energy (eV)

FIG. 20.9 (Left) Components in the intermolecular IQA interaction energies in the 21A0 , S0 and the 11A00 states of (H2O)2 at the ground state equilibrium geometry. The states are arranged in a descending order of EH2OH2O (indicated in purple) from top to bottom. (Right) Change in these contributions for the 21A0 int and the 11A00 excited states with respect to the ground state. The data are reported in eV. From A. Ferna´ndez-Alarco´n, J M. Guevara-Vela, J.L. Casals´ . Martı´n Penda´s, T. Rocha Rinza, Chem. Eur. J. 26 (2020) 17035. Copyright European ChemSainz, A. Costales, E. Francisco, A ical Societies Publishing.

537

5. Alternative topological approaches

EðFÞ ¼ Eð0Þ ðFÞ
b μðFÞ∙F,

(20.76)

Considering the derivative expression for polarizability,  ∂2 EðFÞ , αij ¼
∂Fi ∂Fj F¼0

(20.77)

It is possible to use the expression of energy of Eq. (20.75) to obtain:   ∂2 Eð0Þ ðFÞ ∂2 ðb μðFÞ∙FÞ + , αij ¼
 ∂Fi ∂Fj  ∂Fi ∂Fj F¼0

(20.78)

F¼0

where at the limit of electric field going to zero the second term takes the form    ∂b μðFÞj  ∂2 ðb μðFÞ∙FÞ ∂b μðFÞi  ¼ + ¼ 2αij ,  ∂Fi ∂Fj F¼0 ∂Fi F¼0 ∂Fj 

(20.79)

F¼0

Combining Eqs. (20.78) and (20.79),

 ∂2 Eð0Þ ðFÞ αij ¼  ∂Fi ∂Fj 

,

(20.80)

F¼0

Using the expression (20.80), polarizabilities can be partitioned in atomic contributions using the finite field approach to calculate the electric field perturbed zero order components of the polarizability tensor. To this purpose, Montilla uses the effective atomic energies from the Hirshfield partition scheme over the Hartree-Fock approximation. Where the total energy is calculated by the sum of atomic contributions, X E¼ εA , (20.81) A

where εA ¼ EAA +

1X EAB , 2 B6¼A

(20.82)

Using this definition of energy in the expression of polarizability, we have   X∂2 εð0Þ ðFÞ ∂2 Eð0Þ ðFÞ A ¼ αij ¼   , ∂Fi ∂Fj  ∂Fi ∂Fj  A F¼0

(20.83)

F¼0

where we the atomic contribution to the polarizability is defined as  ð0Þ ∂2 εA ðFÞ A αij ¼  , ∂Fi ∂Fj  F¼0

(20.84)

538

20. Photochemistry: A topological perspective

Eq. (20.84) is the origin-independent energy-based (OIEB) approximation. Such approximation was tested on (XHn) molecular systems, finding that it shows additivity and transferability, as well as predictability in the isotropic polarizability.

6. Final comment The Quantum Chemical Topological (QCT) approaches are extremely relevant to the understanding of photochemical processes, there are a collection of contributions discussed in the present chapter that are the proof of such asseveration. The tools and solutions to photochemical problems addressed by QCT give an opportunity to gain deep knowledge in the understanding of photochemical phenomena as: Polarization of charge density, Transition probabilities, Charge transfer, photodegradation, Conical intersections, hypsochromic and bathochromic effects. Nevertheless, there are many aspects of the photochemistry still unexplored by QCT, as is the case of hyper-polarizabilities, the many electron transition probabilities, among others. There is a great opportunity to contribute with the chemical insight gained by QCT approaches to the understanding of such complex processes, which are waiting to be revealed.

References [1] G.W. Castellan, Physical Chemistry, third ed., Addison-Wesley, Massachusetts, 1983. [2] R. Lomoth, S. Ott, Introducing a dark reaction to photochemistry: photocatalytic hydrogen from [FeFe] hydrogenase active site model complexes, Dalton Trans. 45 (2009) 9952–9959. [3] P. Kla´n, J. Wirz, Photochemistry of Organic Compounds: From Concepts to Practice, Wiley, 2009. [4] A.R. Rodriguez, J.Z. Ruma´n, Espectroscopia, Pearson Education, Madrid, 2004. [5] I.N. Levine, Quantum Chemistry, seventh ed., Pearson, Boston, 1974. [6] W.E. Moerner, D.P. Fromn, Methods of single-molecule fluorescence spectroscopy and microscopy, Rev. Sci. Instrum. 74 (8) (2003) 3597–3619. [7] J. Livet, T.A. Weissman, H. Kang, R.W. Draft, J. Lu, R.A. Bennis, J.R. Sanes, J.W. Lichtman, Transgenic strategies for combinatorial expression of fluorescent proteins in the nervous system, Nature 450 (7166) (2007) 56–62. [8] M.J. Koppe, O.C. Boerman, J.G. Oyen Wim, R.P. Bleichrodt, Peritoneal carcinomatosis of colorectal origin: incidence and current treatment strategies, Ann. Surg. 243 (2) (2006) 212–222. [9] E.A.G. Zagatto, C.C. Oliveira, A. Townshend, P.J. Worsfold (Eds.), Interaction of radiation with the flowing sample, in: Flow Analysis with Spectrophotometric and Luminometric Detection, Elsevier, Amsterdam, 2012, pp. 95–146 (Chapter 4). [10] P. Worsfold, A. Townshend, P. Colin, Bioluminescence, in: Encyclopedia of Analytical Science (Second Edition), second ed., Elsevier, Oxford, 2005, pp. 277–284. [11] D. Escudero, E.D. Laurent, D. Jacquemin, Time-dependent density functional theory: a tool to explore excited states, in: Handbook of Computational Chemistry, Springer International Publishing, Cham, 2017, pp. 927–961. [12] M. Pinnow, Materials for nonlinear optics: chemical perspectives. ACS symposium series no.455. Edited by S. R. Marder, J. E. Sohn and G. D. Stucky. ISBN 0-8412-1939-7. American Chemical Society, Washington, DC, 1991, Acta Polym. 44 (1993) 112, https://doi.org/10.1002/actp.1993.010440216. [13] Y.-R. Shen, The Principles of Nonlinear Optics, Wiley, 2003. [14] J.D. Yujie, Efficient generation of far-infrared radiation from a periodically poled LiNbO3 waveguide based on surface-emitting geometry, J. Opt. Soc. Am. B 28 (5) (2011) 977–981. [15] W.M.K.P. Wijekoon, K.-S. Lee, P.N. Prasad, J.E. Mark, Nonlinear Optical Properties of Polymers: Datasheet from ‘‘Physical Properties of Polymers Handbook’’, SpringerMaterials Springer Science+Business Media LLC, 2007.

References

539

[16] M.D. Levenson, N. Blombergen, Dispersion of the nonlinear optical susceptibilities of organic liquids and solutions, J. Chem. Phys. 60 (4) (1974) 1323–1327. [17] R. Porras-Aguilar, J.C. Ramirez-San-Juan, O. Baldovino-Pantaleo´n, D. May-Arrioja, M.L.A. Carrasco, M.D.I. Castillo, D. Sa´nchez-de-la-Llave, R. Ramos-Garcia, Polarization-controlled contrasted images using dye-doped nematic liquid crystals, Opt. Express 17 (2007) 3417–3423. [18] F. Jensen, Introduction to Computational Chemistry, Wiley, 2007. [19] K.E. Laiding, R.F.W. Bader, Properties of atoms in molecules: atomic polarizabilities, J. Chem. Phys. 93 (10) (1990) 7213–7224. [20] R.F.W. Bader, D. Bayles, G.L. Heard, Properties of atoms in molecules: transition probabilities, Chin. J. Chem. Phys. 112 (23) (2000) 10095–10105. [21] M. Presselt, B. Dirtzek, M. Schmitt, S. Rau, W. Andreas, M. J€ager, U.S. Schubert, J. Popp, A concept to tailor electron delocalization: applying QTAIM analysis to phenyl-terpyridine compounds, J. Phys. Chem. A 114 (50) (2010) 13163–13174. [22] J.N. Latosi
nska, M. Latosi
nska, J. Seliger, V. Zagar, An innovative method for the non-destructive identification of photodegradation products in solid state: 1H-14N NMR-NQR and DFT/QTAIM study of photodegradation of nifedipine (anti-hypertensive) to nitrosonifedipine (potential anti-oxidative), Eur. J. Pharm. Sci. 47 (1) (2012) 97–107. [23] R. Cha´vez-Calvillo, J. Herna´ndez-Trujillo, Quantum molecular dynamics of the topological properties of the electron density: charge transfer in H3(+) and LiF, J. Phys. Chem. A 115 (45) (2011) 13036–13044. [24] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. [25] T. Xu, J. Farrell, R. Momen, A. Azizi, R.K. Steven, S. Jenkins, D.J. Wales, A stress tensor eigenvector projection space for the (H2O)5 potential energy surface, Chem. Phys. Lett. 667 (2017) 25–31. [26] W.J. Huang, R. Momen, A. Azizi, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited states of fulvene, Int. J. Quantum Chem. 118 (2018), e25768, https://doi.org/10.1002/qua.25768. [27] T. Tian, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited state of the ring-opening of cyclohexadiene, Int. J. Quantum Chem. 119 (2019), e25862, https://doi.org/10.1002/qua.25862. [28] X. Bin, A. Azizi, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the photochemical ring-opening reactions of oxirane, Int. J. Quantum Chem. 119 (2019), e25957, https://doi. org/10.1002/qua.25957. [29] K. Nagatoshi, W. Zijlstra Robert, A. van Delden Richard, H. Nobuyuki, L. Feringa Ben, Light-driven monodirectional molecular rotor, Nature 401 (6749) (1999) 152–155. [30] D. Pijper, M.G.M. Jongejan, A. Meetsma, B.L. Feringa, Light-controlled supramolecular helicity of a liquid crystalline phase using a helical polymer functionalized with a single chiroptical molecular switch, J. Am. Chem. Soc. 130 (13) (2008) 4541–4552. [31] D. Pijper, R.A. Van Delden, A. Meetsma, B.L. Feringa, Acceleration of a Nanomotor: electronic control of the rotary speed of a light-driven molecular rotor, J. Am. Chem. Soc. 127 (50) (2005) 17612–17613. [32] T.F. Landaluce, G. London, M.M. Pollard, P. Rudolf, B.L. Feringa, Rotary molecular motors: a large increase in speed through a small change in design, J. Org. Chem. 75 (15) (2010) 5323–5325. [33] M. Ruckenbauer, M. Barbatti, T. M€ uller, H. Lischka, Nonadiabatic excited-state dynamics with hybrid ab initio quantum-mechanical/molecular-mechanical methods: solvation of the pentadieniminium cation in apolar media, J. Phys. Chem. A 114 (25) (2010) 6757–6765. [34] A. Kazaryan, Z. Lan, L.V. Sch€afer, W. Thiel, M. Filatov, Surface hopping excited-state dynamics study of the photoisomerization of a light-driven fluorene molecular rotary motor, J. Chem. Theory Comput. 7 (7) (2011) 2189–2199. [35] J. Canyard, K. Addison, I.A. Heisler, A. Cnossen, W.R. Browne, B.L. Feringa, S.R. Meech, Ultrafast dynamics in the power stroke of a molecular rotary motor, Nat. Chem. 4 (7) (2012) 547–551. [36] L.L. Wang, A. Azizi, R. Momen, T. Xu, S.R. Kirk, Next-generation quantum theory of atoms in molecules for the S1/S0 conical intersections in dynamics trajectories of a light-driven rotary molecular motor, Int. J. Quantum Chem. 120 (2020), e26062, https://doi.org/10.1002/qua.26062. [37] L. Wang, G. Huan, R. Momen, A. Azizi, T. Xu, Kirk, S.R. Kirk, M. Filatov, S. Jenkins, QTAIM and stress tensor characterization of intramolecular interactions along dynamics trajectories of a light-driven rotary molecular motor, J. Phys. Chem. A 121 (25) (2017) 4778–4792.

540

20. Photochemistry: A topological perspective

[38] M.A. Blanco, A.M. Penda´s, E. Francisco, Interacting quantum atoms: a correlated energy decomposition scheme based on the quantum theory of atoms in molecules, J. Chem. Theory Comput. 1 (6) (2005) 1096–1109. ´ .M. Penda´s, T.R. Rinza, [39] A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. Casals-Sainz, A. Costales, E. Francisco, A Photochemistry in real space: batho- and hyspochromism in the water dimer, Chem. Eur. J. 26 (2020) 17035. [40] M. Montilla, J.M. Luis, P. Salvador, Origin-independent decomposition of the static polarizability, J. Chem. Theory Comput. 17 (2) (2021) 1098–1105, https://doi.org/10.1021/acs.jctc.0c00926. [41] L. Gagliardi, L. Roland, G. Karlstr€ om, Local properties of quantum chemical systems: The LoProp approach, J. Chem. Phys. 121 (2004) 4494.

C H A P T E R

1 Introduction to QTAIM and beyond Fernando Cort
es-Guzma´na,*, Juan I. Rodrı´guezb,c,*, and James S.M. Andersona,* a

b

Instituto de Quı´mica, Universidad Nacional Auto´noma de M
exico, M
exico City, M
exico CICATA-Quer
etaro, Instituto Polit
ecnico Nacional, Quer
etaro, M
exico cEscuela Superior de Fı´sica y Matema´ticas, Instituto Polit
ecnico Nacional, M
exico City, M
exico *Corresponding author.

1. Introduction The quantum theory of atoms in molecules (QTAIM), introduced by Richard Bader and co-workers in the 1970s, is a methodology that provides unambiguous definitions for important concepts like a chemical bond, chemical structure, local or atomic properties in a system, etc. These definitions are expressed through mathematical formulae, where the electron density plays a key role [1,2]. Several of Bader’s QTAIM concepts have been expanded in several directions. This evolutionary process continues even now. In the second section of this chapter, the fundamentals of QTAIM are didactically introduced. The extensions and/or generalization of some important QTAIM concepts are discussed in the third section. The fourth section contains information regarding how to formulate a proper quantum substem and that a QTAIM atom satisfies the relations presented.

2. QTAIM 2.1 The electron density topology The Quantum Theory of Atoms in Molecules is a theoretical approach to understanding chemical structure and reactivity on the basis of the molecular or solid electron charge distribution, ρ(r), which is an expectation value of a Dirac observable, which can be acquired either by theoretical (Eq. 1.1) [1,2] or experimental (Eq. 1.2) [3] means

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00021-9

1

Copyright # 2023 Elsevier Inc. All rights reserved.

2

1. Introduction to QTAIM and beyond

ð ð δðr  ri Þ ¼ N ⋯ δðr  r1 Þdw1 dx2 dx3 …dxN jΨðr1 , w1 , x2 , x2 , x3 …xN Þj2 i¼1 ð ð ¼ N ⋯ dw1 dx2 dx3 …dxN jΨðr, w1 , x2 , x2 , x3 …xN Þj2

ρð r Þ ¼

N X

ρð r Þ ¼

1X FðHÞ exp ð2πiH  rÞ 2 H

(1.1) (1.2)

In Eq. (1.1), N is the number of electrons in the system, xi ¼ (ri, ωi) denotes jointly the spatial (ri) and spin (ωi) coordinates of electron i, and Ψ(r1,w1, x2, x3, …, xN) is the state vector of the system. In Eq. (1.2), F(H) are the structure factors and V is the unit-cell volume; the H’s are indices denoting a particular scattering direction corresponding to a crystal plane during a crystallographic experiment. A basic analysis of the electron density can be performed by exploring its topography via the contour, profile, relief, and envelope plots (Fig. 1.1). The origin of this analysis might be traced to a London paper in 1928 where he described the features of the electron density at the homopolar binding [4]. In this way, one can identify regions with large electron density values around the nuclei or between two neighboring nuclei. The topographic analysis is frequently also performed on deformation density (Δρ), which is a field obtained from the subtraction of the promolecuar density from the molecular density. In this way, one can recognize the redistribution of electron density when a molecule is formed from isolated atoms. The next step in the analysis is the topological one.

FIG. 1.1 Topography approach to benzene molecule. Relief (top), envelope (bottom left), and contour (bottom right) plots.

3

2. QTAIM

To characterize the molecular electron density, quantum chemical topological analysis is performed (further details in Section 3). Quantum chemical topological analysis involves the characterization of ρ using its derivatives. —ρ is a vector field with paths attracted, primarily, to the nuclei filling the entire space [5]. The first element of the topological analysis is the determination of the critical points (CP), rc, where the gradient vanishes, —ρ(rc) ¼ 0. The role of each CP in the molecular structure is determined by their local curvature described by the Hessian matrix (Eq. 1.3), a square matrix that contains the second-order partial derivatives of ρ, whose diagonalization produces a set of eigenvectors and eigenvalues that obeys the matrix Eq. (1.4). The eigenvalues define whether a critical point is a minimum, a maximum, or a saddle point. By convention the eigenvalues are ordered so that λ1 < λ2 < λ3. It also determines whether a set of gradient paths originate or terminate at a CP. How to classify and interpret a CP is summarized in Table 1.1. 3 2 2 ∂ρ ∂2 ρ ∂2 ρ 6 ∂x2 ∂x∂x ∂x∂z 7 7 6 6 ∂2 ρ ∂2 ρ ∂2 ρ 7 7 6 T (1.3) rr ρðrÞ ¼ 6 7 6 ∂y∂x ∂y2 ∂y∂z 7 7 6 4 ∂2 ρ ∂2 ρ ∂2 ρ 5 ∂z∂x ∂y∂y ∂z2   rrT ρðrÞ ui ¼ λi ui with ði ¼ 1, 2, 3Þ (1.4) A nuclear attractor is a maximum in all directions. Particularly, this is a point where the gradient paths at nearby points terminate. In general, it coincides with the nuclear position, but in some special systems, an attractor appears in a place where a nucleus is not located. This phenomenon is called a nonnuclear attractor [1,6,52]. Bond critical points (BCPs) are characterized by two negative and one positive curvature, indicating that there are two steepest ascent trajectories of — ρ(r), which originate at this critical point and finish at the TABLE 1.1

Topological features of electron density.

Type

Curvatures

Role

Interpretation

Total number denoted

Maximum (3,3)

λ1 < 0, λ2 < 0, λ3 < 0

Gradient path attractor in three directions

Nuclear attractor Nonnuclear attractor

N+M (nuclear attractors plus nonnuclear attractors)

Saddle point (3,1)

λ1 < 0, λ2 < 0, λ3 > 0

Gradient path attractor in two directions and gradient path source in one direction, internuclear one

Bond critical point

B

Saddle point (3,+1)

λ1 < 0, λ2 > 0, λ3 > 0

Gradient path attractor in one direction and gradient path source in two directions, ring plane

Ring critical point

R

Minimum (3,+3)

λ1 > 0, λ2 > 0, λ3 > 0

Gradient path source in three directions

Cage critical point: C

C

4

1. Introduction to QTAIM and beyond

FIG. 1.2 Molecular graph for benzene. Atomic volumes delimited by the interatomic zero-flux surfaces (green lines). The contour lines (left-hand side) and steepest ascent paths (right-hand side) of the electron density (Eq. 1.1) are also shown in purple.

neighboring nuclei defining a topological object called bond path. The density value at the BCP is also the isovalue of the last contour that connects the two neighboring nuclei. The set of bond paths that define the connectivity within a molecule is called a molecular graph. On the other hand, in the third direction the trajectories of — ρ(r) that terminate at a BCP define another topological object called interatomic surfaces (IAS) which satisfy the zero-flux condition and act as boundaries between atoms (the IAS are also called the zero-flux surfaces, see Section 2.3). In this way, QTAIM defines that two atoms sharing an IAS and a bond path are bonded to each other. Finally, QTAIM defines a topological atom as the basin containing a nuclear attractor delimited by a set of interatomic surfaces. A Ring Critical Point (RCP) is characterized by two positive Hessian eigenvalues associated with two — ρ(r) trajectories which originate at this critical point and finish at the nuclei and BCP involved in a ring structure, defining the ring plane. Finally, a Cage Critical Point (CCP) is a minimum in all directions (that is, a local minimum of ρ), with — ρ(r) paths filling the space of the volume of a cage structure (Fig. 1.2). The critical points obey an important relationship known as the Poincar
e-Hopf relationship. It states that (N + M)  B + R  C ¼ 1 for molecules and the related Morse relationship for periodic systems (N + M)  B + R  C ¼ 0. They are useful in that if the relationships are not satisfied then there are more critical points to be determined.

2.2 Descriptor functions and the chemical bond Several “descriptor” functions can be calculated at each BCP to understand the nature of an interaction. Some local properties are computed directly from its Hessian matrix eigenvalues and others from the contributions of the molecular orbitals to that real space point [7]. The former is related to the way that the electron density behaves at that point. The sum of the eigenvalues gives the Laplacian of electron density (Eq. 1.5), r2ρ(r), describes how concentrated the density is at the BCP. r2ρ(r) < 0 indicates a density concentration in the bonding region associated with a shared interaction or covalent bond. r2ρ(r) around zero describes

5

2. QTAIM

polar interactions, whereas r2ρ(r) > 0 denotes a charge depletion associated with a close interaction as an ionic bond. Ellipticity is defined as E ¼ λ1/λ2  1, and describes the distribution of the density in the plane perpendicular to the bond path, where single and triple bonds give circular distribution with E around zero, whereas a double bond presents a larger distribution in one direction due to the π density, with E > 0. However, the r2ρ(r) and E values at the BCP are the basic descriptors to classify interactions. Sometimes it is more useful to analyze the profile of these quantities in the bonding regions to have the whole bond scenario, as shown in Fig. 1.3 for C–O diatomic molecule. In C–O diatomic, a small change in the BCP position can give a different interpretation about the nature of the bond. Sometimes the BCPs of similar interactions with small changes in their location will yield very different property values. r 2 ρð r Þ ¼

∂2 ρ ∂2 ρ ∂2 ρ + 2 + 2 ∂x2 ∂y ∂z

(1.5)

At each point in space, it is possible to calculate other scalar fields, as the energy densities, where the local virial theorem links the Laplacian of the electron density to the kinetic energy density G(r) (everywhere positive) and the potential energy density V(r) (everywhere negative) [9]. ð 1 2 1 2G + V ¼ r ρðrÞ GðrÞ ¼ N dτrΨ⁎  rΨ (1.6) 4 2 N

N 10 9 8 7 6 5 4 3 2 1 0

0

0.5

1

1.5

2

C

2.5

10 9 8 7 6 5 4 3 2 1 0

0

O

0.5

1

1.5

2

Li

2.5

10 9 8 7 6 5 4 3 2 1 0

10

10

10

5

5

5

0

0

0

–5

–5

–5

–10

0

0.5

1

1.5

2

2.5

–10

0

0.5

1

1.5

2

2.5

–10

0

F

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2 3.5 3 3.5

FIG. 1.3 ρ(r) and r ρ(r) profiles within the internuclear region of the diatomic molecules N–N (left) C–O (center) and Li–F (right). 2

6

1. Introduction to QTAIM and beyond

TABLE 1.2 Classification of the chemical interactions based on the bond critical point properties [8]. ρ(rbcp)

—2ρ(rbcp)

G(rbcp)/ρ(rbcp)

H(rbcp)/ρ(rbcp)

Open shell

Large

≪0

1

≪0

Closed shell

Small

>0

>1

>0

Open shell

Small

Around zero

1. The G(r)/ρ(r) ratio increases as the interatomic surface lies closer to an atomic core, thus it grows with the bond order in homopolar interactions due to the smaller internuclear separations. Analogously, G(r)/ρ(r) is large in polar interactions because the interatomic surface is particularly penetrated into the atomic core of the electropositive atom [8]. The different bonding scenarios are summarized in Table 1.2 as described by the local properties ρ, r2ρ(r), G(r)/ρ(r), and H(r)/ρ(r).

2.3 Integrated or atomic properties According to one of the quantum mechanics’ postulates, all the information (properties) of a quantum system, e.g., a molecule, can be obtained from the Schr€ odinger’s wave function Ψ [11]. The information obtained from Ψ allows for the computation of properties of the molecule as a whole through expectation values, which can be referred to as “global properties.” However, molecules (solids, liquids, etc.) are special quantum systems (QS) that are formed from other QS (atoms, functional groups, etc.). Chemists have always rationalized the properties of a molecule in terms of the properties of the constituent QS’s, i.e., functional groups, or local properties, which have resulted in a fruitful and didactic approach. A simple example: explaining the water’s dipole moment (global property) in terms of the partial charge of oxygen and hydrogen (local properties), gives a simple and clear picture of what is happening at the molecular level. Quantum mechanics, however, does not give an explicit method for computing local properties in a molecule though. One of the greatest contributions of QTAIM to quantum chemistry is that, among other features (vide infra), provides a precise mathematical way for computing the local (subsystem’s) properties in a molecule [1,12]. In this section, the precise QTAIM formalism for determining the local properties of a subsystem in a molecule is introduced. Within QTAIM, a volume is associated with each atom that forms a molecule. Let ΩA be the volume of atom A. The volume of atom A does not overlap with the volume of any other atom.

7

2. QTAIM

The property PA of atom A is computed by the integration of a corresponding density property over the atom volume ΩA [1,12]. ð PA ¼ pðrÞdr (1.7) ΩA

p(r) is the property density corresponding to PA [1,53]. Compared with the expectation value b of the molecule is computed as of the property P ð ð b ⋯ dr1 ⋯drN Ψ⁎ PΨ (1.8) where the integrations are carried over all space. An important feature of the QTAIM atom properties is the additivity. The sum of the atom property over all atoms is equal to the molecular property: X Pmolecule ¼ PA (1.9) A

Similarly, the property of a group of atoms (subsystem), e.g., a functional group, is equal to the property of those atoms that form the group, Pgroup ¼

group X

PA

(1.10)

A

For instance, the atomic charge is computed as (atomic units are used throughout), ð qA ¼ ZA  ρðrÞdr (1.11) ΩA

where ρ(r) and ZA are the molecule’s electron density and the atomic number, respectively. The charge of the molecule is described by Eq. (1.11), which is equal to zero for neutral molecules. X qmolecule ¼ qA (1.12) A

The QTAIM or Bader atomic volumes are regions in real space that enclose each atom, one volume per atom. Bader chose the boundaries of such atomic volumes as the surfaces SΩ that satisfies the so-called zero-flux condition, rρ  nSΩ ¼ 0

(1.13)

where ρ is the system’s electron density (Eq. 1.1) and n is a unit vector normal to the surface SΩ that encompasses the volume ΩA. Bader and coworkers justified the surface of zero-flux condition using Schwinger’s Principle of Stationary action but constrained to a volume. This is how a QTAIM volume is justified as a proper quantum subsystem (more details in Section 4). The union of the atomic volumes covers the whole space. Fig. 1.4 shows the atom volumes in benzene. ℝ 3 ¼ Ω1 [ Ω2 [ Ω 3 [ ⋯ [ ΩM

(1.14)

8

1. Introduction to QTAIM and beyond

FIG. 1.4 QTAIM atomic volumes for benzene.

Another atomic property frequently computed with QTAIM is the energy. The energy of atom A in a molecule, E(A), is determined by the atomic virial theorem which states that in an equilibrium geometry E(A) ¼  T(A), where T(A) is the electronic kinetic energy of atom A. ð ð ð   1 T ðAÞ ¼ KðrÞdr ¼  N dr dτ Ψ⁎ r2 Ψ + Ψr2 Ψ⁎ (1.15) 4 Ω Ω This result is based on the molecular virial theorem which implies that γ ¼ V/T ¼ 2, where V and T equals the total potential and kinetic energies for an equilibrium geometry [13–15]. In practice, the result γ ¼ 2 is in general not exactly satisfied due to the approximations used to solve the electronic problem (see Chapter 2, “An Introduction to Quantum Chemistry”). In order to tackle this deficiency, E(A) is determined by correcting it with a factor, E(A) ¼ (1  γ)T(A); the factor (1  γ) reducing to 1 when γ ¼ 2 [16]. Rodriguez et al. showed a way to compute the QTAIM atomic energies in the context of the density functional theory (DFT) approximation [14,15].

2.4 Attractors and basins: Canonical QTAIM atom As an analogy, imagine being in a landscape with hills and valleys. Imagine you are standing at a particular location and you move up in the direction that has the steepest incline. If you keep walking in this direction eventually you will come to a place where there is no more incline. Almost always when you continually step in this direction you will end up at the top of a mountain. Occasionally you will end up between the two mountains where two directions have the same incline. This could also happen when standing at a point surrounded by mountains. This analogy describes how an atom is typically defined in traditional QTAIM (from hereon called QTAIM), the zero-flux condition in Eq. (1.21). In QTAIM, instead of mountains the electron density is utilized. The direction of the steepest ascent in the direction of the gradient of the density. The steepest ascent paths starting from different points never cross except at critical points. Consider a maximum. Any point whose steepest ascent path terminates at that maximum is part of the attractor basin. This attractor basin is what is traditionally viewed as a QTAIM atom. Every point whose steepest ascent path terminates at a maximum belongs to the atom that includes that maximum. Other points belong to the boundaries between the atoms. Atomic properties in QTAIM are computed by integrating the property density over the basin of that atom. This idea of constructing the atomic surfaces was used by Rodriguez et al. [14,15,17].

3. Beyond QTAIM

9

By construction the boundary of the atom is a surface of zero-flux at every point. This is because the gradient is in the direction of steepest ascent on the surface and the outward normal is orthogonal to the surface. Therefore, if the surface is constructed in the direction of the gradient the flux is zero at every point and thus the zero-flux condition is satisfied (both Eqs. 1.19, 1.21). In general, a volume bound by a surface of zero-flux constructed as above will contain one maximum located at a nucleus. There are also some other unusual “atoms” bound by zero-flux surfaces as in Eq. (1.21). One peculiarity that can occur is that a maximum may not be centered at (or near) a nucleus. In fact, there may be no nucleus within the attractor basin. These attractor basins, atoms, are aptly named nonnuclear attractors (NNAs or 0-NAs) or pseudo atoms. Though it is possible for these to be artifacts in a calculation, there are real examples of these that have been found with highly accurate calculations and experimental measurements [18]. Another peculiarity that can occur is that nuclei occupy the same attractor, only one maximum. These are often referred to as fused atoms, unusual attractors, or multinuclear attractors (MNAs). The most commonly observed case is when a heavier atom and hydrogen atom are in near proximity, two nuclei within one basin, 2-NAs. Physically, the electron from the hydrogen is being pulled closer to the heavier atom resulting in one nucleus. HF+ is the most famous example, but there are others [19,20]. This topic is covered in detail later in the book. Another zero-flux surface that can be constructed that satisfies Eq. (1.21) that includes the nucleus on the boundary, which are explained in detail in the gradient bundles (spindle bundles) chapters in this book.

3. Beyond QTAIM The aim of the QTAIM approach to the molecular systems and chemical reactivity is to give a chemical interpretation to the local and integrated properties of related scalar fields. Within this theoretical framework, unambiguous definitions for chemical bonds, atoms, and chemical structures are given, wherein catastrophe theory allows defining structural change as well, which is fundamental to the breaking and formation of chemical bonds during a reaction mechanism. Nowadays, the bond critical points and bond paths are widely used to characterize chemical interactions to the extent that IUPAC has included these topological features in the definition of a hydrogen bond. In most circumstances, the bonding description offered by the topology of the electron density is concordant with that given by other bonding models such as MO analysis. Nonetheless, there are some instances in which the QTAIM results are not in agreement with those of different chemical bonding schemes. Most of the criticisms of the bonding interpretation of QTAIM come from the existence of unexpected bond paths or the absence of presumed bond paths that are not in accordance with classic bonding models. Distinct scalar fields, for example, the density of the Fermi hole and the source function, have been employed in an attempt to explain these apparent anomalies. However, unlike other theoretical approaches, QTAIM has an experimental counterpart. Coppens developed the multipolar refinement that enables the obtainment of experimental densities from X-ray diffraction [3]. The path traced by Richard Bader can be generalized to other scalar fields different from the electron density giving a complementary way to understand structure and reactivity.

10

1. Introduction to QTAIM and beyond

The quantum chemical topology (QCT) [21] is a general recipe to extract helpful chemical information from scalar and vector fields associated with physical properties through topographic and topological analysis. The general QCT recipe involves at least four steps. 1. The obtention of a scalar or vector field associated with a physical quantity. 2. The topographic or topological analysis of the field. 3. Generation of chemical concepts from local or integrated properties obtained in the previous step. 4. Application of these concepts to solve structure or reactivity problems. Some other fields analyzed by Bader’s group and several groups around the world are listed in Table 1.3 along with their mathematical definition and the concept produced from them. Also, other space and energy partitions are explored to produce alternative definitions of QTAIM concepts or to give a more detailed analysis [41]. Oviedo’s group developed an TABLE 1.3 Some scalar and vector fields studied within the quantum chemical topology. Field

Definition

Some produced concepts

Laplacian of electron density [22–24]

A measure of the concentration or depletion of electron density

Atomic graph Localization of acid/electrophilic and basic/ nucleophilic sites within a molecule

Reduced density gradient [25]

A dimensionless function, used to describe the deviation from a homogeneous electron distribution

Identification of noncovalent interaction

Electron localization function [26,27]

A measure of the likelihood of finding an electron in the neighborhood space of a reference electron located at a given point and with the same spin

Visualization of the core and valence electron, and also shows covalent bonds and lone pairs Localization of acid/electrophilic and basic/ nucleophilic sites within a molecule

Fermi hole [28,29]

It determines the decrease in the probability of finding an electron with the same spin as some reference electron, relative to the given position of the reference electron

Electron localization and delocalization indices Aromaticity measurements

Ehrenfest forces [30,31]

The interaction of an electron with the nuclei and the remaining electrons

Bonding, reactivity, and alternative partitioning to the surface of zero-flux of the electron density

Induced current density [32–34]

The density induced by a static and uniform magnetic field as a perturbation of the ground wave function The electron density times the electron velocity when the system is under an external magnetic field

Measurement of chemical shift, aromaticity and electron delocalization

Electrostatic potential [35,36]

The work carried out in bringing a test positive charge from infinity to the reference point r and has a connection with the energy

The sites of electrophilic attack, cation binding sites, anionic sizes and shapes, lone pairs, π bonds, aromaticity, substituent effects, lone-pair-π interactions

4. Mathematical fundamentals of QTAIM

TABLE 1.3

11

Some scalar and vector fields studied within the quantum chemical topology—cont’d

Field

Definition

Some produced concepts

Kinetic and potential energies densities [37,38]

There are several ways to define these quantities, and each approach uses the definition that gives more chemical information

Identify the polarization of the valence shell by the changes in the topology of energy densities

Fukui function [39,40]

A measure of the potential for the reactivity of molecules based on frontier orbitals

Identification of the reactive sites

alternative to the virial approach to atomic energy taking into account the energetic interaction between atoms, the interacting quantum atoms (IQA) [42]. This approach computes the total energy of a system as a sum of atomic self-energies and the interatomic interaction energies by identifying all of the two-atom contributions. According to this framework, the total energy of a molecule can be expressed as follows: X 1 X X AB E¼ EA E (1.16) net + 2 A A6¼B int A Heidar-Zadeh et al. presented an alternative partition approach of the molecular densities on the basis of the Hirshfeld’s stockholder partitioning [43] following the proposal of Nalewajski and Parr who derived the information-theoretic AIM by minimizing the Kullback-Leibler divergence between the AIM and isolated neutral proatoms [44] ! ρ0A ðrÞ (1.17) ρ ðrÞ ρA ð r Þ ¼ ρ0mol ðrÞ mol The essential features that unite all such methods analyzed by Heidar-Zadeh et al. are (a) a divergence measure for the dissimilarity between the densities of the QTAIM and their proatomic reference densities and (b) a definition for the density of a proatom, together with a procedure for refining/optimizing that density.

4. Mathematical fundamentals of QTAIM 4.1 Definition of an atom There are several definitions of an atom that fall under the umbrella of QTAIM. The most typical will be described first followed by other definitions. Two common elements in all the definitions are that a QTAIM atom is always a hard-wall atom, and it is always bound by a surface of zero flux. 4.1.1 Hard-Wall atoms and QTAIM A hard-wall model of the atom is one where properties are computed is the generalization of the definition by Bader. Instead of the volume associated atom A, ΩA, being bound by a surface of zero-flux the volume can be chosen arbitrarily provided that the volume of atom

12

1. Introduction to QTAIM and beyond

A does not overlap with the volume of any other atom and the union of all the atomic volumes covers the space. The sum of all the hard-wall atomic properties will yield the value of the full chemical system. The property of some group of atoms, for example, a functional group, can be computed by summing the atomic properties of that group of atoms. These atoms are usually chosen so that they border each other though this is not mandatory. The property of a hard-wall atom is carried out using ð ð b ⋯ dr1 ⋯drN Ψ∗ PΨ (1.18) V1

VN

where Vi is a volume that constrains the volume occupied by the electron at ri (* denotes complex conjugate). The volumes do not overlap for each hard-wall atom and the union of all the volumes covers the space. However, the most common form for a hard-wall atom is where only the electron r1 is constrained to a volume Ω. ð ð ð b ⋯ dr1 dr2 ⋯drN Ψ∗ AΨ (1.19) Ω

where this volume can be chosen arbitrarily with the caveat that any atomic volume Ωi, does not overlap with any other atomic volume Ωj. Since they do not overlap the sum of the volumes of each atom, Ωi, then the sum of the atomic contributions yields the result for the entire molecule, in particular, ðð ð Xð ð ð ^ ¼ ^ ⋯ dr1 dr2 ⋯drN Ψ∗ AΨ ⋯ dr1 dr2 ⋯drN Ψ∗ AΨ (1.20) i

Ωi

and, as a consequence, every point in space belongs to some atom or to the boundary between atoms. The atoms in QTAIM can be viewed as a special case of the Hard-Wall Atoms. QTAIM atoms are made up of volumes Ω that are bound by a surface of zero-flux, Eq. (1.13), so that (from the divergence theorem), ð drr2 ρðrÞ ¼ 0 (1.21) Ω

is satisfied for every atomic volume. This is justified by the principle of stationary action for a volume as treated by Bader and coworkers. They showed that Eq. (1.21) is the condition to form an atom as a proper quantum subsystem [45]. The Divergence Theorem (also known as Gauss’ Theorem) which states that the integral over a volume V of the divergence of a vector function of F(r) is equal to the surface integral F(r) dotted with the outward unit normal, n, to the surface, S, that encompasses the volume. Explicitly, ð ðð r  FðrÞdr5 FðrÞ  ndS (1.22) ÐÐ

V

S

where S denotes integration over a closed surface S of the volume V, where n is the outward normal to the surface. The right side of this equation describes the flow amount flowing per unit volume in and out of the surface. This is called the flux and this is the flux used in the QTAIM context. However, for the completeness for a general surface S the flux is defined as

4. Mathematical fundamentals of QTAIM

13

ðð flux ¼

FðrÞ  ndS

(1.23)

S

Notice then using Gauss’ theorem that the integral of the Laplacian of the density, r2ρ(r), is the flux of the gradient of the density, — ρ(r), over the surface of the volume Ω, the surface is denoted ∂ Ω. Explicitly, ð ðð 2 r ρðrÞdr50 ¼ rρðrÞ  ndS (1.24) Ω

∂Ω

From Eq. (1.24), it is sufficient that if Eq. (1.13) is satisfied then Eq. (1.21) is also satisfied. The right equality in Eq. (1.24) is also often referred to as the surface of zero-flux condition and has been explored in some calculations, where Eq. (1.24) is satisfied, but Eq. (1.13) is not satisfied [46,47]. Therefore, this can also be called the surface of the zero-flux condition and, in most cases, the surface of the zero-flux condition being referred to is Eq. (1.12). Eq. (1.12) could be referred to as the surface of zero-flux at every point because the flux on this surface is zero at every point. The definition of the zero-flux condition in Eq. (1.12) is not only aesthetically pleasing, but it is convenient from the practical standpoint of giving a method to determine the surface, the surface must coincide with the direction of — ρ(r). Furthermore, as described in the next section, the closed surfaces that satisfy Eq. (1.12) and contain one and only one attractor (one maximum, usually a nucleus) are not on the boundary of the surface, and all other critical points are on the surfaces producing a unique partitioning. As such, nearly all QTAIM calculations choose the sufficient condition in Eq. (1.12) where the flux is zero at every point on the surface.

4.2 The Bader-stationary action: A proper quantum subsystem One of the issues in quantum mechanics is to address how to find a proper quantum subsystem. The problem was how to assign the quantum mechanical contribution of an atom to the property of molecules and how to compute them. In other words, is there a way to characterize the observable properties of an atom within a molecule? To address this problem, Bader and coworkers supposed that the properties of an open system should be derived from a single variational principle, the principle of stationary action by Schwinger, for a hard-wall atom just as in the total system [48–50]. Analogously to the total system this single variational principle for a hard-wall atom should yield an equation of motion, a force law, and an atomic virial theorem for an atom in a molecule. Furthermore, the atomic equation of motion should have (preferably) the same form and physical content with the corresponding relationships in classical continuum mechanics. Schwinger postulated the action principle in one variational equation. His principle relates the variation of the transformation function, hq2, t2 j q1, t1i, which is the inner product of eigenstates of some Hermitian operators at different times, t1 and t2. The transformation function can be interpreted as the probability amplitude of the change from the state at t1, jq1, t1i to the state at t2, jq2, t2i via a unitary transformation. Schwinger stated that the variation of this transformation function is proportional to the inner product of hq2, t2 j with the variation of the action operator acting on the state jq1, t1i, namely

14

1. Introduction to QTAIM and beyond

D E ^ jq1 t1 δhq2 , t2 j q1 t1 i ¼ i=ħ q2 , t2 j δW

(1.25)

In other words, the variation of the transformation is proportional the inner product of the same two states with the action of a single infinitesimal transformation generator, the variab [2,51]. These transformations may be described by the unitary tion of the action operator, δW Fðt2 Þ which act on the eigenstates leading to the fundamental transformations b Fðt1 Þ and b operator relationship b ¼b δW Fðt2 Þ  b Fðt1 Þ

(1.26)

which is the operator principle of stationary action. The leading variation in the action is then the three equations equivalently.   E D i 1h   Fðt2 ÞΨi  Ψjb δW ¼ Ψjb Fðt1 ÞΨ + hc 2 ð t2 i 1 i hD h b bi E Ψj H, F Ψ + hc δW ¼ t1 2 ħ  +# ð t2 "* 1 db F δW ¼ Ψj Ψ + hc (1.27) dt  t1 2 or for Hamilton principle δW ¼ 0, where the action is defined through the Lagrangian ð t2 ð ð W ¼ dt ⋯ dr1 ⋯drN L (1.28) t1

where L is the Lagrangian density. Schwinger’s Principle of Stationary action holds for than the two times t1 and t2. If time is extended to t3 the contributions from t1 to t2 and t2 to t3 are additive, explicitly ð t3 ð ð ð t2 ð ð ð t3 ð ð dt ⋯ dr1 ⋯drN L ¼ dt ⋯ dr1 ⋯drN L + dt ⋯ dr1 ⋯drN L (1.29) t1

t1

t2

and the variations are additive as well. Bader and coworkers were motivated to use the approach of Schwinger to find a proper quantum subsystem. They returned to the Lagrangian density L used by Schwinger, Dirac, and Feynman, namely   N X 1 ∂Ψ∗ ħ2 ∗ ∂Ψ  iħΨ L ¼ iħΨ r j Ψ∗  r j Ψ (1.30)  VΨ∗ Ψ  2 ∂t ∂t 2mj j¼1 To find a proper quantum subsystem, they suggested rather than compute the Lagrangian density evaluated over all space they took the Lagrangian density over a piece of space, a volume, Ω. This volume must be continuously deformable over time. In particular, all particles except one were evaluated over all space and the last particle is varied over some unknown volume. Therefore, the action of a subsystem, WΩ, is chosen as ð ð ð t2 ð (1.31) W Ω ¼ dt dr1 ⋯ dr2 ⋯drN L t1

Ω

4. Mathematical fundamentals of QTAIM

15

where the Lagrangian density is chosen as in Eq. (1.27). Also, Bader and coworkers acknowledged that if the volume Ω is extended to all space the traditional analysis of Schwinger must be recaptured. In Schwinger’s treatment, the fundamental equation is that the leading varib must for all space are equal to equivalently, Eq. (1.24) or for ations in action over all space, δW the Hamilton principle δW ¼ 0. An important ansatz made by Schwinger was recognizing the relationship between the variation of the wave-function and infinitesimal generators. He identified that the variations in the wave-function are associated with the action of generators of infinitesimal unitary transformations on the wave-function. When these operators act on a state function they induce changes in the dynamical properties of the system. It is this identification that enabled Schwinger to transform the variation of the action integral into the principle of stationary action. Dirac [54] postulated the connection between these generators and those of classical mechanics. In other words, these generators are the quantum mechanical analog of the generators of infinitesimal contact transformations in classical mechanics. Dirac attributed the correspondence between the property generators are responsible for the fundamental similarity in the structure of classical and quantum mechanics. This is what allowed Schwinger to state the principle of stationary action and furthermore that is applicable to both classical and quantum mechanics. Bader and coworkers used this idea in formulating the physics of a subsystem. It is through this principle that one is able to formulate how to compute the properties of a subsystem in quantum mechanics of some total system, an open system. In other words, find the properties of an atom in a molecule. The infinitesimal generator as defined by Bader and coworkers just as by Schwinger is i δΨ ¼  b FΨ ħ

(1.32)

where b F is a Hermitian operator. Returning to the action for a volume if the Schr€ odinger equation is fulfilled notice that ð t2 ð t2 ð ð WΩ ¼ dtL ¼ dt dr dr2 dr3 ⋯drN L t1

t1

Ω

 2  ð ð ħ ¼ dt dr dr2 dr3 ⋯drN  r2 ðΨ∗ ΨÞ 4 t1 Ω ð t2

(1.33)

where the identity r2(AB) ¼ (r2A)B + 2 r A  r B + A r2B was used. If the action vanishes for the subspace then the action would be stationary. This motivates the constraint that all the spatial variations of the term on the second line of Eq. (1.30) must vanish, ð δ

 2  ð   ħ dr dr2 dr3 ⋯drN  r2 Ψ∗ Ψ ¼ 0 4 Ω ð

 2  ðð ð  2  ð    ħ ħ  dr dr2 dr3 ⋯drN  r2 Ψ∗ δΨ + hc + δS dr2 dr3 ⋯drN  r Ψ∗ δΨ dS 4 4 Ω

ð

 2  ð ðð ð       ħ2  ħ  r  rΨ∗ δΨ + r  Ψ∗ rδΨ + hc + δS dr2 dr3 ⋯drN  r Ψ∗ δΨ  n dS dr dr2 dr3 ⋯drN 4 4 Ω (1.34)

¼

¼

16

1. Introduction to QTAIM and beyond

Notice that this is the variation of the zero-flux condition. This is the key condition in showing what contiguous volumes form a proper quantum subsystem. The variation of the action yields

 ð t2 ð  iħ ∂δΨ ∂Ψ∗ ħ2 ∗ ∗ ∗ δWΩ ¼ dt dr Ψ  ∂Ψ  rΨ  rδΨ  VΨ δΨ + hc 2 ∂t ∂t 2 t1 Ω  2  ðð ð ħ + δS dr2 dr3 ⋯drN  rðΨ∗ ΨÞ  n dS 4

(1.35)

where the variations of time and position are taken as zero. In  order to rewrite the Lagrangian density in terms of the Hamiltonian, the identity rj Ψ⁎  rj Ψ ¼ 12 Ψ∗ r2j Ψ  Ψr2j Ψ⁎ + r2j ðΨ Ψ Þ yields the variation of the action in the form ð ð ð δW Ω ¼ dr1 ⋯ dr2 dr3 ⋯drN Ω

2 3  N i ∗ 2 h X   1 ∂δΨ ∂Ψ ħ 4 iħΨ∗  iħδΨ Ψ∗ r2j Ψ  δΨr2j Ψ∗ + r2j Ψ∗ δΨ + hc5  VΨ∗ δΨ  2 ∂t ∂t 4mj j¼1 2 3  N i ∗ 2 h X   1 ∂Ψ ∂Ψ ħ  iħΨ Ψ∗ r2j Ψ  Ψr2j Ψ∗ + r2j Ψ∗ Ψ 5δS  VΨ∗ Ψ  + 4 iħΨ∗ 2 ∂t ∂t 4m j j¼1 (1.36) And if the wave-function satisfies the Schr€ odinger equation the leading variation in the action yields 2 3   N i 2 h X 1 ∂δΨ 1 ħ 4 iħΨ∗ Ψ∗ r2j δΨ + r2j ðΨ∗ δΨÞ 5 + hc  VΨ∗ δΨ  ð ð ð 2 ∂t 2 4mj j¼1 3 δWΩ ¼ dr1 ⋯ dr2 dr3 ⋯drN 2 N 2 X Ω ħ r2 ðΨ∗ ΨÞ5δS +4 4mj j j¼1 (1.37) The next step involves two critical substitutions. The first is the relation between the variation of the wave-function and the unitary infinitesimal generator, b F, as in Equation which was the crucial suggestion from Schwinger. The second is to use the constraint in equation, the variations of the zero-flux condition must vanish, to eliminate variation of the surface " # N 2 N 2    ∗  P P ħ ħ 2 r2j Ψ∗ Ψ δS, and the term,  4m 4mj rj Ψ δΨ + hc term. The variation in the action j j¼1

becomes

j¼1

References



22 ib

ð ð ð ∂  FΨ Ψ 66 1 1 ib ħ * * 6 6  VΨ  FΨ Ψ δW Ω ¼ dr1 ⋯ dr2 dr3 ⋯drN 44 iħΨ ∂t 2 2 ħ Ω

17

(1.38)

3 3

N 2 X ħ i 7 7 Ψ* r2j  b + FΨ Ψ5 + hc5 ħ 4m j j¼1 To simplify the first term on the right-hand side of the equal sign, the product rule for the time derivative yields 22

ð ð ð b 1 ib 66 1 * b ∂Ψ 1 * ∂F * + Ψ Ψ  VΨ  FΨ Ψ δW Ω ¼ dr1 ⋯ dr2 dr3 ⋯drN 44 Ψ F (1.39) 2 ∂t 2 ∂t 2 ħ Ω

+

N X j¼1





3

3

ħ i 7 7 Ψ* r2j  b FΨ Ψ5 + hc5 ħ 4mj 2

The Schr€ odinger equation allows the time derivative of the wave-function to be rewritten as the Hamiltonian acting on the wave-function. The equation can be written in terms of the commutator of the Hamiltonian with the generator. Explicitly in terms of the Hamiltonian and the commutator are "" #

# ð ð ð 1 ∗ ∂b F 1 ∗b b 1 b ib Ψ Ψ + Ψ FHΨ  ΨH  F Ψ + hc δW Ω ¼ dr1 ⋯ dr2 dr3 ⋯drN 2 ∂t 2 2 ħ Ω " " # # ð ð ð 1 ∗ ∂b F 1 ∗ i h b bi ¼ dr1 ⋯ dr2 dr3 ⋯drN Ψ Ψ+ Ψ H, F Ψ + hc (1.40) 2 ∂t 2 ħ Ω which the Heisenberg equation of motion is the key equation, where all relationships in classical mechanics and quantum mechanics can be derived. Note the commutator notation h i b F b ¼H bF bF bH. b was used H,

References [1] R.F.W. Bader, Atoms in Molecules: A Quantum Theory (International Series of Monographs on Chemistry), first ed., Oxford University Press, USA, 1994. [2] R.F. Bader, Principle of stationary action and the definition of a proper open system, Phys. Rev. B Condens. Matter 49 (1994) 13348–13356, https://doi.org/10.1103/physrevb.49.13348. [3] P. Coppens, X-Ray Charge Densities and Chemical Bonding, Oxford University Press, 1997, https://doi.org/ 10.1093/oso/9780195098235.001.0001. [4] R. Eisenschitz, F. London, On the ratio of the van der waals forces and the homo-polar binding forces. Quantum chemistry: classic scientific papers, World Sci. 8 (2000) 336–368, https://doi.org/10.1142/9789812795762_0022.

18

1. Introduction to QTAIM and beyond

[5] R.F.W. Bader, S.G. Anderson, A.J. Duke, Quantum topology of molecular charge distributions. I, J. Am. Chem. Soc. 101 (1979) 1389–1395, https://doi.org/10.1021/ja00500a006. [6] W.L. Cao, C. Gatti, P.J. MacDougall, R.F.W. Bader, On the presence of non-nuclear attractors in the charge distributions of Li and Na clusters, Chem. Phys. Lett. 141 (1987) 380–385, https://doi.org/10.1016/0009-2614(87) 85044-3. [7] R.F.W. Bader, H. Ess
en, The characterization of atomic interactions, J. Chem. Phys. 80 (1984) 1943–1960, https:// doi.org/10.1063/1.446956. [8] P. Macchi, Chemical bonding in transition metal carbonyl clusters: complementary analysis of theoretical and experimental electron densities, Coord. Chem. Rev. 238–239 (2003) 383–412, https://doi.org/10.1016/S00108545(02)00252-7. [9] R.F.W. Bader, A bond path: a universal indicator of bonded interactions, J. Phys. Chem. A 102 (1998) 7314–7323, https://doi.org/10.1021/jp981794v. [10] D. Cremer, E. Kraka, Chemical bonds without bonding Electron density ? Does the difference Electron-density analysis suffice for a description of the chemical bond? Angew. Chem. Int. Ed. 23 (1984) 627–628, https://doi. org/10.1002/anie.198406271. [11] L.I. Schiff, Quantum Mechanics, third ed., McGraw-Hill Education, 1968. [12] C.F. Matta, R.J. Boyd (Eds.), The Quantum Theory of Atoms in Molecules, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2007, https://doi.org/10.1002/9783527610709. [13] I. Levine, Quantum Chemistry, seventh ed., Pearson, Boston, 2013. [14] J.I. Rodrı´guez, P.W. Ayers, A.W. G€ otz, F.L. Castillo-Alvarado, Virial theorem in the Kohn-Sham densityfunctional theory formalism: accurate calculation of the atomic quantum theory of atoms in molecules energies, J. Chem. Phys. 131 (2009), 021101, https://doi.org/10.1063/1.3160670. [15] J.I. Rodrı´guez, R.F.W. Bader, P.W. Ayers, C. Michel, A.W. G€ otz, C. Bo, A high performance grid-based algorithm for computing QTAIM properties, Chem. Phys. Lett. 472 (2009) 149–152, https://doi.org/10.1016/j.cplett.2009.02.081. [16] F. Cort
es-Guzma´n, R.F.W. Bader, Transferability of group energies and satisfaction of the virial theorem, Chem. Phys. Lett. 379 (2003) 183–192, https://doi.org/10.1016/j.cplett.2003.07.021. [17] J.I. Rodrı´guez, A.M. K€ oster, P.W. Ayers, An efficient grid-based scheme to compute QTAIM atomic properties without explicit calculation of zero-flux surfaces, J. Comput. Chem. 30 (2009) 1082–1092. [18] L.A. Terrabuio, T.Q. Teodoro, C.F. Matta, R.L.A. Haiduke, Nonnuclear attractors in heteronuclear diatomic systems, J. Phys. Chem. A 120 (2016) 1168–1174, https://doi.org/10.1021/acs.jpca.5b10888. [19] M.T. Carroll, R.F.W. Bader, An analysis of the hydrogen bond in BASE-HF complexes using the theory of atoms in molecules, Mol. Phys. 65 (1988) 695–722, https://doi.org/10.1080/00268978800101351. [20] C. Foroutan-Nejad, S. Shahbazian, Atomic basins with more than a single nucleus: a computational fact or a mathematical artifact? J. Mol. Struct. THEOCHEM 894 (2009) 20–22, https://doi.org/10.1016/j.theochem. 2008.09.038. [21] P.L.A. Popelier, Quantum chemical topology: on bonds and potentials, in: D.J. Wales (Ed.), Intermolecular Forces and Clusters I, Springer-Verlag, Berlin/Heidelberg, 2005, pp. 1–56, https://doi.org/10.1007/b135617. [22] R.F.W. Bader, P.J. MacDougall, C.D.H. Lau, Bonded and nonbonded charge concentrations and their relation to molecular geometry and reactivity, J. Am. Chem. Soc. 106 (1984) 1594–1605, https://doi.org/10.1021/ ja00318a009. [23] F. Cortes-Guzman, R. Bader, Complementarity of QTAIM and MO theory in the study of bonding in donoracceptor complexes, Coord. Chem. Rev. 249 (2005) 633–662, https://doi.org/10.1016/j.ccr.2004.08.022. [24] N.O.J. Malcolm, P.L.A. Popelier, The full topology of the Laplacian of the electron density: scrutinising a physical basis for the VSEPR model, Faraday Discuss. 124 (2003) 353–363. discussion 393 https://doi.org/10.1039/ b211650m. [25] E.R. Johnson, S. Keinan, P. Mori-Sa´nchez, J. Contreras-Garcı´a, A.J. Cohen, W. Yang, Revealing noncovalent interactions, J. Am. Chem. Soc. 132 (2010) 6498–6506, https://doi.org/10.1021/ja100936w. [26] Y. Grin, A. Savin, B. Silvi, The ELF perspective of chemical bonding, in: G. Frenking, S. Shaik (Eds.), The Chemical Bond: Fundamental Aspects of Chemical Bonding, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2014, pp. 345–382, https://doi.org/10.1002/9783527664696.ch10. [27] A. Savin, R. Nesper, S. Wengert, T.F. F€assler, ELF: the electron localization function, Angew. Chem. Int. Ed. 36 (1997) 1808–1832, https://doi.org/10.1002/anie.199718081. [28] R.F.W. Bader, A. Streitwieser, A. Neuhaus, K.E. Laidig, P. Speers, Electron delocalization and the fermi hole, J. Am. Chem. Soc. 118 (1996) 4959–4965, https://doi.org/10.1021/ja953563x.

References

19

[29] R. Ponec, C. Gatti, Do the structural changes defined by the electron density topology necessarily affect the picture of the bonding? Inorg. Chem. 48 (2009) 11024–11031, https://doi.org/10.1021/ic901197b. [30] R.F.W. Bader, D.-C. Fang, Properties of atoms in molecules: caged atoms and the ehrenfest force, J. Chem. Theory Comput. 1 (2005) 403–414, https://doi.org/10.1021/ct049839l. [31] A. Martı´n Penda´s, J. Herna´ndez-Trujillo, The Ehrenfest force field: topology and consequences for the definition of an atom in a molecule, J. Chem. Phys. 137 (2012) 134101, https://doi.org/10.1063/1.4755326. [32] H. Fliegl, D. Sundholm, S. Taubert, J. Jus
elius, W. Klopper, Magnetically induced current densities in aromatic, antiaromatic, homoaromatic, and nonaromatic hydrocarbons, J. Phys. Chem. A 113 (2009) 8668–8676, https:// doi.org/10.1021/jp9029776. [33] P.W. Fowler, E. Steiner, R.W.A. Havenith, L.W. Jenneskens, Current density, chemical shifts and aromaticity, Magn. Reson. Chem. 42 (2004), https://doi.org/10.1002/mrc.1445. Spec no: S68–78. [34] T.A. Keith, R.F.W. Bader, Topological analysis of magnetically induced molecular current distributions, J. Chem. Phys. 99 (1993) 3669–3682, https://doi.org/10.1063/1.466165. [35] P. Balanarayan, R. Kavathekar, S.R. Gadre, Electrostatic potential topography for exploring electronic reorganizations in 1,3 dipolar cycloadditions, J. Phys. Chem. A 111 (2007) 2733–2738, https://doi.org/10.1021/jp067247o. [36] S.R. Gadre, C.H. Suresh, N. Mohan, Electrostatic potential topology for probing molecular structure, bonding and reactivity, Molecules 26 (2021), https://doi.org/10.3390/molecules26113289. [37] P. Carpio-Martı´nez, J.E. Barquera-Lozada, A.M. Penda´s, F. Cort
es-Guzma´n, Laplacian of the hamiltonian kinetic energy density as an indicator of binding and weak interactions, ChemPhysChem 21 (2020) 194–203, https:// doi.org/10.1002/cphc.201900769. [38] D.I. Ramı´rez-Palma, F. Cort
es-Guzma´n, From the Linnett-Gillespie model to the polarization of the spin valence shells of metals in complexes, Phys. Chem. Chem. Phys. 22 (2020) 24201–24212, https://doi.org/10.1039/ d0cp02064h. [39] P. Fuentealba, E. Florez, W. Tiznado, Topological analysis of the Fukui function, J. Chem. Theory Comput. 6 (2010) 1470–1478, https://doi.org/10.1021/ct100022w. [40] J.L. Gazquez, A. Cedillo, B. Go´mez, A. Vela, Molecular fragments in density functional theory, J. Phys. Chem. A 110 (2006) 4535–4537, https://doi.org/10.1021/jp056421q. [41] I. Mayer, Energy partitioning schemes: a dilemma, Faraday Discuss. 135 (2007) 439–450. discussion 489 https:// doi.org/10.1039/b609463e. [42] E. Francisco, A. Martı´n Penda´s, M.A. Blanco, A molecular energy decomposition scheme for atoms in molecules, J. Chem. Theory Comput. 2 (2006) 90–102, https://doi.org/10.1021/ct0502209. [43] F. Heidar-Zadeh, P.W. Ayers, T. Verstraelen, I. Vinogradov, E. V€ ohringer-Martinez, P. Bultinck, Informationtheoretic approaches to atoms-in-molecules: hirshfeld family of partitioning schemes, J. Phys. Chem. A 122 (2018) 4219–4245, https://doi.org/10.1021/acs.jpca.7b08966. [44] R.F. Nalewajski, R.G. Parr, Information theory, atoms in molecules, and molecular similarity, Proc. Natl. Acad. Sci. U. S. A. 97 (2000) 8879–8882, https://doi.org/10.1073/pnas.97.16.8879. [45] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, Oxford, 1990. [46] F. Heidarzadeh, S. Shahbazian, The quantum divided basins: a new class of quantum subsystems, Int. J. Quantum Chem. 111 (2011) 2788–2801, https://doi.org/10.1002/qua.22629. [47] F.H. Zadeh, S. Shahbazian, Toward a fuzzy atom view within the context of the quantum theory of atoms in molecules: quasi-atoms, Theor. Chem. Accounts 128 (2011) 175–181, https://doi.org/10.1007/s00214-010-0811-x. [48] J. Schwinger, The theory of quantized fields, I. Phys. Rev. 82 (1951) 914–927, https://doi.org/10.1103/ PhysRev.82.914. [49] S. Srebrenik, R.F.W. Bader, T.T. Nguyen-Dang, Subspace quantum mechanics and the variational principle, J. Chem. Phys. 68 (1978) 3667–3679, https://doi.org/10.1063/1.436225. [50] S. Srebrenik, R.F.W. Bader, Towards the development of the quantum mechanics of a subspace, J. Chem. Phys. 63 (1975) 3945, https://doi.org/10.1063/1.431834. [51] R.F.W. Bader, S. Srebrenik, T.T. Nguyen-Dang, Subspace quantum dynamics and the quantum action principle, J. Chem. Phys. 68 (1978) 3680–3691, https://doi.org/10.1063/1.436226. [52] J.S.M. Anderson, L. Massa, C.F. Matta, Non-nuclear maxima and the universality of Bright Wilson’s justification of the first Hohenberg Kohn theorem revisited, Chem. Phys. Lett. 780 (2021), https://doi.org/10.1016/j. cplett.2021.138940. [53] J.S.M. Anderson, P.W. Ayers, J.I. Rodriguez Hernandez, How ambiguous is the local kinetic energy? J. Phys. Chem. A 114 (33) (2010) 8884–8895, https://doi.org/10.1021/jp1029745. [54] P.A.M. Dirac, The Principles of Quantum Mechanics, fourth ed., Oxford University Press, London, 1958.

C H A P T E R

2 An introduction to quantum chemistry David C. Thompsona,* and Juan I. Rodrı´guezb,c,* a

Chemical Computing Group, Montreal, QC, Canada bCICATA-Quer
etaro, Instituto Polit
ecnico Nacional, Quer
etaro, M
exico cEscuela Superior de Fı´sica y Matema´ticas, Instituto Polit
ecnico Nacional, Mexico City, Mexico *Corresponding author.

1. What is quantum chemistry? The fundamental unit of chemistry is the electron. Indeed, chemistry could be considered the science of understanding how electrons organize in space. Chemists are not usually concerned with the abstract association of electrons, but more in their organization within atoms (surrounding atomic nuclei), among atoms (as molecules, or discrete groups of atomic nuclei “bonded” together), and as supra-molecular forms (e.g., peptides and proteins). Quantum mechanics is a foundational theory of the physical world that allows the exploration of subatomic and atomic scale phenomena and, accordingly, quantum chemistry is the application of the methods of quantum mechanics to problems of chemical interest. What do the laws of quantum mechanics tell us about the organization of electrons in space for atomic, molecular, and supramolecular species? Moreover, what are the associated properties of such systems? These are important questions, the answering of which provides useful insight to problems such as (including, but not limited to): drug discovery [1–3], biocatalysis [4,5], materials design [6,7] and, more recently, quantum computing [8]. A foundational quantum mechanical description of a system can be found in the nonrelativistic time (t) dependent Schr€ odinger equation: b ¼ iħ ∂Ψ HΨ ∂t

(2.1)

b and Ψ are the Hamiltonian operator and wave-function, respectively. A solution to Here H Eq. (2.1), a determination of Ψ , describes all of the properties of a system (according to a fundamental postulate of quantum mechanics [9,10]). This is illustrated in Fig. 2.1, where a number of examples of “Quantum Systems” are presented for illustrative purposes.

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00012-8

21

Copyright # 2023 Elsevier Inc. All rights reserved.

22

2. An introduction to quantum chemistry

FIG. 2.1 Schr€odinger’s equation and examples of quantum systems.

2. The molecular equation To practically apply quantum mechanics to problems of chemical interest, it is necessary to make some simplifying assumptions. First, let us consider that the system is a molecule (a collection of nuclei and electrons) and that this molecular system is isolated: no external electromagnetic fields, in the absence of solvent, to be considered in the gas phase. Under these simplifying considerations, Eq. (2.1) can be re-written in its stationary form, such that there is now no explicit time-dependence in the potential: b ¼ EΨ HΨ

(2.2)

An extra simplification can be introduced: let us suppose that the nuclei in the molecule can be considered as charged point masses with no “internal structure.” Under such conditions, the Hamiltonian operator in Eq. (2.2) for a molecule with M nuclei and N electrons can be written in atomic units (a.u.) as: XM XN 1 1 ^ ¼T ^ +V ^ ¼ H r2! r2!  A¼1 2M i¼1 2 ri A RA (2.3) XM1 XM XM XN XN1 XN Z 1 Z Z  + ! A  +  ! A B!   ! A¼1 i¼1  i¼1 j>i ! A¼1 B>A  !   ri  rj  R A  r i  RA  RB  ! where ! ri is the vector coordinate of electron i. RA , MA, and ZA are the vector coordinates, mass and atomic number of nucleus A, respectively (see Fig. 2.2). The system’s kinetic energy operator b is equal to the two first Laplacian sums in the right-hand side of Eq. (2.3); the potential energy T b equal to the rest of the terms. Eq. (2.3) is termed the molecular Hamiltonian operator. operator Vis Eqs. (2.2) and (2.3) define a partial differential equation for the unknown Ψ , which succinctly represent the n-body problem in quantum mechanics. There is one atomic system (M ¼ 1 and N ¼ 1, the hydrogen atom) and a small number of model systems (including realistic electron-electron interactions) (e.g., [11,12]) for which analytic solutions are known so, in general, Eqs. (2.2) and (2.3) have to be solved by approximate methods.

2. The molecular equation

23

FIG. 2.2 Vector coordinates of the charged ! point-mass nuclei ( RA , blue dots); illustrative distribution of vector coordinates of electrons (green dots) in a molecule.

2.1 The Born-Oppenheimer approximation The Born-Oppenheimer (BO) approximation [13] allows for the decoupling of nuclear and electronic motions, and is a result of the atomic nucleus being far more massive than an electron (a proton’s mass is roughly 2000 times greater than an electron’s mass). Accordingly, the movement of a nuclei is much slower than that of electrons, which has been confirmed by experiment even at room (and lower) temperatures [14,15]. Specifically, within the BO approximation the nuclei are treated as fixed, such that: !  ! RA ¼ const; A ¼ 1,…,M (2.4) where the index A refers to the nuclei within the system. Eq. (2.4) specifies that each nucleus position is a constant vector so that the kinetic energy associated with the nuclei is equal to zero (first term in the right-hand side of Eq. 2.3 in the molecular Hamiltonian), such that: X XM XN XN1 XN Z 1 b ¼  N 1 r2!   ! A  +   H ! i¼1 2 A¼1 i¼1  i¼1 j>i !  ! ri  rj  r i  RA  ri  XM1 XM ZA ZB  (2.5) + ! A¼1 B>A  !  RA  RB  Another mathematical consequence of the BO approximation is that the nucleus-nucleus interaction term is equal to a constant: XM1 XM ZA ZB  Enuc ≡ A¼1 (2.6) ! ¼ const B>A  !  RA  RB 

24

2. An introduction to quantum chemistry

Applying, 2 3 X X X X X N 1 M N ZA N1 N 1 6   +   + Enuc 7 4 i¼1 r2!  5Ψ ¼ EΨ !  A¼1 i¼1  ! i¼1 j>i !  2 ri  r r i j R  r  2

A

i

3 X X X X X N 1 M N Z N1 N 1 6  ! A  + ! !7 4 i¼1 r2!  5 Ψ ¼ E0 Ψ   A¼1 i¼1 i¼1 j>i  r 2 ri  rj  i R  r  A

(2.7)

i

where E0 ¼ E  Enuc

(2.8)

3. The electronic structure problem The coordinates of the nuclei in the Hamiltonian (Eq. 2.7) can be considered to be parameters (like ZA) since they are constant vectors, and are not variable coordinates in a differential equation. Therefore, Eq. (2.7) represents the Schr€ odinger equation for N electrons in an (external) potential generated by the fixed charged nuclei: " # XN 1 !  XN1 XN 1 ! 2   Ψ elec ¼ Eelec Ψ elec (2.9)  i¼1 r! + V ext r1 , …, rN + i¼1 j>i ! 2 ri rj  ri  ! where the external potential is defined as: XM XN ZA  V ext ¼  A¼1 i¼1  !   R A  ri 

(2.10)

This formally defines the electronic Hamiltonian operator under the BO approximation: XN 1 XN1 XN XM XN ZA 1 b elec ≡      + r2  H (2.11) i¼1 2 ! A¼1 i¼1 ! i¼1 j>i !  ! ri r  r  ! R  r  A

i

i

j

and its corresponding electronic energy: Eelec ≡E0 ¼ E  Enuc

(2.12)

so that the total or molecular energy is defined (under the BO approximation) as: ETot ≡E ¼ Eelec + Enuc

(2.13)

Eq. (2.9) can be written in a compact eigenvalue-eigenvector representation: b elec Ψ elec ¼ Eelec Ψ elec H

(2.14)

such that solving Eq. (2.14) amounts to deriving solutions to the “Electronic Structure Problem” (subsequent to the BO approximation). The electronic wave-function, Ψ elec, is only

3. The electronic structure problem

25

a function of the electron coordinates but it depends parametrically on the nuclear coordi! b elec ), nates (as the parameters RA appear in the Hamiltonian H  !    ! !  ! ! ! ! ! !  Ψ elec ¼ Ψ elec X1 , …, XN ¼ Ψ elec X1 , …, XN ; R1 , …, RM ¼ Ψ elec X1 , …, XN ; RA (2.15) ! Similarly, the electronic energy also depends on the parameters RA (it is a constant with ! respect to Xi ), ! !  ! !  RA Eelec ¼ Eelec R1 , …, RM ¼ Eelec RA ¼ Eelec (2.16) The solution of the electronic problem allows one to obtain the molecule’s electronic structure for a fixed nuclear geometry. As such, this fixed geometry serves as an input to the electronic structure problem. But, how then to choose an “appropriate” input nuclear geometry from among the seemingly infinite possible nuclear arrangements? This problem is addressed later in the current work, in Section 5. For now, let us suppose that a reasonable fixed nuclear geometry is known, and explore ways of determining solutions to the electronic structure problem.

3.1 Determination of the many-electron wave-function A first step in the determination of a many-electron wave-function satisfying Eq. (2.14), is one of representation. Electrons have spin (artificially introduced in this nonrelativistic treatment). Accordingly, a many-electron wave-function must be constructed such that upon exchange of two identical electrons, the total electronic wave-function must be antisymmetric, i.e., if the space and spin coordinates of two identical electrons are interchanged, then the total wave-function changes its sign. A representation that accounts for this behavior has the structure of a determinant:  ! !  !     χ 1 X1 χ 2 X1 ⋯ χ K X1     !  !   ! ! !  1  χ X!2 χ 2 X2 ⋯ χ K X2  (2.17) Ψ elec X1 , X2 , …, XN ¼ pffiffiffiffiffiffi  1   N!  ⋮  ⋮ ⋱ ⋮ !  !  !    χ 1 XN χ 2 XN ⋯ χ K XN  This object being termed a “Slater determinant.” Wherein χ i above is the associated “spin orbital” of electron i. Systematic variation of χ i, subject to their remaining orthonormal, can be performed in a mathematically rigorous way using a self-consistent methodology first proposed by Hartree [16] and subsequently refined by Fock [17] and Slater [18] to include electron exchange (correlation between particles of like spin). The resulting Hartree-Fock wave-function is the “best” single determinantal representation constructed out of K Hartree-Fock orbitals which arise from minimizing the expectation value of the total energy of the system with respect to χ i. More practical considerations relating to self-consistent field methodologies are described in detail below. The equations to solve in Hartree-Fock theory have the canonical form:  !  ! (2.18) f ðiÞχ Xi ¼ Ei χ Xi

26

2. An introduction to quantum chemistry

where f(i) represents an effective one-electron “Fock” operator. In atomic units, this has the form: M X 1 ZA f ðiÞ ¼  r2i  + vHF ðiÞ 2 r iA A¼1

(2.19)

To solve this set of equations, we introduce a suitable set of basis functions in which our spin orbitals can be expanded, and reduce this to an exercise in linear algebra. Because of this, basis functions are usually chosen for mathematical convenience, such that the resulting integrals and integral derivatives appearing in the matrix equations can be calculated efficiently. To this end, Gaussian functions [19], Slater-type orbitals [20], and other tractable functions [21–23] have all been used as primitive basis function candidates. Detailed treatment of the solution of the Hartree-Fock equations can be found in the following classic text on electronic structure theory [24], wherein the interested reader will find it applied to both restricted (closed shell spin-paired) and unrestricted (open shell spinunpaired) systems. A canonical example of the general form for a Hartree-Fock program can be found here [25].

3.2 Beyond a single determinant The single determinantal description described above does not include the effects of correlation between opposite-spin electrons, and it is this “chemical glue” [26], and its correct description, that is the focus of modern post-Hartree-Fock explorations of chemical physics. To account for this additional “correlation”, methods have been developed that involve expressing the electronic wavefunction as a combination of Slater determinants, Ψ μ:  ! ! ! X Cμ Ψ μ (2.20) Ψ elec X1 , X2 , …, XN ¼ μ

Those additional determinants representing differing electronic configurations with respect to a “reference.” While this reference could be the Hartree-Fock determinant, it does not have to be. Allowing electrons to explore these alternate configurational representations permits electrons of opposite spin to avoid each other (thus incorporating exactly what is missing from the Hartree-Fock description). There are three principal theoretical methods for including these correlated states: (a) Configuration interaction—here the wavefunction is described as a linear expansion, with the Hartree-Fock determinant as a reference. It is conceptually quite straightforward to understand, however there are theoretical considerations (size consistency) to be aware of when using a truncated expansion. (b) Coupled cluster—here an exponential operator (the cluster operator) is used to generate excited determinantal states. (c) Møller-Plesset perturbation theory—here many-body perturbation theory is applied to the Hartree-Fock solution, yielding tractable contributions that account for higher-order correlation between electrons.

4. Density functional theory

27

If Hartree-Fock is the best single-determinantal description for the electronic wavefunction, and the above-mentioned methods are useful extensions beyond this to begin to account for opposite-spin electron correlation, it is conceptually useful to consider the best multideterminantal description of an electronic wavefunction. According to Lowdin’s theorem [27] the exact wavefunction, within a finite basis set representation, can be described as a linear combination of all possible excited determinants from a ground state reference. This “Full Configuration Interaction” (FCI) method scales factorially with the number of electrons, and remains impractical for all but the smallest of systems. Some recent benchmarking work provides a useful summary and review of recent progress related to FCI approaches (see, e.g., [28,29]). In the following section, we move away from wave-function based solutions to the electronic problem, and instead re-focus our attention to solutions described through the electron density ρ(r).

4. Density functional theory Density functional theory (DFT) is a variational method for solving Eq. (2.14). Within the DFT formalism the function of interest is changed from Ψ elec to the electron density, which is formally defined in terms of Ψ elec: ðð ð     X1=2 ! !  2 ! ! ! ! !  … Ψ ρ r ≡N r , m , X …, X d X d X …d X (2.21)   2 N s1 2 3 N elec m ¼1=2 s1

The density is a real scalar function that only depends on three spatial variables, which is more manageable (both conceptually and practically) than the wave-function that depends on all 4N electron coordinates. Hohenberg and Kohn (HK) described the electronic problem in terms of ρ(r) through the following two theorems [30,31]: HK Theorem 2.1 The ground state’s electron density determines all the properties of the electronic system. HK Theorem 2.2 The electronic energy is a functional of ρ, Eelec ¼ Eelec ½ρ

(2.22)

The ground state electron density ρ0 minimizes such a functional, min Eelec ½ρ E0elec ¼ Eelec ½ρ0  ¼ |ffl{zffl}

(2.23)

ρ

Such that E0elec is the ground state energy of the electronic Eq. (2.14). The first theorem ensures that obtaining the density is enough to describe the system even without knowing the wave-function explicitly. The second theorem provides a way to obtain the ground state density by minimizing the energy functional (Eq. 2.22). The challenge however, is that the exact functional form is unknown. In the Kohn-Shan formulation, the energy functional is decomposed accordingly: Eelec ½ρ ¼ T ½ρ + J ½ρ + V ext ½ρ + Exc ½ρ

(2.24)

28

2. An introduction to quantum chemistry

where T[ρ], J[ρ], Vext[ρ], and Exc[ρ] are the kinetic energy, Coulomb, external, and exchangecorrelation (xc) functionals, respectively. The Coulomb and external functionals are known:    0 ! ! !0 ð ð ρ ! r ρ r drdr 1   (2.25) J ½ρ ¼ ! !0  2 R3 R3 r  r  and

ð V ext ½ρ ¼

R3

    ! ! ! ρ r vext r d r

(2.26)

!   P !  ZA  respectively, and where, for a molecule, vext r ¼  M A¼1  ! !. Exact expressions for T[ρ]  RA  r  and Exc[ρ] are not known. Kohn and Sham (KS) approximated the kinetic energy functional, T[ρ], as the Hartree-Fock independent electron kinetic energy functional [31]: XN ð ! r2 ! ! T½ρ ’ T½ψ 1 ,…, ψ N  ¼  i¼1 ψ ∗i X ψ X dX (2.27) 2 i where ψ i are an orthonormal set of spin-orbitals. As can be seen from (2.27), T[ρ] is not explicitly a functional of the density but of the spin orbitals. As such, the KS methodology can be considered as a hybrid density-orbital functional theory. The KS energy functional is obtained by substituting Eqs. (2.25)–(2.27) in Eq. (2.24):    0 ! ! ! !0 XN ð ! r2 ! ! ð ! ! ! 1 ð ð ρ r ρ r d r d r KS ∗   Eelec ¼  i¼1 ψ i X ψ X dX + ρ r vext r d r + ! !0  2 R3 R3 2 i R3 r  r  + Exc ½ρ (2.28) According to the second HK theorem, the KS functional (Eq. 2.28) must be minimized with respect to the orbitals, ψ i, to obtain the ground state density. This minimization must be performed subject to the condition: ð   ! ! ρ r dr ¼ N (2.29) R3

using the technique of variational calculus. The corresponding Euler-Lagrange equations are known as the Kohn-Sham equations: c f KS ψ i ¼ Ei ψ i ; i ¼ 1,2,…,N

(2.30)

  2 ! c r f KS ¼ + vext r + bJ + V xc 2

(2.31)

c where f KS is the Kohn-Sham operator:

4. Density functional theory

29

bJ is the Coulomb operator:   ! 2 !  !  XN ð ψ j X 0  dX 0 ^Jψ i ≡ ψ i X   ! ! 0  j6¼i r r 

(2.32)

and V xc ¼ δEδρxc is the xc potential (the variational derivative of Exc). Mathematically, the KS equations can be classified as a set of N integro-differential nonlinear coupled equations, which are very difficult to solve. The solutions of the KS equations, which are the orbitals that minimize the energy functional, are the so-called Kohn-Sham molecular spin-orbitals ψ i and their corresponding Kohn-Sham orbital energies, Ei. All the properties of the system can be obtained from the KS molecular orbitals. From Eqs. (2.31) and (2.32), we see that the KS operator is a one-electron operator. The KS formalism allows one to pseudo-separate the electronic problem into a set of Schr€ odinger-like equations, one for each electron state (orbital). Each of these one-electron equations takes into account the interaction with the other electrons in an “averaged way” via the KS potential:   ! vKS ¼ vext r + bJ + V xc (2.33) which is mean-field or effective potential in the KS operator (2.31), c r + vKS f KS ¼ 2 2

(2.34)

4.1 Solution of the Kohn-Sham equations We note that the KS potential depends on the orbitals via the Coulomb operator bJ and xc potential vxc since they are defined in terms of the spin-orbitals (see for instance Eq. 2.32); as such the KS operator depends on the spin-orbitals ψ i. This means that solutions of the KS equations are needed to construct the KS operator. However, we can only solve the KS equations if this operator is known! From a practical perspective, the KS equations are solved using the self-consistent-field (SCF) method or algorithm: b Step 1. Consider N known spin-orbitals, ψ (0) i , as guess or initial function set to construct J and vxc. c Step 2. Substitute bJ and vxc of Step 1 into f KS and solve the KS equations to obtain a “new” (1) solution set, ψ i . If these “new” orbitals are equal to the initial ones ψ (0) i , then the SCF process has converged and the solutions of the KS equations are ψ (1) i ; otherwise set ð0Þ

ψi

ð1Þ

¼ ψ i ; i ¼ 1,…,N

(2.35)

and return to Step 1. This is repeated until convergence is reached (we note that in practice there is no guarantee that the SCF procedure will converge). Self-consistency suggests that the orbitals used to construct the mean-field vKS are the same as, or are “consistent” with, the orbitals that result from solving the KS equations.

30

2. An introduction to quantum chemistry

DFT is an extremely successful approach used widely to explore a variety of chemical and physical phenomena. The interested reader is encouraged to consult the following reference for a deeper dive introduction into DFT [32]. There are around 50,000 papers published each year that report the results of KS DFT calculations [33]. There are over 300 approximations to the xc functional Exc[ρ] [34], and it is estimated that roughly 30% of all super computer use is devoted to solving the requisite equations necessary for DFT calculations [35].

5. The nuclear problem As mentioned n ! earlier, theoelectronic structure problem is to be solved at fixed nuclear geometry RA , A ¼ 1, …, M under the BO approximation. Given this, which nuclear geometry should be used? Separating the electronic and nuclear components: ! ! ! ! ! !  !  !  Ψ ¼ Ψ X1 , …, XN , R1 , …, RM ¼ Ψ elec X1 , …, XN Ψ nuc R1 , …, RM

(2.36)

and substituting Eq. (2.36) into Eqs. (2.2) and (2.3), one arrives at a “nuclear equation” through separation of variables 2 3   X X X ! ! M 1 M1 M Z ZB 7 6  A ! 5Ψ nuc ¼ EΨ nuc (2.37) r2! + Eelec R1 , …, RM + 4 A¼1 A¼1 B>A !  2MA RA RA  RB  This represents a Schr€ odinger-like equation for nuclei, subject to the potential: !  ! ! XM1 XM ! !  !  1  Unuc R1 , …, RM ¼ ETot R1 , …, RM ¼ Eelec R1 , …, RM + ! A¼1 B>A  !  RA  RB  (2.38) where the electronic energy plays the role of an attractive potential for stable molecules. Considering Eqs. (2.14) and (2.38), the BO approximation allows one to separate the molecular Schr€ odinger equation (Eqs. 2.2, 2.3) into two Schr€ odinger-like equations, one for the electrons (2.14) and the other for the nuclei (2.38). Strictly, this is actually a pseudo-separation of variables since the molecular problem itself is not physically separable.

5.1 Potential energy surfaces To recap, according to Eq. (2.3), the total energy of the molecule can be expressed as: ! ! ! XM1 XM !  !  !  1  Unuc R1 , …, RM ¼ ETot R1 , …, RM ¼ Eelec R1 , …, RM + ! A¼1 B>A  !  RA  RB  (2.39)

31

5. The nuclear problem

As such Unuc is not only the potential for the nuclei in Eq. (2.38) but also the total energy of the molecule under the BO approximation. This function is called the Potential Energy Surface (PES), and is a function of 3M variables. The PES is a scalar field, or hyper-surface: ! !  Unuc R1 , …, RM : R3M ! R (2.40) ! !  whereby a given nuclear structure, represented by the positional vector R1 , …, RM is ! !  actually a point in the hyperspace R3M: R1 , …, RM  R3M . Structures with a lower value ! ! !  !  of energy Unuc R1 , …, RM ¼ ETot R1 , …, RM are expected to be “stable”. Indeed, the global minimum of the PES corresponds to the ground state nuclear structure of a stable molecule under the BO approximation. In the following section, the concept of nuclear stability is formally introduced; during this discussion, we will see that only the minima of the PES are actually stable nuclear structures.

5.2 The Hellmann-Feynman theorem and molecular stability The Hellmann-Feynman theorem posits: The effective force on a nucleus in a molecule is equal to the sum of the Coulomb forces due to the other nuclei (point charges) and the electron density (continuous charge density). Such effective force is equal to minus the gradient of the PES [15].

This is stated mathematically as: ! FA

 h! i ! !

 ð ZA ρ ! r RA  r d r ∂Unuc ∂Unuc ∂Unuc ¼ rRA Unuc ¼  , , ¼ !  ∂RAx ∂RAy ∂RAz !3  R3 RA  r  h! !i ZA ZB RA  RB XM + ; 8A !  B¼1,B6¼A ! 3  RA  RB 

(2.41)

  ! ! where F A is an effective force on nucleus A (whose coordinate is RA ¼ RAx , RAy , RAz ) and

0    ! !0 ! ρ r is the electron density, Eq. (2.21). Note that if R1 , …, RM  R3M is a minimum of the PES, then: !0 !0 rUnuc

ðR 1 ,…,RM Þ  ! ∂Unuc ∂Unuc ∂Unuc ∂Unuc ∂Unuc ∂Unuc ∂Unuc ∂Unuc ∂Unuc  0 ¼0 , , , , , , …, , , ¼ 0 ! ! R1x R1y R1z R2x R2y R2z RMx RMy RMz R ,…, R 1

M

(2.42)

32

2. An introduction to quantum chemistry

It follows from Eq. (2.42) that if a molecule has a structure that is a minimum of the PES, then the effective force on each nucleus in the molecule vanishes according to Eq. (2.41): ! FA

!

¼ 0 ; A ¼ 1,…,M

(2.43)

Accordingly, a stable structure of a molecule is a minimum of the PES and vice versa. The global minimum of the PES is the ground-state structure of the molecule (under the BO approximation). Other local minima are structural isomers, while geometries representing the other critical points of the PES (maxima and saddle points) are classified as metastable structures. It can be demonstrated that not all 3M nuclear coordinates are independent due to conservation theorems. The independent nuclear coordinates (degrees of freedom) are equal to 3M-6 for nonlinear molecules and 3M-5 for linear molecules [15]. Under the BO approximation, the molecular “nuclear problem” can be approached as the search for critical points on the PES (the minima being of particular interest). The nuclear problem can be reframed as an optimization problem, albeit an especially complex one. Since the PES is not known beforehand, it has to be constructed “on the fly” through solving the electronic Schr€ odinger equation and using Eq. (2.37). Given this, a global optimization at the quantum mechanical level is not feasible for all but the smallest of systems.

5.3 Geometry optimization From a practical perspective, within the field of quantum chemistry, geometry optimization a local optimization of the PES. Given an initial guess geometry

0 represents  ! !0 R1 , …, RM R3M , the process of geometry optimization will (in principle) identify the nearest local minimum (see Fig. 2.3). In general, the following canonical

0algorithm  is used to locally explore a PES: ! !0 Step 1: An initial point is given R1 , …, RM and the electronic problem is solved, using

0  ! !0 Eq. (2.14) to obtain Eelec R1 , …, RM and the PES ! ! XM1 XM !  !  Z ZB  A !  Unuc R1 , …, RM ≡Eelec R1 , …, RM + A¼1 B>A !  R A  RB 

!0

!0

(2.44)



Step 2: Determine if R1 , …, RM is a minimum of the PES (or critical point), that is, check if:

0 

 ! !0 ! ∂Unuc ∂Unuc ∂Unuc  ¼0 , , …, (2.45) rUnuc R1 , …, RM ¼ ∂R1x ∂R1y ∂RMz !R0 , …, !R0 1

M

33

5. The nuclear problem



!0

!0



FIG. 2.3 Representation of a simple 1-dimensional PES. The initial point R1 , …, RM (blue dot) in the geometry optimization procedure is highlighted.

In practice, the condition expressed by Eq. (2.45) is approximated through:

0   ! !0   ∂Unuc R1 , …, RM     < E8i ¼ 1,2,…,M; ν ¼ x,y,z    ∂Riν    

for E  0, E > 0. If Eq. (2.46) is satisfied, the procedure converges and represents a minimum. Otherwise, go to Step 

3.  Step 3: Find

set

!0

!1

!1

!0

R1 , …, RM

!0

R1 , …, RM

¼

!1

!1

R1 , …, RM

!0



R1 , …, RM

!0

in a vicinity of R1 , …, RM such that

1 

0  ! !1 ! !0 Unuc R1 , …, RM < Unuc R1 , …, RM



!0

(2.46)

(2.47)

 and return to Step 1.

To perform Step 3, that is, to find a point in the vicinity of

!0

!0

R1 , …, RM

 such that the en-

ergy (Unuc) is decreased, standard optimization methods can be used (e.g., Newton-Raphson, Conjugate gradient, etc.). The geometry optimization procedure performs multiple function evaluations of Unuc in the trajectory toward the minimum (see Fig. 2.3), the electronic problem must be solved on each point (a so-called single-point calculation). In practice, Eq. (2.45) is typically the main criterion for the geometry optimization procedure. In most quantum chemistry software packages, it is called the gradient or force criterion, and will have a typical value between 104 and 106Hartree/Angstrom.

34

2. An introduction to quantum chemistry

6. Quantum chemistry software packages While a comprehensive review of the current state of the electronic structure software community is out of scope for the current work, we will provide a number of expansive reviews and resources for the interested student to engage with. In the introduction, a number of fields of endeavor were highlighted, fields that could potentially benefit from insight gained through quantum chemistry calculations. Because of this, unsurprisingly, there exist a number of commercial packages that sit alongside an ever-increasing number of academic and Open-Source contributions. Wikipedia contains a reasonable list of quantum chemistry software packages [36] and a simple search for “quantum chemistry” on GitHub (the popular online software development and version control platform) returns 534 repository hits (as of January 14, 2022, where a hit was indicated by the presence of the keywords “quantum chemistry”), highlighting the extent of development and interest. A historical look at the development and growth of these software packages can be found here [37]. This is an important contribution that attempts to contextualize the tensions exist between Open Source and commercial software development, discussing innovation, software sustainability, and maintenance. The paper serves as an introduction to the online repository: https://atomistic.software/ which seeks to: … be a comprehensive list of all major atomistic simulation engines, with annual updates going forward.

This is a curtailed list, which avoids the long tail seen when exploring GitHub repositories through using an arbitrary cutoff of 1 year of 100 (or more) academic citations (as measured by Google Scholar). In addition, a recent Special Issue of the Journal of Chemical Physics provided a forum to highlight many important contributions to modern electronic structure theory, and saw the creation of a special section of the journal devoted to Electronic Structure software [38].

7. Concluding remarks The field of quantum chemistry is relatively young, arising from the application of quantum mechanics (a development of the early 20th century) to problems of chemical interest. Because of this, it is “neither physics nor chemistry” [39]. The following contribution has sought to provide the motivated student with an introduction to the theory of quantum chemistry. The principal approximations were described, as were a variety of methods for solving the electronic structure problem. For brevity, we provided a canonical view of those methods—beginning with determinantal wave-function based approaches—while then introducing the method of density functional theory. The Born-Oppenheimer approximation

References

35

and nuclear motion were briefly covered which then introduced the reader to the concept of a potential energy surface. The panoply of commercial and open-source quantum chemistry tools were presented, and a brief review of the current state of the commercial and open source marketplace was provided. It is the authors hope that this short review is of use to the reader, who is now equipped to engage in the vibrant field of quantum chemistry; a subject of practical use and interest across a variety of fields of human activity.

References [1] O.A. Arodola, M.E. Soliman, Quantum mechanics implementation in drug-design workflows: does it really help? Drug Des. Devel. Ther. 11 (2017) 2551–2564, https://doi.org/10.2147/DDDT.S126344. [2] D.B. Boyd, Quantum mechanics in drug design: methods and applications, Drug Inf. J. 17 (1983) 121–131. [3] D. Mucs, R.A. Bryce, The application of quantum mechanics in structure-based drug design, Expert Opin. Drug Discovery 8 (2013) 263–276, https://doi.org/10.1517/17460441.2013.752812. [4] M.P. Gleeson, D. Gleeson, QM/MM calculations in drug discovery: a useful method for studying binding phenomena? J. Chem. Inf. Model. 49 (2009) 670–677, https://doi.org/10.1021/ci800419j. [5] M.E.S. Lind, F. Himo, Quantum chemistry as a tool in asymmetric biocatalysis: limonene epoxide hydrolase test case, Angew. Chem. 125 (2013) 4661–4665, https://doi.org/10.1002/ange.201300594. [6] K. N
emeth, Materials design by quantum-chemical and other theoretical/computational means: applications to energy storage and photoemissive materials, Int. J. Quantum Chem. (2014), https://doi.org/10.1002/qua.24616. [7] T. Weymuth, M. Reiher, Inverse quantum chemistry: concepts and strategies for rational compound design, Int. J. Quantum Chem. 114 (2014) 823–837, https://doi.org/10.1002/qua.24687. [8] Y. Cao, J. Romero, J.P. Olson, M. Degroote, P.D. Johnson, M. Kieferova´, I.D. Kivlichan, T. Menke, B. Peropadre, N.P.D. Sawaya, S. Sim, L. Veis, A. Aspuru-Guzik, Quantum Chemistry in the Age of Quantum Computing, 2018. ArXiv181209976 Quant-Ph. [9] J. Autschbach, Quantum Theory for Chemical Applications: From Basic Concepts to Advanced Topics, first ed., Oxford University Press, 2020, https://doi.org/10.1093/oso/9780190920807.001.0001. [10] I.N. Levine, Quantum Chemistry, sixth ed., Pearson Prentice Hall, 2009. [11] P.-F. Loos, P.M.W. Gill, A tale of two electrons: correlation at high density, Chem. Phys. Lett. 500 (2010) 1–8, https://doi.org/10.1016/j.cplett.2010.09.019. [12] M. Taut, Two electrons in an external oscillator potential: particular analytic solutions of a coulomb correlation problem, Phys. Rev. A 48 (1993) 3561–3566, https://doi.org/10.1103/PhysRevA.48.3561. [13] M. Born, R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 389 (1927) 457–484, https://doi.org/ 10.1002/andp.19273892002. [14] M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, 1995. [15] I.N. Levine, Molecular Spectroscopy, John Wiley & Sons, Inc, 1975. [16] D.R. Hartree, The wave mechanics of an atom with a non-coulomb central field. Part I. theory and methods, Math. Proc. Camb. Philos. Soc. 24 (1928) 89–110, https://doi.org/10.1017/S0305004100011919. [17] V. Fock, Nherungsmethode zur Lsung des quantenmechanischen Mehrkrperproblems, Z. Phys. 61 (1930) 126–148, https://doi.org/10.1007/BF01340294. [18] J.C. Slater, The self consistent field and the structure of atoms, Phys. Rev. 32 (1928) 339–348, https://doi.org/ 10.1103/PhysRev.32.339. [19] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, G.A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A.V. Marenich, J. Bloino, B.G. Janesko, R. Gomperts, B. Mennucci, H.P. Hratchian, J.V. Ortiz, A.F. Izmaylov, J.L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V.G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J.A. Montgomery Jr., J.E. Peralta, F. Ogliaro, M.J. Bearpark, J.J.

36

[20] [21] [22] [23] [24] [25] [26] [27] [28]

[29]

[30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

2. An introduction to quantum chemistry

Heyd, E.N. Brothers, K.N. Kudin, V.N. Staroverov, T.A. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A.P. Rendell, J.C. Burant, S.S. Iyengar, J. Tomasi, M. Cossi, J.M. Millam, M. Klene, C. Adamo, R. Cammi, J.W. Ochterski, R.L. Martin, K. Morokuma, O. Farkas, J.B. Foresman, D.J. Fox, Gaussian~16 Revision C.01, 2016. M.A. Watson, N.C. Handy, A.J. Cohen, Density functional calculations, using Slater basis sets, with exact exchange, J. Chem. Phys. 119 (2003) 6475–6481, https://doi.org/10.1063/1.1604371. P.M.W. Gill, P.-F. Loos, D. Agboola, Basis functions for electronic structure calculations on spheres, J. Chem. Phys. 141 (2014), 244102, https://doi.org/10.1063/1.4903984. F.A. Pahl, N.C. Handy, Plane waves and radial polynomials: a new mixed basis, Mol. Phys. 100 (2002) 3199–3224, https://doi.org/10.1080/00268970210133206. D.C. Thompson, A. Alavi, A comparison of Hartree–Fock and exact diagonalization solutions for a model two-electron system, J. Chem. Phys. 122 (2005), 124107. A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications, New York, NY, 1996. C. Froese Fischer, General Hartree-Fock program, Comput. Phys. Commun. 43 (1987) 355–365, https://doi.org/ 10.1016/0010-4655(87)90053-1. J.M.L. Martin, Electron correlation: nature’s weird and wonderful chemical glue, Isr. J. Chem. (2021), https:// doi.org/10.1002/ijch.202100111. I. Mayer, L€ owdin’s pairing theorem and some of its applications, Mol. Phys. 108 (2010) 3273–3278, https://doi. org/10.1080/00268976.2010.508473. Y. Damour, M. V
eril, F. Kossoski, M. Caffarel, D. Jacquemin, A. Scemama, P.-F. Loos, Accurate full configuration interaction correlation energy estimates for five- and six-membered rings, J. Chem. Phys. 155 (2021), 134104, https://doi.org/10.1063/5.0065314. J.J. Eriksen, T.A. Anderson, J.E. Deustua, K. Ghanem, D. Hait, M.R. Hoffmann, S. Lee, D.S. Levine, I. Magoulas, J. Shen, N.M. Tubman, K.B. Whaley, E. Xu, Y. Yao, N. Zhang, A. Alavi, G.K.-L. Chan, M. Head-Gordon, W. Liu, P. Piecuch, S. Sharma, S.L. Ten-no, C.J. Umrigar, J. Gauss, The Ground State Electronic Energy of Benzene, 2020. ArXiv200802678 Phys. P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864–B871, https://doi.org/ 10.1103/PhysRev.136.B864. W. Kohn, L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (1965) A1133–A1138, https://doi.org/10.1103/PhysRev.140.A1133. R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, International Series of Monographs on Chemistry, Oxford University Press, 1994. K. Burke, J. Kozlowski, Lies My Teacher Told Me About Density Functional Theory: Seeing Through Them with the Hubbard Dimer, 2021. ArXiv210811534 Cond-Mat Physicsphysics. P. Morgante, R. Peverati, The devil in the details: a tutorial review on some undervalued aspects of density functional theory calculations, Int. J. Quantum Chem. 120 (2020), https://doi.org/10.1002/qua.26332. B. Kalita, K. Burke, Using Machine Learning to Find New Density Functionals, 2021. ArXiv211205554 Phys. Wikipedia search for “Quantum Chemistry” n.d. https://en.wikipedia.org/wiki/List_of_quantum_chemistry_ and_solid-state_physics_software (Accessed 14 January 2022). L. Talirz, L.M. Ghiringhelli, B. Smit, Trends in atomistic simulation software usage [article v1.0], Living J. Comput. Mol. Sci. 3 (2021), https://doi.org/10.33011/livecoms.3.1.1483. C.D. Sherrill, D.E. Manolopoulos, T.J. Martı´nez, A. Michaelides, Electronic structure software, J. Chem. Phys. 153 (2020), 070401, https://doi.org/10.1063/5.0023185. K. Gavroglu, A. Simo˜es, Neither Physics nor Chemistry: A History of Quantum Chemistry, MIT Press, 2011.

C H A P T E R

3 New high-performance QTAIM algorithms: From organic photovoltaics to catalyst materials Juan I. Rodrı´gueza,b,*, H
ector D. Morales-Rodrı´gueza, Emiliano Dorantes-Herna´ndeza, and ´ lvarez-Gonzagaa Omar A. A a

Escuela Superior de Fı´sica y Matema´ticas, Instituto Polit
ecnico Nacional, M
exico City, M
exico b CICATA-Quer
etaro, Instituto Polit
ecnico Nacional, Quer
etaro, M
exico *Corresponding author.

1. Introduction The Bader’s quantum theory of atoms in molecules (QTAIM) allows to precisely determine the quantum properties of a subgroup of atoms (e.g., functional groups, protein reactive sites, monomers, etc.) that form stable systems like molecules, solids, nanomaterials, etc. QTAIM also provides quantitative information about the chemical bond for each pair of bonded atoms in the system [1,2]. Historically, QTAIM was mainly applied to small molecules for developing “conceptual” chemistry because of the QTAIM’s demanding CPU time. Using the standard software/algorithms, a QTAIM calculation for a molecule typically took about the same time than solving the electronic structure problem (SCF procedure) in the best-case scenario [3,4]. The other handicap for extending the applications of QTAIM was the use of “external” nonuser-friendly software. The wave function or density needed to be obtained first from another source to be input in the standard QTAIM software. Recently, however, a new generation of high-performance grid-based QTAIM algorithms/ software has been developed [4–7]. These algorithms decrease the QTAIM computing time in several orders of magnitude. One of these methods was implemented inside the highperformance quantum chemistry computational package AMS (www.scm.com) [4,6]. Thus, QTAIM can now be used in real-world applications that usually involve hundreds or Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00005-0

37

Copyright # 2023 Elsevier Inc. All rights reserved.

38

3. New high-performance QTAIM algorithms

thousands of atoms. In this chapter, we briefly introduce the high-performance grid-based QTAIM algorithms. The computational efficiency of these algorithms is discussed. Then some real-world applications that used these algorithms are described.

2. QTAIM standard algorithms/software Within QTAIM, an atomic volume Ωa is associated with each atom A in a molecule (or solid). A property PA (e.g., charge, energy, polarizability, etc.) of atom A in the system is obtained as an expectation value of a corresponding property density p. In practice, PA is obtained as the integral of p over the corresponding atomic volume Ωa [1,2] (see Chapter 1), ð   ! ! PA ¼ p r dr: (3.1) Ωa

The properties obtained by formula (3.1) are the so-called QTAIM integrated or atom properties. The boundary of the atomic volumes Ωa are the zero-flux surfaces, which are surfaces SΩ in real space such that [1,2], rρ  b njSΩ ¼ 0,

(3.2)

where ρ is the system’s electron density and b n is a unit vector normal to the surface SΩ (see Chapters 1 and 2). The bottleneck of the QTAIM standard software was the construction of each atomic surface SΩ, which was needed for the integrals (2) to be performed (one integral per atom) [8–17]. Obviously, the larger the system, the more surfaces to be constructed. The numerical algorithms used to construct the zero-flux surfaces SΩ are quite sophisticated and ingenious but cumbersome and computationally demanding [8–17].

3. New generation of high-performance QTAIM algorithms In the high-performance QTAIM approaches, the atomic integration in Eq. (3.1) is carried out without the explicit construction of the atomic zero-flux surfaces (Eq. 3.2) [3–5,7,18]. Instead, an integration grid in real space is partitioned into subsets ωA. Each subset ωA is composed of all grid points contained in the atomic volume Ωa so that Eq. (3.1) is reduced to a simple quadrature over the grid points in ωA [3–5,7,18], ð     X ! ! ! PA ¼ p r dr ¼ wi p r i (3.3) Ωa

!

r i ωA

The key point is that there exists a way to associate each grid point to its corresponding atomic volume Ωa without knowing SΩ [3,4]! Following the steepest-density-ascent path from ! a grid point r i will end at the atom nucleus A the point belongs to. In principle, the steepest ascent path must be constructed for each grid point. In practice, however, there are means to avoid such path construction for several points. The approach adopted to perform this procedure is what characterize the differences of the high-performance methods. Overall, the CPU time is proportional to the grid size, but, because the zero-flux surfaces (Eq. 3.2) are

39

3. New generation of high-performance QTAIM algorithms

TABLE 3.1 Molecule H2O Fe(C5H5)2 C70

CPU timing for obtaining the QTAIM integrated properties for a set of representative molecules. Sym C2v D5h D5

Max. jLΩj 4

2.5  10

3

1.4  10

3

4.3  10

Toriginal

Tvec

Tvec+sym

T

15.55

1.32

0.37

0.11

4060

711.1

40.9

10.5

35,675

6948

737.9

257.2

Toriginal is the time of our previous grid-based version which was already faster than original algorithms (see ref. [18]). Tvec is the CPU time for a vectorized algorithm; Tvec+sym is the CPU time for vectorized algorithm and exploiting the molecular symmetry and T is the CPU time for the vectorized, symmetry adapted and parallel (4 processors used) version. Reproduced from Rodriguez, J.I., Bader, R.F.W., Ayers, P.W., Michel, C., Gotz, A., Bo, C., 2009. Chem. Phys. Lett. 472, 149.

not explicitly obtained, the CPU time of these grid-based methods is smaller than the CPU time of the standard methods by one to three orders of magnitude [3,4,18]. We implemented a high-performance method that filter several grid points that can be unambiguously assigned to a nearby atom and other points are so far from the atoms that they contribute minimally to the atomic properties. (See refs. [3,4,18] for details.) Vectorization and parallelization of the steepest-ascent path construction procedure and exploiting the molecule symmetry decreased several orders of magnitude as can be seen in Table 3.1 for a set of representative molecules [4]. As for obtaining the electron density topology, we adopted several of the gridbased procedures to also speed-up the determination of the density critical points [6]. The parallelization efficiency of the high-performance QTAIM method is shown in Fig. 3.1. So our QTAIM method effectively exploits the so-called distributed computing. We have developed a complete QTAIM high-performance module in the Amsterdam Modeling Suite

FIG. 3.1 Parallelization performance of our grid-based QTAIM high-performance method for the calculation of the electron density QTAIM topology for carbon nanotubes C294 and C294H18 as implemented in ADF. Reproduced from Rodriguez, J.I., 2013. J. Comp. Chem. 34, 681.

40

3. New high-performance QTAIM algorithms

(AMS. www.scm.com), which is complemented with the AMS powerful graphical user interface (GUI). Recently a similar grid-based algorithm was implemented on graphics processing units (GPU’s) which accelerated the CPU QTAIM implementation [19].

4. QTAIM real-world applications QTAIM is a powerful tool for studying and designing new materials in several technological applications. In this section, we introduce three of such applications where QTAIM plays a key role for understanding the new phenomena and/or properties.

4.1 Organic photovoltaics Organic photovoltaics (OPV) represent a clean, renewable, transparent, flexible and lowcost technology alternative to the silicon-based solar cells [20–22]. For years, the main handicap for the OPV to reach the commercial level has been its low efficiency [20–22]. Recently, however, OPV have reached efficiencies exceeding 13% [23–26]. In the so-called bulk-heterojunction OPV (BH-OPV), the active layer is formed by an inhomogeneous mixed of donors (usually a semiconductor polymer) and acceptors (usually a fullerene or an organic molecule). When a photon impacts the BH-OPV active layer, an exciton is created at the donor. The exciton diffuses until reaching the donor-acceptor junction, which provokes its dissociation generating thus the free charge carriers. The nature of the donor-acceptor pair has a strong impact on the OPV efficiency and other properties [20]. Quantum chemistry computational simulations have been used as a complementary tool to experiments for finding mechanism for increasing the OPV efficiency [27–31]. In particular, QTAIM calculations have been used to understand the stable structures of some donor-acceptor pairs and how they influence the OPV electronic and charge transfer properties [31,32]. Fig. 3.2A shows the stable structure of the donor-acceptor pair formed by the of poly(3-hexylthiophene) (P3HT) as the donor, and the [6,6]-phenyl-C 61-butyric acid methyl ester (PCBM) as the acceptor. The OPV whose active layer is a blend of P3HT and PCBM is one of the most popular and studied solar cell because of many interesting features (see reference [32]). Using QTAIM, it was determined that the main stabilizing factor of this supramolecular system is the Van der Waals (vdW) force between the P3HT (donor) and the PCBM (acceptor) [27,33,34]. This was confirmed by analyzing the QTAIM topology of the electron density (see the molecular graph of the P3HT/PCBM system in Fig. 3.2B). According to QTAIM, there is no covalent bonds between the P3HT and PCBM [33,34]. However, there are 37 intermolecular noncovalent bonds which are confirmed by the presence of the corresponding bond critical points (BCP). All these 37 intermolecular bonds can be classified as weak bonds because the value of the Laplacian of the electron density is greater than zero and have a small value of the electron density (ρ < 0.01 a.u.), with an internuclear distance that is significantly longer than the sum of the vdW radio of the corresponding atoms. Notice that the low values of the density produce the reddish color on the intermolecular bond paths (Fig. 3.2B). QTAIM can discriminate which functional group is participating in the global vdW interaction. For instance, it was determined that 83.8% of the total number of the noncovalent intermolecular bonds

4. QTAIM real-world applications

41

FIG. 3.2 Stable structure (A) for the P3HT/PCBM system obtained at the DFT/PBE-D3/TZP level of theory (see ref. [27]). Molecular graph (B) of the stable structure (A) (see ref. [33,34]). Nuclear critical points (NCP), bond critical points (BCP), and ring critical points (RCP) are in gray, red, and green, respectively. Bond path color scale is according to the value of the electron density: from blue/high to green/low.

are between atoms in the P3HT side chains and the PCBM (see refs. [33,34]). The impact of the P3HT/PCBM topological features on the charge transfer properties the OPV cell can be seen in Fig. 3.3. When vdW interaction are properly taken into account in the DFT calculations, the computed charge transfer rates might increase by 3 orders of magnitude, getting closer to the experimental reported rates (see Fig. 3.3) [31,32].

4.2 Nanostructures as wires For the molecular electronics to become a real technology is necessary first to construct every part of a circuit (wire, transistor, battery, logical gate, etc.) using only one or few molecules [35–38]. Using a single molecule as a wire has been the research topic of many experimental and theoretical works [35,39–50]. To model the lead-molecule-lead array, we proposed a simplified model system formed by two gold nanoclusters linked by a dithiol (see Fig. 3.4).

42

3. New high-performance QTAIM algorithms

FIG. 3.3

Charge transfer process in the P3HT/PCBM OPV. Up (below) is the P3HT/PCBM structure obtained with (without) considering van der Waals interaction in the DFT model. HOMO-LUMO orbitals and charge separation rates (kCS) are shown for each system as computed in ref. [31]. Here e and h+ represent an electron and a hole, respectively.

FIG. 3.4 Optimized structure (A) and molecular graph (B) of the gold nanocluster complex Au4SC3H6SAu4. The optimized structure was obtained at the relativistic ZORA-DFT/PBE/TPZ level of theory (see refs. [33,34]). Nuclear critical points (NCP), bond critical points (BCP) and ring critical points (RCP) are in gray, red, and green, respectively. Bond path color scale is according to the value of the electron density: from blue/high to green/low.

The structural, topological, electronic and optical properties were calculated the gold cluster complexes Au4SCnH2nSAu4 as a function of the dithiol size (n ¼ 2–5) [33,34,51]. A DFT-QTAIM studied was carried out to characterize the bonding between the organic and the metallic parts [33,34]. Here we briefly describe Au4SC3H6SAu4 as a representative

4. QTAIM real-world applications

43

model system (see Fig. 3.4). Overall, it was determined that sulfur acts as a bridge between the organic and metallic parts. Sulfurforms  a covalent   bond with carbon. However, it is bonded ! to gold by a closed-shell bond r2 ρ r BCP > 0 with some electrostatic character (qS ¼ ˚ . It was determined that sulfur 0.103e and qAu ¼ 0.082 e), with a bond distance around 2.2 A can be bonded with one (right-hand side) or two (left-hand side) gold atoms. In the latter case, the AudAu bond in the gold cluster is broken when the two gold atoms form bonds with sulfur. Notice that QTAIM predicts the SdS bond formation (Fig. 3.4B) which is not depicted in ball-and-stick molecular structure Fig. 3.4A [33,34]. It is because the bonds in the molecular structure are depicted based on a geometric criterion only while QTAIM predicts bonds from a quantum mechanics criterion. Actually, this SdS bond forms a closed trajectory (ring) with the SdC and CdC bonds in the dithiol, which is confirmed by the presence of the ring critical point (RCP) (see Fig. 3.4B). See references [31,51] for the detailed description of this system.

4.3 Catalyst materials Hydrogen fuel is a clean energy source that might soon replace gasoline in automobiles. Finding more efficient methods for obtaining hydrogen from water is still a challenging problem [52,53]. Optimizing the hydrogen evolution reaction (HER) process is the target of many research groups [54–56]. Recently, monolayer (2D) materials based on molybdenum disulfide (MoS2) have been investigated as catalyst for the HER, which are cheaper that the Pt-based catalyst [57–61]. Molybdenum disulfide is a layered semiconductor crystal, which belongs to transition metal dichalcogenide group [62]. Each monolayer is formed by three sublayers: one Mo sublayer is sandwiched between two S sublayers (see Fig. 3.5). In MoS2, each Mo atom is strongly bonded to three S atoms from each S sublayer, forming a trigonal prism with S atoms placed at the vertices and one Mo in the center. Sulfur atoms in the same sublayer also interact weakly with other S atoms belonging to a first-neighbor monolayer. The distribution of monolayers in the crystal yields two bulk MoS2 configurations: hexagonal (2H) and rhombohedral (3R). Bulk 2H has a bandgap energy of 1.23 eV, while monolayer has a bandgap of 1.85 eV [62–66]. A MoS2 monolayer primitive cell contains only one Mo atom and two S atoms. Lattice ˚ , while in bulk MoS2 lattice constant c is constants in the monolayer plane are a ¼ b ¼ 3.16 A ˚ 12.294 A [63,64,66]. In this section, five MoS2-based materials are studied: the pure MoS2 bulk and monolayer systems (see Fig. 3.5); the MoS2 monolayer doped with Co (Co/ MoS2) and Zn (Zn/ MoS2); and the MoS2 monolayer with vacancies (Vac/MoS2) (see Fig. 3.6). For the doped and vacancy systems, a 4  4 supercell is considered. For these systems, a Mo is substituted by Co, or Zn, or a vacancy (see Fig. 3.6). The geometry of all systems was fully optimized using the BAND-software, which is the Amsterdam Modeling Suite (AMS) software branch for periodic systems. A single point calculation was performed at the optimized geometry to obtain the QTAIM properties using our highperformance algorithm [4,6]. For the DFT calculations, the generalized gradient approximation Perdew-Burke-Ernzerhof (GGA-PBE) [67] was used along with a triple-ζ Slater-type basis set with a polarization function (TZP). Relativistic effects were considered by using the scalar zeroth order regular approximation (scalar ZORA) [68]. The van der Waals interactions were considered using the dispersion Grimme (D3) approximation [69]. According to the QTAIM bond information shown in Table 3.2 and Fig. 3.7, all the bonds presented for pure and doped MoS2-based materials are noncovalent since r2ρ > 0 [1,2]. For pure MoS2 bulk and monolayer, QTAIM predicts six ModS bonds per each Mo, where two

FIG. 3.5 Primitive cell (A) and optimized geometry (C) of the MoS2 bulk. Primitive cell (B) and optimized geometry (D) of the 2D MoS2 monolayer.

FIG. 3.6 Optimized structures of MoS2 system (A) with vacancy (Vac/MoS2), (B) doped with Zn (Zn/MoS2) and (C) doped with Co (Co/MoS2).

45

4. QTAIM real-world applications

TABLE 3.2

Bond information for all MoS2-based materials.

System

Atom 1 [charge]

Atom 2 [charge]

B. Dist.

λ1b

λ2b

λ3b

ρb

—2ρb

3D MoS2

Mo(2) [0.308]

S(3) [0.154]

2.418

0.081

0.076

0.282

0.082

0.125

S(3) [0.154]

S(5) [0.154]

3.499

0.006

0.006

0.038

0.009

0.026

2D MoS2

Mo(14) [0.308]

S(45) [0.154]

2.420

0.080

0.076

0.283

0.081

0.127

Vac/MoS2

Mo(13) [0.346]

S(16) [0.150]

2.431

0.079

0.077

0.279

0.080

0.122

Mo(13) [0.346]

S(45) [0.159]

2.386

0.092

0.088

0.291

0.088

0.111

S(45) [0.159]

S(46) [0.159]

3.224

0.011

0.007

0.059

0.015

0.041

Mo(10) [0.322]

S(18) [0.153]

2.441

0.073

0.070

0.277

0.077

0.134

Mo(10) [0.322]

S(41) [0.188]

2.416

0.083

0.080

0.279

0.083

0.116

Zn(48) [0.336]

S(41) [0.188]

2.540

0.033

0.031

0.178

0.044

0.114

Mo(3) [0.331]

S(18) [0.158]

2.419

0.077

0.073

0.278

0.082

0.128

Mo(3) [0.331]

S(23) [0.086]

2.411

0.083

0.076

0.284

0.084

0.125

Co(16) [0.110]

S(23) [0.086]

2.250

0.085

0.074

0.303

0.084

0.143

Zn/MoS2

Co/MoS2

Bond distances are in Angstrom and the rest of quantities are in a.u.

FIG. 3.7 Topological properties of MoS2-based systems.

46

3. New high-performance QTAIM algorithms

bonds, corresponding to S atoms one below the other, have distances 1.29% higher than the other four ModS atoms, yielding differences in topological values up to 10% with respect to values in Table 3.2. Additionally, interlayer SdS bonds are predicted between two adjacent monolayers in the pure MoS2 bulk. Each S atom is bonded to three S atoms belonging to an adjacent monolayer (Fig. 3.8A). The bond distance for the interlayer SdS bonds is equal to

FIG. 3.8 Crystal graphs of (A) 3D MoS2, (B) 2D MoS2, (C) Vac/MoS2, (D) Zn/MoS2 and (E) Co/MoS2.

5. Conclusions

47

˚ and a density value lower than 0.01 at the BCP (see Table 3.2), which indicates that SdS 3.49 A is a week vdW bond. As for Vac/MoS2 monolayer, the S atoms neighboring the vacancy loose one ModS bond and forms two SdS bonds each other (Fig. 3.8C). The bond paths corresponding to these new SdS bonds form an equilateral triangle, which is confirmed by the presence of a ring critical point (RCP). Notice that this phenomenon appears in both S-atom sublayers (see Fig. 3.8A). The nature of these SdS bonds is similar the SdS bonds formed in the S-atoms sublayer (see Table 3.2 and Fig. 3.7). The ModS bonds neighboring the vacancy conserve the same features than without the vacancy (see Table 3.2 and Fig. 3.7). In the case of Zn/MoS2 system, QTAIM predicts six ZndS bonds, which shows that Zn has the same coordination number than the substituted Mo. However, as shown in Fig. 3.7, these ZndS bonds are weaker than ModS bonds. The density Hessian eigenvalues for the ZndS are closer to zero and electron density is smaller than for the ZndS bonds. The ZndS bond distance is about 5% longer than for ModS bonds. In case of Co/MoS2 monolayer, Fig. 3.7 shows that the topology of the electron density has the same general features for both doped systems. Partial charge analysis shows that S atoms has negative charge (0.154e) and Mo atoms positive charge (0.308e), which is consistent with the atomic electronegativity scale. Modifying the pure MoS2 by doping or a vacancy produce only small charge redistribution around the doped/vacancy zone mainly (see Fig. 3.9C–E). The most substantial change is experienced by Mo in the Vac/MoS2 system, which get a positive charge equal to 0.346e. Interestingly, the S neighboring atom to the vacancy conserve their original partial charge by the oxidation of the Mo atoms bonded to it. This process produces a strong electrophilic zone around the Mo atoms close to the vacancy. In the contrary, the inclusion of Zn atom yields a nucleophilic zone around the S atoms bonded to it (see Fig. 3.9D). In Fig. 3.9E, it is shown that Co atoms are reduced by the six S atoms in its neighborhood. It is well known that the ability to accept additional electron density and redistribute the charge is essential for improving the catalytic activity of materials. Our DFT-QTAIM calculations confirmed these conditions in the MoS2 materials studied here.

5. Conclusions In this chapter, we have briefly described the new generation of high-performance gridbased QTAIM algorithms. These methods reduce the CPU time for a typical QTAIM calculation for up to 3 orders of magnitude. This CPU timing can be further decreased proportionally with the number of processors used in the algorithms’ parallel version [4,6]. The parallel CPU time scaling is quasi-linear with the system size [4,6]. The GPU’s version of some of these algorithms have been developed and tested recently [19]. Using these QTAIM highperformance algorithms, systems with hundreds or thousands of atoms can be studied allowing to apply QTAIM to real-world applications.

48

3. New high-performance QTAIM algorithms

FIG. 3.9 Partial charges of (A) 3D MoS2, (B) 2D MoS2, (C) Vac/MoS2, (D) Zn/MoS2 and (E) Co/MoS2.

References [1] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. [2] C.F. Matta, R.J. Boyd (Eds.), The Quantum Theory of Atoms in Molecules. From Solid State to DNA and Drug Design, Wiley-VCH, Weinheim, 2007. [3] J.I. Rodriguez, Efficient Grid-Based Techniques for Density Functional Theory (Ph.D. dissertation), McMaster University, Hamilton, ON, Canada, 2008. [4] J.I. Rodriguez, R.F.W. Bader, P.W. Ayers, C. Michel, A. Gotz, C. Bo, Chem. Phys. Lett. 472 (2009) 149. [5] G. Henkelman, A. Arnalsson, H. Jonsson, Comput. Mater. Sci. 36 (2006) 354. [6] J.I. Rodriguez, J. Comput. Chem. 34 (2013) 681. [7] E. Sanville, S.D. Kenny, R. Smith, G. Henkelman, J. Comput. Chem. 28 (2007) 899. [8] F.W. Biegler-Konig, T.T. Nguyen-Dang, Y. Tal, R.F.W. Bader, A.J. Duke, J. Phys. B: At. Mol. Phys. 14 (1981) 2739. [9] F.W. Biegler-Koning, R.F.W. Bader, T.H. Tang, J. Comput. Chem. 3 (1982) 317. [10] J. Cioslowski, Chem. Phys. Lett. 194 (1992) 73. [11] J. Cioslowski, A. Nanayakkara, M. Challacombe, Chem. Phys. Lett. 203 (1993) 137.

References

49

[12] J. Cioslowski, B.B. Stefanov, Mol. Phys. 84 (1995) 707. [13] T.A. Keith, Molecules in Magnetic Fields (Ph.D. dissertation), McMaster University, Hamilton, ON, Canada, 1993. [14] P.L.A. Popelier, Mol. Phys. 87 (1996) 1169. [15] P.L.A. Popelier, Comput. Phys. Commun. 108 (1998) 180. [16] M. Rafat, P.A.L. Popelier, J. Comput. Chem. 28 (2007) 2602. [17] B.B. Stefanov, J. Cioslowski, J. Comput. Chem. 16 (1995) 1394. [18] J.I. Rodriguez, A.M. Koster, P.W. Ayers, A. Santos-Valle, A. Vela, G. Merino, J. Comput. Chem. (2009), https:// doi.org/10.1002/jcc.21134. in press. [19] R. Hernandez-Esparza, S.M. Mejia-Chica, A.D. Zapata-Escobar, A. Guevara-Garcia, A. Martinez-Melchor, J.M. Hernandez-Perez, R. Vargas, J. Garza, J. Comput. Chem. 35 (2014) 2272. [20] C.J. Brabec, V. Dyakonov, J. Parisi, N.S. Sariciftci (Eds.), Organic Photovoltaics: Concepts and Realization, Springer, Berlin, 2003. [21] S.S. Sun, N.S. Sariciftci (Eds.), Organic Photovoltaics: Mechanisms, Materials, and Devices, Taylor & Francis Group, Boca Raton, Florida, 2005. [22] http://www.npr.org/blogs/thetwo-way/2014/11/10/363023227/solar-bike-path-opens-this-week-inthenetherlands, Last accessed on 3/30/2020, 2014. [23] L. Hong, R. Yu, H. Yao, Nat. Commun. 9 (2018) 4645. [24] R. Wang, X. Zhang, D. Zhang, Nano Energy 56 (2018) 494. [25] S. Xie, R. Wang, D. Zhang, Nano Energy 51 (2018) 736. [26] H. Yao, W. Zhao, S. Li, J. Am. Chem. Soc. 139 (21) (2017) 7148. [27] I. Gutierrez-Gonzalez, J.I. Rodriguez, B. Molina-Brito, A.W. Gotz, F.L. Castillo-Alvarado, Chem. Phys. Lett. 612 (2014) 234, https://doi.org/10.1016/j.cplett.2014.08.030. [28] A.W. Gotz, J.I. Rodriguez, F.L. Castillo-Alvarado, D.E. Trujillo-Gonzalez, Int. J. Quantum Chem. 119 (2019), e25883. [29] T. Liu, D. Cheung, A. Troisi, Phys. Chem. Chem. Phys. 13 (2011) 21461. [30] T. Liu, A. Troisi, J. Phys. Chem. C 115 (5) (2011) 2406, https://doi.org/10.1021/jp109130y. [31] J.P. Martinez, J.I. Rodriguez, D.E. Trujillo-Gonzalez, A.W. Gotz, F. Castillo-Alvarado, J. Phys. Chem. C 121 (2017) 20134, https://doi.org/10.1021/acs.jpcc.7b05107. [32] J.I. Rodriguez, A. G€ otz, Structure, electronic, and charge transfer properties of organic photovoltaics from density functional theory methods, in: J.K. Roy, S. Kar, J. Leszczynski (Eds.), Development of Solar Cells, Theory and Experiments, Springer, USA, 2021. [33] J.I. Rodriguez, C. Matta, C. Uribe, A.W. G€ otz, F.L. Castillo-Alvarado, B. Molina-Brito, Chem. Phys. Lett. 644 (2016) 157, https://doi.org/10.1016/j.Cplett.2015.11.052. [34] J.I. Rodriguez, E.A. Uribe, M.I. Baltazar-Mennndez, J. Autschbach, F.L. Castillo-Alvarado, I. Gutierrez-Gonzalez, Size evolution relativistic DFT-QTAIM study on the gold cluster complexes Au 4-S-CnH 2n -S0 -Au 4 0 (n ¼ 2–5), Chem. Phys. Lett. 660 (2016) 287–294. [35] S.V. Aradhya, L. Venkataraman, Single-molecule junctions beyond electronic transport, Nat. Nanotechnol. 8 (6) (2013) 399–410. http://www.nature.com/doifinder/10.1038/nnano.2013.91. [36] R.M. Metzger, Unimolecular electronics, Chem. Rev. 115 (11) (2015) 5056–5115, https://doi.org/10.1021/ cr500459d. 25950274. [37] M. Ratner, A brief history of molecular electronics, Nat. Nanotechnol. 8 (6) (2013) 378–381. http://www.nature. com/doifinder/10.1038/nnano.2013.110. [38] L. Sun, Y.A. Diaz-Fernandez, T.A. Gschneidtner, F. Westerlund, S. Lara-Avila, K. Moth-Poulsen, Singlemolecule electronics: from chemical design to functional devices, Chem. Soc. Rev. 43 (21) (2014) 7378–7411, https://doi.org/10.1039/C4CS00143E. [39] R. Baer, D. Neuhauser, Ab initio electrical conductance of a molecular wire, Int. J. Quantum Chem. 91 (3 SPEC) (2003) 524–532. [40] M. Brandbyge, J.L. Mozos, P. Ordejon, J. Taylor, K. Stokbro, Density-functional method for nonequilibrium electron transport, Phys. Rev. B: Condens. Matter Mater. Phys. 65 (16) (2002) 1654011–16540117. [41] P.S. Damle, A.W. Ghosh, S. Datta, Unified description of molecular conduction: from molecules to metallic wires, Phys. Rev. B: Condens. Matter Mater. Phys. 64 (20) (2001) 1–4. [42] P.A. Derosa, J.M. Seminario, Electron transport through single molecules: scattering treatment using density functional and green function theories, J. Phys. Chem. B 105 (2) (2001) 471–481.

50

3. New high-performance QTAIM algorithms

[43] Z.Q. Fan, Z.H. Zhang, W. Tian, X.Q. Deng, G.P. Tang, F. Xie, Altering regularities on resistances of doped Au-alkanedithiol-Au junctions, Org. Electron. 14 (10) (2013) 2705–2710, https://doi.org/10.1016/j. orgel.2013.07.018. [44] J.R. Heath, M.A. Ratner, Molecular electronics improvements in our understanding of how molecules transport charge, and how they interface to the macroscopic world, are fueling new devices and applications, Phys. Today 56 (May) (2003) 43. [45] J. Heurich, J.C. Cuevas, W. Wenzel, G. Schon, Electrical transport through single-molecule junctions: from molecular orbitals to conduction channels, Phys. Rev. Lett. 88 (25) (2002) 4. [46] Y. Isshiki, T. Nishino, M. Kiguchi, Fluctuation in Interface and electronic structure of single- molecule junctions investigated by current versus bias voltage characteristics, J. Am. Chem. Soc. 140 (2018) 3760–3767. [47] C. Li, I. Pobelov, T. Wandlowski, A. Bagrets, A. Arnold, F. Evers, Charge transport in single AujalkanedithioljAu junctions: coordination geometries and conformational degrees of freedom, J. Am. Chem. Soc. 130 (17) (2008) 318–326. http://arxiv.org/abs/0802.2407. [48] J. Li, G. Speyer, O.F. Sankey, Conduction switching of photochromic molecules, Phys. Rev. Lett. 93 (24) (2004) 1–4. [49] E. Lortscher, Wiring molecules into circuits, Nat. Nanotechnol. 8 (6) (2013) 381–384, https://doi.org/10.1038/ nnano.2013.105. [50] A. Nitzan, M.A. Ratner, Electron transport in molecular wire junctions, Science 300 (5624) (2003) 1384–1389. http://science.sciencemag.org/content/300/5624/1384. [51] J.I. Rodriguez, U.A. Vergara-Beltran, J. Autschbach, Chem. Phys. Lett. 732 (2019), 136625. [52] M. Ji, J. Wang, Review and comparison of various hydrogen production methods based on costs and life cycle impact assessment indicators, Int. J. Hydrog. Energy 46 (2021) 38612–38635. [53] J.A. Turner, Sustainable hydrogen production, Science 305 (2004) 972–974. [54] J.K. Norskov, T. Bligaard, A. Logadottir, J.R. Kitchin, J.G. Chen, S. Pandelov, U. Stimming, Trends in the exchange current for hydrogen evolution, J. Electrochem. Soc. 152 (3) (2005) J23–J26. [55] Z.W. Seh, J. Kibsgaard, C.F. Dickens, I. Chorkendorff, J.K. Norskov, T.F. Jaramillo, Combining theory and experiment in electrocatalysis: insights into materials design, Science 355 (2017) 146. [56] Y. Zheng, Y. Jiao, M. Jaroniec, S.Z. Qiao, Advancing the electrochemistry of the hydrogen-evolution reaction through combining experiment and theory, Angew. Chem. Int. Ed. 54 (1) (2015) 52–65. [57] B. Hinnemann, P.G. Moses, J. Bonde, K.P. Jorgensen, J.H. Nielsen, S. Horch, I. Chorkendorff, J.K. Norskov, Biomimetic hydrogen evolution: Mos2 nanoparticles as catalyst for hydrogen evolution, J. Am. Chem. Soc. 127 (2005) 5308–5309. [58] T.H.M. Lau, X.W. Lu, J. Kulhav y, S. Wu, L. Lu, T.S. Wu, R. Kato, J.S. Foord, Y.L. Soo, K. Suenaga, et al., Transition metal atom doping of the basal plane of Mos 2 monolayer nanosheets for electrochemical hydrogen evolution, Chem. Sci. 9 (21) (2018) 4769–4776. [59] P. Liu, J. Zhu, J. Zhang, K. Tao, D. Gao, P. Xi, Active basal plane catalytic activity and conductivity in zn doped mos2 nanosheets for efficient hydrogen evolution, Electrochim. Acta 260 (2018) 24–30. [60] Morales-Rodriguez, H.D., Vargas, R., Autschbach, J., Rodriguez, J.I., n.d.. “Activation of the Basal Plane of MoS 2 Monolayer for Hydrogen Evolution: A DFT-QTAIM Study” (in preparation). [61] T. Sun, J. Wang, X. Chi, Y. Lin, Z. Chen, X. Ling, C. Qiu, Y. Xu, L. Song, W. Chen, et al., Engineering the electronic structure of mos2 nanorods by n and mn dopants for ultra-efficient hydrogen production, ACS Catal. 8 (8) (2018) 7585–7592. [62] C. Klein, J.D. Dana, C.S. Hurlbut, Manual of Mineral Science, twenty-second ed., Wiley, 2002. [63] U. Krishnan, M. Kaur, K. Singh, M. Kumar, A. Kumar, A synoptic review of MoS 2: synthesis to applications, in: Superlattices and Microstructures, vol. 128, Academic Press, 2019, pp. 274–297, https://doi.org/10.1016/j. spmi.2019.02.005. [64] A.K. Singh, P. Kumar, D.J. Late, A. Kumar, S. Patel, J. Singh, 2D layered transition metal dichalcogenides (MoS2): synthesis, applications and theoretical aspects, Appl. Mater. Today 13 (2018) 242–270, https://doi.org/10.1016/ j.apmt.2018.09.003. Elsevier Ltd. [65] A. Splendiani, et al., Emerging photoluminescence in monolayer MoS2, Nano Lett. 10 (4) (2010) 1271–1275, https://doi.org/10.1021/nl903868w. [66] J.A. Wilson, A.D. Yoffe, The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties, Adv. Phys. 18 (73) (1969) 193–335, https://doi.org/10.1080/ 00018736900101307.

References

51

[67] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868. [68] E. van Lenthe, J.G. Snijders, E.J. Baerends, The zero-order regular approximation for relativistic effects: the effect of spin-orbit coupling in closed shell molecules, J. Chem. Phys. 105 (15) (1996) 6505–6516, https://doi.org/ 10.1063/1.472460. [69] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132 (2010), https://doi.org/ 10.1063/1.3382344.

C H A P T E R

4 Structural and bond evolutions during a chemical reaction Pablo Carpio-Martı´nez and Fernando Cort
es-Guzma´n Instituto de Quı´mica, Universidad Nacional Auto´noma de M
exico, M
exico City, M
exico

1. Introduction In 1979 Bader, Nguyen-Dang and Tal presented the definition of molecular structure and its evolution, incorporating catastrophe theory into the topological analysis to understand structural changes [1]. Catastrophe theory describes the evolution of a system when gradually changing parameters produce sudden effects (called catastrophes) due to unexpected discontinuities [2]. Within the context of chemical change, such parameters might be related to the nuclear configurations associated with an evolving variable such as time or an internal reaction coordinate. The definition of chemical change is one of the most important contributions of QTAIM as it provides deep insight into the structural changes involved in a reaction mechanism. This theoretical approach can determine the exact point where a bond is either broken or formed during a reaction [3]. Bader et al. used the electron density and its gradient (see Eq. 4.1) to accurately define the chemical structure and their changes subject to interactions between chemical species [4]. In general, the fields generated from the one-electron density (as the Laplacian of electron density, kinetic energy densities, spin densities, among others) present catastrophe processes, which allows one to follow structural changes of the reactants along a specific reaction path. However, this is not the case for the fields obtained from the two-electron density. Despite the efforts to analyze a reaction mechanism with two-electron density properties (such as the electron delocalization between two atoms) it is impossible to determine the bond breaking point between two chemical species without defining an arbitrary limit first. This latter is mainly due to the smooth and continuous nature of the functions. Thus, it is necessary to use an arbitrary delocalization value to define the absence of bonding. Alternatively, catastrophe theory might provide valuable information on the evolution of chemical bonds by analyzing two functions, i.e., the Laplacian of the electron density and the electron

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00016-5

53

Copyright # 2023 Elsevier Inc. All rights reserved.

54

4. Structural and bond evolutions during a chemical reaction

delocalization function. Both functions exhibit how the atomic valence shell changes during a reaction or a conformational change. In this chapter, we review a few novel perspectives concerning the structural evolution of molecules and their simultaneous applications to different systems and reactions. —ρðrÞ ¼

∂ρðrÞ ^ ∂ρðrÞ ^ ∂ρðrÞ ^ i+ k¼0 j+ ∂x ∂y ∂z

(4.1)

2. Energetic evolution IUPAC defines a reaction mechanism as A detailed description of the process leading from the reactants to the products of a reaction, including a characterization as complete as possible of the composition, structure, energy, and other properties of reaction intermediates, products, and transition states [5]. The energetic evolution of a reaction is related to the energy changes from the reactants to the products through the transition states on the potential energy surface of the reacting system. From the theoretical standpoint, this energetic evolution is determined by the following steps: (i) calculating the potential energy surface in terms of the atomic positions; (ii) obtaining the transition state, reactants, and products; and (iii) unequivocally determining the reaction path connecting the relevant stable points and the transition state point [6]. The transition state is a saddle point with positive curvature in all degrees of freedom except the one which corresponds to the crossing of the barrier for which it is negative. Each transition state is related to an activated complex, which is defined as the collection of intermediate structures in a chemical reaction that persists while bonds are breaking and new bonds are forming. It, therefore, does not have any specific/defined representation, but rather a range of unstable configurations. Thus a collection of atoms passes through clearly defined products and reactants [7]. The objective of analyzing structural evolution is to determine the connectivity between atoms/structures and their corresponding configurational changes in the activated complex.

3. Geometry and structure In the study of chemical change, one needs to define two important concepts and make the differences between them; the first one is molecular geometry and the second is molecular structure. The geometry of a molecular system is defined as the set of nuclear coordinates, i.e., X ¼ X(X1, X2, X3, …, XN) which represent the exact position in the space of all nuclei. Any infinitesimal change in the coordinate values leads to a different geometry, X0 . To more accurately define the molecular structure, we need to use a connectivity or bonding scheme, as the approach developed by Richard Bader [8]. The Theory of Atoms in Molecules defines chemical structure based on the topology of the electron density, which is summarized in terms of its extrema or critical points (where —ρ ¼ 0) and the gradient paths associated with them. The Hessian matrix (matrix of second-order partial derivatives of the electron density with respect

55

3. Geometry and structure

Diagonalization process of the Hessian matrix of ρ. Hessian matrix of ρ evaluated at the critical point rc (left), diagonalization of the Hessian matrix by a unitary transformation (middle) and diagonalized Hessian matrix (right).

FIG. 4.1

to position coordinates) provides a way to distinguish between a local minimum, a local maximum, or a saddle point. This matrix can be diagonalized by a unitary transformation (see Fig. 4.1), where U is a unitary matrix constructed from a set of three eigenvalue equations Aui ¼ λiui in which ui is the ith column vector (eigenvector) in U. U1AU ¼ Λ transforms the Hessian into its diagonal form, in which λ1, λ2, and λ3 are the curvatures of the density [9]. Critical points are classified according to their rank, ω and signature σ. The rank is the number of nonzero curvatures of electron density at a critical point. The signature is the algebraic sum of the signs of the curvatures. The four types of stable critical points, having three nonzero eigenvalues, are shown in Table 4.1. A set of gradient trajectories is associated with each critical point. They start at infinity or at other critical points with positive curvatures and terminate at other critical points with negative curvatures. In the bond critical points, a pair of trajectories originate at each critical point and terminate at the neighboring nuclei, defining a line of maximum density, called a bond path [4]. The linked network of bond paths defines a molecular graph that recovers the molecular structure [10]. Within the QTAIM bonding scheme, for any configuration X, there is a charge density ρ(r, X) and its associated gradient vector field —ρ(r, X). Then, any X0 nuclear configuration in the neighborhood of another configuration should possess the same structure while it has a different geometry. The same nuclei should be identically linked to the same network of bond paths in both X and X0 . Then, two molecular graphs are homeomorphic (topologically equivalent), if they have the same connectivity, i.e., they are continuously mapped into each other (one TABLE 4.1 Critical point

Features of the critical points of electron density. Chemical structure element

Features

Extreme

(3,3)

Three negative curvatures

ρ is a local maximum

Nuclear critical point (NCP)

(3,1)

Two negative curvatures

ρ is a maximum in a plane defined by two axis but a minimum along the third axis which is perpendicular to the plane

Bond critical point (BCP)

(3,+1)

Two positive curvatures

ρ is a minimum by two axis but a maximum along the third axis which is perpendicular to the plane

Ring critical point (RCP)

(3,+3)

Three positive curvatures

ρ is a local minimum

Cage critical point (CCP)

56

4. Structural and bond evolutions during a chemical reaction

by one) [11]. On the other hand, the molecular graph undergoes discontinuous and abrupt changes if the nuclei displace into critical configurations viz. where the connectivity changes. In this scenario, one can either form or break a bond and consequently change one structure into another. In this way, it is possible to define a structural region as a set of molecular geometries with the same connectivity. The mathematical definition of a critical point, —ρ(r) ¼ 0, is in some cases too restrictive to define all types of chemical interactions including weakly interacting systems. Lane et al. [12] showed that the value λ2ρ, corresponding to an attractive hydrogen bonding interaction, presents the same shape in two cases where a BCP is present or absent. The only difference arises in the minimum values of the reduced density gradient. In some cases, it goes down to zero, whereas, in others, it is positive and small. The relaxation of the critical point criteria seems to be necessary yet, it involves the problem of defining an acceptable deviation range from zero. As expected, the theoretical level induces changes in the description of weak interactions. Thus, diffusion functions and the inclusion of dispersion are needed to obtain the right molecular graph. This fact raises questions and provokes confusion on the interpretation of the structure and its change [13].

4. Catastrophe theory A catastrophe is defined by a set of critical points of a function generated by a germ function, which is mathematically unstable because their first derivative and higher derivatives are zero [14]. When a germ function is perturbed, a series of stable and unstable classes of functions are generated. The topology of functions in different classes is unique. The points in catastrophe space at the boundaries between points belonging to the stable classes correspond to transition structures between stable functions. These points are called catastrophe points, and when a function in one stable class is smoothly varied and crosses the boundary between two stable classes, the topological character of the function suddenly changes into a function of the second stable class [1]. To illustrate a catastrophe, we used a hypothetical planar triatomic molecule A-B-C (see Fig. 4.2), as Melnick and Whitehead did [2]. The position of nuclei and the electron bond paths in any triatomic charge density distribution are described by Eq. (4.2), where the germ function used is V ¼ y2 z  13 z3 and the rest is the perturbation, with control parameters u, υ and ω, which define the three-dimensional control space. f(y, z), y and z form the three dimensions of the electronic charge distribution of configuration space. 1 f ðy, zÞ ¼ y2 z  z3 + wy2 + wy + vz 3

(4.2)

Each point in configuration space corresponds to a unique geometrical configuration of the nuclei. The configuration space is partitioned into several regions, where the movement from one point to another is controlled by the parameters u, υ, and ω. A smooth variation in triatomic structure (lengthening or shortening of bonds or variation of bond angle) can fall in an unstable point, called a catastrophe point, associated with a transition structure, where bonds suddenly appear or disappear.

57

5. Structural evolution and chemical change

a)

b)

FIG. 4.2 (A) Configuration space of the three-atomic planar molecule with control parameters u, υ, and ω.

(B) Cross-section of the space where ω ¼ 0, showing the catastrophe set for the A-A-A system. (C) Cross-section of the space, where ω 6¼ 0, showing the catastrophe set for the B-A-B system [2].

When ω ¼ 0, the configuration space is partitioned into three structural regions. Points that form the partitioning lines correspond to transitions between two stable structures; thus, as u and υ are smoothly varied, a catastrophe line is reached, and a transition structure results. When ω 6¼ 0, there are four structural regions corresponding to relatively stable triatomic structures. A structure is stable when a small variation of the control parameters u and/or υ does not change the basic structure. The value of the ω parameter depends on the electronic properties of the systems and is related to the irreducible representation within the molecular symmetry group [2].

5. Structural evolution and chemical change The structural evolution of a chemical reaction may be defined as the study of a chemical reaction mechanism partitioning nuclear configuration space into a finite number of structural regions. This approach establishes the bond breaking and bond formation sequence in a reaction mechanism by the catastrophe points within a reaction path. There are only two types of catastrophe processes used to understand chemical changes: bifurcation and conflict mechanisms. The former is associated with the ring-opening or closure, with the occurrence or annihilation of a BCP and an RCP, and the latter with the group migration, where bond goes through an unstable point between two atoms [1]. Fig. 4.3 shows examples of these two catastrophe mechanisms. The expression 13 x3 + ax is an example of the bifurcation mechanism, controlled by the parameter a. This function is almost linear with a < 0, but it finds a catastrophe point at a ¼ 0, and from this point, a maximum and a minimum appear. The expression 14 x4 + 12 x2 + bx is an example of the conflict mechanism, where b is the control parameter. At b ¼ 0, this function is unstable, and any change to positive or negative values causes the appearance of a minimum.

58

4. Structural and bond evolutions during a chemical reaction

FIG. 4.3 Mathematical examples of bifurcation and conflict mechanisms. (A) Bifurcation mechanism represented by the function x3 + ax, a ¼ 5 (dashed), a ¼ 0 (solid line), and a ¼  5 (dotted line). (B) Conflict mechanism represented by the function f ðxÞ ¼ 14 x4 + 38 x2 + ax, a ¼  5 (dashed), a ¼ 0 (solid line), and a ¼ 5 (dotted line).

A ring structure is formed or destroyed in a bifurcation catastrophe process, where a BCP and an RCP appear or annihilate. In contrast, the conflict catastrophe consists of two nuclei competing for a single BCP, resolved by an infinitesimal distortion of the conflict geometry. The conflict process is associated with an atom or a group migration, occurring through an unstable structure. Fig. 4.4 shows these two processes for a three-group system. Each of these processes is associated with an evolution diagram that describes the evolution of a catastrophe process (see Fig. 4.5). It depicts the change in the electron density values as a function of the evolution coordinate, i.e., time or internal reaction coordinate (IRC). In the bifurcation catastrophe process, one can see that two CP appear at a specific evolution point, a BCP, and an RCP, with the same electron density value, which differs as the evolution goes forward. The evolution diagram for a conflict catastrophe shows an abrupt change in the electron density value associated with the BCP change.

FIG. 4.4 (A) Conflict and (B) bifurcation catastrophe mechanisms [15].

59

6. Evolution of electron density

BCP Ubcp

Ubcp

BCP_a

RCP

BCP_b

a)

Evolution coordinate

b)

Evolution coordinate

FIG. 4.5 Evolution diagrams for conflict (A) and bifurcation (B) mechanisms.

6. Evolution of electron density There are various reaction mechanisms elucidated based on the analysis catastrophe process of topological properties of electron density. Zheng et al. have contributed to the structural evolution analysis of several isomerization reactions: • • • • •

Transposition of HXO2 as the chlorous acid: HOOX ! HOXO ! HXO2 (X ¼ Cl, Br, I) [16]. The stability and isomerizations of CH3SO isomers [17]. The reactions of INCO to IOCN, INCS to ISCN, and INCSe to ISeCN [18]. The reactions of FNCO to FOCN, FNCS to FSCN, and FNCSe to FSeCN [19]. The reactions of BrNCO to BrOCN, BrNCS to BrSCN, and BrNCSe to BrSeCN [20].

The authors found both catastrophe mechanisms in isomerization reactions and showed that the energetic barriers can be split into structural regions along the internal reaction coordinate as shown in Fig. 4.6 for the HClO2 isomerization. Organic reactions have been described in terms of the catastrophe mechanism. Herna´ndezTrujillo et al. analyzed the electron density evolution along the iminol-amide tautomerism and keto-enol prototropisms of guanine and thymine, respectively [21]. Heard et al. analyzed the HF, HCl, HBr, and HOH elimination reactions of halohydrocarbons and halohydroalcohols [22]. Contini et al. analyzed the cycloaddition reaction of the morpholino enamines of N-methylpiperidone and N-methyl tropinone with sulfonylazides [23]. Xu et al. gave detailed reaction mechanisms of the simplest Criegee intermediate CH2OO and its derivatives with methane, which play a key role in controlling the atmospheric budget of hydroxyl radical, organic acids, and secondary organic aerosols [24]. Grabowski found that the tetrel bond–σ-hole bond is a preliminary stage of the SN2 reaction of the F4C⋯NCH complex, afterward a bifurcation catastrophe process produces the CdN σ bond involved in the nucleophilic substitution [25]. Rode et al. investigated the [2 + 2] cycloaddition reaction paths using the changes of the ellipticity at the BCP [26]. As for coordination chemistry, there are several examples where topological analysis was shown to be useful in the study of reaction mechanisms. Macchi and Sironi analyzed the evolution along with the terminal to bridging conversion path of [FeCo(CO)8] [27]. Our group has studied the structural evolution of the mechanism of olefin insertion during the hydroformylation reaction catalyzed by a carbonyl cobalt complex, where a sequence of three consecutive catastrophe steps (bifurcation, conflict, bifurcation) allow the olefin rotation, the

60

4. Structural and bond evolutions during a chemical reaction

O

O

O O Cl

H O

Cl

HOOCl

O

H

O

Cl

S1a = –1.32

H

Cl

O

STS(S1a = –0.65)

ETS

H

H

O

H

O H Cl

O

Cl

O

O

H

S1a = +0.02

HOClO

S1b = –0.43

O

Cl

O ETS

O STS(S1b = –0.62)

H

H

Cl

O

S1b = –0.83

H

O

Cl

Cl

O

O

O

Cl HClO2

O

FIG. 4.6 Maps of gradient vector field of electron density of isomerization reaction of HClO2 (+ represents nuclear of atomic; ▲ represents bond critical point;  represents the ring critical point) [16].

migration of the cobalt atom between the olefin carbon atoms and the transfer of hydride from the metal to a carbon atom, respectively [28]. Lei et al. studied the alkene insertion step in Rh-Yanphos catalyzed hydroformylation [29]. Vastine et al. identified and characterized the possible CarbonHydrogen bond activation mechanisms through the molecular graph of the transition states [30]. Shen et al. investigated the mechanism of dimethyl carbonate synthesis on Cu-exchanged zeolite β by analyzing the molecular graph [31]. Guilleme et al. provided insight into the mechanism of the axial ligand exchange reaction between chlorosubphthalocyanines and phenols [32]. There are some examples where the structural evolution is analyzed in terms of the time instead of the internal reaction coordinate. Our group also analyzed the time evolution of the molecular structure of [Fe{C(CH2)3}(CO)3] through Born-Oppenheimer molecular dynamics to illustrate the changing behavior of the molecular graph of an electronic system (see Fig. 4.7). Bond paths between the metallic core and C(CH2)3 are uninterruptedly formed and broken, suggesting a “hopping” ligand over the Fe atom and thus defining the temporal evolution of bond paths and showing that the changes in ellipticity are a good indicator of a bond breaking [34]. Herna´ndez-Trujillo et al. studied the time evolution of the topological properties of the electron density during quantum molecular dynamics of the dynamical nature of the H+3 and LiF [35]. Teixeira et al. explored the oxygen-transfer step of the Sharpless epoxidation using an assortment of Born-Oppenheimer molecular dynamics coupled to topological analysis of samples taken from the MD trajectories [36]. The reaction mechanisms

61

6. Evolution of electron density

7

'E /kcal·mol–1

5 4

0.2 0.076

H CH2 OC Co CH 2 OC CO

0.18 0.072

H a OC Co CH2 OC CH2 CO b H CH2 OC Co CH 2 OC CO

3 2 1 0

r(r) /a.u.

Ha CH2 CO OC H2C b CO

6

0.16 0.068 90 100 110 0.14

Ub(Co-H) Ub(Co-C(a)) Ub(Co-C(b)) Ur Ub(C(a)-H)

0.12 0.1 A

0.08

B

0.06

H a OC Co CH2 OC CH2 CO b

90 110 130 150 170 T/degrees

A

C

0.04 110 130 150 170 –1.5 T/degrees

–1.3 –0.5 0.3 1.1 1.9 Rx /Bohr·amu–1/2

(a)

–0.7 0.1 0.9 Rx /Bohr·amu–1/2

(b)

(c) FIG. 4.7 (A) Energetic profile, (B) evolvement of ρ(r) in selected bond critical points as well as the formed ring critical point, and (C) connectivity of the four structural regions of the ethylene insertion in the CodH bond during the hydroformylation of this olefin. A, B, and C in (B) denote catastrophe points, and the dashed line in the first structure of (C) indicates the axis for the rotation of the olefin throughout the process. Red and yellow points denote (3,1) and (3,+1) points of the electron density [33].

which involve excited states have been also studied on the basis of their structural evolution. Our group detailed the photo-induced hydrogen atom transfer in salicylideneaniline [37]. Some local properties derived from the electron density are also useful to detect the structural change as metallicity, the stress tensor polarizability, and the stress tensor eigenvalues. The evolution of these properties presents dramatic changes in the points where a bond is formed or broken and they have been used to analyze the noncompetitive/competitive torquoselectivity (stereoselectivity in electrocyclic reactions) of cyclobutene and changes of the tetrasulfur tetranitride S4N4 molecular graphs [38,39]. There exist other scalar fields that can complement the information pertaining to structural evolution in a chemical process. For instance, Andr
es et al. studied the mechanism for the NH3 + LiH ! LiNH2 + H2 reaction combining the topology of electron density and its reduced gradient (Eq. 4.3) as defined by the noncovalent interactions (NCI) index. This combined approach identifies the evolution from weak to strong interactions, recovering the bonding patterns along the reaction pathway [40]. sðrÞ ¼

jrρðrÞj 2ð3π 2 Þ1=3 ρðrÞ4=3

(4.3)

62

4. Structural and bond evolutions during a chemical reaction

7. Evolution of the Laplacian of electron density In 1984, Bader, MacDougall, and Lau presented the Laplacian of electron density as a tool to understand the chemical structure and reactivity [41]. This scalar function (Eq. 4.4) has the property of locating where the density is locally concentrated, where r2ρ(r < 0), and locally depleted, r2ρ(r > 0). r 2 ρð r Þ ¼

∂ 2 ρð r Þ ∂ 2 ρð r Þ ∂ 2 ρð r Þ + + ∂x2 ∂y2 ∂z2

(4.4)

Based on r2ρ(r), It is possible to distinguish charge concentrations (CCs) regions and charge depletion (CDs) regions associated with base and acid Lewis sites. r2ρ(r) provides theoretical support for the Valence Shell Electron Pair Repulsion (VSEPR) model, introduced by Gillespie [42]. Bader proposed the Atomic Graph (AG) as a topological object that contains the electron polarization within the valence shell. An AG is a polyhedron composed of the set of critical points and the gradient paths that link them. These AG can be useful in the identification of functional groups [43]. An AG undergoes a catastrophe process that changes its shape. Popelier has made a great contribution to understanding the topology of r2ρ(r) and its changes [44]. Popelier examined the changes in three valence shell charge concentrations and three depletions graphs within the umbrella inversion of the ammonia molecule, as a function of the angle between the C3 axis and a hydrogen atom. Through the use of planar graphs, the transition mechanisms can be easily rationalized. He found that the transition state presents a distorted planar geometry. The transitions between structures in the valence shell charge concentration and charge depletion graphs do not occur simultaneously. In this process, it is possible to identify a bifurcation mechanism that substitutes a charge concentration with a charge depletion, as shown in Fig. 4.8 [45]. Our group analyzed AG changes during several chemical reactions to identify the modifications of local charge concentrations and depletions in individual steps to gain new insights into these processes. Then we were able to define VSCC evolution as the description of changes of the electron density concentrations and depletions on the VSCC of an atom along with a chemical reaction. The VSCC evolution describes the formation, annihilation, and relocation of charge concentrations and depletions as a consequence of the atomic interactions leading to the formation or breaking of chemical bonds. It also shows the expansion or contraction of the valence shell charge concentration in agreement with the change in coordination number of a carbon atom. The VSCC evolution provides

FIG. 4.8 Pictorial representation of the mechanism of transition from VSCC I to VSCC II. Filled triangles represent (3,1) CPs, filled circles (3,3) CPs, and open triangles (3,+1) CPs. The triplets in brackets indicate the signs of the eigenvalues. Exhaustive discussion on this mechanism can be found in ref. [44].

8. Electron localization function and its evolution

63

information complementary to that given by the analysis of the structural evolution, in virtue of the fact that the former can describe changes in atomic interactions within a single structural region [33]. The analysis of the evolution of the Laplacian is very useful in excited states. The photoinduced electron transfer with a molecule provokes dramatic changes of the AG and then induces a geometrical modification. In the case of nucleic acid purine, the photoexcitation produces an increment of electron population with a concomitant stabilization of an atom that presents the largest geometrical deformation during the evolution toward the conical intersection. The Laplacian of the electron density around this atom shows a charge concentration perpendicular to the ring plane just after the vertical excitation [46]. We also proposed an alternative explanation for the photoinduced structural change in copper(I) complexes, based on the changes of copper AG. After photoexcitation of a CIC (S0 ! S1), a metal-to-ligand charge transfer stabilizes the ligand and destabilizes the metal. To counterbalance the atomic destabilization, the valence shell of the copper center is polarized during the deactivation path. This polarization increases the magnitude of the intra-atomic nuclear-electron interactions within the copper atom and provokes the flattening of the structure to obtain the geometry with the maximum interaction between the charge depletions of the metal and the charge concentrations of the ligand [47].

8. Electron localization function and its evolution The electron localization function (ELF) provides a measure of electron localization in atoms and molecules for a single determinantal wave function, built from Hartree Fock or Kohn Sham orbitals φi as shown in Eq. (4.5). Introduced by Becke and Edgecombe, this function is often interpreted as how efficient the Pauli repulsion is at a given point in space [48]. On the basis of Bader’s theory of atoms in molecules, Silvi and Savin pioneered the topological analysis of the gradient field of ELF [49] leading to basins of attractors that can be interpreted as either bonds or lone pairs. In general, two types of basins can be found in a molecule, i.e., core basins and valence basins. The former surrounds the nuclei with atomic number Z > 2, whereas the latter is characterized by the number of atomic valence shells to which they participate. According to Krokidis, Noury, and Silvi [50], the molecular graph resulting from the ELF gradient field and critical points provides a complete representation of bonding in a molecule, accounting for bonds, lone pairs, and their organization around the nuclei. Moreover, ELF in conjunction with Bond Evolution Theory (BET) and Catastrophe Theory (CT), provides a powerful tool to describe the flow of electron density in a chemical reaction. This synergistic combination of theoretical tools has shown an outstanding connection with more familiar concepts encountered in chemistry like curly arrows, bond formation/breaking, molecular mechanisms, etc. [51]. ELF ¼ 1+ D¼

1X 1 jrρj2 jrφi j2  2 i 8 ρ

1
2

(4.5)

D Dh

Dh ¼

3  2 5=3 5=3 3π ρ 10

64

4. Structural and bond evolutions during a chemical reaction

Within the context of CT and BET, Krokidis et al. analyzed the ammonia inversion process, the breaking of the ethane CdC bond, and the breaking of the dative bond in NH3BH3. They also studied the proton transfer in the protonated water dimer (H5O2)+ where they observed topological changes in the structure of ELF during the process at a given OdO distance. These changes are associated (i) first with the breaking of the OH covalent bond and (ii) with its reformation with the second oxygen. This yields two catastrophes showing that the proton transfer is always a succession of three topological structures [52]. Polo et al. studied the steps of the reaction mechanism for the 1,3-dipolar cycloaddition between fulminic acid and acetylene. Their findings suggest that the cycloaddition process involves seven catastrophes (five-fold-type and two cusp-type catastrophes) and eight structural stability domains. The first step of the reaction leads to the activated complex which is formed through a cusp catastrophe. Then, the ring closure is a dative process in which the oxygen atom provides an electron pair by means of a cusp catastrophe [53]. Similarly, Berski et al. analyzed Diels-Alder-type reactions showing that the formation of six-membered rings requires 10 (or 11) steps separated by fold and cusp catastrophes [54–56]. Adjieufack and coworkers focused on the decomposition of glycerol carbonate revealing nonconcerted processes with the same number (four) of structural stability domains for each reaction pathway [57]. CT has not been limited to the analysis and description of organic reactions. Kegl and coworkers, for example, studied the unconventional back-donation in Ni(PH3)2(η2-CO2) complexes and explained the bent character of the NidC bond path [58]. Andr
es et al. analyzed the reaction pathway associated with the decomposition of stable planar hyper-coordinate carbon species, CN3Mg+3 [59]. The ELF topological analysis has shown to be a powerful tool to study the evolution of chemical species along a reaction path. Yet, the identification of weak interactions remains difficult. To overcome this issue, Gilet et al. introduced the ELF/NCI cross interpretative approach that enables one to follow the evolution of chemical species in a wide range of interactions. These approaches can work in a synergistic fashion providing information on the chemical species close to the formation of chemical bonds [60]. Fang et al. demonstrated the applicability of the ELF/NCI topological approach to study the reactions of DNA polymerase λ and the ε subunit of DNA polymerase, allowing the prediction of the transition state from structures located along with the reaction coordinate, differential metal coordination, and energy barrier differences [61]. Similarly, Zahedi et al. studied the cheletropic decarbonylation of unsaturated cyclic ketones cyclohepta-3,5-dien-1-one CHD, cyclopent3-en-1-one CPE, and bicyclo[2.2.1]hept-2-en-7-one BCH. They showed that these processes take place along different topological stability domains (11, 8, and 8 for CHD, CPE, and BCH, respectively) that can be represented by the sequence of turning points [62].

9. Evolution of the molecular electrostatic potential The molecular electrostatic potential (MESP) provides a description of the charge distribution around a molecule generated by its nuclei and electrons [63,64]. In atomic units, the MESP at a point r is defined as Z X ZA ρðr0 Þdr0 V ðrÞ ¼  (4.6) jRA  rj j r0  rj A

10. Reaction evolution in terms of integrated properties

65

Using the MESP, Balanarayan et al. [65] studied the electron reorganization during the reactions (i) HCNO + HCCH and (ii) NNCH2 + HCCH. The reaction mechanisms derived from the MESP topography are single and consistent. In addition, the MESP topography has a close connection with the valence electron behavior through a topographical manifestation of lone pairs and π-bonds. The derived reaction mechanisms also connect well with classical chemical concepts.

10. Reaction evolution in terms of integrated properties 10.1 Local vs. integrated properties Even though CT provides deep insights into the structural changes involved in a chemical reaction mechanism, there are some criticisms regarding its use because in some instances the resulting topological results are in disagreement with other chemical-bonding schemes, such as molecular orbital theory. Some of the critiques come from the existence (or absence) of unexpected/presumed [66,67] bond paths that are not in agreement with classic bonding models (e.g., Lewis structures) or with other bond definitions. Distinct scalar fields—for example, the density of the Fermi hole and the source function—have been employed in an attempt to explain these apparent anomalies [68,69]. Previously, we mentioned that relaxing the mathematical condition for a critical point allows the identification of weak interactions. Some people attribute this problem to the structural stability (within the context of CT) of some scalar fields, which depend on control-space variables such as nuclear coordinates [70]. However, it was shown that the notion of a dynamic molecular graph consisting of the whole set of different structures provides valuable insights in the description of different chemical bonds, especially those for which the analysis on the basis of static structures leads to controversial conclusions [34]. Garcı´a-Revilla et al. proposed to find alternative integrated properties to define a molecular graph to avoid catastrophe processes. This would lead to circumventing the restrictive/ astringent mathematical definition of local topological features. These alternative features are listed below [70]: 1. 2. 3. 4.

To To To To

be invariant under orbital transformations. be based on some real-space measure of local energetic stabilization. be able to retrieve standard chemical graphs in isolated molecules. vary continuously under geometrical rearrangements.

In general, the atomic properties describe changes involved in chemical reactions (as the atomic charges) but few of them detect the point (or set of points) where the bonds break or form. The integrated properties present cross points as a sign of structural change, where the properties of one bond decrease while the properties of the other increase. A prominent example of this behavior is the magnitude of the atomic quadrupole moments during the reduction of carbonyl groups, as shown in Fig. 4.9. In the axial attack of a hydride to cyclohexanone, jQ(C)j and jQ(O)j show a similar increasing curve, whereas jQ(H)j exhibits a monotonic decreasing behavior. This latter with two intersections around the transition state: first jQ(H)j crosses jQ(O)j and then jQ(C)j. These atomic quadrupole changes denote the formation of the CdH bond and the transition from carbonyl to alcohol functional groups. In the next sections,

66

4. Structural and bond evolutions during a chemical reaction OH O +

OH

1.2 C O H

1

|Q|

0.8 0.6 0.4 0.2 1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

d(C-H)

FIG. 4.9

The magnitude of the atomic quadrupole moment of the atoms involved in the nucleophilic axial attack on cyclohexanone by lithium aluminum hydride [71].

we describe some quantities that fulfill the sought features and have been used to analyze structural changes in reactions and bond evolution [71].

10.2 Localization and delocalization indices Bader et al. defined a measure of the number of electrons localized on an atom as λ(A), and similarly, the number of electrons shared with other atoms as δ(A, B). Mathematically, these quantities (defined in Eq. 4.7) can spot the double integration of the exchange-correlation density over the basins of either one or two different atoms in the molecule (not necessarily sharing a bond path). The localization and delocalization indices add up to the atomic and molecular population [72,73]. Z Z ρxc ðr1 , r2 Þdr1 dr2 λðAÞ ¼ ΩA ΩA

Z

δðA, BÞ ¼ 2 N¼

X A

Z

ΩA ΩB

λðAÞ +

ρxc ðr1 , r2 Þdr1 dr2

XX

(4.7)

δðA, BÞ

A B6¼A

The delocalization index can be used to estimate the formal bond order of a chemical bond. This index is defined as parameter-free, intuitive, and slightly independent on the level of theory. The delocalization index is also able to detect the subtler bonding effects that underpin most practical organic and inorganic chemistry [74,75]. The delocalization index can also be used to define aromaticity in several ways [76–80]. The evolution of the localization and the delocalization indices were used to study the electron-pair reorganization taking place during the reaction path of organic reactions as

10. Reaction evolution in terms of integrated properties

67

intramolecular rearrangements, nucleophilic substitution, electrophilic addition, and DielsAlder cycloaddition [81,82]. The bond-forming and bond-breaking processes can be detected by a delocalization crossing point, i.e., the point where the delocalization of the breaking bond decreases, whereas it increases in the forming bond. The crossing points appear around the transition states but they are not conditional to the presence of the catastrophes of the electron density. For instance, in the transposition from C-N-H to H-CN, the catastrophe point appears at 0.75, before the transition state (TS), and the crossing point occurs at the TS. One can say that the local property detects the structural change earlier than the integrated one as it needs the average over the points within the basin. In the case of the concerted DielsAlder reaction, the delocalization index is able to detect the aromaticity of the TS, revealing that there is not a simple correlation between interatomic distance and electron delocalization. In this way, Matito et al. analyzed the changes in aromaticity along the reaction path of the Diels-Alder reaction between ethene and 1,3-butadiene using several definitions including those based on the delocalization indices, confirming the existence of an aromatic transition state [83]. Werstiuk et al. develop a path to analyze the rearrangement of nonclassical cation using the complementary information from the change in molecular graphs, delocalization index, and the atomic basin visualization (QTAIM-DI-VISAB) [84–87]. They showed that a free singlet tertbutylmethylene does not possess a pentacoordinate methyl group and, therefore, is not bridged. Delocalization does play a pivotal role in stabilizing these carbenes by favoring the nonsymmetrical conformations, but they are far from being any sort of bridged species. Werstiuk also characterized the bonding of the nonclassical 7-norbornyl cation and its rearrangement transitions states, showing that this cation actually exhibits the molecular graph of the bicyclo[3.2.0] heptyl cation at its equilibrium geometry [87]. They also found that, contrary to conclusions reached in earlier studies, the transition state for the degenerate rearrangement of 1,5-hexadiene is not aromatic and that the driving force for the very facile Cope rearrangement of semibullvalene is caused by the stabilization of individual atoms as well as electronic delocalization, not by the release of strain in the three-membered ring [86]. Finally, they characterize the bonding of the 9-barbaralyl cation and related cations, and the rearrangement transition states. In these cases, they unambiguously defined the bonding and their homoaromaticity [84].

10.3 IQA analysis and chemical reactions The interacting quantum atoms (IQA) divide the total energy of a molecule into a sum of atomic self-energies and interaction energies between all the atoms of the molecule [88]. This can be achieved by writing down all of the terms containing the contributions related to a given atom A, A AA AA EA net ¼ T + V en + V ee :

(4.8)

Next, one can write down the interatomic interaction energies by identifying all of the two-atom contributions, AB AB AB AB EAB int ¼ V nn + V en + V ne + V ee

(4.9)

68

4. Structural and bond evolutions during a chemical reaction

with A 6¼ B. The above equation contains all the interaction potential energies of particles belonging to atoms A and B, respectively. According to this framework, the total energy of a molecule can be expressed as follows: X 1 X X AB EA E : (4.10) E¼ net + 2 A A6¼B int A These energies contain all the intra-atomic energetic terms as the sum of pairwise additive interaction energies between atoms. This methodology has shown to be useful in the quantitative analysis of the roles of deformation, classical, and nonclassical interactions in different bonding regimes [89–91]. By means of the IQA scheme, Lo´pez-Ferna´ndez studied the Curtius rearrangement reaction finding two distinguishable phases when the rearranged atom is H: the first one corresponds to the separation of N2, and the second one to the N-H/C-H bond rearrangement [92]. Recently, Thacker et al. introduced the Relative Energy Gradient (REG) method [93] with two-fold objective (i) to determine subsets of partitioned energies that best represent the total behavior of the system and (ii) to extract chemical information from an energetically partitioned system. In general, one takes a few snapshots along a relevant trajectory in which the system evolves. Then the REG method searches for correlations between the total energy and the atomic energies. This latter is applied energy barrier-wise in such a way that the PES is separated into segments (called barriers) defined by the turning points in the total energy. Finally, the REG method ranks all IQA energy terms. The terms with the largest REG values are more relevant than IQA terms with small REG values. This method has been applied successfully by Thacker and coworkers to determine the mechanism of peptide hydrolysis in the aspartic active site of the enzyme HIV-1 protease [94].

10.4 Force analysis of chemical processes In a diatomic molecule AB, the Ehrenfest forces are the forces exerted on the electron density in atom A by nuclei A and B, and the forces exerted on the electron density in atom A by the density of both atoms. Herna´ndez-Trujillo and Bader studied the evolution of the electron densities of two separated atoms in terms of the Ehrenfest forces of several diatomic molecules (from closed-shell, with and without charge transfer, through polar-shared, to equally shared interactions) [95]. The authors found that the Ehrenfest forces acting on each atom in the second-row homonuclear diatomics and in the diatomic hydrides pull it toward its neighbor. The contribution to the atomic Ehrenfest force with the greatest magnitude is the force exerted by the nucleus of the neighboring atom which draws the atoms together. Jara-Cort
es et al. analyzed the bond evolution in terms of the atomic contributions to the forces on the nuclei, i.e., the Hellmann-Feynman forces. This approach quantifies the relationship between the atomic electronic reorganization and the evolution of functional group interactions with the forces exerted on the nuclear framework during a chemical transformation. Using the IQA approach scheme, the forces driving a chemical process are locally assigned to atoms or functional group contributions. The interatomic component of the forces can be ascribed as bonding forces; their exchange-correlation and electrostatic contributions reveal the nature of the interactions affecting the forces on the nuclei. This method was used to analyze the chemical interactions involved in the formation of ground and

Bibliography

69

excited state diatomic molecules, the prototropism of formamide, the Diels-Alder cycloaddition of 1,3-butadiene with ethylene, and the Jahn-Teller effect of hydrated transition metal complexes [96]. Likewise, Barrales-Martı´nez et al. used the reaction force (i.e., the projection of the Hellmann-Feynman forces acting on the nuclei of a molecular system onto a suitable reaction coordinate) to identify and quantify the chemical entities that drive or retard a chemical reaction [97].

11. Perspectives The reaction mechanism is a fundamental concept widely used in chemistry. Yet, there is no theoretical approach that unequivocally allows one to describe structural changes in a chemical reaction. In this chapter, we reviewed the most used theoretical approaches that enable the identification of points in chemical space where bonds break/form in a chemical reaction. These series of methodologies were presented on the basis of the local and integrated properties of scalar and vector fields, as defined by QTAIM. In this way, the structural changes can be pin-point in different stages of a reaction path, linking the reactant structures to those of the products. The electron density, its Laplacian, and ELF are ubiquitous functions that provide insightful and predictive information regarding the structure of molecules when they evolve in different chemical processes and environments. These theoretical approaches gained great attention during the past decade, next, they gained powerful applicability with the advent of Thom’s Catastrophe theory. A prominent example is embodied by the Bonding Evolution Theory, in which bond breaking and forming processes found a more solid physical rationale. The study of bonding evolution has gone far beyond these scalar fields. For instance, the synergistic combination of ELF with NCI allows the identification of points in which molecules start/begin to interact (points that could not be identified by the use of ELF solely). With the aim of overcoming some of the abrupt discontinuities (associated with the catastrophe processes in the aforementioned scalar fields), some integrated properties have been proposed. These methodologies allow the description of structural changes in molecules during a chemical reaction finding the crossing points of continuum properties of chemical bonds. The main purpose of the ideas presented herein is not to choose between either the local or the integrated schemes to study structural changes and molecular mechanisms. On the contrary, our objective is to emphasize the theoretical progress that has been made to better understand how molecules change, form, or interact. Finally, this chapter aims to inspire and leave the door open for new theoretical methodologies that approach us to more sophisticated, yet more realistic, models that match experimental observations.

Bibliography [1] [2] [3] [4]

R.F.W. Bader, T.T. Nguyen-Dang, Y. Tal, J. Chem. Phys. 70 (1979) 4316. S.J. Melnick, M.A. Whitehead, J. Chem. Phys. 76 (1982) 1433. R.F.W. Bader, Y. Tal, S.G. Anderson, T.T. Nguyen-Dang, Isr. J. Chem. 19 (1980) 8. R.F.W. Bader, J. Phys. Chem. A 102 (1998) 7314.

70

4. Structural and bond evolutions during a chemical reaction

[5] A.D. Mcnaught, Compendium of Chemical Terminology: IUPAC Recommendations, second ed., Blackwell Science, Oxford [England], 1997. [6] K. Fukui, Acc. Chem. Res. 14 (1981) 363. [7] H. Eyring, J. Chem. Phys. 3 (1935) 107. [8] T.T. Nguyen-Dang, R.F.W. Bader, Physica A 114 (1982) 68. [9] K. Collard, G.G. Hall, Int. J. Quantum Chem. 12 (1977) 623. [10] R.F.W. Bader, C.F. Matta, F. Cort
es-Guzma´n, Organometallics 23 (2004) 6253. [11] R.F.W. Bader, T.T. Nguyen-Dang, Y. Tal, Rep. Prog. Phys. 44 (1981) 893. [12] J.R. Lane, J. Contreras-Garcı´a, J.-P. Piquemal, B.J. Miller, H.G. Kjaergaard, J. Chem. Theory Comput. 9 (2013) 3263. [13] M. Jabło
nski, Symmetry, 2021. [14] R. Gilmore, Catastrophe Theory for Scientists and Engineers, Courier Corporation, 1993. [15] R.F.W. Bader, T.H. Tang, Y. Tal, F.W. Biegler-Koenig, J. Am. Chem. Soc. 104 (1982) 940. [16] X. Li, Y. Zeng, L. Meng, S. Zheng, J. Phys. Chem. A 111 (2007) 1530. [17] X. Li, H. Fan, L. Meng, Y. Zeng, S. Zheng, J. Phys. Chem. A 111 (2007) 2343. [18] Y. Zeng, S. Zheng, L. Meng, Inorg. Chem. 43 (2004) 5311. [19] Y. Zeng, S. Zheng, L. Meng, J. Phys. Chem. A 108 (2004) 10527. [20] Y. Zeng, L. Meng, S. Zheng, J. Mol. Struct. THEOCHEM 684 (2004) 103. [21] L. Mejı´a-Mazariegos, J. Herna´ndez-Trujillo, Chem. Phys. Lett. 482 (2009) 24. [22] C.L. Parworth, M.K. Tucker, B.E. Holmes, G.L. Heard, J. Phys. Chem. A 115 (2011) 13133. [23] A. Contini, E. Erba, RSC Adv. 2 (2012) 10652. [24] K. Xu, W. Wang, W. Wei, W. Feng, Q. Sun, P. Li, J. Phys. Chem. A 121 (2017) 7236. [25] S.J. Grabowski, Phys. Chem. Chem. Phys. 16 (2014) 1824. [26] J.E. Rode, J.C. Dobrowolski, Chem. Phys. Lett. 449 (2007) 240. [27] P. Macchi, Coord. Chem. Rev. 238–239 (2003) 383. [28] J.P. Salinas-Olvera, R.M. Go´mez, F. Cort
es-Guzman, J. Phys. Chem. A 112 (2008) 2906. [29] X.-J. Du, Y.-H. Tang, X. Zhang, M. Lei, Chin. Chem. Lett. 24 (2013) 1083. [30] B.A. Vastine, M.B. Hall, J. Am. Chem. Soc. 129 (2007) 12068. [31] Y. Shen, Q. Meng, S. Huang, S. Wang, J. Gong, X. Ma, RSC Adv. 2 (2012) 7109. [32] J. Guilleme, L. Martı´nez-Ferna´ndez, D. Gonza´lez-Rodrı´guez, I. Corral, M. Ya´n˜ez, T. Torres, J. Am. Chem. Soc. 136 (2014) 14289. [33] F. Cort
es-Guzma´n, R.M. Go´mez, T. Rocha-Rinza, M.A. Sa´nchez-Obrego´n, J.M. Guevara-Vela, J. Phys. Chem. A 115 (2011) 12924. [34] F. Cort
es-Guzma´n, T. Rocha-Rinza, J.M. Guevara-Vela, G. Cuevas, R.M. Go´mez, Chem. Eur. J. 20 (2014) 5665. [35] R. Cha´vez-Calvillo, J. Herna´ndez-Trujillo, J. Phys. Chem. A 115 (2011) 13036. [36] F. Teixeira, R. Mosquera, A. Melo, C. Freire, M.N.D.S. Cordeiro, Inorg. Chem. 56 (2017) 2124. [37] L. Guti
errez-Arzaluz, F. Cort
es-Guzma´n, T. Rocha-Rinza, J. Peo´n, Phys. Chem. Chem. Phys. 17 (2015) 31608. [38] R. Momen, A. Azizi, A. Morales-Bayuelo, M. Pazhoohesh, X. Ji, J. Chem. Phys. 155 (2021), 204305. [39] Y. Xu, T. Xu, D. Jiajun, S.R. Kirk, S. Jenkins, Int. J. Quantum Chem. 116 (2016) 1025. [40] J. Andr
es, S. Berski, J. Contreras-Garcı´a, P. Gonza´lez-Navarrete, J. Phys. Chem. A 118 (2014) 1663. [41] R.F.W. Bader, P.J. MacDougall, C.D.H. Lau, J. Am. Chem. Soc. 106 (1984) 1594. [42] N.O.J. Malcolm, P.L.A. Popelier, Faraday Discuss. 124 (2003) 353. [43] P.L.A. Popelier, J. Burke, N.O.J. Malcolm, Int. J. Quantum Chem. 92 (2003) 326. [44] P. Popelier, Coord. Chem. Rev. 197 (2000) 169. [45] N.O.J. Malcolm, P.L.A. Popelier, J. Phys. Chem. A 105 (2001) 7638. [46] L. Guti
errez-Arzaluz, D. Ramı´rez-Palma, F. Buitro´n-Cabrera, T. Rocha-Rinza, F. Cort
es-Guzma´n, J. Peon, Chem. Phys. Lett. 683 (2017) 425–430. [47] L. Gutierrez-Arzaluz, D.I. Ramı´rez-Palma, L.G. Ramı´rez-Palma, J.E. Barquera-Lozada, J. Peo´n, F. CortesGuzman, Chem. Eur. J. 25 (2018) 775. [48] A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. [49] B. Silvi, A. Savin, Nature 371 (1994) 683. [50] X. Krokidis, S. Noury, B. Silvi, J. Phys. Chem. A 101 (1997) 7277. [51] J. Andr
es, P. Gonza´lez-Navarrete, V.S. Safont, B. Silvi, Phys. Chem. Chem. Phys. 19 (2017) 29031.

Bibliography

[52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97]

71

X. Krokidis, R. Vuilleumier, D. Borgis, B. Silvi, Mol. Phys. 96 (1999) 265. V. Polo, J. Andres, R. Castillo, S. Berski, B. Silvi, Chem. Eur. J. 10 (2004) 5165. P. Gonza´lez-Navarrete, L.R. Domingo, J. Andr
es, S. Berski, B. Silvi, J. Comput. Chem. 33 (2012) 2400. S. Berski, J. Andr
es, B. Silvi, L.R. Domingo, J. Phys. Chem. A 110 (2006) 13939. S. Berski, J. Andr
es, B. Silvi, L.R. Domingo, J. Phys. Chem. A 107 (2003) 6014. A.I. Adjieufack, V. Li
egeois, I. Mbouombouo Ndassa, B. Champagne, RSC Adv. 11 (2021) 10083. T. K
egl, R. Ponec, L. Kolla´r, J. Phys. Chem. A 115 (2011) 12463. J. Andr
es, L. Gracia, P. Gonza´lez-Navarrete, V.S. Safont, Comput. Theor. Chem. 1053 (2015) 17. N. Gillet, R. Chaudret, J. Contreras-Garcίa, W. Yang, B. Silvi, J.-P. Piquemal, J. Chem. Theory Comput. 8 (2012) 3993. D. Fang, R. Chaudret, J.-P. Piquemal, G.A. Cisneros, J. Chem. Theory Comput. 9 (2013) 2156. E. Zahedi, S. Shaabani, A. Shiroudi, J. Phys. Chem. A 121 (2017) 8504. E. Scrocco, J. Tomasi, New Concepts II, Springer Berlin Heidelberg, Berlin, Heidelberg, 1973, pp. 95–170. J.S. Murray, P. Politzer, WIREs Comput. Mol. Sci. 1 (2011) 153. P. Balanarayan, R. Kavathekar, S.R. Gadre, J. Phys. Chem. A 111 (2007) 2733. F. Cort
es-Guzma´n, J. Herna´ndez-Trujillo, G. Cuevas, J. Phys. Chem. A 107 (2003) 9253. L.J. Farrugia, C. Evans, M. Tegel, J. Phys. Chem. A 110 (2006) 7952. R. Ponec, C. Gatti, Inorg. Chem. 48 (2009) 11024. E. Matito, J. Poater, F.M. Bickelhaupt, M. Sola`, J. Phys. Chem. B 110 (2006) 7189. M. Garcı´a-Revilla, E. Francisco, P.L.A. Popelier, A. Martı´n Penda´s, ChemPhysChem 14 (2013) 1211. D.I. Ramı´rez-Palma, C.R. Garcı´a-Jacas, P. Carpio-Martı´nez, F. Cort
es-Guzma´n, Phys. Chem. Chem. Phys. 22 (2020) 9283. X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. R.F.W. Bader, A. Streitwieser, A. Neuhaus, K.E. Laidig, P. Speers, J. Am. Chem. Soc. 118 (1996) 4959. C.L. Firme, O.A.C. Antunes, P.M. Esteves, Chem. Phys. Lett. 468 (2009) 129. ´ . Martı´n Penda´s, P.L.A. Popelier, Chem. Sci. 9 (2018) 5517. C. Outeiral, M.A. Vincent, A J. Poater, X. Fradera, M. Duran, M. Sola`, Chem. Eur. J. 9 (2003) 400. E. Matito, M. Duran, M. Sola`, J. Chem. Phys. 122 (2005) 14109. P. Bultinck, M. Rafat, R. Ponec, B. Van Gheluwe, R. Carbo´-Dorca, P. Popelier, J. Phys. Chem. A 110 (2006) 7642. E. Matito, P. Salvador, M. Duran, M. Sola`, J. Phys. Chem. A 110 (2006) 5108. C.F. Matta, J. Herna´ndez-Trujillo, J. Phys. Chem. A 107 (2003) 7496. J. Poater, M. Sola`, M. Duran, X. Fradera, J. Phys. Chem. A 105 (2001) 2052. X. Fradera, J. Poater, S. Simon, M. Duran, M. Sola`, Theor. Chem. Accounts 108 (2002) 214. E. Matito, J. Poater, M. Duran, M. Sola`, J. Mol. Struct. THEOCHEM 727 (2005) 165. N.H. Werstiuk, Can. J. Chem. 88 (2010) 1195. D.A. Poulsen, N.H. Werstiuk, J. Chem. Theory Comput. 2 (2006) 75. E.C. Brown, R.F.W. Bader, N.H. Werstiuk, J. Phys. Chem. A 113 (2009) 3254. N.H. Werstiuk, J. Chem. Theory Comput. 3 (2007) 2258. J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A. Martin Pendas, Molecules 25 (2020) 4028, https://doi.org/ 10.3390/molecules25174028. E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Theory Comput. 2 (2006) 90. M.A. Blanco, A. Martı´n Penda´s, E. Francisco, J. Chem. Theory Comput. 1 (2005) 1096. A.M. Penda´s, E. Francisco, M.A. Blanco, J. Comput. Chem. 26 (2005) 344. J.L. Lo´pez-Ferna´ndez, R.A. Mosquera, A.M. Gran˜a, Int. J. Quantum Chem. 120 (2020). J.C.R. Thacker, P.L.A. Popelier, Theor. Chem. Accounts 136 (2017) 86. J.C.R. Thacker, M.A. Vincent, P.L.A. Popelier, Chem. Eur. J. 24 (2018) 11200. J. Herna´ndez-Trujillo, R.F.W. Bader, J. Phys. Chem. A 104 (2000) 1779. J. Jara-Cort
es, E. Leal-Sa´nchez, J. Herna´ndez-Trujillo, J. Phys. Chem. A 124 (2020) 6370. ´ .M. Penda´s, ChemPhysChem 22 (2021) 1976. C. Barrales-Martı´nez, S. Guti
errez-Oliva, A. Toro-Labb
e, A

C H A P T E R

5 The MC-QTAIM: A framework for extending the “atoms in molecules” analysis beyond purely electronic systems Shant Shahbazian Department of Physics, Shahid Beheshti University, Tehran, Iran

1. Introduction The idea that matter is composed of atoms, i.e., indivisible ingredients, goes far back in history to the age of ancient Greeks, but laid dormant, though not completely forgotten, for more than two millennia [1]. In the modern era, it was revived triumphantly through Dalton’s atomic theory which attributes a distinct atom to each chemical element, and from then on the theory dominates the modern chemistry and physics [2,3]. Its central role in modern thinking is best described by Feynman that leaves no room for further comments: “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words?" I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied [4]. With such a respectable history, it is probably shocking to a non-expert observer that the concept of an “atom in a molecule” is a controversial topic subject for theoretical community and a source of disputes and discussions [5,6], as well as a room for creative theoretical work [7,8]. The main problem with the concept of an atom in a molecule is simply the fact that “Daltonian” atom is currently not anymore the atom conceived by Leucippus and

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00017-7

73

Copyright # 2023 Elsevier Inc. All rights reserved.

74

5. The MC-QTAIM

Democritus; its status as the “indivisible” was lost to the elementary particles [9]. In fact, to current chemists and condensed matter physicists, the true atoms are electrons and nuclei. The nuclei are not literally indivisible but beyond nuclear physics and chemistry, for almost all other practical purposes, it is safe to ignore their internal structure and may be treated as if they are “effectively” elementary. In that sense, at the most fundamental level a molecule or a crystal is conceived to be composed of electrons and nuclei and the formulation of basic quantum equations is based on their properties and interactions. Accordingly, as emphasized masterfully by Laughlin and Pines [10], the “theory of everything” for almost all of chemistry and condensed matter physics is Schr€ odinger’s equation governing electrons and nuclei. This triggers a dichotomy in the abovementioned disciplines since in the electronic Schr€ odinger equation there is no mention of Daltonian atoms, whereas the phenomenological models and the general language of these disciplines are full of the jargon of the Daltonian atomic theory. For a chemist, a water molecule is composed of two hydrogen and one oxygen atoms, while for a condensed matter physicist, the crystalline salt is composed of sodium and chlorine ions. In this jargon, atoms bear electric charges and dipoles, they interact and may bond to each other; in other words, they are seemingly well-characterized entities with properties and interaction modes. Phenomenological models of molecules and condensed phases like molecular mechanics and molecular dynamics are the computational incarnation of this viewpoint [11,12]. To remedy this dichotomy, and to reconcile the language of the “theory of everything” with the language of the Daltonian atomic theory, many researchers have tried to suggest methodologies to “extract” atoms in molecules (AIM) from the solutions of the electronic Schr€ odinger equation [13–16]. The basic premise of these methodologies is that AIM and their properties are somehow “buried” in the electronic wavefunctions [16], reduced densities matrices [15], electron densities [13,14], and if these are “mined” properly, AIM can be retrieved. Since the basic principles of quantum mechanics are silent on how this mining should be realized, the proposed methodologies are inherently “heuristic” and this is the origin of the controversies and disputes around the best way of introducing the concept of AIM. A typical response to this problem is dismissing the whole issue as a “pseudo-problem” and assuming that the concept of AIM while useful, to be intrinsically vague and not amenable to a rigorous theoretical analysis within quantum mechanics [5]. Indeed, there are other seemingly vague concepts that despite their usefulness, may defy a rigorous analysis and their omission could undermine the basic structures of modern scientific disciplines. “Species” in taxonomy and evolutionary biology and “consciousness” in neuroscience and psychology are illustrative examples [17,18]. However, as discussed elsewhere [6], it is hard, if even possible in principle, to demonstrate the intrinsic vagueness of a concept conclusively [19]. There are indeed historical examples where a seemingly vague concept has been transformed into a rigorous entity through introduction of a novel theoretical framework where the “affinity” may serve as a classic example. The concept of affinity in chemistry has a tortious history and has been a vague concept for centuries [20]. Finally, it was placed on a rigorous basis by de Donder through employing then newly proposed concept of the chemical potential within modern thermodynamics [21]. Hopefully, the final fate of the concept of AIM is also yet to be determined but in the meantime, the mentioned heuristic methodologies may serve as temporary theoretical frameworks to analyze the properties and interaction modes of AIM as far as possible.

2. Revealing AIM beyond purely electronic systems

75

For more than a decade, the present author and his coworkers were particularly focused on the framework of the quantum theory of atoms in molecules (QTAIM), which was championed by Bader and coworkers since the mid-70s [14,22–27]. Our primary focus was on the mathematical foundations of the QTAIM itself [28,29]. Eventually, the research program shifted to generalization of the methodology to be applicable to the “multi-component” quantum systems, which are molecular systems containing other quantum particles apart from electrons [30–36]. The central question was to check whether a generalized version of the QTAIM may reveal AIM in molecular systems where electrons are not the sole quantum particles. Although the origins of this problem has been reviewed some time ago [37], in the second section of this contribution a brief survey is done to demonstrate why this is a legitimate question. Interestingly, contemplating various physical systems as composed of real or effective elementary particles, and at the same time perceiving them as composed of Daltonian-type atoms is far more widespread than usually conceived by chemists [38]. Some examples will be discussed in the second section. Our generalized methodology, called the multi-component QTAIM (MC-QTAIM), was applied to ab initio wavefunctions of various so-called “exotic” molecular species composed of electrons and other elementary particles like positrons and muons revealing the underlying AIM structure in these species [39–47]. Novel concepts and features emerged from these studies including the regional positron affinities [40], the positronic bond [47], and the positive/ negative muon’s capability/incapability to shape its own atomic basin [42,44,45]. Also, when applied to the purely electronic systems, the MC-QTAIM analysis recovers the results of the QTAIM analysis, demonstrating that the former encompasses the latter just as a special case [32]. In the second and third sections of this chapter, a comparative analysis is also done on the formulation of the QTAIM and the MC-QTAIM and the basic ideas behind these methodologies are articulated. In this articulation a new theoretical ingredient of the QTAIM, i.e., property fluctuation of AIM, is introduced which goes beyond the well-established formalism of the usual particle number fluctuation of AIM [14,48,49]. At the next step, this new ingredient will be extended within the context of the MC-QTAIM. Particularly, the original idea of Bader that AIM are somehow “open” entities amenable to particle and property exchange [14], is quantified by this new ingredient in a firm manner. Finally, in the fourth section some future opportunities for theoretical developments of the MC-QTAIM and even beyond are briefly discussed.

2. Revealing AIM beyond purely electronic systems 2.1 AIM as conceived within the QTAIM Within context of the QTAIM, each atom in a molecule is conceived as a 3-dimensional basin in real-space enclosed by 2-dimensionl boundaries [14]. These boundaries are the zero-flux surfaces emerging from the local zero-flux equation of the gradient of the electron
 Ð !
! !
! ! density: rρ r :n r ¼ 0 [14]. Note that: ρ r ¼ N dτ0 Ψ∗ Ψ , where Ψ is the electronic wavefunction of a purely electronic system and N is the number of electrons while dτ0 implies

76

5. The MC-QTAIM

summing over spin variables of all electrons and
integrating over spatial coordinates of all ! ! electrons except one arbitrary electron and n r is the unit vector normal to the surface. Bader called these basins interchangeably atomic basins, AIM, topological atoms, or quantum atoms [14], but in this chapter they are called atomic basins. In a recent review paper, the origin of these basins has been scrutinized; thus, the details are not reiterated herein and only the main points are reemphasized [29]. Let us start from the fact that molecular electron densities are to a good approximation the sum of electron densities of the constituent atoms, and the atomic densities, i.e., electron density of free atoms, deform only marginally in molecules [50–52]. Particularly, the main trait of an atomic density, namely its maximum at the nucleus and “monotonic” decaying away from nucleus [53,54], retain within molecules. This “robustness” of the atomic densities within molecular environment guarantees that molecular densities usually have a simple topography around equilibrium geometries, containing a local maximum at each nucleus. In the language of topological analysis of electron densities [14], a (3,3) critical point (CP) is located at each nucleus and all zero-flux surfaces that do not cross these CPs are the boundaries of AIM, partitioning the real-space exhaustively into atomic basins. All gradient paths within an atomic basin, originating at infinity or from other types of CPs on its boundaries [14], are ultimately terminating at (3, 3) CPs. Thus, an atomic basin is composed of a (3, 3) CP and its basin of “attraction” and the nucleus and all the electronic population within the basin of attraction belong to that basin. Whether these basins are appropriate representatives of Daltonian atoms is in principle a disputable matter but within the context of the QTAIM, this equivalence is an axiom. Interestingly, not the monotonicity nor the robustness of atomic densities have been proven rigorously from the first principles of quantum mechanics yet [55–61], and the following “conjecture” was proposed some time ago to justify the equivalence [29]: n o ! For a molecular system containing N electrons, with position vectors r i and inter-electronic distances  o n! o n ! !  rij ¼  r i  r j  , and Q clamped nuclei with atomic numbers {Zα}, with position vectors Rα and inter!  n ! o  nuclear distances Rαβ ¼ Rα  Rβ  , described by the following electronic Hamiltonian in atomic units: ^ ¼ H

 X Q N X N N X X X 1 N 2 1 Z ^ ext ,V ^ ext ¼  ! α  ri + +V  !  2 r ij α R i i j>i i  α  r i

There is always a critical distance between each pair of nuclei, denoted as {Rcαβ}, that for geometries for o !
! n derived from the ground elecwhich the inter-nuclear distances are larger than {Rcαβ}, rρ r ; Rαβ > Rcαβ tronic state contains just Q numbers of (3, 3) CPs located at the position of the nuclei.

Perhaps, this conjecture may rightfully be called the “Bader’s conjecture” (BC); although it was not explicitly stated by him in the original literature of the QTAIM, it was tacitly assumed throughout developments of the methodology [62]. Indeed a wealth of computational studies on electron densities of numerous molecules and solids as well as experimental charge densities derived from the X-ray diffraction data point to the validity of the BC [14,63–66]. One may hope that someday BC will be proven and its status will be elevated to “Bader’s theorem”. By the way, let us briefly review some points regarding the current status of this conjecture.

2. Revealing AIM beyond purely electronic systems

77

b ext is crucial in this conjecture since if one replaces Coulomb’s potential First, the role of V with another arbitrary potential the whole conjecture may fall apart where a vivid example is 2 N Q 
! b ext ¼  P P ω2 =2 Rα  ! the harmonic trap potential: V r [67]. In this case, Schr€ odinger’s i α i

α

equation is analytically solvable and it is straightforward to demonstrate that for this particular external potential the number of (3, 3) CPs is irrelevant to the number of nuclei [67]. On the other hand, as discussed elsewhere [68], it is possible to modify electron-nucleus Coulomb potential by taking into account the finite size of nucleus [69]; the electron density resulting from the short-range modified electron-nucleus potential yet conforms to the BC. Between these two extremes, there are various external potentials, e.g., magnetic interactions or relativistic corrections, which one may add to or replace with the Coulomb’s point charge potential. Naturally, the question arises for which type of potentials the conjecture retains and at the best of author’s knowledge, this is an open and untouched problem [29,67]. The reverse reasoning leads to: for all external potentials fulfilling the BC the resulting basins are chemically welldefined regions within the context of the QTAIM since an atomic basin is attributed to each nucleus. In fact, the BC reveals the basic principle behind the concept of AIM within the context of the QTAIM; an atom in a molecule is a region with high “clustering” of electrons in the real-space. In other words, as far as the real-space clustering is justified as a criterion to define Daltonian atoms, the BC may lead to a proper partitioning scheme for that system. This rational is also the basis of all subsequent discussions on the partitioning of systems beyond the purely electronic systems into atomic basins. second point is to stress that only the electron density of the electronic ground state,
The

 ! ρ r ; Rαβ , is the target of the conjecture, which stems from the fact that almost all original QTAIM computational studies were confined to these densities [14]. However, more recent studies by Bader himself [70], as well as others [71–75], point to the fact that at least for lowenergy excited electronic states the local zero-flux equation is yet applicable and yields reasonable atomic basins. This may be reflected by encompassing low-energy excited electronic states into the conjecture, resolving the previously stated limitation to the ground state. However, since according to the best of author’s knowledge, no systematic study has yet been conducted to check the validity of the conjecture for the whole spectrum of the excited electronic states of a molecule, this reasonable modification may be postponed. Indeed, the monotonicity of atomic electron densities is lost for the excited atomic states and thus it is very doubtful that the BC is applicable to high-energy excited molecular electronic states. This is because of the fact that the electron densities of high-energy excited states are exceedingly deformed in comparison with the ground-state electron density. One may conclude that most probably the BC is applicable to the ground and low-energy excited electronic states although before final settlement, further computational studies must be done to have a better perception on the true borderline between low- and high-energy electronic excited states. This point will be discussed again when considering possible applications of the MC-QTAIM to nonelectronic systems. The third point regarding BC is the “conditional” truth of the equivalence of the number of (3, 3) CPs and the number of nuclei for geometries for which the inter-nuclear distances are larger than a critical distance: {Rαβ > Rcαβ}. This conditional truth stems from the fact that “non-nuclear” (3, 3) CPs, located in between nuclei, instead of being located at nuclei,

78

5. The MC-QTAIM

appear if the inter-nuclear distance decreases sufficiently [76–84]. The associated basins are usually called “pseudo-atoms” and to be fair, this was noticed by Bader himself [85], but it was usually treated more as an exception rather than the rule [14]. However, since the illuminating study of Pendas and coworkers it is known that for sufficiently close inter-nuclear distances pseudo-atoms may appear between the atoms of a system [86,87], and even forming “quasi-molecules” when bonded to each other [88,89]. Even more, it was shown by Bader himself that for very close inter-nuclear distances the two atomic basins and the pseudo-atom in between merge and the two nuclei then reside in a single basin containing a single nonnuclear (3, 3) CP [90]. Such a basin is usually called the united-atom limit, once again first discovered by Bader and coworkers [62], and curiously there are rare but illustrative examples that they may appear even at or around equilibrium geometries [91]. In contrast to these examples, it is fair to say that for most considered molecules at the ambient conditions the criterion: {Requlibrium > Rcαβ} is satisfied, and this explains the general success of the local αβ zero-flux equation in delineating reasonable atomic boundaries. However, the discussed counterexamples demonstrate that the electron clustering in the real-space does not have a “universal” pattern and one must be prepared to accept that the number of atomic basins varies sometimes considerably, upon large geometrical variations of the nuclei. Whether this may be interpreted as a weakness of the BC to recover Daltonian atoms or conversely, a hint for extension of the original concept of Daltonian atom is a matter of taste and possibly dispute. Accordingly, in the extension of the QTAIM beyond purely electronic systems we need first to answer the basic question of how atomic basins must be defined in these systems. Nevertheless, let us first briefly review the evidence that demonstrates seeking Daltonian-type atoms beyond the purely electronic systems is a worthwhile enterprise.

2.2 AIM beyond purely electronic systems The previous discussion reveals an intermediate level of organization of the electronic matter, which manifests itself in the real-space clustering of electrons around nuclei resulting in new “composite” entities, i.e., AIM. In the case of electronic matter, the origin of this organization is the dominant interaction of the clamped nuclei and electrons. Accordingly, the question emerges whether there are other examples in nature when a similar organization emerges through other mechanisms manifesting itself by some type of real-space clustering. Interestingly, the answer is affirmative [92], and herein some of better-known examples at the atomic and subatomic levels are considered briefly which are relevant to possible future extensions of the QTAIM. There are three main classes of systems, apart from the purely electronic matter, where quantum systems may organize themselves into a molecular-like structure and reveal a sort of intermediate level of organization. These classes include: (1) hadronic molecules, (2) nuclear molecules, and (3) exotic molecules. While in this chapter, only the technical details of the latter class is considered, a brief account of all these classes are given in this subsection.

2. Revealing AIM beyond purely electronic systems

79

At the most fundamental level, matter is composed of quarks and leptons and the hadrons are the group of composite particles made solely of quarks though this is by part a simplification since apart from quarks, gluons are also an important ingredient of hadrons [9,93]. Quarks have fractional electric charges, one-third and two-third of the electron’s charge, and there are six “flavors” (or types) of them classified as three families, also sometimes called generations (the usual stable matter around us is solely composed of the first/lightest of these generations) [9]. The nature of forces responsible for hadrons’ formation are quite distinct from the Coulomb forces acting within molecules and are based on a property called color charges of quarks, to be distinguished from the electric charges [9]. The forces between quarks are called color/strong forces and are described by the theory of quantum chromodynamics (QCD) (details of color charges and their relationship to the inter-quark interaction are not considered herein and may be found in relevant textbooks) [9,94]. In fact, it is more appropriate to think of quark systems like the usual atomic systems [95]; an atom is composed of a handful of interacting elementary particles but has an infinite numbers of states, which are observable in the atomic spectroscopies. In the case of a quark system, i.e., a handful of interacting quarks, each state is conceived as a new hadron and because of this rationale, searching for and systematizing of hadrons are called hadron spectroscopy [96,97]. Originally, hadrons were assumed to be only two-quark (mesons) or three-quark (baryons) systems where the familiar ingredients of nuclei, protons and neutrons, are examples of the latter group, while other hadrons are inherently unstable and decay to more stable particles/states [9,94]. However, more recently examples of tetraquark and pentaquark systems, also called exotic hadrons, have been discovered [98–102], and the rules governing their construction is an active field of research [103,104]. A subset of these multiquark systems are called hadronic molecules since they are organized not just as an amorphous mixture of quarks but have sub-hadronic structures [105]; in simplest form this can be a di-hadronic structure for a tetraquark system [102,106]. Accordingly, a molecular tetraquark may be conceived as if it is composed of two tightly bonded mesons and it is indeed desirable to deduce this structure from ab initio QCD calculations. Hence, starting from quarks, their color properties and their interaction rules given by the QCD, various theoretical and computational methodologies have been utilized to deduce the molecular nature of the exotic hadrons [107–113]. Particularly relevant to our objective is ab initio lattice-QCD computational studies [114–116], where the derived real-space color and energy densities are employed to visualize the real-space state of the normal mesons [117,118], and the exotic hadrons [119–125]. The topographies of these densities are reminiscent of those of the electron densities but at the best of author’s knowledge, no topological analysis has been conducted yet on these densities. We will not consider further this area in this contribution but it is tempting to seek for an extended BC-like conjecture to derive the “atomic” structure of the molecular hadrons, which is a completely uncharted territory, and worth exploring. The next examples considered herein are nuclear molecules as a special subset of nuclei [126–135]. In principle, nuclei are also composed of quarks but at the usual densities encountered in nuclei, they are effectively composed of protons and neutrons, collectively called nucleons, bonded by strong nuclear forces. The details of strong forces between nucleons are generally more complicated than those operative between quarks since nucleons as hadrons, in contrast to quarks, do not carry color charges and are “colorless” [9,100]. For comparison

80

5. The MC-QTAIM

note that electrically charged particles within atoms are interacting through the universal Coulomb interactions. However, the atoms themselves are electrically neutral and their interactions, though also electromagnetic in nature, are described by “effective” force laws, which are indeed more complicated and less universal than the Coulomb interactions [136]. Thus, the ab initio “theory of everything” of nuclei is Schr€ odinger’s equation, or a relativistic extension of it, governing the many-body system composed of nucleons interacting through effective strong forces [137–142]. Apart from the ab initio [143–155], and the second-quantized based methods [156,157], various models have been also proposed to unravel certain aspects or properties of nuclei [158]. Examples include the vibrational model [159,160], the rotational model [161], the shell model [162,163], the interacting boson model [164–167], and other collective models [168,169]. However for most nuclei, this many-body system is usually conceived as a droplet-like entity, usually called liquid-drop model of nucleus [170]. Within context of this model a nucleus is conceived as an amorphous system with a spherical or deformed shape, however, there are a subset of nuclei that defies this picture and are called nuclear molecules. Accordingly, a new model of nuclei was proposed in the 1930s where it was assumed that certain light nuclei carrying 4n nucleons (n ¼ 2, 3, …): 8Be, 12C, 16O, 20Ne, 24Mg, 28 Si, …, are effectively composed of the alpha-particle (Helium-4 nucleus: 4He) clusters at their ground state [171]. Since then the α cluster model found to be applicable not only to the stable ground states of certain light nuclei but also to various excited states formed during nuclear reactions [172,173], α-decay process [174], and even for certain heavy nuclei [175]. Particularly, in contrast to the excited electronic states of molecules discussed previously, α clustering is even more pronounced for the excited nuclear states in the threshold of nuclear disintegration as it is usually systemized by the Ikeda diagram [174]. Of particular interest in the class of α clustering models is the close-packed spherically arranged geometrical model of α clustered nuclei, championed by Hafstad and Teller [176], which was also pursued by Pauling [177]. Assuming a “bond” between each close-packed αdα contact and attributing a single “bond energy” parameter to all αdα bonds, this simple model is capable of reproducing the nuclear binding energy to a good accuracy [176]. The simplest quantum mechanical treatment of this geometrical model is to omit the nucleons altogether and treating each α cluster as a structureless point-particle interacting through the effective αdα potentials [178]. However, in practice, this “coarse-grained” quantum model had a limited success and even for 12C, composed of just three α clusters [176], it not easy to reproduce experimentally observed nuclear states and their properties without invoking three-body potential energy terms [179,180]. The need for three-body (or in general n-body) potential energy terms, which are not reducible to the usual two-body interaction potentials, is reminiscent of the genuine n-body inter-atomic potentials appearing in the theory of intermolecular forces [136]. This reveals the fact that the internal structure of α clusters is not intact within the nucleus and “internal deformations” are induced upon interactions with other α clusters. A natural step for improvement is to conceive the nucleus as if it is composed of nucleons, but at the same time to group the nucleons into cluster subsets in the theoretical description. The resonating group method (RGM), first proposed by Wheeler in 1937 [181], is the first attempt in this direction where the total wavefunction of the nucleus is the antisymmetrized product of the cluster and the inter-cluster wavefunctions [182]. The variables of a cluster wavefunction are the relative positions of the nucleons of that cluster, which somehow describe cluster’s

2. Revealing AIM beyond purely electronic systems

81

“internal” structure. On the other hand, each inter-cluster wavefunction describes the relative position of a pair of clusters using the relative position of the center of masses of the pair as the proper variable. Another similar and simpler approach, first proposed by Margenau [183], and elaborated further by Hill and Wheeler [184], is usually called Brink or Brink-Bloch model (See particularly Brink’s contribution in Ref. 185 [171,185]. In this model, the basic wavefunction is the Slater determinant of the cluster wavefunctions, each localized in a certain point of the real-space; based on the Hill-Wheeler method, these wavefunctions can be superposed using a weight function where the relative position of cluster pairs act as its variable. This elaborated form of Brink’s wavefunction is transformable into corresponding RGM wavefunction revealing the tight connection of these two methods [186,187]. Brink’s model has been used extensively for ab initio calculations on the cluster structures of many nuclei and one of the outcomes of these calculations is the nucleon densities [188–194]. The topographies of these densities are similar to the usual electronic densities promoting the idea that the topological analysis may reveal the underlying cluster structure in these densities. Although, to the best of author’s knowledge, like the case of the Hadronic molecules discussed previously, no such study has been conducted yet. Surprisingly, some of these nucleon densities are easily reproducible also by the deformed harmonic oscillator model that is usually used as a simple model of the nuclear energy levels [195,196]. It is even more fascinating to perform the topological analysis on the unbiased nucleon densities, which are obtained without the assumption of the cluster structure a priori in the ab initio nuclear wavefunctions, derived directly from the “theory of everything”. One of the simplest unbiased ab initio methods that starts from single nucleons, using a Slater determinant wavefunction composed of the one-nucleon wavefunctions, is the antisymmetrized molecular dynamics (AMD) [197,198], which recovers the clustered nucleon densities [199]. The derived clustered structures were further confirmed with more intricate ab initio studies that take various inter-nuclear correlations and even the relativistic effects into account [200,201]. All these open up a new domain for developing a novel methodology, the “quantum theory of clusters in nuclei” (QTCIN), to derive the cluster structure of nuclei from ab initio nuclear wavefunctions. Indeed, recent applications of the electron localization function (ELF) [202–204], to nuclei [205–207], called nucleon localization function (NLF) in this new context [208], have demonstrated the usefulness of concepts developed originally for the electronic matter in studying the nuclear matter. Finally, let us mention the intriguing possibility of the “condensation” of α clusters in the nuclei [209–212], as a kind of the Bose-Einstein condensation [213], which is currently an active field of research [214–217]. The present author has discussed briefly the possibility of the topological analysis of a bosonic system some time ago [218], and clustered nuclei may be a real opportunity to apply this idea. All in all, the detailed origin of the nuclear clustering is yet considered an open problem [219–221], and a version of the QTCIN, if developed properly, may help to clarify the very nature of this phenomenon from the analysis of the nuclear ab initio calculations [222,223]. The last class considered herein is the exotic molecules, which will be categorized into four main subclasses in this contribution [224–228]. The first subclass includes “pure antimatter” which is derived by replacing all composing quantum particles of a molecule by corresponding antiparticles. The second/third subclass is the set of exotic molecules containing usually one but rarely more negatively/positively charged particles in addition

82

5. The MC-QTAIM

to electrons and nuclei. The forth subclass is called hereafter “truly exotic species” where at least one of the main ingredients of usual molecules, i.e., electrons or nuclei, are totally absent from the system. Let us consider each subclass briefly. Pure antimatter is composed of the anti-particles; an antiparticle has exactly the same properties of its corresponding particle except from the sign of its electric charge [9]. The simplest and probably the most and best studied antimatter system is the antihydrogen atom composed of one positron, e+ [9], the antiparticle of electron, and one antiproton, p [229,230]. Based on basic symmetries of the standard model of particle physics [9], the antihydrogen atom is conceived as the exact “mirror image” of the hydrogen atom having the same properties and energy spectrum [231]. While predicted theoretically a long time ago [231], this antiatom has been “synthesized” only recently using accelerator beams [232–235], and its optical spectrum has been studied intensively since then [236–241]. In principle, nothing prevents the production of heavier antinuclei [242,243], and indeed the antihelium nuclei have been produced recently [244,245]. However, gathering antinuclei and positrons for synthesizing antiatoms, and their subsequent trapping and spectroscopic studies, all pose real technological challenges. In fact, while antiatoms are as stable as atoms, since matter and antimatter annihilate upon coming together [9], the storage and study of antiatoms, in a world made of atoms, require considerable efforts. Thus, it is not hard to imagine that synthesizing even the simplest antimatter molecules, e.g., the antimatter hydrogen molecule composed of two antihydrogen atoms, will be a real experimental achievement. On the other hand, if the basic symmetries of the standard model are universally correct, the world of antimatter is quite boring for a theorist since it is the exact mirror image of the matter world. As will be discussed in subsequent section, this makes the extension of the QTAIM to the antimolecules quite straightforward. The second subclass includes exotic atoms where a heavy negatively charged particle is attached to an atom or a molecule. Examples are muon, μ, mμ  207me, pion, mpion  273me, kaon, mkaon  966me, and antiproton; both pion and kaon are mesons while muon belongs to the lepton family and is a heavier congener of electron [9]. Some atomic members of the second subclass have been known for a long time which include pionic [246], muonic [247,248], Kaonic [249–251], and antiprotonic [252,253], atoms. None of these atoms are stable since not only pion and kaon are intrinsically unstable, with half-life less than a microsecond [9] but also all these particles may participate in the “weak” or the strong nuclear interactions and finally being absorbed by the nucleus of the atom [254–258]. In fact, they are usually captured initially in orbitals with high atomic quantum numbers and then fall into lower energy orbitals, the lowest one with an orbital radius comparable to the size of nucleus, through the de-excitation process and upon emitting photons in X-ray wavelength [259–263]. This is easily understandable since from the basic hydrogen atom problem in quantum mechanics, it is well-known that the mean distance of the particle revolving around the nucleus is inversely proportional to its mass [264]. Consequently, the mean distances for the heavy negatively charged particles are hundreds of time smaller than the mean electron-nucleus distances. Accordingly, the spectroscopy of these exotic atoms may yield unique information about properties of nuclei as well as the forces operative within nuclei that are hard to be reached from other sources [265–269]. One may conclude that in the ground and probably some of the lowlying excited states the exotic atoms are not solely “Columbic” systems and the mentioned intricate interactions of the negatively charged particle and nucleus must be taken into

2. Revealing AIM beyond purely electronic systems

83

account even in simplified quantum models of these atoms [258,270]. The same is true when the negatively charged particles are captured by molecules [271], though in present contribution, for reasons to be disclosed in the next subsection, these intricacies are all ignored. The third subclass includes all exotics where any one of the positively charged elementary particles is added to the usual atoms and molecules, though in practice, the positronic and muonic systems are the best known and most studied species from this subclass [272–274]. In fact, through various positron annihilation spectroscopies [275–277], the positron emission tomography [278,279], and the muon spin resonance spectroscopies (μSR) [280–290], a wealth of information is available on the positronic and muonic species particularly in condensed phases [291–313]. Of particular interest to chemists are muonic organic, organometallic, and biochemical molecules where a muonium atom, composed of an electron and a positive muon, μ+, bonds to a molecule making a radical species with an unpaired electron, which is studied through the μSR [314–335]. Even more interestingly, recent computational studies reveal that both e+ and μ+ may participate directly in novel and unprecedented forms of bonding including the vibrational bonding [336,337], and the one- and two-positron bonds [47,338–342]. Thus, our focus on the developments of the MC-QTAIM were on these species and more details will be disclosed in the subsequent subsection. The subclass of the truly exotic species contains a heterogeneous set of systems where a package of particles through their attractive interactions make bound states, and the backbone is no longer a given usual atom or molecule. In this brief survey, we will cherry-pick only a handful of members of this set to disclose some of their interesting traits. Note that confirming the stability of a few-body quantum system, even when the interactions are solely the familiar Coulomb interactions, is not a trivial task theoretically and the stability/instability of some proposed species has been uncertain for decades [343]. Let us start from systems composed exclusively from electrons and positrons which were proposed theoretically as stable species by Wheeler in 1946 [344–346]. The positronium atom, Ps, composed of an electron and a positron, is the prime example which was discovered experimentally by Deutsch in 1951 [347,348], and is one of the most studied systems of this subclass [349–351]. Since then the positronium negative ion, Ps, Ps plus an electron [352–354], and also PsH, Ps plus a hydrogen atom [355,356], were also discovered experimentally though based on our proposed classification scheme, PsH belongs to the third subclass. Nevertheless, probably the most interesting discovery of this field was the production of what is called the “molecular positronium”, Ps2, composed of two electrons and two positrons [354,357–359]. Systems composed of electrons and positrons, as well as from heavier leptons, are “clean” Columbic systems [360,361]. However, this is not the case when the constituents of two-body exotic atoms are solely mesons or hadrons, e.g., pion and kaon [362], where the strong interactions are also operative in addition to the Coulomb interactions [363]. Protonium atom, composed of p and p+, is an interesting example of these systems, which has been produced in accelerators and studied through the de-excitation process and the X-ray spectrum [257,364–368], similar to the processes detailed previously for the pionic and kaonic atoms. Nevertheless, probably the most interesting examples, like the case of the third subclass, are those exotic molecular-like species where the particle responsible for “bonding” is not an electron. The first example is a Columbic three-body species composed of a μ and two of hydrogen isotopes, i.e., proton, deuterium and tritium, and is a practically a very compact diatomic molecule [369], which has an important role also in the muon-catalyzed fusion [370,371]. In fact,

84

5. The MC-QTAIM

this species is “isomorphic” to the hydrogen ion molecule where μ now plays the role of bonding agent instead of electron. The next example is a three-body system, which is not a purely Columbic system and is composed of two protons and a negatively charged kaon, where the latter now plays the role of bonding agent through both Coulomb and strong nuclear interactions [372,373]. All these examples and even more complex examples [374–376], reveal that clustering and the intermediate level of organization in few- and many-body quantum systems are commonplace and it is desirable to describe this organization within the context of a unified scheme.

2.3 AIM as conceived in the MC-QTAIM Hopefully, the previous discussions convinced the skeptical reader that the extension of the QTAIM’s paradigm beyond the purely electronic systems is due and the subject is worth pursuing. As emphasized in the introduction, from the three main discussed classes only a subclass of the exotic species, namely, the positronic and muonic systems have been considered computationally within the context of the MC-QTAIM [39–47]. Like the case of the QTAIM, the first step in the analysis is deciphering the inter-atomic boundaries and this is done through the local zero-flux equation of the gradient of the Gamma density:

 ! ! ! ! rΓðsÞ r :n r ¼ 0 [32]. The Gamma density is the mass-scaled sum of the one-particle den

 P
 s ! ! ! ðm1 =mn Þρn r , where the sities of all quantum particles of the system: ΓðsÞ r ¼ ρ1 r + n¼2

(usually electron or positron). Each onesubscript "1" is given to the lightest quantum
 particle Ð 0 ∗ ! particle density is defined as follows: ρn r ¼ Nn dτn Ψ Ψ, where Ψ is the wavefunction of a multi-component quantum system, while mn and Nn are the mass and the number of the particles of n  th subset, respectively, and dτ0n implies summing over spin variables of all quantum particles and integrating over spatial coordinates of all quantum particles except one arbitrary particle belonging to the n  th subset. The contribution of the one-particle density of each particle to the Gamma density is scaled according to its inverse mass relative to the lightest particle. The superscript s is called the “cardinal number”; the QTAIM is a special case of the MC-QTAIM where s ¼ 1, while for the positronic and muonic species s ¼ 2 (this version may also be called the two-component QTAIM or the TC-QTAIM) [31]. The purely antimatter molecules are also single-component cases, s ¼ 1, and the resulting local zero-flux equation,

 ! ! ! ! rρpositron r :n r ¼ 0, is reminiscent of the local zero-flux equation of the QTAIM where
 ! ρpositron r is the counterpart of the electron density. In fact, this reveals the previously emphasized complete symmetry of the matter and the antimatter molecular structures and more generally the complete symmetry in matter and antimatter chemistries. The original motivation for the introduction of the Gamma density was the extension of the regional virial theorem and the subsystem vibrational procedure of the QTAIM to the multicomponent quantum systems within the context of the MC-QTAIM [32]. Nevertheless, the ultimate test of the effectiveness of the corresponding local zero-flux equation was its success in delineating reasonable inter-atomic boundaries in the positronic and muonic species

85

2. Revealing AIM beyond purely electronic systems

[39–47]. Based on this success, an extended version of the BC is proposed for the twocomponent systems as follows: n o ! For an exotic molecular system containing Ne electrons, with position vectors r i and inter-electronic dis o n! o n ! !  tances rij ¼  r i  r j  , and q clamped nuclei each with atomic numbers {Zα}, with position vectors Rα and ! n ! o  inter-nuclear distances Rαβ ¼ Rα  Rβ  , and Np quantum particles, with a unit of positive charge and with a  o n o n ! ! ! mass mp(mp me), and with position vector q k and inter-particle distances qkl ¼  q k  q l  , described by the following Hamiltonian in atomic units: b ¼ H

 X  X Np Np Np X Np Ne X Ne Ne X X X X 1 Ne 2 1 1 1 1 b ext ,   + ri +  r2k +  +V  ! !   2 2m r q p ij  q r i i i k k  i k jii lik kl k b ext ¼  V

Q Ne X X i

α

Np X Q X Z Z ! α  + ! α  ! !     α Rα  q k  Rα  r i  k

There is always a critical mass,mc, and a critical distance between each pair of nuclei, {Rcαβ}, that above these
n o ! ð2Þ ! contains just P ¼ Q + Np numbers of (3, 3) CPs where critical values mp > mc, rΓgroundstate r ; Rαβ > Rcαβ
n o ! ð2Þ ! Q of them are located at the position of the clamped nuclei, whereas for mp < mc, rΓgroundstate r ; Rαβ > Rcαβ contains just Q numbers of (3, 3) CPs at the location of the clamped nuclei.

This conjecture is hereafter called the extended Bader’s conjecture (EBC) and apart from the previously raised points regarding the BC, this extended version needs some extra clarifications.



 ! ! ! ! In the case of the positronic and muonic species: ρ r r r r , thus: ¼ ρ and ρ ¼ ρ 1 e 2 P

 

  ! ! ð2Þ ! ð2Þ ! r ¼ ρe r + me =mp ρp r . Γ r is a “combined” density [41], and this means that Γ the appearance of (3, 3) CPs is a convoluted act of the two one-particle densities and
the  ! relative masses of the constituent particles, all contributing to the topography of Γð2Þ r .
 ! Apart from their direct contribution to Γð2Þ r , the masses are present in the Hamiltonian and so they influence indirectly the topography of the one-particle densities through shaping the multi-component wavefunction. Deriving the exact mass-dependence of the one-particle densities is not an easy task analytically.
By the way, computational studies reveal that for !

mp  me, e.g., the positronic species, ρp r is a very diffuse and flat function thus contributes
 ! marginally to the main topographical features of Γð2Þ r . In this limit, the number of (3, 3)

 ! ! CPs of Γð2Þ r and ρe r are both equal to Q and all are located at the clamped nuclei that

means the positrons are unable to shape their own atomic basins. Indeed, a wealth of independent studies on the positronic densities [338,339,377–394], apart from ours [40,47], conform to this picture. In other words, the positron is contained
within one (or rarely two)  !

atomic basins shaped by the clamped nuclei and effectively, Γð2Þ r only reveals the electron clustering [40,47]. The other extreme is: mp > > me, captured mathematically by the limit: me/mp ! 0, is easy to be studied analytically if one assumes that each heavy quantum particle

86

5. The MC-QTAIM


 ! acts like a harmonic oscillator [32]. In this limit ρp r is a very localized delta-like function

 ! ! and it is possible to demonstrate analytically that: lim me =mp !0 Γð2Þ r ! ρ r [32], where
 ! ρ r is the usual electron density used within the context of the QTAIM (not to be confused
 ! with ρe r ). Practically, this limit yields the clamped proton in the adiabatic view and the
 ! number of (3, 3) CPs of Γð2Þ r is equal to Q + NP since there are now effectively Q + NP
 ! clamped nuclei, and the EBC “reduces” to the BC. Once again, in this limit also Γð2Þ r captures effectively only the electron clustering surrounding the clamped nuclei and the localized quantum particles. One may conclude that somewhere between: mp  me and mp > > me, the positive quantum particle acquires the capacity to shape its own atomic basin. The exact numerical value of this critical mass is not fixed and changes among various species. The phenomenon has been termed the “topological structural transformation” since the Q-atomic basin system transforms into Q + NP-atomic basin species [43]. However, since we know that proton is almost always capable of forming its own atomic basin in molecules, then: mpositron < mc < mproton, and interestingly, muon’s mass is in this range. Our computational studies on the muonic molecules [42,44,45], neglecting some special cases [43], revealed that the positive muon is indeed capable of forming its own atomic basin, thus for many molecular species: mc < mμ. It is remarkable that apart from some very simple positronic species with one or no nuclei, which are hard to be classified either as atom or as molecule [395–410], based on majority of both experimental [411–414], and theoretical literature [377–394], molecular structure are not seriously altered upon the addition of positron, whereas for the muonic molecules a wealth of experimental and theoretical studies lead to the picture that μ+ is the “light radioisotope” of hydrogen [415–428], and forms its own atomic basin. All these conform well to the predictions derived from the EBC and since the only constraint of the conjecture is mp me, the EBC must be applicable to every real or hypothetical particle as far as the dominant interactions are the Coulomb interactions. Probably the next natural choice is a molecule containing the heavier tau lepton, mtau  3477me, as the last known member of the lepton family [9]. However, because of its much smaller lifetime compared with muons,  1013s vs.  106s [9], there are no unambiguous experimentally detected “tauonic” atoms and molecules. Nevertheless, if observed in future, the EBC predicts concretely that tau, because of its large mass, which is almost equal to that of deuterium, must be capable to form its own atomic basin in yet hypothetical tauonic molecules and act as a “heavy radioisotope” of hydrogen. Let us stress that it seems reasonable to extend the EBC to encompass multi-component systems

 !

containing s-type of quantum particles, replacing ΓðsÞ r

!

with Γð2Þ r

in the conjecture.

There are indeed computational and theoretical evidence supporting this extension [42], however we prefer to relegate this possibility before further studies. The curious reader may wonder why the EBC does not cover the case of the exotic molecules containing the negatively charged quantum particles. In fact, it is feasible to propose a more extended version of the EBC to encompass these species as well, however, as articulated below, the resulting extension is of no chemical significance. Let’s start from considering the

87

2. Revealing AIM beyond purely electronic systems

situation with a computationally studied example, namely, a five-body system composed of a μ, a point α particle, a proton, and two electrons [42]. Our ab initio calculations clearly revealed that this system is effectively a diatomic species composed of a very compact cluster of α + μ, which bonds to proton through a two-electron bond. This “internal clusterization” is easily verifiable geometrically since the mean (α, μ) distance is more than 100 times smaller than the mean (α, p+) and (μ, p+) distances. Also, the virial theorem, Ddictating the balance E D E beb b [429], tween the expectation values of the kinetic and potential energies, T ¼ ð1=2Þ V retains for the cluster revealing the “decoupling” of the cluster’s dynamics from rest of the system. Accordingly, one may replace the cluster with a hypothetical point particle with a unit of positive charge (the net charge of the cluster), and a mass of mcluster ¼ mα + mμ, and redo the ab initio calculations on this four-body system. The resulting electronic structure is virtually indistinguishable from that of the original five-body system and is quite similar to a normal hydrogen molecule [42]. In a subsequent highly accurate ab initio study, Frolov considered the internal clusterization in abμ e2 and abcμ e2 systems where a, b, c stand for any of the three hydrogen isotopes and e for electron [430]. Various computed geometrical mean values in these systems revealed that in the case of abμ e2 species, each system is composed of a very compact cluster of abμ (vide supra) where the two electrons revolve around the cluster; practically resembling a hydrogen anion or a hydride. In the case of abcμ e2 species, the same study revealed the very compact cluster of abμ is bonded to the lighter isotope c through the two-electron bond, similar to a hydrogen molecule. Interestingly, since the compact abμ is itself a diatomic species, like the Russian dolls, abcμ e2 structure is effectively a compact diatomic system within a more spatially extended diatomic structure [430]. Since μ is the lightest negatively charged particle apart from electron, in the case of other negatively charged particles even more compact clusters are formed and all these are in line with the previously discussed traits of the negatively charged particle contained within exotic atoms. Based on these examples, the main role of a negatively charged particle in molecules is “screening” the nuclear charge and diminishing the effective atomic number of its centering nucleus by one unit, as seen by electrons. In the case of μ, Reyes and coworkers have derived the very localized negative muon’s one-particle density around the clamped nucleus by ab initio calculations [431–433]. They properly resemble this extreme nuclear charge screening as a kind of “alchemical transmutation” and the same is true for the role of other negatively charged particles in the exotic atoms and molecules. In such cases, instead of the deducing inter-atomic boundaries through the local zero-flux equation of the MC-QTAIM, one may instead use the local zero-flux equation of the QTAIM. The proper effective Hamiltonian for molecules with charge-screened clamped nuclei is the usual electronic Hamiltonian where the atomic number of each nucleus, with a negatively charged particle revolving around, changes from Z to Z  1. In next step, one may derive the electron density from the ab initio computed effective ground-state electronic wavefunction and then incorporate it into the local zero-flux equation of the QTAIM. What if one
insists
to use the local zero-flux
 !

equation of the MC-QTAIM in such situations using Γð2Þ r

!

¼ ρe r

!

+ ðme =mN ÞρN r (the

subscript N stands for negative)? Evidently, since the negatively charged particle is localized
 !

around the clamped nucleus, its one-particle density, ρN r , is a delta-like function centered

 ! ! at the nucleus where the one-particle electron density, ρe r , is also maximum. Thus, ρN r

88

5. The MC-QTAIM


 ! only affects Γð2Þ r at and immediately around the nucleus by elevating the height of the
 ! Gamma density without changing its topography. In fact, ρe r solely dictates the topogra
 ! phy of Γð2Þ r while the electron density derived from the previously mentioned effective
 ! model accurately reproduces ρe r . To sum up, the number of atomic basins does not vary before and after the addition of the negatively charged particles to the molecular system and only the properties of the atomic basin containing the particle vary because of the variation in the nuclear charge. The only remaining technical difficulty is the possible effect of the extra terms in the Hamiltonian, describing the absorption of the negatively charged particles into the nucleus [270,434], on the Gamma density, which must clarified in future studies. Finally, let us briefly mention the case of the truly exotic systems where because of some technical obstacles, the effectiveness of the current version of the EBC for these species is less clear than for the other above-considered exotic molecules. The main problem in dealing with the truly exotic systems is the fact that the mass ratio of the constituent particles is usually not as small as that of the mass ratio of electrons to nuclei. Thus, clamping certain particles in these systems and making an adiabatic approximation from the outset for ab initio calculations, similar to the Born-Oppenheimer (BO) procedure for usual molecules [435–437], is not legitimate per se. The only safe way of dealing with these systems is to treat all constituent particles as quantum particles, i.e., each having both kinetic and potential energy operators in the Hamiltonian, and to solve Schr€ odinger’s equation in a fully non-BO scheme [438–440]. Apart from the fact that non-BO calculations are more computationally demanding than those within the adiabatic approximation, there are also serious “qualitative” differences between the properties and symmetries of wavefunctions derived within and beyond the BO paradigm [441–456]. Herein, we will not discuss these differences and leave this subject to a future study where all details will be reviewed (Shahbazian, under preparation). However, let us just stress that in deriving both BC and EBC the presence of the “laboratoryfixed” framework is a necessary condition per se which is absent beyond the adiabatic approximation. Thus, future developments of new methodologies of deriving AIM structure from molecular non-BO wavefunctions, either usual or exotic, must be based on the “body-fixed” quantities and concepts, as is also evident from the recent non-BO studies [454–456].

3. AIM properties beyond purely electronic systems 3.1 AIM properties as conceived within the QTAIM Within the context of the QTAIM, each atomic basin, Ω, receives its share of molecular D E D E P Q b , as follows: A b ¼ properties, i.e., usually molecular expectation values, A AðΩk Þ (herek¼1

after it is assumed that the number
ofbasins in a molecule is equal to the number of clamped Ð ! ! b nuclei) [14,26,27]. AðΩk Þ ¼ d r A r , is the contribution of k  th basin from property A Ωk

3. AIM properties beyond purely electronic systems

89

h i
 Ð ! b is the property density [14,457]. For the case of the one-electron and, A r ¼ dτ0 Re Ψ∗ AΨ
 N ! b ¼ Pm b i, e.g., the kinetic energy, the property density, M r ¼ operators, M i¼1

0  ! ! !  b r ρð1Þ r , r !r ¼!r 0 , is expressed using the reduced spinless first-order density matrix m
0 

0  Ð ! PÐ ! ! ! ! ! ! ! (1-RDM): ρð1Þ r , r ¼ N d r 2 … d r N Ψ∗ r 1 , r 2 , … Ψ r 1 , r 2 , … [14,458]. In the origspins N N b ¼ P Pb inal formulation of the QTAIM [14], for the two-electron operators, G gij , e.g., the i¼1 j>i
 Ð !
! !  ! g r 1, r 2 electron-electron repulsion term, the property density, G r ¼ ð1=2Þ d r 2b
 ! ! ρð2Þ r 1 , r 2 , is expressed by the diagonal part of the reduced spinless second-order density
 Ð ! PÐ ! ! ! matrix (2-RDM): ρð2Þ r 1 , r 2 ¼ N ðN  1Þ d r 3 … d r N Ψ∗ Ψ [14,458]. However, as spins

demonstrated by Popelier, Penda´s and coworkers in their proposed “interacting quantum atoms” energy partitioning scheme [459–463], it is much more informative to partition the expectation values of the two-electron properties into intra- and inter-basin contributions: Q Q Q ^ ¼ P GðΩk Þ + P P GðΩk Ωl Þ. Assuming that the operators do not include spatial hGi k¼1

k¼1 l>k

derivatives, these contributions are
also expressible employing the diagonal part of the Ð ! Ð ! ! !  ð2Þ
! !  Ð ! Ð !
! !  g r 1, r 2 ρ g r 1, r 2 r 1 , r 2 , G ð Ωk , Ωl Þ ¼ d r 1 d r 2 b 2-RDM: GðΩk Þ ¼ ð1=2Þ d r 1 d r 2b Ωk Ωk Ωk Ωl
 ! ! ρð2Þ r 1 , r 2 . Generally, the inter-basin contribution is a quantitative measure of the interaction and/or communication between the two atomic basins Ωk and Ωl. In one sense, the whole QTAIM analysis aims for “chemical interpretation” of the computed basin and inter-basin properties as well as their variations in various molecules or the conformers of the same molecule [14,26,27]. Interestingly, when it came to the electron population in atomic basins, Bader bypassed his original density-based formalism and introduced the concept of the electron number fluctuation [14,48,49]. This is the basis of definition of the so-called electron delocalization index used to quantify the degree of covalency and the bond orders (vide infra) [464,465]. Let us consider some details of this procedure briefly. b ¼ Nb Incorporating A 1 into the above formalism yields the mean basin electron population:
Q Ð ! ! P N ð Ωk Þ ¼ d r ρ r , N ¼ N ðΩk Þ [14,466]. Nevertheless, with some theoretical arguments, Ωk

k¼1

Bader conceived atomic basins as “open subsystem” with “non-vanishing fluctuations” of properties (See page 171 in Ref. 14). Pursuing this line of thought and in order to introduce electron number fluctuation, the basin “electron number distribution” was introduced:   Ð ! Ð ! Ð ! P ∗ N Ð ! Pm ð Ωk Þ ¼ d r 1… d r m d r m+1 … drN Ψ Ψ, and because of the normalizam Ωk spins Ωk R3 Ωk R3 Ωk tion of wavefunction one arrives at:

N P

m¼0

Pm ðΩk Þ ¼ 1 [14,48,49]. Each Pm(Ωk) is the probability

90

5. The MC-QTAIM

of observing an “image” of molecule in which there is an m-electron cluster,0  m  N, in the k  th basin while the rest of N  m electrons are in R3  Ωk. It is straightforward to demonstrate that the electron population is also directly derivable as the mean value of this distriN P bution: N ðΩk Þ ¼ mPm ðΩk Þ [48]. At the next step, the electron fluctuation is introduced as m¼1

the variance of the distribution as follows: ΛðΩk , N Þ ¼ N 2 ðΩk Þ  ½N ðΩK Þ2 ¼ 

N P

N P

m 2 Pm ð Ω k Þ 

m¼1

2 mPm ðΩk Þ

[48]. Alternatively, it is straightforward to derive Λ(Ωk, N) directly from " #2 Ð ! Ð ! ð2Þ
! !  Ð !
! Ð !
! r 1, r 2 + d r ρ r  dr ρ r [14,48,49]. the 2-RDM: ΛðΩk , N Þ ¼ d r 1 d r 2 ρ m¼1

Ωk

Ωk

Ωk

Ωk

Generally:Λ(Ωk, N) > 0, but for the whole system as a closed system:Λ(R3, N) ¼ 0, which is in line with the fact that while the electrons fluctuate between basins, the total number of electrons is a conserved quantity. This line of reasoning has never been extended to include other properties apart from electron number thus the mentioned “non-vanishing fluctuations” of properties has never been quantified. Based on this background, we may introduce the general idea of property fluctuations of atomic basins for the one-electron properties and then considering some of its immediate ramifications. In quantum mechanics, the variance of a property, corresponding to a quantum state, is   D 2 E D E2 b  M b , and it is only null for the constants of computed as follows: Λ R3 , M ¼ M motion, i.e., the properties commuting with the Hamiltonian of molecule [467]. Assuming R3 to be divided into an atomic basin,Ω1, and the complement region,Ω2 ¼ R3  Ω1, it is straightforward to decompose the total variance into the basin variances and the inter-basin covariance:   Λ R3 , M ¼ ΛðΩ1 , MÞ + ΛðΩ2 , MÞ + 2CovM ðΩ1 , Ω2 Þ ΛðΩk , MÞ ¼ M2 ðΩk Þ  ½MðΩk Þ2 ,CovM ðΩ1 , Ω2 Þ ¼ MðΩ1 , Ω2 Þ  MðΩ1 ÞMðΩ2 Þ where

ð M ð Ωk Þ ¼ Ωk

ð +

! dr1

Ωk

ð

! dr2

h Re

h

0 i ! 2 ! ð1Þ ! b r ρ d r Re m r , r  !0 ! r ¼ r !

2




! b1 r 1 m



! b2 r 2 m

Ωk

ð M ð Ω1 , Ω 2 Þ ¼ Ω1

! dr1

ð Ω2





0 i ! 0 ! ! ð2Þ ! r 1 , r 2 , r 1 , r 2  !0 ! ρ ri ¼ ri




0 i ! ! ! 0 ! ! ð2Þ ! b b m d r 2 Re m1 r 1 2 r 2 ρ r 1 , r 2 , r 1 , r 2  !0 ! ri ¼ ri !

h

ð
0 

0  Xð ! ! ! 0 ! ! ! ! ! 0 ! ! ! ! ρð2Þ r 1 , r 2 , r 1 , r 2 ¼ NðN  1Þ d r 3 … d r N Ψ∗ r 1 , r 2 , r 3 … Ψ r 1 , r 2 , r 3 … spins




! 0 ! 0 ! ! ρð2Þ r 1 , r 2 , r 1 , r 2



91

3. AIM properties beyond purely electronic systems

in these expressions is the general non-diagonal form of the 2-RDM. Also, it is easy to check that as a matter of consistency, one recovers the previously derived b ¼ 1 in these equations. For constants of motion: expression for Λ(Ωk, N) by incorporating m Λ(Ω1, M) + Λ(Ω2, M) ¼  2CovM(Ω, Ω0 ), which asserts that the total system is closed in regard to the property M. All these results are easily extendable to an exhaustive partitioning of R3 into Q Q P Q   P P Q atomic basins, [Ωk ¼ R3 , finally yielding: Λ R3 , M ¼ Λ ð Ωk , M Þ + 2 CovM ðΩk , Ωl Þ. k

k¼1

k¼1 l>k

In the case of the number of electrons, the index of electron number delocalization is introduced as follows: δN(Ωk, Ωl) ¼ 2 jCovN(Ωk, Ωl)j [464], which is a measure of the inter-basin electron sharing. In the same spirit, one may propose the index of property delocalization: δM(Ωk, Ωl) ¼ 2 jCovM(Ωk, Ωl)j, which contains the original electron number delocalization as a special case. However, in contrast to the electron number distribution, there is no analogous property distribution that could be used to compute the mean,M(Ωk), and variance,Λ(Ωk, M), simultaneously. In order to compute M(Ωk), the following distribution may be introduced:   h
 i Ð ! Ð ! Ð ! P N Ð ! ! b r 1 Ψ . This is a generald r 1… d r m d r m+1 … drN Re Ψ∗ m M m ð Ωk Þ ¼ m Ωk 3 3 spins Ωk R Ωk R Ωk ized form of {Pm(Ωk)}, but in contrast to {Pm(Ωk)}, {Mm(Ωk)} has the physical dimension of the D E D E N b Since M b =N ¼ P Mm ðΩk Þ, therefore each Mm(Ωk) is a contribution of M b =N property M. m¼0

attributed to an “image” in which m-electrons are in Ωk while the rest of N  m electrons are N P in R3  Ωk. It is also straightforward to demonstrate that: MðΩk Þ ¼ mMm ðΩk Þ ¼ m¼1 Ð !
! d r M r , thus recovering the basin properties originally deduced from the formulation

Ωk

discussed previously. However, as stressed, M2 ðΩk Þ 6¼

N P

m2 Mm ðΩk Þ, and Λ(Ωk, M) is “not”

m¼1

the variance of {Mm(Ωk)} distribution. This result reveals the special trait of the electron number b ¼m b2 ¼ b distribution since in this case: m 1, while in general for an interacting system of particles b b i is not a constant of motion and thus there is no even when M is a constant of motion, each m simple relationship between Mm(Ωk) and Pm(Ωk). Let us finally stress that the role of M(Ωk, Ωl) for the one-electron properties is similar to that of G(Ωk, Ωl) for the two-electron properties; both yield the inter-basin contributions although, the latter appears when computing the basin properties, while the former only appears when computing the basin property fluctuations. In fact, δM(Ωk, Ωl) is gauging the amount of “openness” of atomic basins relative to various properties so this index is the quantitative manifestation of Bader’s original idea of the property fluctuation.

3.2 AIM properties as conceived within the MC-QTAIM The same idea of property partitioning is also applicable within the context of the MC-QTAIM to the s-component molecular systems, containing

s P n¼1

N n ¼ Nt quantum

92

5. The MC-QTAIM


 ! particles, and partitioned into P atomic basins through the zero-flux equation of ΓðsÞ r and s Nn b ¼ P Pm b n,i, the expectation values are the EBC. In the case of the one-particle properties, M n¼1 i¼1

D E P P s b ¼ ~ ðΩk Þ ¼ P Mn ðΩk Þ is the total basin contri~ ðΩk Þ, where M partitioned as follows: M M n¼1
k¼1 Ð ! ! bution and Mn ðΩk Þ ¼ d r Mn r is the contribution of n  th subset of particles to the k  th Ωk

 ! ! bn r basin [32,36]. The corresponding property density is as follows: Mn r ¼ m

0 
0  Ð ! !0 ! ð 1Þ ! !  ð1Þ ! ! ρn r , r !r ¼!r 0 , in which ρn r , r ¼ N n dτ0n Ψ∗ …, r n,1 , …, r n,i , … r n,Nn , …
 ! ! ! Ψ …, r n,1 , …, r n,i , … r n,Nn , … is the generalized form of the 1-RDM for the n  th subset of
particles where its diagonal part yields the previously introduced one-particle density,  !

ρn r

[32,33]. In the case of the two-particle operators, there are two types of operators

namely, those acting between members of the same component and!those operative between ! Nn P Nn Nn P Nm s s P s P P P P b¼ b b gn,ij + gnm,ij : members of two different components, G n¼1

n¼1 m>n

i¼1 j>i

i¼1 j¼1

Employing the rational of partitioning the expectation value into intra- and inter-basin D E P P P P ~ ðΩk Þ ¼ ~ ð Ωk Þ + P P G ~ ðΩk , Ωl Þ, where G b ¼ G contributions one arrives at: G k¼1 k¼1 l>k   s s s s P P ~ ðΩk , Ωl Þ ¼ P Gn ðΩk , Ωl Þ + P Gnm ðΩk , Ωl Þ . Assuming Gn ðΩk Þ + Gnm ðΩk Þ and G n¼1

m>n

n¼1

m>n

that the two-particle operators do not contain spatial derivatives,
 the
termsin brackets are Ð ! Ð ! ! ! ð2Þ ! ! gnn r 1 , r 2 ρn r 1 , r 2 , Gnm ðΩk Þ ¼ computed as follows: Gn ðΩk Þ ¼ ð1=2Þ d r 1 d r 2 b Ωk Ωk



 Ð ! Ð ! Ð ! Ð ! ! ! ! ! ð2Þ ! ! ð2Þ ! ! gnm r n , r m ρnm r n , r m , Gn ðΩk , Ωl Þ ¼ d r 1 d r 2 b gnn r 1 , r 2 ρn r 1 , r 2 , drn drm b Ωk Ωk Ωl

 Ωk
 Ð ! Ð ! ! ! ð2Þ ! ! ð2Þ ! ! Gnm ðΩk , Ωl Þ ¼ d r n d r m b gnm r n , r m ρnm r n , r m , in which ρn r 1 , r 2 ¼ N n ðN n  1Þ Ωk
Ωl  Ð 000 Ð 00 ∗ ð2Þ ! ! dτn Ψ Ψ, and ρnm r n , r m ¼ Nn N m dτn Ψ∗ Ψ, which is the pair density. In these integrations, dτn00 implies summing over spin variables of all quantum particles and integrating over spatial coordinates of all quantum particles except two arbitrary particles, denoted herein as particles “1” and “2”, belonging to the n  th subset. While, dτn000 implies summing over spin variables of all quantum particles and integrating over spatial coordinates of all quantum particles except two particles, one belongs to the n  th and the other to the m  th sub
arbitrary  ð2Þ !

!

sets. Also, ρn r 1 , r 2 is the generalized form of the 2-RDM for the n  th subset of particles,
 ð2Þ ! ! whereas ρnm r n , r m , which emerges because of the “distinguishability” of the two particles belonging to two different components, is a novel pair density foreign to the purely electronic systems [32,33]. In addition, it is straightforward to demonstrate that for the singlecomponent systems, all these equations reduce to the partitioning schemes of the QTAIM

93

3. AIM properties beyond purely electronic systems

discussed in the previous subsection. Let us now consider the case of the particle fluctuation within context of the MC-QTAIM. b ¼b Incorporating m 1 into the one-particle property density yields the basin populations for P P s Ð !
! P each type of particles: N n ðΩk Þ ¼ d r ρn r , Nt ¼ Nn ðΩk Þ [31,33]. In order to introduce Ωk

k¼1 n¼1

the particle number fluctuation, the basin particle number distribution for each subset is de  Ð ! Ð ! Ð Ð ! Nn Ð ! d r n,1 … d r n,m d r n,m+1 … d r n,N dωn Ψ∗ Ψ, where fined as follows: Pnm ðΩk Þ ¼ m Ωk 3 3 Ωk R Ωk R Ωk dωn implies summing over spin variables of all quantum particles and integrating over spatial coordinates of all quantum particles except the n  th subset while the normalization condiNn P tion of the multi-component wavefunctions implies: Pnm ðΩk Þ ¼ 1 [33]. Each Pnm(Ωk) is the m¼0

probability of observing an “image” of molecule in which there is an m-particle cluster from the n  th subset,0  m  Nn, in the k  th basin while the rest of Nn  m particles are in R3  Ωk. The basin populations of each type of particles are derived as the mean value of each Nn P mPnm ðΩk Þ, while the particle fluctuation is introduced as the varidistribution:N n ðΩk Þ ¼ m¼1

ance

of

these

Λn ðΩk , N n Þ ¼ N 2n ðΩk Þ  ½N n ðΩK Þ2 ¼

distributions:

N Pn

Nn P m¼1

m2 Pnm ðΩk Þ 

2 mPnm ðΩk Þ . The variance may alternatively be deduced from the generalized form of m¼1 Ð ! Ð ! ð2Þ
! !  the 2-RDM as follows: Λn ðΩk , NÞ ¼ d r 1 d r 2 ρn r 1 , r 2 + N n ðΩk Þ  ½N n ðΩk Þ2 [33]. For Ωk

Ωk

the particles effectively confined into a single basin, the corresponding variance is null: Λn(Ωk, Nn)  0 [33,35], but in general: Λn(Ωk, Nn) > 0, while: Λn(R3, Nn) ¼ 0, where the latter is the result of particle number conservation. It is straightforward to demonstrate that: P P P P P P Λ n ð Ωk , N n Þ + 2 CovN where the inter-basin covariances, n ðΩk , Ωl Þ ¼ 0, k¼1 k¼1 l>k Ð ! Ð ! ð2Þ
! !  CovN d r 1 d r 2 ρn r 1 , r 2  N n ðΩk ÞN n ðΩl Þ, are used to introduce the delocalizan ð Ωk , Ωl Þ ¼ Ωk

Ωl

N tion index for each subset of particles: δN n (Ωk, Ωl) ¼ 2 jCovn (Ωk, Ωl)j [33,35]. Apart from the abovementioned distributions, a novel type of joint distribution may also be introduced for the multi-component systems, which has not been considered previously in the MC-QTAIM literature. The joint basin particle number distribution for a couple of subsets    Ð ! Ð ! Ð ! Nq Ð ! Nn is defined as follows: Pnq ð Ω Þ ¼ d r n,1 … d r n,m d r n,m+1 … d r n,Nn k mp p Ωk m Ωk R3 Ωk R3 Ωk Ð ! Ð ! Ð ! Ð ! Ð d r q,1 … d r q,p d r q,p+1 … d r q,Nq dωnq Ψ∗ Ψ, where dωnp implies summing over spin

Ωk

Ωk

R3 Ωk

R3 Ωk

variables of all quantum particles and integrating over spatial coordinates of all quantum particles except the spatial coordinates of the n  th and the q  th subsets. Each Pnq mp(Ωk) is the joint probability of observing an “image” of a molecule in which there is an m-particle cluster from the n  th subset,0  m  Nn, and a p-particle cluster from the q  th subset,0  p  Nq,

94

5. The MC-QTAIM

in the k  th basin while the rest of Nn  m and Nq  p particles are in R3  Ωk. The previously introduced basin particle number distributions are derivable from the joint distribution as Nq Nn P P q Pnq Pnq follows: Pnm ðΩk Þ ¼ mp ðΩk Þ , Pp ðΩk Þ ¼ mp ðΩk Þ , while the normalization of the p¼1

m¼1

wavefunction implies that:

Nq Nn P P m¼1 p¼1

Pnq mp ðΩk Þ ¼ 1. The covariance of the joint distribution is for-

mally defined as follows: CovN nq ðΩk Þ ¼

Nq Nn P P m¼1 p¼1

  ðm  N n ðΩk ÞÞ p  N q ðΩk Þ Pnq mp ðΩk Þ [468], which

after some mathematical manipulations may alternatively be expressed using the pair density Ð ! Ð ! ð2Þ
! !  Ð !
! Ð !
! N as follows: Covnq ðΩk Þ ¼ d r n d r q ρnq r n , r q  d r ρn r d r ρq r . It is straightforward Ωk

P P

to demonstrate that:

Ωk

CovN nq ðΩk Þ + 2

k¼1

P P P P k¼1 l>k

Ωk

Ωk

N

Covnq ðΩk , Ωl Þ ¼ 0, where the novel inter-basin coN

variances are defined as follows: Covnq ðΩk , Ωl Þ ¼ " # Ð ! Ð ! ð2Þ
! !  Ð ! Ð ! ð2Þ
! !  ð1=2Þ d r n d r q ρnq r n , r q  Nn ðΩk ÞN q ðΩl Þ + d r n d r q ρnq r n , r q  N q ðΩk ÞN n ðΩl Þ . Ωk

Ωl

Ωl

Ωk

N Covnq ðΩk ,

and Ωl Þ may found proper chemical interpretations like the case Whether N of Covn (Ωk, Ωl) is an open problem that needs further theoretical and computational investigations. At this stage of development, we may introduce the basin property fluctuations. Like the case of the QTAIM discussed previously, our starting point is the variance of a   D 2 E D E2 b  M b . Assuming one-particle property for the total molecular system: Λ R3 , M ¼ M CovN nq(Ωk)

R3 to be divided into Q atomic basins, [Ωk ¼ R3, it is straightforward to decompose the total k

variance into various basin and inter-basin covariances: " # " # P s P s X s X X X  3  X M Λn ðΩk , MÞ + 2 Covnq ðΩk Þ Λ R ,M ¼ k¼1

n¼1

"

+2

P X P s X X k¼1 l>k

k¼1

# CovM n ð Ωk ,

n¼1

Ωl Þ + 4

n¼1 q>n

" P X P s X s X X k¼1 l>k

# M Covnq ðΩk ,

Ωl Þ

n¼1 q>n

where Λn ðΩk , MÞ ¼ M2n ðΩk Þ  ½Mn ðΩk Þ2 , CovM n ðΩk , Ωl Þ ¼ Mn ðΩk , Ωl Þ  Mn ðΩk ÞMn ðΩl Þ ð h

0 i ! ! 2 ! 2 ð1Þ ! b n r ρn r , r  !0 ! Mn ðΩk Þ ¼ d r Re m r ¼ r Ωk

ð

!

ð

dr1

+ Ωk

Ωk

h


0 i ! ! ! ! ! 0 ! ! b n r 2 ρðn2Þ r 1 , r 2 , r 1 , r 2  !0 ! bn r 1 m d r 2 Re m ri ¼ ri

3. AIM properties beyond purely electronic systems

ð M n ð Ωk , Ωl Þ ¼

!

ð

dr1 Ωk

Ωl

h


0 i ! ! ! ! ! 0 ! ! b n r 2 ρðn2Þ r 1 , r 2 , r 1 , r 2  !0 ! bn r 1 m d r 2 Re m ri ¼ ri

95

ð
0 

0 0  ! ! ! ! ! !0 ! ! ! ! ρðn2Þ r 1 , r 2 , r 1 , r 2 ¼ N n ðNn  1Þ dτ00n Ψ∗ r n,1 , r n,2 , r n,3 … Ψ r n,1 , r n,2 , r n,3 … and CovM nq ðΩk Þ ¼ Mnq ðΩk Þ  Mn ðΩk ÞMq ðΩk Þ,



 M Covnq ðΩk , Ωl Þ ¼ ð1=2Þ Mnq ðΩk , Ωl Þ  Mn ðΩk ÞMq ðΩl Þ + ð1=2Þ Mnq ðΩl , Ωk Þ  Mn ðΩl ÞMq ðΩk Þ ð ð h


i ! ! ! ! ! ! b q r q ρðnq2Þ r n , r q bn r n m Mnq ðΩk Þ ¼ d r n d r q Re m Ωk

Ωk

ð

! drn

Mnq ðΩk , Ωl Þ ¼ Ωk

ð

Mnq ðΩl , Ωk Þ ¼


ð2Þ !0 !0 ! ! ρn r 1 , r 2 , r 1 , r 2



Ωl

! drn

ð

h


i ! ! ! ! ! b q r q ρðnq2Þ r n , r q bn r n m d r q Re m

Ωl

ð

h


i ! ! ! ! ! b q r q ρðnq2Þ r n , r q bn r n m d r q Re m

Ωk

in these expressions is the non-diagonal form of the previously introduced generalized 2-RDM for the n  th subset of particles. Interestingly, the contribuM

tions from joint distributions, i.e., CovM nq(Ωk) and Covnq ðΩk , Ωl Þ, emerge automatically from b is not a constant of motion, the partitioning of the total variance. As far as the property M 3 the total variance is non-zero, Λ(R , M) > 0, however, even if it is a constant of motion, Λ(R3, M) ¼ 0, it not possible to separate the one-particle and joint contributions into two groups and the sum of contributions of each group is not separately equal to zero. This stems from the fact that particles from different subsets may exchange properties thus in contrast to the number of particles, there is no “local” conservation law for the properties of each subset. Accordingly, it is not possible to deduce a conservation law for the sum of the variances and covariances associated to the n  th subset of particles nor the joint distribution of the n  th, and the p  th, subsets of particles. By the way, the sum of the b ¼1 all contributions of particle number fluctuations is recovered if one incorporates m M into the property fluctuation expression. One may propose: δM n (Ωk, Ωl) ¼ 2 jCovn (Ωk, Ωl)j, as the index of property delocalization for the n  th subset that contains the previously defined particle number delocalization as a special case. However, as detailed previously, it is important to realize that δM n (Ωk, Ωl), in contrast to δM(Ωk, Ωl), is “not” the sole contribution M

of the inter-basin property fluctuation and the role of Covnq ðΩk , Ωl Þ must be also taken N

into account. Like the case of CovN nq(Ωk) and Covnq ðΩk , Ωl Þ, future theoretical and computational studies may shed some light on possible chemical interpretations of CovM nq(Ωk) and M

Covnq .

96

5. The MC-QTAIM

4. Conclusion The primary goal of an AIM-based partitioning scheme is to propose an explanation for the origin of stability of a molecular system in comparison with sum of its constituent atoms or compared with other isomers/conformers of the molecule. This explanation is “coarsegrained” by its nature since from the viewpoint of the “theory of everything”, the ultimate source of the stability of all quantum few- and many-body systems are the stabilizing interactions between the constituent elementary particles. The effectiveness of such coarsegrained explanation critically depends on its ability to “locate” the origin of stability to one or at most to a small number of interactions between atomic basins. In other words, the spatially “non-local” explanation of stability within context of the “theory of everything”, which spreads throughout the molecule, is replaced by a spatially “localized” explanation. Whether this is feasible for a special molecular system or a group of systems in a comparative study is a matter of computational considerations and is beyond the theoretical analysis of the foundations of a partitioning methodology. This is also the case for the introduced property delocalization index but taking the fact that deducing the inter-basin contribution of the twoelectron interactions had a huge success in unrevealing the origin of the local stabilizing interactions [462], one may reasonably expect that the proposed index to be informative as well. Also, the idea of clusterization in few-body and many-body quantum systems need further studies beyond the previous reports. Let us stress that a system with well-separated clusters of particles may be considered in an “adiabatic” paradigm as discussed by Frolov [430]. The Hamiltonian of a clustered system may be divided into the cluster Hamiltonians, each containing the internal dynamics of the cluster plus an extra potential energy term simulating the effect of other clusters. After solving Schr€ odinger’s equation for each cluster, similar to the usual BO procedure [435,436], the dynamical couplings between clusters may also be taken into account as “non-adiabatic” corrections. This tacitly implies that there is a link between the “degree of clusterization” at the ground state and the energy spectrum of the system, which needs further scrutiny in future theoretical studies. Accordingly, beyond calculating the average inter-particle distances, which is practical only for systems composed of distinguishable particles, it is desirable to seek for more universal “indices of clusterization”. While the BC and EBC are examples in this regard, in cases where the clusterization has multiple layers, and clusters are formed independently at different spatial scales [42,430], novel procedures and concepts must be introduced. The most desirable procedure is to infer the clusterization from the Hamiltonian itself, i.e., the properties of quantum particles and their modes of interaction, rather than the deriving them from analyzing the wavefunctions or from the computed expectation values. The MC-QTAIM analysis of the exotic molecules started almost a decade ago in our laboratory, and arguably the most important achievement in the initial phase of developments was the demonstration that μ+ has the capacity to form its own atomic basin [42,44,45]. This adds a new type of atom in a molecule, an exotic one, to the known AIM, but, the recent discovery of the positron bond [338], and its subsequent MC-QTAIM analysis [47], revealed the capacity of the MC-QTAIM analysis in tracing and quantifying the “exotic bonds” as well. The fact that agents other than electrons may act as the bonding glue in the exotic species, has the massage that the usual chemical bonds, discovered in the purely electronic systems

References

97

with all their ramifications, are just a single subclass among a large class of yet to be identified exotic bonds. All these support the viewpoint that concepts of AIM and bonds, although originally invented in chemistry, are also applicable to few-body quantum systems that bear no apparent resemblance to the usual molecules. A large amount of theoretical developments and computational applications remains to be done in this area as well as in the case of the Hadronic and nuclear molecules mentioned previously.

Acknowledgments The author is grateful to Mohammad Goli for his constructive comments on a previous draft.

References [1] A.G. van Melsen, From Atomos to Atom: The History of the Concept Atom, Dover Publications, Mineola, NY, 2004. [2] S.G. Brush, Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Onsager, first ed., Princeton University Press, Princeton, N.J, 1983. [3] A.J. Rocke, Chemical Atomism in the Nineteenth Century: From Dalton to Cannizzaro, Ohio State University Press, Columbus, 1984. [4] R.P. Feynman, R.B. Leighton, M. Sands, Six Easy Pieces: Essentials of Physics Explained by Its Most Brilliant Teacher, fourth ed., Basic Books, New York, 2011. [5] R.G. Parr, P.W. Ayers, R.F. Nalewajski, J. Phys. Chem. A 109 (2005) 3957. [6] S. Shahbazian, Found. Chem. 16 (2014) 77. [7] P.L. Ayers, R.J. Boyd, P. Bultinck, M. Caffarel, R. Carbo´-Dorca, M. Causa´, J. Cioslowski, J. Contreras-Garcia, D.L. ´ . Martı´n Penda´s, P.L.A. Popelier, K. Cooper, P. Coppens, C. Gatti, S. Grabowsky, P. Lazzeretti, P. Macchi, A Ruedenberg, H. Rzepa, A. Savin, A. Sax, W.H.E. Schwarz, S. Shahbazian, B. Silvi, M. Sola`, V. Tsirelson, Comput. Theor. Chem. 1053 (2015) 2. [8] J. Andres, P.W. Ayers, R.A. Boto, R. Carbo´-Dorca, H. Chermette, J. Cioslowski, J. Contreras-Garcı´a, D.L. ´ .M. Penda´s, E. Matito, I. Mayer, A.J. Cooper, G. Frenking, C. Gatti, F. Heidar-Zadeh, L. Joubert, A Misquitta, Y. Mo, J. Pilme, P.L.A. Popelier, M. Rahm, E. Ramos-Cordoba, P. Salvador, W.H.E. Schwarz, S.  Shahbazian, B. Silvi, M. Sola`, K. Szalewicz, V. Tognetti, F. Weinhold, E.-L. Zins, J. Comput. Chem. 40 (2019) 2248. [9] D. Griffiths, Introduction to Elementary Particles, second ed., Wiley-VCH, Weinheim, 2008. [10] R.B. Laughlin, D. Pines, Proc. Natl. Acad. Sci. 97 (2000) 28. [11] U. Burkert, N.L. Allinger, Molecular Mechanics, American Chemical Society, Washington, DC, 1982. [12] C.J. Cramer, Essentials of Computational Chemistry: Theories and Models, second ed., Wiley, Chichester, West Sussex, England; Hoboken, NJ, 2004. [13] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press USA, New York, NY, 1994. [14] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, 1994. [15] F. Weinhold, C.R. Landis, Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, first ed., Cambridge University Press, Cambridge, UK; New York, 2005. [16] I. Mayer, Bond Orders and Energy Components: Extracting Chemical Information from Molecular Wave Functions, CRC Press, Boca Raton, 2016. [17] D.J. Chalmers, The Conscious Mind: In Search of a Fundamental Theory, Revised ed. edition, Oxford University Press, New York, 1997. [18] W. Kunz, Do Species Exist?: Principles of Taxonomic Classification, first ed., Wiley-Blackwell, Weinheim, Germany: Chichester, 2012. [19] G. Ronzitti (Ed.), Vagueness: A Guide, Springer, Netherlands, 2011. [20] M.G. Kim, Affinity, That Elusive Dream: A Genealogy of the Chemical Revolution, The MIT Press, Cambridge, Mass, 2008.

98

5. The MC-QTAIM

[21] D. Kondepudi, I. Prigogine, Modern Thermodynamics: From Heat Engines to Dissipative Structures, first ed., John Wiley & Sons, Chichester; New York, 1998. [22] R.F.W. Bader, T.T. Nguyen-Dang, Y. Tal, Rep. Prog. Phys. 44 (1981) 893. [23] R.F.W. Bader, T.T. Nguyen-Dang, Adv. Quantum Chem. 14 (1981) 63. [24] R.F.W. Bader, Acc. Chem. Res. 18 (1985) 9. [25] R.F.W. Bader, Chem. Rev. 91 (1991) 893. [26] P.L. Popelier, Atoms in Molecules: An Introduction, first ed., Prentice Hall, Harlow, 2000. [27] C.F. Matta, R.J. Boyd, The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, Wiley-VCH, Weinheim, 2007. [28] S. Shahbazian, Int. J. Quantum Chem. 111 (2011) 4497. [29] S. Shahbazian, Int. J. Quantum Chem. 118 (2018), e25637. [30] M. Goli, S. Shahbazian, Theor. Chem. Accounts 129 (2011) 235. [31] M. Goli, S. Shahbazian, Theor. Chem. Accounts 131 (2012) 1208. [32] M. Goli, S. Shahbazian, Theor. Chem. Accounts 132 (2013) 1362. [33] M. Goli, S. Shahbazian, Theor. Chem. Accounts 132 (2013) 1410. [34] M. Goli, S. Shahbazian, Comput. Theor. Chem. 1053 (2015) 96. [35] M. Goli, S. Shahbazian, ChemPhysChem 17 (2016) 3875. [36] M. Gharabaghi, S. Shahbazian, J. Chem. Phys. 146 (2017) 154106. [37] S. Shahbazian, Found. Chem. 15 (2013) 287. [38] R. Hefferlin, Molecular Taxonomy: String, Quark, Hadron, Nuclear, Atomic and Chemical Molecule Periodic or Invariant Systems, LAP Lambert Academic Publishing, Saarbr€ ucken, Germany, 2011. [39] P. Nasertayoob, M. Goli, S. Shahbazian, Int. J. Quantum Chem. 111 (2011) 1970. [40] M. Goli, S. Shahbazian, Int. J. Quantum Chem. 111 (2011) 1982. [41] F. Heidar Zadeh, S. Shahbazian, Int. J. Quantum Chem. 111 (2011) 1999. [42] M. Goli, S. Shahbazian, Phys. Chem. Chem. Phys. 16 (2014) 6602. [43] M. Goli, S. Shahbazian, Phys. Chem. Chem. Phys. 17 (2015) 245. [44] M. Goli, S. Shahbazian, Phys. Chem. Chem. Phys. 17 (2015) 7023. [45] M. Goli, S. Shahbazian, Chem. A Eur. J. 22 (2016) 2525. [46] M. Goli, S. Shahbazian, Phys. Chem. Chem. Phys. 20 (2018) 16749. [47] M. Goli, S. Shahbazian, ChemPhysChem 20 (2019) 831. [48] R.F.W. Bader, M.E. Stephens, Chem. Phys. Lett. 26 (1974) 445. [49] R.F.W. Bader, M.E. Stephens, J. Am. Chem. Soc. 97 (1975) 7391. [50] M.A. Spackman, E.N. Maslen, J. Phys. Chem. 90 (1986) 2020. [51] Z.A. Keyvani, S. Shahbazian, M. Zahedi, Chem. A Eur. J. 22 (2016) 5003. [52] Z.A. Keyvani, S. Shahbazian, M. Zahedi, ChemPhysChem 17 (2016) 3260. [53] H. Weinstein, P. Politzer, S. Srebrenik, Theor. Chim. Acta 38 (1975) 159. [54] J.C. Angulo, R.J. Ya´n˜ez, J.S. Dehesa, E. Romera, Int. J. Quantum Chem. 58 (1996) 11. [55] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, Phys. Rev. A 16 (1977) 1782. [56] P.W. Ayers, R.G. Parr, Int. J. Quantum Chem. 95 (2003) 877. [57] R.G. Parr, J. Chem. Sci. 117 (2005) 613. [58] W. Kohn, A. Yaniv, Proc. Natl. Acad. Sci. 75 (1978) 5270. [59] E. Prodan, W. Kohn, Proc. Natl. Acad. Sci. 102 (2005) 11635. [60] E. Prodan, Phys. Rev. B 73 (2006), 085108. [61] S. Fias, F. Heidar-Zadeh, P. Geerlings, P.W. Ayers, Proc. Natl. Acad. Sci. 114 (2017) 11633. [62] R.F.W. Bader, Y. Tal, S.G. Anderson, T.T. Nguyen-Dang, Isr. J. Chem. 19 (1980) 8. [63] P. Coppens, X-Ray Charge Densities and Chemical Bonding, International Union of Crystallography, Chester, England: Oxford; New York, 1997. [64] T.S. Koritsanszky, P. Coppens, Chem. Rev. 101 (2001) 1583. [65] C. Gatti, P. Macchi (Eds.), Modern Charge-Density Analysis, Springer, Netherlands, 2012. [66] P. Macchi, Crystallogr. Rev. 19 (2013) 58. [67] H. Joypazadeh, S. Shahbazian, Found. Chem. 16 (2014) 63. [68] P. Nasertayoob, S. Shahbazian, Int. J. Quantum Chem. 110 (2010) 1188. [69] D. Andrae, Phys. Rep. 336 (2000) 413.

References

[70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117]

99

R.F.W. Bader, D. Bayles, G.L. Heard, J. Chem. Phys. 112 (2000) 10095. Y.-G. Wang, K.B. Wiberg, N.H. Werstiuk, J. Phys. Chem. A 111 (2007) 3592. ´ .M. Penda´s, L. Gonza´lez, R.A. Mosquera, Phys. Chem. Chem. Phys. 16 (2014) 9249. D. Ferro-Costas, A ´ .M. Penda´s, R.A. Mosquera, Phys. Chem. Chem. Phys. 17 (2015) 26059. D. Ferro-Costas, E. Francisco, A L. Gutierrez-Arzaluz, D.I. Ramı´rez-Palma, L.G. Ramı´rez-Palma, J.E. Barquera-Lozada, J. Peon, F. CortesGuzma´n, Chem. A Eur. J. 25 (2019) 775. ´ .M. Penda´s, T. RochaA. Ferna´ndez-Alarco´n, J.L. Casals-Sainz, J.M. Guevara-Vela, A. Costales, E. Francisco, A Rinza, Phys. Chem. Chem. Phys. 21 (2019) 13428. C. Gatti, P. Fantucci, G. Pacchioni, Theor. Chim. Acta 72 (1987) 433. J. Cioslowski, J. Phys. Chem. 94 (1990) 5496. K.E. Edgecombe, R.O. Esquivel, V.H. Smith, F. M€ uller-Plathe, J. Chem. Phys. 97 (1992) 2593. K.E. Edgecombe, J. Vedene, H. Smith, F. M€ uller-Plathe, Z. Naturforsch. 48a (1993) 127. G.I. Bersuker, C. Peng, J.E. Boggs, J. Phys. Chem. 97 (1993) 9323. P. Mori-Sa´nchez, A.M. Penda´s, V. Luan˜a, Phys. Rev. B 63 (2001) 125103. G.K.H. Madsen, P. Blaha, K. Schwarz, J. Chem. Phys. 117 (2002) 8030. V. Luan˜a, P. Mori-Sa´nchez, A. Costales, M.A. Blanco, A.M. Penda´s, J. Chem. Phys. 119 (2003) 6341. L.A. Terrabuio, T.Q. Teodoro, M.G. Rachid, R.L.A. Haiduke, J. Phys. Chem. A 117 (2013) 10489. W.L. Cao, C. Gatti, P.J. MacDougall, R.F.W. Bader, Chem. Phys. Lett. 141 (1987) 380. A.M. Penda´s, M.A. Blanco, A. Costales, P.M. Sa´nchez, V. Luan˜a, Phys. Rev. Lett. 83 (1999) 1930. A. Costales, M.A. Blanco, A. Martı´n Penda´s, P. Mori-Sa´nchez, V. Luan˜a, J. Phys. Chem. A 108 (2004) 2794. M. Miao, R. Hoffmann, J. Am. Chem. Soc. 137 (2015) 3631. M. Miao, R. Hoffmann, J. Botana, I.I. Naumov, R.J. Hemley, Angew. Chem. Int. Ed. 56 (2017) 972. J. Herna´ndez-Trujillo, F. Cortes-Guzma´n, D.-C. Fang, R.F.W. Bader, Faraday Discuss. 135 (2006) 79. C. Foroutan-Nejad, S. Shahbazian, J. Mol. Struct. (THEOCHEM) 894 (2009) 20. R.F.G. Ruiz, A.R. Vernon, Eur. Phys. J. A 56 (2020) 136. G.E. Brown, M. Rho, Phys. Today 36 (1983) 24. F. Halzen, A.D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, first ed., Wiley, New York, 1984. E.D. Bloom, G.J. Feldman, Sci. Am. 246 (1982) 66. N. Isgur, G. Karl, Phys. Today 36 (1983) 36. J.L. Rosner, J. Phys. G Nucl. Part Phys. 34 (2007) S127. E.S. Swanson, Phys. Rep. 429 (2006) 243. S.L. Olsen, Phys. Today 67 (2014) 56. M. Karliner, J.L. Rosner, T. Skwarnicki, Annu. Rev. Nucl. Part. Sci. 68 (2018) 17. Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, S.-L. Zhu, Prog. Part. Nucl. Phys. 107 (2019) 237. A. Ali, L. Maiani, A.D. Polosa, Multiquark Hadrons, first ed., Cambridge University Press, 2019. J.-M. Richard, Few-Body Syst. 57 (2016) 1185. M.R. Shepherd, J.J. Dudek, R.E. Mitchell, Nature 534 (2016) 487. F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, B.-S. Zou, Rev. Mod. Phys. 90 (2018), 015004. A.K. Rai, J.N. Pandya, P.C. Vinodkummar, Nucl. Phys. A 782 (2007) 406. C.-Y. Wong, Phys. Rev. C 69 (2004), 055202. M. Suzuki, Phys. Rev. D 72 (2005) 114013. Y.-R. Liu, X. Liu, W.-Z. Deng, S.-L. Zhu, Eur. Phys. J. C 56 (2008) 63. X. Liu, Y.-R. Liu, W.-Z. Deng, S.-L. Zhu, Phys. Rev. D 77 (2008), 034003. N. Lee, Z.-G. Luo, X.-L. Chen, S.-L. Zhu, Phys. Rev. D 84 (2011), 014031. J.M.M. Hall, W. Kamleh, D.B. Leinweber, B.J. Menadue, B.J. Owen, A.W. Thomas, R.D. Young, Phys. Rev. Lett. 114 (2015) 132002. J.-B. Cheng, Y.-R. Liu, Phys. Rev. D 100 (2019), 054002. C. Rebbi, Sci. Am. 248 (1983) 54. D.H. Weingarten, Sci. Am. 274 (1996) 116. H. Wittig, in: H. Schopper (Ed.), Particle Physics Reference Library: Volume 1: Theory and Experiments, Springer International Publishing, Cham, 2020, pp. 137–262. M. Cardoso, N. Cardoso, P. Bicudo, Phys. Rev. D 81 (2010), 034504.

100 [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169]

5. The MC-QTAIM

N. Cardoso, M. Cardoso, P. Bicudo, Phys. Lett. B 710 (2012) 343. N. Cardoso, M. Cardoso, P. Bicudo, Phys. Rev. D 84 (2011), 054508. M. Cardoso, N. Cardoso, P. Bicudo, Phys. Rev. D 86 (2012), 014503. P. Bicudo, N. Cardoso, M. Cardoso, Prog. Part. Nucl. Phys. 67 (2012) 440. N. Cardoso, P. Bicudo, Phys. Rev. D 87 (2013), 034504. P. Bicudo, M. Cardoso, Phys. Rev. D 94 (2016), 094032. P. Bicudo, N. Cardoso, M. Cardoso, Phys. Rev. D 98 (2018) 114507. L. M€ uller, O. Philipsen, C. Reisinger, M. Wagner, Phys. Rev. D 100 (2019), 054503. A.H. Wuosmaa, R.R. Betts, M. Freer, B.R. Fulton, Annu. Rev. Nucl. Part. Sci. 45 (1995) 89. M. Freer, A.C. Merchant, J. Phys. G Nucl. Part Phys. 23 (1997) 261. R.R. Betts, A.H. Wuosmaa, Rep. Prog. Phys. 60 (1997) 819. B.R. Fulton, Contemp. Phys. 40 (1999) 299. M. Freer, C. R. Phys. 4 (2003) 475. W. von Oertzen, M. Freer, Y. Kanada-En’yo, Phys. Rep. 432 (2006) 43. M. Freer, Rep. Prog. Phys. 70 (2007) 2149. H. Horiuchi, K. Ikeda, K. Kat o, Prog. Theor. Phys. Suppl. 192 (2012) 1. M. Freer, H.O.U. Fynbo, Prog. Part. Nucl. Phys. 78 (2014) 1. M. Freer, H. Horiuchi, Y. Kanada-En’yo, D. Lee, U.-G. Meißner, Rev. Mod. Phys. 90 (2018), 035004. A.J. Stone, The Theory of Intermolecular Forces, second ed., Clarendon Press, oxford, 2013. R.J.N. Phillips, Rep. Prog. Phys. 22 (1959) 562. M.J. Moravcsik, Rep. Prog. Phys. 35 (1972) 587. R. Machleidt, K. Holinde, C. Elster, Phys. Rep. 149 (1987) 1. R. Machleidt, Phys. Rep. 242 (1994) 5. R. Machleidt, I. Slaus, J. Phys. G Nucl. Part Phys. 27 (2001) R69. E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Rev. Mod. Phys. 81 (2009) 1773. J.W. Negele, Rev. Mod. Phys. 54 (1982) 913. J.W. Negele, Phys. Today 38 (1985) 24. M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Mod. Phys. 75 (2003) 121. D. Vretenar, A.V. Afanasjev, G.A. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101. D.J. Dean, Phys. Today 60 (2007) 48. J.E. Drut, R.J. Furnstahl, L. Platter, Prog. Part. Nucl. Phys. 64 (2010) 120. M. Baldo, L. Robledo, P. Schuck, X. Vin˜as, J. Phys. G Nucl. Part Phys. 37 (2010), 064015. T. Niksˇic, D. Vretenar, P. Ring, Prog. Part. Nucl. Phys. 66 (2011) 519. J. Dobaczewski, J. Phys. G Nucl. Part. Phys. 43 (2016), 04LT01. S. Shen, H. Liang, W.H. Long, J. Meng, P. Ring, Prog. Part. Nucl. Phys. 109 (2019) 103713. N. Schunck, Energy Density Functional Methods for Atomic Nuclei, IOP publishing Ltd, 2019. G. Colo`, Adv. Phys. X 5 (2020) 1740061. J. Meng, P. Zhao, AAPPS Bull. 31 (2021) 2. A.B. Migdal, Nuclear Theory: The Quasiparticle Method, first ed., Benjamin, 1968. J.-P. Blaizot, G. Ripka, Quantum Theory of Finite Systems, The MIT Press, Cambridge, Mass, 1985. M. Greiner, Nuclear Models, Springer, Berlin, 2009. A.M. Green, Rep. Prog. Phys. 28 (1965) 113. G.F. Bertsch, Sci. Am. 248 (1983) 62. A. Bohr, Rev. Mod. Phys. 48 (1976) 365. I. Ragnarsson, S. G€ osta Nilsson, R.K. Sheline, Phys. Rep. 45 (1978) 1. I. Talmi, in: J.W. Negele, E.W. Vogt (Eds.), Advances in Nuclear Physics, vol. 27, Springer US, Boston, MA, 2003, pp. 1–275. A. Arima, F. Iachello, Annu. Rev. Nucl. Part. Sci. 31 (1981) 75. F. Iachello, Am. Sci. 70 (1982) 294. R.F. Casten, D.H. Feng, Phys. Today 37 (1984) 26. F. Iachello, A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge, 1987. B.R. Mottelson, Science 193 (1976) 287. G. Scharff-Goldhaber, C.B. Dover, A.L. Goodman, Annu. Rev. Nucl. Part. Sci. 26 (1976) 239.

References

[170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217]

101

S.S.M. Wong, Introductory Nuclear Physics, second ed., Wiley-VCH, New York, 1999. D.M. Brink, J. Phys. Conf. Ser. 111 (2008), 012001. D.A. Bromley, Sci. Am. 239 (1978) 58. W. Greiner, J.Y. Park, W. Scheid, Nuclear Molecules, World Scientific Publishing Company, Singapore; River Edge, N.J, 1995. K. Ikeda, N. Takigawa, H. Horiuchi, Prog. Theor. Phys. Suppl. E68 (1968) 464. Z. Ren, B. Zhou, Front. Phys. 13 (2018) 132110. L.R. Hafstad, E. Teller, Phys. Rev. 54 (1938) 681. L. Pauling, Science 150 (1965) 297. Y. Kanada-En’yo, D. Lee, Phys. Rev. C 103 (2021), 024318. J.L. Visschers, R. Van Wageningen, Phys. Lett. B 34 (1971) 455. D.V. Fedorov, A.S. Jensen, Phys. Lett. B 389 (1996) 631. J.A. Wheeler, Phys. Rev. 52 (1937) 1107. K. Wildermuth, Y.C. Tang, A Unified Theory of the Nucleus, Academic Press, New York, 1977. H. Margenau, Phys. Rev. 59 (1941) 37. D.L. Hill, J.A. Wheeler, Phys. Rev. 89 (1953) 1102. C. Bloch (Ed.), Many-Body Description of Nuclear Structure and Reactions 1966 Proceedings of the International School of Physics Course XXXVI, Academic Press, 1966. H. Horiuchi, Prog. Theor. Phys. Suppl. 62 (1977) 90. Y. Funaki, H. Horiuchi, A. Tohsaki, Prog. Part. Nucl. Phys. 82 (2015) 78. W. Bauhoff, H. Schultheis, R. Schultheis, Phys. Lett. B 95 (1980) 5. W. Bauhoff, H. Schultheis, R. Schultheis, Phys. Rev. C 22 (1980) 861. W. Bauhoff, H. Schultheis, R. Schultheis, Phys. Lett. B 106 (1981) 278. W. Bauhoff, H. Schultheis, R. Schultheis, Phys. Rev. C 29 (1984) 1046. S. Marsh, W.D.M. Rae, Phys. Lett. B 180 (1986) 185. J. Zhang, W.D.M. Rae, Nucl. Phys. A 564 (1993) 252. J. Zhang, W.D.M. Rae, A.C. Merchant, Nucl. Phys. A 575 (1994) 61. M. Freer, R.R. Betts, A.H. Wuosmaa, Nucl. Phys. A 587 (1995) 36. R. Canavan, M. Freer, J. Phys. G Nucl. Part Phys. 47 (2020), 095102. Y. Kanada-En’yo, M. Kimura, H. Horiuchi, C. R. Phys. 4 (2003) 497. Y. Kanada-En’yo, M. Kimura, A. Ono, Prog. Theor. Exp. Phys. 2012 (2012). Y. Kanada-En’yo, H. Horiuchi, Phys. Rev. C 68 (2003), 014319. R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandharipande, Phys. Rev. C 62 (2000), 014001. J.-P. Ebran, E. Khan, T. Niksˇic, D. Vretenar, Nature 487 (2012) 341. A.D. Becke, K.E. Edgecombe, J. Chem. Phys. 92 (1990) 5397. B. Silvi, A. Savin, Nature 371 (1994) 683. A. Savin, R. Nesper, S. Wengert, T.F. F€assler, Angew. Chem. Int. Ed. 36 (1997) 1808. P.-G. Reinhard, J.A. Maruhn, A.S. Umar, V.E. Oberacker, Phys. Rev. C 83 (2011), 034312. C.L. Zhang, B. Schuetrumpf, W. Nazarewicz, Phys. Rev. C 94 (2016), 064323. P. Jerabek, B. Schuetrumpf, P. Schwerdtfeger, W. Nazarewicz, Phys. Rev. Lett. 120 (2018), 053001. B. Schuetrumpf, C. Zhang, EPJ Web Conf. 163 (2017), 00050. G. R€ opke, A. Schnell, P. Schuck, P. Nozie`res, Phys. Rev. Lett. 80 (1998) 3177. A. Tohsaki, H. Horiuchi, P. Schuck, G. R€ opke, Phys. Rev. Lett. 87 (2001) 192501. Y. Funaki, A. Tohsaki, H. Horiuchi, P. Schuck, G. R€ opke, Phys. Rev. C 67 (2003), 051306. H. Horiuchi, J. Phys. Conf. Ser. 321 (2011), 012042. A.J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Illustrated edition, Oxford University Press, Oxford; New York, 2006. Y. Funaki, H. Horiuchi, W. von Oertzen, G. R€ opke, P. Schuck, A. Tohsaki, T. Yamada, Phys. Rev. C 80 (2009), 064326. Y. Funaki, M. Girod, H. Horiuchi, G. R€ opke, P. Schuck, A. Tohsaki, T. Yamada, J. Phys. G Nucl. Part Phys. 37 (2010), 064012. P. Schuck, Y. Funaki, H. Horiuchi, G. R€ opke, A. Tohsaki, T. Yamada, Phys. Scr. 91 (2016) 123001. J.-P. Ebran, M. Girod, E. Khan, R.D. Lasseri, P. Schuck, Phys. Rev. C 102 (2020), 014305.

102

5. The MC-QTAIM

[218] S. Shahbazian, in: R. Chauvin, C. Lepetit, B. Silvi, E. Alikhani (Eds.), Applications of Topological Methods in Molecular Chemistry, Springer International Publishing, 2016, pp. 89–100. [219] J. Okołowicz, W. Nazarewicz, M. Płoszajczak, Fortschr. Phys. 61 (2013) 66. [220] J.-P. Ebran, E. Khan, T. Niksˇic, D. Vretenar, Phys. Rev. C 87 (2013), 044307. [221] J.-P. Ebran, E. Khan, T. Niksˇic, D. Vretenar, J. Phys. G Nucl. Part Phys. 44 (2017) 103001. [222] K. Kravvaris, A. Volya, Phys. Rev. Lett. 119 (2017), 062501. [223] D.M. Rodkin, Y.M. Tchuvil’sky, Phys. Lett. B 788 (2019) 238. [224] D. Horva´th, Nucl. Inst. Methods Phys. Res. B 87 (1994) 273. [225] L.M. Simons, D. Abbot, B. Bach, R. Bacher, A. Badertscher, P. Bl€ um, P. DeCecco, J. Eades, J. Egger, K. Elsener, D. Gotta, P. Hauser, K. Heitlinger, D. Horva´th, F. Kottmann, E. Morenzoni, J. Missimer, J.J. Reidy, R. Siegel, D. Taqqu, D. Viel, Nucl. Inst. Methods Phys. Res. B 87 (1994) 293. [226] E. Friedman, Nucl. Phys. A 639 (1998) 511c. [227] P. Indelicato, Phys. Scr. 2004 (2004) 20. [228] D. Horva´th, in: A. Vertes, S. Nagy, Z. Klencsa´r, R.G. Lovas, F. R€ osch (Eds.), Handbook of Nuclear Chemistry, Springer US, Boston, MA, 2011, pp. 1485–1513. [229] J. Eades, F.J. Hartmann, Rev. Mod. Phys. 71 (1999) 373. [230] J.-M. Richard, Front. Phys. 8 (6) (2020) 1. [231] M. Charlton, J. Eades, D. Horva´th, R.J. Hughes, C. Zimmermann, Phys. Rep. 241 (1994) 65. [232] G. Baur, G. Boero, A. Brauksiepe, A. Buzzo, W. Eyrich, R. Geyer, D. Grzonka, J. Hauffe, K. Kilian, M. LoVetere, M. Macri, M. Moosburger, R. Nellen, W. Oelert, S. Passaggio, A. Pozzo, K. R€ ohrich, K. Sachs, G. Schepers, T. Sefzick, R.S. Simon, R. Stratmann, F. Stinzing, M. Wolke, Phys. Lett. B 368 (1996) 251. [233] M. Amoretti, C. Amsler, G. Bonomi, A. Bouchta, P. Bowe, C. Carraro, C.L. Cesar, M. Charlton, M.J.T. Collier, M. Doser, V. Filippini, K.S. Fine, A. Fontana, M.C. Fujiwara, R. Funakoshi, P. Genova, J.S. Hangst, R.S. Hayano, M.H. Holzscheiter, L.V. Jørgensen, V. Lagomarsino, R. Landua, D. Lindel€ of, E.L. Rizzini, M. Macrı`, N. Madsen, G. Manuzio, M. Marchesotti, P. Montagna, H. Pruys, C. Regenfus, P. Riedler, J. Rochet, A. Rotondi, G. Rouleau, G. Testera, A. Variola, T.L. Watson, D.P. van der Werf, Nature 419 (2002) 456. [234] ATRAP Collaboration, G. Gabrielse, N.S. Bowden, P. Oxley, A. Speck, C.H. Storry, J.N. Tan, M. Wessels, D. Grzonka, W. Oelert, G. Schepers, T. Sefzick, J. Walz, H. Pittner, T.W. H€ansch, E.A. Hessels, Phys. Rev. Lett. 89 (2002), 213401. [235] G.B. Andresen, M.D. Ashkezari, M. Baquero-Ruiz, W. Bertsche, P.D. Bowe, E. Butler, C.L. Cesar, S. Chapman, M. Charlton, A. Deller, S. Eriksson, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, A. Gutierrez, J.S. Hangst, W.N. Hardy, M.E. Hayden, A.J. Humphries, R. Hydomako, M.J. Jenkins, S. Jonsell, L.V. Jørgensen, L. Kurchaninov, N. Madsen, S. Menary, P. Nolan, K. Olchanski, A. Olin, A. Povilus, P. Pusa, F. Robicheaux, E. Sarid, S.S. el Nasr, D.M. Silveira, C. So, J.W. Storey, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, Y. Yamazaki, Nature 468 (2010) 673. [236] M. Ahmadi, B.X.R. Alves, C.J. Baker, W. Bertsche, E. Butler, A. Capra, C. Carruth, C.L. Cesar, M. Charlton, S. Cohen, R. Collister, S. Eriksson, A. Evans, N. Evetts, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, A. Gutierrez, J.S. Hangst, W.N. Hardy, M.E. Hayden, C.A. Isaac, A. Ishida, M.A. Johnson, S.A. Jones, S. Jonsell, L. Kurchaninov, N. Madsen, M. Mathers, D. Maxwell, J.T.K. McKenna, S. Menary, J.M. Michan, T. Momose, J.J. Munich, P. Nolan, K. Olchanski, A. Olin, P. Pusa, C.Ø. Rasmussen, F. Robicheaux, R.L. Sacramento, M. Sameed, E. Sarid, D.M. Silveira, S. Stracka, G. Stutter, C. So, T.D. Tharp, J.E. Thompson, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, Nature 541 (2017) 506. [237] M. Ahmadi, B.X.R. Alves, C.J. Baker, W. Bertsche, E. Butler, A. Capra, C. Carruth, C.L. Cesar, M. Charlton, S. Cohen, R. Collister, S. Eriksson, A. Evans, N. Evetts, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, A. Gutierrez, J.S. Hangst, W.N. Hardy, M.E. Hayden, C.A. Isaac, A. Ishida, M.A. Johnson, S.A. Jones, S. Jonsell, L. Kurchaninov, N. Madsen, M. Mathers, D. Maxwell, J.T.K. McKenna, S. Menary, J.M. Michan, T. Momose, J.J. Munich, P. Nolan, K. Olchanski, A. Olin, P. Pusa, C.Ø. Rasmussen, F. Robicheaux, R.L. Sacramento, M. Sameed, E. Sarid, D.M. Silveira, S. Stracka, G. Stutter, C. So, T.D. Tharp, J.E. Thompson, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, Nature 548 (2017) 66. [238] M. Ahmadi, B.X.R. Alves, C.J. Baker, W. Bertsche, A. Capra, C. Carruth, C.L. Cesar, M. Charlton, S. Cohen, R. Collister, S. Eriksson, A. Evans, N. Evetts, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, J.S. Hangst, W.N. Hardy, M.E. Hayden, E.D. Hunter, C.A. Isaac, M.A. Johnson, J.M. Jones, S.A. Jones, S. Jonsell, A. Khramov, P. Knapp, L. Kurchaninov, N. Madsen, D. Maxwell, J.T.K. McKenna, S. Menary, J.M. Michan, T.

References

[239]

[240]

[241]

[242] [243] [244]

[245]

103

Momose, J.J. Munich, K. Olchanski, A. Olin, P. Pusa, C.Ø. Rasmussen, F. Robicheaux, R.L. Sacramento, M. Sameed, E. Sarid, D.M. Silveira, D.M. Starko, G. Stutter, C. So, T.D. Tharp, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, Nature 561 (2018) 211. E. Widmann, C. Amsler, S. Arguedas Cuendis, H. Breuker, M. Diermaier, P. Dupre, C. Evans, M. Fleck, A. Gligorova, H. Higaki, Y. Kanai, B. Kolbinger, N. Kuroda, M. Leali, A.M.M. Leite, V. M€ackel, C. Malbrunot, V. Mascagna, O. Massiczek, Y. Matsuda, D.J. Murtagh, Y. Nagata, A. Nanda, D. Phan, C. Sauerzopf, M.C. Simon, M. Tajima, H. Spitzer, M. Strube, S. Ulmer, L. Venturelli, M. Wiesinger, Y. Yamazaki, J. Zmeskal, Hyperfine Interact. 240 (2018) 5. M. Ahmadi, B.X.R. Alves, C.J. Baker, W. Bertsche, A. Capra, C. Carruth, C.L. Cesar, M. Charlton, S. Cohen, R. Collister, S. Eriksson, A. Evans, N. Evetts, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, P. Granum, J.S. Hangst, W.N. Hardy, M.E. Hayden, E.D. Hunter, C.A. Isaac, M.A. Johnson, J.M. Jones, S.A. Jones, S. Jonsell, A. Khramov, P. Knapp, L. Kurchaninov, N. Madsen, D. Maxwell, J.T.K. McKenna, S. Menary, J.M. Michan, T. Momose, J.J. Munich, K. Olchanski, A. Olin, P. Pusa, C.Ø. Rasmussen, F. Robicheaux, R.L. Sacramento, M. Sameed, E. Sarid, D.M. Silveira, C. So, D.M. Starko, G. Stutter, T.D. Tharp, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, The ALPHA Collaboration, Nature 578 (2020) 375. C.J. Baker, W. Bertsche, A. Capra, C. Carruth, C.L. Cesar, M. Charlton, A. Christensen, R. Collister, A.C. Mathad, S. Eriksson, A. Evans, N. Evetts, J. Fajans, T. Friesen, M.C. Fujiwara, D.R. Gill, P. Grandemange, P. Granum, J.S. Hangst, W.N. Hardy, M.E. Hayden, D. Hodgkinson, E. Hunter, C.A. Isaac, M.A. Johnson, J.M. Jones, S.A. Jones, S. Jonsell, A. Khramov, P. Knapp, L. Kurchaninov, N. Madsen, D. Maxwell, J.T.K. McKenna, S. Menary, J.M. Michan, T. Momose, P.S. Mullan, J.J. Munich, K. Olchanski, A. Olin, J. Peszka, A. Powell, P. Pusa, C.Ø. Rasmussen, F. Robicheaux, R.L. Sacramento, M. Sameed, E. Sarid, D.M. Silveira, D.M. Starko, C. So, G. Stutter, T.D. Tharp, A. Thibeault, R.I. Thompson, D.P. van der Werf, J.S. Wurtele, Nature 592 (2021) 35. W. Greiner, Int. J. Mod. Phys. E 05 (1996) 1. J. Chen, D. Keane, Y.-G. Ma, A. Tang, Z. Xu, Phys. Rep. 760 (2018) 1. R. Arsenescu, C. Baglin, H.P. Beck, K. Borer, A. Bussie`re, K. Elsener, P. Gorodetzky, J.P. Guillaud, S. Kabana, R. Klingenberg, G. Lehmann, T. Linden, K.D. Lohmann, R. Mommsen, U. Moser, K. Pretzl, J. Schacher, R. Spiwoks, J. Tuominiemi, M. Weber, New J. Phys. 5 (2003) 1. H. Agakishiev, M.M. Aggarwal, Z. Ahammed, A.V. Alakhverdyants, I. Alekseev, J. Alford, B.D. Anderson, C.D. Anson, D. Arkhipkin, G.S. Averichev, J. Balewski, D.R. Beavis, N.K. Behera, R. Bellwied, M.J. Betancourt, R.R. Betts, A. Bhasin, A.K. Bhati, H. Bichsel, J. Bielcik, J. Bielcikova, B. Biritz, L.C. Bland, I.G. Bordyuzhin, W. Borowski, J. Bouchet, E. Braidot, A.V. Brandin, A. Bridgeman, S.G. Brovko, E. Bruna, S. Bueltmann, I. Bunzarov, T.P. Burton, X.Z. Cai, H. Caines, M. Calderon, D. Cebra, R. Cendejas, M.C. Cervantes, Z. Chajecki, P. Chaloupka, S. Chattopadhyay, H.F. Chen, J.H. Chen, J.Y. Chen, L. Chen, J. Cheng, M. Cherney, A. Chikanian, K.E. Choi, W. Christie, P. Chung, M.J.M. Codrington, R. Corliss, J.G. Cramer, H.J. Crawford, A.D. Leyva, L.C. De Silva, R.R. Debbe, T.G. Dedovich, A.A. Derevschikov, R.D. de Souza, L. Didenko, P. Djawotho, S.M. Dogra, X. Dong, J.L. Drachenberg, J.E. Draper, J.C. Dunlop, L.G. Efimov, M. Elnimr, J. Engelage, G. Eppley, M. Estienne, L. Eun, O. Evdokimov, R. Fatemi, J. Fedorisin, R.G. Fersch, P. Filip, E. Finch, V. Fine, Y. Fisyak, C.A. Gagliardi, D.R. Gangadharan, F. Geurts, P. Ghosh, Y.N. Gorbunov, A. Gordon, O.G. Grebenyuk, D. Grosnick, S.M. Guertin, A. Gupta, S. Gupta, W. Guryn, B. Haag, O. Hajkova, A. Hamed, L.-X. Han, J.W. Harris, J.P. Hays-Wehle, M. Heinz, S. Heppelmann, A. Hirsch, E. Hjort, G.W. Hoffmann, D.J. Hofman, B. Huang, H.Z. Huang, T.J. Humanic, L. Huo, G. Igo, P. Jacobs, W.W. Jacobs, C. Jena, F. Jin, J. Joseph, E.G. Judd, S. Kabana, K. Kang, J. Kapitan, K. Kauder, H.W. Ke, D. Keane, A. Kechechyan, D. Kettler, D.P. Kikola, J. Kiryluk, A. Kisiel, V. Kizka, S.R. Klein, A.G. Knospe, D.D. Koetke, T. Kollegger, J. Konzer, I. Koralt, L. Koroleva, W. Korsch, L. Kotchenda, V. Kouchpil, P. Kravtsov, K. Krueger, M. Krus, L. Kumar, P. Kurnadi, M.A.C. Lamont, J.M. Landgraf, S. LaPointe, J. Lauret, A. Lebedev, R. Lednicky, J.H. Lee, W. Leight, M.J. LeVine, C. Li, L. Li, N. Li, W. Li, X. Li, X. Li, Y. Li, Z.M. Li, M.A. Lisa, F. Liu, H. Liu, J. Liu, T. Ljubicic, W.J. Llope, R.S. Longacre, W.A. Love, Y. Lu, E.V. Lukashov, X. Luo, G.L. Ma, Y.G. Ma, D.P. Mahapatra, R. Majka, O.I. Mall, R. Manweiler, S. Margetis, C. Markert, H. Masui, H.S. Matis, Y.A. Matulenko, D. McDonald, T.S. McShane, A. Meschanin, R. Milner, N.G. Minaev, S. Mioduszewski, M.K. Mitrovski, Y. Mohammed, B. Mohanty, M.M. Mondal, B. Morozov, D.A. Morozov, M.G. Munhoz, M.K. Mustafa, M. Naglis, B.K. Nandi, T.K. Nayak, P.K. Netrakanti, L.V. Nogach, S.B. Nurushev, G. Odyniec, A. Ogawa, K. Oh, A. Ohlson, V. Okorokov, E.W. Oldag, D. Olson, M. Pachr, B.S. Page, S.K. Pal, Y.

104

[246] [247] [248] [249] [250]

[251] [252] [253] [254] [255] [256] [257] [258] [259] [260]

[261]

[262] [263]

[264] [265] [266] [267] [268] [269]

5. The MC-QTAIM

Pandit, Y. Panebratsev, T. Pawlak, H. Pei, T. Peitzmann, C. Perkins, W. Peryt, P. Pile, M. Planinic, M.A. Ploskon, J. Pluta, D. Plyku, N. Poljak, J. Porter, A.M. Poskanzer, B.V.K.S. Potukuchi, C.B. Powell, D. Prindle, C. Pruneau, N.K. Pruthi, P.R. Pujahari, J. Putschke, H. Qiu, R. Raniwala, S. Raniwala, R.L. Ray, R. Redwine, R. Reed, H.G. Ritter, J.B. Roberts, O.V. Rogachevskiy, J.L. Romero, L. Ruan, J. Rusnak, N.R. Sahoo, I. Sakrejda, S. Salur, J. Sandweiss, E. Sangaline, A. Sarkar, J. Schambach, R.P. Scharenberg, A.M. Schmah, N. Schmitz, T.R. Schuster, J. Seele, J. Seger, I. Selyuzhenkov, P. Seyboth, N. Shah, E. Shahaliev, M. Shao, M. Sharma, S.S. Shi, Q.Y. Shou, E.P. Sichtermann, F. Simon, R.N. Singaraju, M.J. Skoby, N. Smirnov, D. Solanki, P. Sorensen, H.M. Spinka, B. Srivastava, T.D.S. Stanislaus, D. Staszak, S.G. Steadman, J.R. Stevens, R. Stock, M. Strikhanov, B. Stringfellow, A.A.P. Suaide, M.C. Suarez, N.L. Subba, M. Sumbera, X.M. Sun, Y. Sun, Z. Sun, B. Surrow, D.N. Svirida, T.J.M. Symons, A.S. de Toledo, J. Takahashi, A.H. Tang, The STAR Collaboration, Nature 473 (2011) 353. M. Camac, A.D. McGuire, J.B. Platt, H.J. Schulte, Phys. Rev. 88 (1952) 134. V.L. Fitch, J. Rainwater, Phys. Rev. 92 (1953) 789. S. Koslov, V. Fitch, J. Rainwater, Phys. Rev. 95 (1954) 291. G.R. Burleson, D. Cohen, R.C. Lamb, D.N. Michael, R.A. Schluter, T.O. White, Phys. Rev. Lett. 15 (1965) 70. M. Iwasaki, R.S. Hayano, T.M. Ito, S.N. Nakamura, T.P. Terada, D.R. Gill, L. Lee, A. Olin, M. Salomon, S. Yen, K. Bartlett, G.A. Beer, G. Mason, G. Trayling, H. Outa, T. Taniguchi, Y. Yamashita, R. Seki, Phys. Rev. Lett. 78 (1997) 3067. C. Curceanu, C. Guaraldo, M. Iliescu, M. Cargnelli, R. Hayano, J. Marton, J. Zmeskal, T. Ishiwatari, M. Iwasaki, S. Okada, D.L. Sirghi, H. Tatsuno, Rev. Mod. Phys. 91 (2019), 025006. J.S. Cohen, AIP Conf. Proc. 500 (2000) 540. T. Yamazaki, N. Morita, R.S. Hayano, E. Widmann, J. Eades, Phys. Rep. 366 (2002) 183. J. H€ ufner, Phys. Rep. 21 (1975) 1. C.J. Batty, E. Friedman, A. Gal, Phys. Rep. 287 (1997) 385. D.F. Measday, Phys. Rep. 354 (2001) 243. E. Klempt, F. Bradamante, A. Martin, J.-M. Richard, Phys. Rep. 368 (2002) 119. T. Yamazaki, S. Hirenzaki, R.S. Hayano, H. Toki, Phys. Rep. 514 (2012) 1. D. Gotta, Prog. Part. Nucl. Phys. 52 (2004) 133. DEAR Collaboration, G. Beer, A.M. Bragadireanu, M. Cargnelli, C. Curceanu-Petrascu, J.-P. Egger, H. Fuhrmann, C. Guaraldo, M. Iliescu, T. Ishiwatari, K. Itahashi, M. Iwasaki, P. Kienle, T. Koike, B. Lauss, V. Lucherini, L. Ludhova, J. Marton, F. Mulhauser, T. Ponta, L.A. Schaller, R. Seki, D.L. Sirghi, F. Sirghi, J. Zmeskal, Phys. Rev. Lett. 94 (2005), 212302. M. Bazzi, G. Beer, L. Bombelli, A.M. Bragadireanu, M. Cargnelli, G. Corradi, C. Curceanu (Petrascu), A. d’Uffizi, C. Fiorini, T. Frizzi, F. Ghio, B. Girolami, C. Guaraldo, R.S. Hayano, M. Iliescu, T. Ishiwatari, M. Iwasaki, P. Kienle, P.L. Sandri, A. Longoni, V. Lucherini, J. Marton, S. Okada, D. Pietreanu, T. Ponta, A. Rizzo, A.R. Vidal, A. Scordo, H. Shi, D.L. Sirghi, F. Sirghi, H. Tatsuno, A. Tudorache, V. Tudorache, O.V. Doce, E. Widmann, J. Zmeskal, Phys. Lett. B 704 (2011) 113. A. Knecht, A. Skawran, S.M. Vogiatzi, Eur. Phys. J. Plus 135 (2020) 777. A. Hirtl, D.F. Anagnostopoulos, D.S. Covita, H. Fuhrmann, H. Gorke, D. Gotta, A. Gruber, M. Hennebach, P. Indelicato, T.S. Jensen, E.-O.L. Bigot, Y.-W. Liu, V.E. Markushin, J. Marton, M. Nekipelov, J.M.F. dos Santos, L.M. Simons, T. Strauch, M. Trassinelli, J.F.C.A. Veloso, J. Zmeskal, Eur. Phys. J. A 57 (2021) 70. D. McIntyre, C. Manogue, J. Tate, Quantum Mechanics: A Paradigms Approach, first ed., Pearson, Boston, 2012. M. Hori, A. So´ter, D. Barna, A. Dax, R. Hayano, S. Friedreich, B. Juha´sz, T. Pask, E. Widmann, D. Horva´th, L. Venturelli, N. Zurlo, Nature 475 (2011) 484. M. Hori, H. Aghai-Khozani, A. So´ter, D. Barna, A. Dax, R. Hayano, T. Kobayashi, Y. Murakami, K. Todoroki, H. Yamada, D. Horva´th, L. Venturelli, Science 354 (2016) 610. F. Ficek, P. Fadeev, V.V. Flambaum, D.F. Jackson Kimball, M.G. Kozlov, Y.V. Stadnik, D. Budker, Phys. Rev. Lett. 120 (2018) 183002. M. Hori, H. Aghai-Khozani, A. So´ter, A. Dax, D. Barna, Nature 581 (2020) 37. J.J. Krauth, K. Schuhmann, M.A. Ahmed, F.D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D.S. Covita, A.J. Dax, M. Diepold, L.M.P. Fernandes, B. Franke, S. Galtier, A.L. Gouvea, J. G€ otzfried, T. Graf, T.W. H€ansch, J. Hartmann, M. Hildebrandt, P. Indelicato, L. Julien, K. Kirch, A. Knecht, Y.-W. Liu, J. Machado, C.M.B.

References

[270] [271] [272] [273] [274] [275] [276] [277] [278] [279] [280] [281] [282] [283] [284] [285] [286] [287] [288] [289] [290] [291] [292] [293] [294] [295] [296] [297] [298] [299] [300] [301] [302] [303] [304] [305] [306] [307] [308] [309] [310] [311] [312] [313] [314]

105

Monteiro, F. Mulhauser, B. Naar, T. Nebel, F. Nez, J.M.F. dos Santos, J.P. Santos, C.I. Szabo, D. Taqqu, J.F.C.A. Veloso, J. Vogelsang, A. Voss, B. Weichelt, R. Pohl, A. Antognini, F. Kottmann, Nature 589 (2021) 527. A. Deloff, Fundamentals in Hadronic Atom Theory, Illustrated edition, World Scientific Publishing Company, River Edge, N.J, 2003. L.I. Ponomarev, Annu. Rev. Nucl. Sci. 23 (1973) 395. V.I. Goldanskii, V.G. Firsov, Annu. Rev. Phys. Chem. 22 (1971) 209. C.M. Surko, F.A. Gianturco (Eds.), New Directions in Antimatter Chemistry and Physics, 2001st ed., Springer, Dordrecht; Boston, 2001. K. Nagamine, Introductory Muon Science, Cambridge University Press, Cambridge, 2003. C.M. Surko, G.F. Gribakin, S.J. Buckman, J. Phys. B: At. Mol. Opt. Phys. 38 (2005) R57. G.F. Gribakin, J.A. Young, C.M. Surko, Rev. Mod. Phys. 82 (2010) 2557. K. S€ uvegh, T. Marek, A. Vertes, S. Nagy, Z. Klencsa´r, R.G. Lovas, F. R€ osch, Handbook of Nuclear Chemistry, Springer US, Boston, MA, 2011, pp. 1461–1484. A. Del Guerra, N. Belcari, M. Bisogni, Riv. Nuovo Cim. 39 (2016) 155. J.R. Dilworth, S.I. Pascu, Chem. Soc. Rev. 47 (2018) 2554. R.H. Heffner, D.G. Fleming, Phys. Today 37 (1984) 38. P.W. Percival, H. Fischer, Chem. Phys. 16 (1976) 89. J.H. Brewer, Philos. Transact. A Math. Phys. Eng. Sci. 350 (1995) 313. E. Roduner, Appl. Magn. Reson. 13 (1997) 1. S.J. Blundell, Contemp. Phys. 40 (1999) 175. S.J. Blundell, Chem. Rev. 104 (2004) 5717. J.H. Brewer, Phys. Procedia 30 (2012) 2. I. McKenzie, Annu. Rep. Prog. Chem. Sect. C: Phys. Chem. 109 (2013) 65. K. Yokoyama, P. Murahari, P. Heathcote, L. Nuccio, J.S. Lord, N.A. Morley, A.J. Drew, Phys. Scr. 88 (2013), 068511. L. Nuccio, L. Schulz, A.J. Drew, J. Phys. D Appl. Phys. 47 (2014) 473001. J.H. Brewer, Hyperfine Interact. 230 (2015) 35. B.D. Patterson, Rev. Mod. Phys. 60 (1988) 69. S.F.J. Cox, R.C. Newman, R.A. Catlow, E.A. Davis, A.M. Stoneham, M. Symons, S.F.J. Cox, A.M. Stoneham, M.C.R. Symons, Philos. Transact. A Math. Phys. Eng. Sci. 350 (1995) 171. K. Iwata, G.F. Gribakin, R.G. Greaves, C.M. Surko, Phys. Rev. Lett. 79 (1997) 39. V.G. Storchak, N.V. Prokof’ev, Rev. Mod. Phys. 70 (1998) 929. K. Iwata, G.F. Gribakin, R.G. Greaves, C. Kurz, C.M. Surko, Phys. Rev. A 61 (2000), 022719. G.F. Gribakin, Phys. Rev. A 61 (2000), 022720. S.J. Gilbert, L.D. Barnes, J.P. Sullivan, C.M. Surko, Phys. Rev. Lett. 88 (2002), 043201. L.D. Barnes, S.J. Gilbert, C.M. Surko, Phys. Rev. A 67 (2003), 032706. L.D. Barnes, J.A. Young, C.M. Surko, Phys. Rev. A 74 (2006), 012706. G.F. Gribakin, C.M.R. Lee, Phys. Rev. Lett. 97 (2006) 193201. J.A. Young, C.M. Surko, Phys. Rev. A 77 (2008), 052704. J.A. Young, C.M. Surko, Phys. Rev. A 78 (2008), 032702. S.F.J. Cox, Rep. Prog. Phys. 72 (2009) 116501. G.F. Gribakin, C.M.R. Lee, Eur. Phys. J. D 51 (2009) 51. J.R. Danielson, J.J. Gosselin, C.M. Surko, Phys. Rev. Lett. 104 (2010) 233201. B. Zgardzi nska, T. Goworek, Chem. Phys. 405 (2012) 32. F. Tuomisto, I. Makkonen, Rev. Mod. Phys. 85 (2013) 1583. P. Dalmas de Reotier, A. Maisuradze, A. Yaouanc, J. Phys. Soc. Jpn. 85 (2016), 091010. Y. Koike, T. Adachi, J. Phys. Soc. Jpn. 85 (2016), 091006. J.E. Sonier, J. Phys. Soc. Jpn. 85 (2016), 091005. K. Shimomura, T.U. Ito, J. Phys. Soc. Jpn. 85 (2016), 091013. F. Pratt, J. Phys. Soc. Jpn. 85 (2016), 091008. K. Wang, L. Schulz, M. Willis, S. Zhang, A.J. Misquitta, A.J. Drew, J. Phys. Soc. Jpn. 85 (2016), 091011. K. Nagamine, F.L. Pratt, S. Ohira, I. Watanabe, K. Ishida, S.N. Nakamura, T. Matsuzaki, Phys. B Condens. Matter. 289–290 (2000) 631.

106 [315] [316] [317] [318] [319] [320] [321] [322] [323] [324] [325] [326] [327] [328] [329] [330] [331] [332] [333] [334] [335] [336] [337] [338] [339] [340] [341] [342] [343] [344] [345] [346] [347] [348] [349] [350] [351] [352] [353] [354] [355] [356] [357]

5. The MC-QTAIM

I. McKenzie, J.-C. Brodovitch, P.W. Percival, T. Ramnial, J.A.C. Clyburne, J. Am. Chem. Soc. 125 (2003) 11565. K. Nagamine, E. Torikai, J. Phys. Condens. Matter 16 (2004) S4797. R.M. Macrae, Phys. B Condens. Matter. 374–375 (2006) 307. U.A. Jayasooriya, R. Grinter, P.L. Hubbard, G.M. Aston, J.A. Stride, G.A. Hopkins, L. Camus, I.D. Reid, S.P. Cottrell, S.F.J. Cox, Chem. A Eur. J. 13 (2007) 2266. B.M. McCollum, T. Abe, J.-C. Brodovitch, J.A.C. Clyburne, T. Iwamoto, M. Kira, P.W. Percival, R. West, Angew. Chem. Int. Ed. 47 (2008) 9772. B.M. McCollum, J.-C. Brodovitch, J.A.C. Clyburne, A. Mitra, P.W. Percival, A. Tomasik, R. West, Chem. A Eur. J. 15 (2009) 8409. B.M. McCollum, J.-C. Brodovitch, J.A.C. Clyburne, P.W. Percival, R. West, Phys. B Condens. Matter. 404 (2009) 940. R. West, P.W. Percival, Dalton Trans. 39 (2010) 9209. A. Mitra, J.-C. Brodovitch, C. Krempner, P.W. Percival, P. Vyas, R. West, Angew. Chem. Int. Ed. 49 (2010) 2893. P.W. Percival, J.-C. Brodovitch, M. Mozafari, A. Mitra, R. West, R.S. Ghadwal, R. Azhakar, H.W. Roesky, Chem. A Eur. J. 17 (2011) 11970. P.W. Percival, B.M. McCollum, J.-C. Brodovitch, M. Driess, A. Mitra, M. Mozafari, R. West, Y. Xiong, S. Yao, Organometallics 31 (2012) 2709. I. McKenzie, Phys. Chem. Chem. Phys. 16 (2014) 10600. R. West, K. Samedov, P.W. Percival, Chem. A Eur. J. 20 (2014) 9184. R. West, K. Samedov, A. Mitra, P.W. Percival, J.-C. Brodovitch, G. Langille, B.M. McCollum, T. Iwamoto, S. Ishida, C. Jones, J. Li, Can. J. Chem. 92 (2014) 508. M. Kobayashi, I. Tanaka, T. Kiyotani, N. Niimura, JPS Conf. Proc. 8 (2015), 033006. J.A. Wright, J.N.T. Peck, S.P. Cottrell, A. Jablonskyt_e, V.S. Oganesyan, C.J. Pickett, U.A. Jayasooriya, Angew. Chem. Int. Ed. 55 (2016) 14580. S. Ito, Y. Ueta, K. Koshino, K.M. Kojima, I. McKenzie, K. Mikami, Angew. Chem. Int. Ed. 57 (2018) 8608. I. McKenzie, Can. J. Chem. 96 (2018) 358. A.D. Pant, Y. Sugawara, I. Yanagihara, G.P. Khanal, I. Shiraki, W. Higemoto, K. Shimomura, K. Ishida, F.L. Pratt, E. Torikai, K. Nagamine, JPS Conf. Proc. 8 (2015), 033007. L. Chandrasena, K. Samedov, I. McKenzie, M. Mozafari, R. West, D.P. Gates, P.W. Percival, Angew. Chem. Int. Ed. 58 (2019) 297. Y. Sugawara, JPS Conf. Proc. 25 (2019), 011008. 1. D.G. Fleming, J. Manz, K. Sato, T. Takayanagi, Angew. Chem. Int. Ed. 53 (2014) 13706. C.J. Rhodes, R.M. Macrae, Sci. Prog. 98 (2015) 12. J. Charry, M.T.d.N. Varella, A. Reyes, Angew. Chem. Int. Ed. 57 (2018) 8859. F. Moncada, L. Pedraza-Gonza´lez, J. Charry, M.T.d.N. Varella, A. Reyes, Chem. Sci. 11 (2019) 44. S. Ito, D. Yoshida, Y. Kita, M. Tachikawa, J. Chem. Phys. 153 (2020) 224305. D. Bressanini, J. Chem. Phys. 154 (2021), 224306. D. Bressanini, J. Chem. Phys. 155 (2021), 054306. E.A.G. Armour, J.-M. Richard, K. Varga, Phys. Rep. 413 (2005) 1. J.A. Wheeler, Ann. N. Y. Acad. Sci. 48 (1946) 219. A. Ore, Phys. Rev. 70 (1946) 90. A.P.J. Mills, Riv. Nuovo Cim. 34 (2011) 151. M. Deutsch, Phys. Rev. 82 (1951) 455. H.C. Corben, S. DeBenedetti, Sci. Am. 191 (1954) 88. A. Rich, Rev. Mod. Phys. 53 (1981) 127. D.B. Cassidy, Eur. Phys. J. D 72 (2018) 53. A.P. Mills, AIP Conf. Proc. 2182 (2019), 030001. A.P. Mills, Phys. Rev. Lett. 46 (1981) 717. Y. Nagashima, Phys. Rep. 545 (2014) 95. M. Emami-Razavi, J.W. Darewych, Eur. Phys. J. D 75 (2021) 188. D.M. Schrader, F.M. Jacobsen, N.-P. Frandsen, U. Mikkelsen, Phys. Rev. Lett. 69 (1992) 57. D.M. Schrader, F.M. Jacobsen, N.-P. Frandsen, U. Mikkelsen, Phys. Rev. Lett. 69 (1992) 2880. D.B. Cassidy, A.P. Mills Jr., Nature 449 (2007) 195.

References

[358] [359] [360] [361] [362]

[363] [364] [365] [366] [367]

[368]

[369] [370] [371] [372] [373] [374] [375] [376] [377] [378] [379] [380] [381] [382] [383] [384] [385] [386] [387] [388] [389] [390]

107

D.B. Cassidy, S.H.M. Deng, A.P. Mills, Phys. Rev. A 76 (2007), 062511. D.B. Cassidy, T.H. Hisakado, H.W.K. Tom, A.P. Mills, Phys. Rev. Lett. 108 (2012) 133402. M.I. Eides, H. Grotch, V.A. Shelyuto, Phys. Rep. 342 (2001) 63. S.J. Brodsky, R.F. Lebed, Phys. Rev. Lett. 102 (2009) 213401. DIRAC Collaboration, B. Adeva, L. Afanasyev, Y. Allkofer, C. Amsler, A. Anania, S. Aogaki, A. Benelli, V. Brekhovskikh, T. Cechak, M. Chiba, P. Chliapnikov, P. Doskarova, D. Drijard, A. Dudarev, D. Dumitriu, D. Fluerasu, A. Gorin, O. Gorchakov, K. Gritsay, C. Guaraldo, M. Gugiu, M. Hansroul, Z. Hons, S. Horikawa, Y. Iwashita, V. Karpukhin, J. Kluson, M. Kobayashi, V. Kruglov, L. Kruglova, A. Kulikov, E. Kulish, A. Kuptsov, A. Lamberto, A. Lanaro, R. Lednicky, C. Marin˜as, J. Martincik, L. Nemenov, M. Nikitin, K. Okada, V. Olchevskii, M. Pentia, A. Penzo, M. Plo, P. Prusa, G. Rappazzo, A.R. Vidal, A. Ryazantsev, V. Rykalin, J. Saborido, J. Schacher, A. Sidorov, J. Smolik, F. Takeutchi, L. Tauscher, T. Trojek, S. Trusov, T. Urban, T. Vrba, V. Yazkov, Y. Yoshimura, M. Zhabitsky, P. Zrelov, Phys. Rev. Lett. 117 (2016), 112001. L. Nemenov, in: S.G. Karshenboim, F. Bassani, F.S. Pavone, M. Inguscio, T.W. H€ansch (Eds.), The Hydrogen Atom: Precision Physics of Simple Atomic Systems, Springer, Berlin, Heidelberg, 2001, pp. 223–245. C.J. Batty, Rep. Prog. Phys. 52 (1989) 1165. M. Augsburger, D.F. Anagnostopoulos, G. Borchert, D. Chatellard, J.-P. Egger, P. El-Khoury, H. Gorke, D. Gotta, P. Hauser, P. Indelicato, K. Kirch, S. Lenz, T. Siems, L.M. Simons, Hyperfine Interact. 118 (1999) 59. D. Gotta, Hyperfine Interact. 118 (1999) 35. N. Zurlo, M. Amoretti, C. Amsler, G. Bonomi, C. Carraro, C.L. Cesar, M. Charlton, M. Doser, A. Fontana, R. Funakoshi, P. Genova, R.S. Hayano, L.V. Jørgensen, A. Kellerbauer, V. Lagomarsino, R. Landua, E. Lodi Rizzini, M. Macrı`, N. Madsen, G. Manuzio, D. Mitchard, P. Montagna, L.G. Posada, H. Pruys, C. Regenfus, A. Rotondi, G. Testera, D.P. Van der Werf, A. Variola, L. Venturelli, Y. Yamazaki, Hyperfine Interact. 172 (2006) 97. L. Venturelli, M. Amoretti, C. Amsler, G. Bonomi, C. Carraro, C.L. Cesar, M. Charlton, M. Doser, A. Fontana, R. Funakoshi, P. Genova, R.S. Hayano, L.V. Jørgensen, A. Kellerbauer, V. Lagomarsino, R. Landua, E.L. Rizzini, M. Macrı`, N. Madsen, G. Manuzio, D. Mitchard, P. Montagna, L.G. Posada, H. Pruys, C. Regenfus, A. Rotondi, G. Testera, D.P. Van der Werf, A. Variola, Y. Yamazaki, N. Zurlo, Nucl. Inst. Methods Phys. Res. B 261 (2007) 40. A.M. Frolov, D.M. Wardlaw, Eur. Phys. J. D 63 (2011) 339. S.E. Jones, Nature 321 (1986) 127. J. Rafelski, S.E. Jones, Sci. Am. 257 (1987) 84. T. Yamazaki, Y. Akaishi, Phys. Rev. C 76 (2007), 045201. T. Yamazaki, Prog. Theor. Phys. Suppl. 170 (2007) 138. A. Dote, H. Horiuchi, Y. Akaishi, T. Yamazaki, Phys. Rev. C 70 (2004), 044313. T. Yamazaki, A. Dote, Y. Akaishi, Phys. Lett. B 587 (2004) 167. A. Dote, H. Horiuchi, Y. Akaishi, T. Yamazaki, Phys. Lett. B 590 (2004) 51. H.A. Kurtz, K.D. Jordan, Int. J. Quantum Chem. 14 (1978) 747. H.A. Kurtz, K.D. Jordan, J. Chem. Phys. 75 (1981) 1876. C. Kao, P.E. Cade, J. Chem. Phys. 80 (1984) 3234. M.J. Puska, A.P. Seitsonen, R.M. Nieminen, Phys. Rev. B 52 (1995) 10947. K. Strasburger, J. Chem. Phys. 111 (1999) 10555. K. Strasburger, J. Chem. Phys. 114 (2001) 615. M. Tachikawa, R.J. Buenker, M. Kimura, J. Chem. Phys. 119 (2003) 5005. R.J. Buenker, H.-P. Liebermann, V. Melnikov, M. Tachikawa, L. Pichl, M. Kimura, J. Phys. Chem. A 109 (2005) 5956. R.J. Buenker, H.-P. Liebermann, M. Tachikawa, L. Pichl, M. Kimura, Nucl. Inst. Methods Phys. Res. B 247 (2006) 47. F.A. Gianturco, J. Franz, R.J. Buenker, H.-P. Liebermann, L. Pichl, J.-M. Rost, M. Tachikawa, M. Kimura, Phys. Rev. A 73 (2006), 022705. M. Tachikawa, J. Phys. Condens. Matter 19 (2007) 365235. M. Tachikawa, Y. Kita, R.J. Buenker, Phys. Chem. Chem. Phys. 13 (2011) 2701. K. Koyanagi, Y. Kita, M. Tachikawa, Eur. Phys. J. D 66 (2012) 121. K. Koyanagi, Y. Kita, Y. Shigeta, M. Tachikawa, ChemPhysChem 14 (2013) 3458.

108 [391] [392] [393] [394] [395] [396] [397] [398] [399] [400] [401] [402] [403] [404] [405] [406] [407] [408] [409] [410] [411] [412] [413] [414] [415] [416] [417] [418] [419] [420] [421] [422] [423] [424] [425] [426] [427] [428] [429] [430] [431] [432] [433] [434] [435] [436] [437]

5. The MC-QTAIM

K. Koyanagi, Y. Takeda, T. Oyamada, Y. Kita, M. Tachikawa, Phys. Chem. Chem. Phys. 15 (2013) 16208. Y. Sugiura, K. Suzuki, T. Takayanagi, Y. Kita, M. Tachikawa, J. Comput. Chem. 39 (2018) 2060. Y. Sugiura, T. Takayanagi, Y. Kita, M. Tachikawa, Eur. Phys. J. D 73 (2019) 162. K. Ono, T. Oyamada, Y. Kita, M. Tachikawa, Eur. Phys. J. D 74 (2020) 89. J.M. Rost, D. Wintgen, Phys. Rev. Lett. 69 (1992) 2499. N. Jiang, D.M. Schrader, Phys. Rev. Lett. 81 (1998) 5113. G.G. Ryzhikh, J. Mitroy, K. Varga, J. Phys. B: At. Mol. Opt. Phys. 31 (1998) 3965. S.L. Saito, Nucl. Inst. Methods Phys. Res. B 171 (2000) 60. F. Rolim, J.P. Braga, J.R. Mohallem, Chem. Phys. Lett. 332 (2000) 139. J. Mitroy, M.W.J. Bromley, G.G. Ryzhikh, J. Phys. B: At. Mol. Opt. Phys. 35 (2002) R81. D. Bressanini, G. Morosi, J. Chem. Phys. 119 (2003) 7037. L.J. Dunne, J.N. Murrell, Int. J. Quantum Chem. 96 (2004) 512. F. Rolim, T. Moreira, J.R. Mohallem, Braz. J. Phys. 34 (2004) 1197. J. Mitroy, Phys. Rev. Lett. 94 (2005), 033402. J.-Y. Zhang, J. Mitroy, Phys. Rev. A 76 (2007), 014501. C.M. Surko, Nature 449 (2007) 153. D. Assafra˜o, H.R.J. Walters, J.R. Mohallem, Nucl. Inst. Methods Phys. Res. B 266 (2008) 491. R. Heyrovska, Nat. Preced. 1 (2011). D. Bressanini, Phys. Rev. A 97 (2018), 012508. D. Bressanini, Phys. Rev. A 99 (2019), 022510. H.J. Ache, Angew. Chem. Int. Ed. 11 (1972) 179. Y.C. Jean, P.E. Mallon, D.M. Schrader, Principles and Applications of Positron and Positronium Chemistry, World Scientific, 2003. S.J. Brawley, S. Armitage, J. Beale, D.E. Leslie, A.I. Williams, G. Laricchia, Science 330 (2010) 789. X. Ma, F. Wang, Radiat. Phys. Chem. 89 (2013) 14. D.C. Walker, Muon and Muonium Chemistry, Cambridge University Press, 1983. D.C. Walker, Acc. Chem. Res. 18 (1985) 167. E. Roduner, The Positive Muon as a Probe in Free Radical Chemistry: Potential and Limitations of the ΜSR Techniques, Springer-Verlag, Berlin Heidelberg, 1988. L.A. Schaller, C. Petitjean (Eds.), Muonic Atoms and Molecules, Birkh€auser Basel, 1993. E. Roduner, Chem. Soc. Rev. 22 (1993) 337. D.C. Walker, J. Chem. Soc. Faraday Trans. 94 (1998) 1. C.J. Rhodes, J. Chem. Soc. Perkin Trans. 2 (2002) 1379. D.G. Fleming, D.J. Arseneau, O. Sukhorukov, J.H. Brewer, S.L. Mielke, G.C. Schatz, B.C. Garrett, K.A. Peterson, D.G. Truhlar, Science 331 (2011) 448. C.J. Rhodes, Sci. Prog. 95 (2012) 101. J.S. M€ oller, P. Bonfa`, D. Ceresoli, F. Bernardini, S.J. Blundell, T. Lancaster, R.D. Renzi, N. Marzari, I. Watanabe, S. Sulaiman, M.I. Mohamed-Ibrahim, Phys. Scr. 88 (2013), 068510. N.J. Clayden, Phys. Scr. 88 (2013), 068507. R.M. Macrae, Sci. Prog. 96 (2013) 237. J. Aldegunde, P.G. Jambrina, E. Garcı´a, V.J. Herrero, V. Sa´ez-Ra´banos, F.J. Aoiz, Mol. Phys. 111 (2013) 3169. K. Ghandi, A. MacLean, Hyperfine Interact. 230 (2015) 17. I. Levine, Quantum Chemistry, seventh ed., Pearson, Boston, 2013. A.M. Frolov, Eur. Phys. J. D 72 (2018) 222. F. Moncada, D. Cruz, A. Reyes, Chem. Phys. Lett. 539–540 (2012) 209. F. Moncada, D. Cruz, A. Reyes, Chem. Phys. Lett. 570 (2013) 16. E. Posada, F. Moncada, A. Reyes, J. Phys. Chem. A 118 (2014) 9491. J.G. Muga, J.P. Palao, B. Navarro, I.L. Egusquiza, Phys. Rep. 395 (2004) 357. M. Born, R. Oppenheimer, Ann. Phys. 389 (1927) 457. M. Baer, Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections, first ed., Wiley-Interscience, Hoboken, N.J, 2006. H. Mustroph, ChemPhysChem 17 (2016) 2616.

References

109

[438] Y. Suzuki, K. Varga, Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems, 1998th ed., Springer, Berlin; New York, 1998. [439] S. Bubin, M. Pavanello, W.-C. Tung, K.L. Sharkey, L. Adamowicz, Chem. Rev. 113 (2013) 36. [440] J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, K. Varga, Rev. Mod. Phys. 85 (2013) 693. [441] R.G. Woolley, Adv. Phys. 25 (1976) 27. [442] H. Essen, Int. J. Quantum Chem. 12 (1977) 721. [443] H. Primas, Chemistry, Quantum Mechanics and Reductionism: Perspectives in Theoretical Chemistry, Springer Berlin Heidelberg, Berlin, Heidelberg, 1983. [444] S.J. Weininger, J. Chem. Educ. 61 (1984) 939. [445] R.G. Woolley, J. Chem. Educ. 62 (1985) 1082. [446] R.G. Woolley, Chem. Phys. Lett. 125 (1986) 200. [447] H.J. Monkhorst, Int. J. Quantum Chem. 72 (1999) 281. [448] B.T. Sutcliffe, R.G. Woolley, Phys. Chem. Chem. Phys. 7 (2005) 3664. [449] M. Cafiero, L. Adamowicz, Chem. Phys. Lett. 387 (2004) 136. [450] B.T. Sutcliffe, R.G. Woolley, Chem. Phys. Lett. 408 (2005) 445. [451] U. M€ uller-Herold, J. Chem. Phys. 124 (2006), 014105. [452] M. Cafiero, L. Adamowicz, Int. J. Quantum Chem. 107 (2007) 2679. [453] U. M€ uller-Herold, Eur. Phys. J. D 49 (2008) 311. [454] E. Ma´tyus, J. Hutter, U. M€ uller-Herold, M. Reiher, Phys. Rev. A 83 (2011), 052512. [455] E. Ma´tyus, J. Hutter, U. M€ uller-Herold, M. Reiher, J. Chem. Phys. 135 (2011) 204302. [456] E. Ma´tyus, Mol. Phys. 117 (2019) 590. [457] R.F.W. Bader, J. Phys. Chem. A 111 (2007) 7966. [458] R. McWeeny, Methods of Molecular Quantum Mechanics, second ed., Academic Press, London, 1992. [459] P.L.A. Popelier, D.S. Kosov, J. Chem. Phys. 114 (2001) 6539. [460] M.A. Blanco, A. Martı´n Penda´s, E. Francisco, J. Chem. Theory Comput. 1 (2005) 1096. [461] E. Francisco, A. Martı´n Penda´s, M.A. Blanco, J. Chem. Theory Comput. 2 (2006) 90. ´ . Martı´n Penda´s, Molecules 25 (2020) 4028. [462] J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A [463] A.F. Silva, L.J. Duarte, P.L.A. Popelier, Struct. Chem. 31 (2020) 507. [464] X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. [465] X. Fradera, J. Poater, S. Simon, M. Duran, M. Sola`, Theor. Chem. Accounts 108 (2002) 214. [466] R.F.W. Bader, P.F. Zou, Chem. Phys. Lett. 191 (1992) 54. [467] J.J. Sakurai, J.J. Napolitano, Modern Quantum Mechanics, second ed., Pearson, Boston, 2010. [468] D.C. Montgomery, G.C. Runger, Applied Statistics and Probability for Engineers, seventh ed., Wiley, 2018.

C H A P T E R

6 Theory developments and applications of next-generation QTAIM (NG-QTAIM) Xing Nie, Yong Yang, Tianlv Xu, Steven R. Kirk, and Samantha Jenkins Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research and Key Laboratory of Resource National and Local Joint Engineering Laboratory for New Petro-chemical Materials and Fine Utilization of Resources, College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha, Hunan, People’s Republic of China

In this contribution, the historical context of next-generation quantum theory of atoms in molecules (NG-QTAIM) is provided along with recent theoretical developments and a selection of applications. NG-QTAIM is currently the only vector-based quantum chemical theory since all other quantum chemical theories are scalar-based. As a consequence, NG-QTAIM can be used to investigate directional phenomena that include chiral discrimination, conical intersections, tunneling pathways of the flip rearrangement between permutation-inversion isomers and competitive ring opening reactions. Areas of suggested future development include molecular device environments such as laser irradiation and the prediction of asymmetric synthetic reactions toward the stereocontrol of chiral products are discussed. The latter will be undertaken by tracking the NG-QTAIM chirality of intermediate structures, where the Cahn-Ingold-Prelog (CIP) classifications are inaccessible due to the achiral nature of the intermediate species and/or products and reactants.

1. The vector-based perspective of chemical bonding 1.1 Historical context The most facile, i.e., preferred direction of electron charge density accumulation determines the direction of bond motion [1] from the electron-preceding perspective where a Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00014-1

111

Copyright # 2023 Elsevier Inc. All rights reserved.

112

6. Theory developments and applications of next-generation QTAIM

change in the electronic charge density distribution that defines a chemical bond results in a change in atomic positions [2]. Bone and Bader later proposed that if a structure is slightly perturbed the preferred motion of the electrons coincides with the motion of the atoms [3] that was then confirmed [4,5] that indeed the direction in which electrons find it easiest to move coincided with the preferred direction of motion of the atoms [4]. This led on from an earlier investigation that compared the easy directions of phonon modes of the cubic ice phase (ice Ic), at the bottom of the rotational band, with the easy directions of charge density accumulation: the e2 eigenvectors of the Hessian of the total charge density distribution and associated eigenvalues [6]. Water molecules, connected by hydrogen bonding, rotate more easily at the bottom of the rotational band as they minimally distort the hydrogen-bond network for rotational motion compared with any other location in the rotational band. The total electronic charge-density distribution of a relaxed structure quantifies where the electrons have accumulated and where they will prefer to relocate if they are perturbed that subsequently will be followed by the atomic nuclei. Since the phonon normal modes are obtained by performing harmonic distortions it can be seen that the information concerning the relocation of the atoms after being perturbed must already be contained within the relaxed structure; i.e., in the total charge-density distribution. A previous geometrical analysis of the projected density of states [7] provided evidence that it was possible to predict features in the projected density of states from the relaxed structure. In particular, the total electronic charge-density distribution of various phases of ice was used to explore the information that was obtainable for the most or least preferred directions of motion of atoms undergoing phonon vibrations based on earlier phonon calculations [7]. This was undertaken by examination of the eigenvectors of the Hessian of the total charge density distribution and combining with the intramolecular rotational projections at the bottom of the rotational band.

1.2 The need for a non-scalar, physics-based perspective of chemical bonding Conventional scalar-based chemistry cannot probe energetically degenerate global energy minima or local minima. Using next-generation QTAIM (NG-QTAIM) provides the first directional 3-D interpretation of chemical bonding, including bond-flexing, bond-twist and bond-anharmonicity. Unique developments of NG-QTAIM include: stereochemistry, distinguishing and quantifying isotopomers, tunneling pathways of the flip rearrangement between permutation-inversion isomers corresponding to the energetically degenerate global energy minima. NG-QTAIM has been used to distinguish properties relating to the S0 and S1 degenerate excited states at a conical intersection. The 3-D normal modes of vibration were presented in the 3-D Uσ-space. In addition, NG-QTAIM can be used, with or without nuclear motion present, to visualize the response to laser irradiation. The chemical origin of chirality, the missing chirality-helicity equivalence, was found using the stress tensor trajectory Tσ(s) in the space U(s). Agreement was found with the naming schemes of S and R stereoisomers from optical (laser) experiments. Essential insights into molecular switch functioning not available from the energy barrier or any scalar measures and a new measure to assess the degree of purity of the axial bond torsion for the design of rotary molecular motors are also provided. Next-generation QTAIM (NG-QTAIM) provides the directional and 3-D perspective needed to adapt to the needs of new molecular environments such as devices.

2. Origins of the NG-QTAIM bond-path framework set B

113

Conventional quantum theory of atoms in molecules (QTAIM) [8] that only uses scalar quantities is insufficient to distinguish S and R stereoisomers at the energy minimum, i.e., torsion θ ¼ 0.0° and can at best quantify the asymmetry of the charge density distribution in the form of the bond critical point ellipticity ε. Next-generation QTAIM (NG-QTAIM) [9] is a vector-based and directional quantum mechanical theory constructed within QTAIM [8] using the stress tensor, can differentiate the S and R stereoisomers for all values of the torsion θ, 180.0°  θ  +180.0°. The quantum stress tensor is directly related to the Ehrenfest force by the virial theorem and hence provides a physical explanation of the low frequency normal modes that accompany structural rearrangements [5]. The form of the stress tensor is ambiguous and consequently other forms of the stress tensor exist [10]. We use Bader’s formulation of the stress tensor [11] and NG-QTAIM more effectively distinguished the S and R stereoisomers of lactic acid [12] than conventional QTAIM.

1.3 Overview of the basic NG-QTAIM concepts The basic concepts of NG-QTAIM fall into two categories. Firstly, the scaled mapping the eigenvectors along the entire bond-path that connects the edges of the nuclear basins of bonded nuclei. This enables a representation of bonding, the bond-path framework set B that enables the admixture of double or single or ionic character to be visualized along an entire bond-path, rather than only returning a single number at the BCP [1,13–28]. The next step of this development process was to quantify the wrapping of the bond-path framework set B along the entirety of the bond-path, this enables the planarity and or twisting of B to be determined which is important in particular for excited state ring opening reactions [29–31]. Secondly, the mapping of the bond critical point (BCP) eigenvectors in a stress tensor trajectory space Uσ with stress tensor trajectories Tσ(s). To map out a non-zero extent in stress tensor trajectory space Uσ requires some perturbation of the BCP to enable the most preferred direction of charge density accumulation to be sampled. This perturbation could be of the form of pair of clockwise (180.0°  θ  0°) and counterclockwise (0.0°  θ  180.0°) directions of torsion, alternatively, a normal mode distortions where no suitable torsion bond is present [32].

2. Origins of the NG-QTAIM bond-path framework set B 2.1 Theory background: Construction of the NG-QTAIM bond-path framework set B The first developments of Next-Generation QTAIM (NG-QTAIM) were uncovered by an investigation into the response of the electronic charge density to the electronic excitation and double bond torsion in fulvene [33]. First, we present some background on QTAIM and the stress tensor σ(r). QTAIM that utilizes higher derivatives of ρ(rb) in effect, acting as a “magnifying lens” on the ρ(rb) derived properties of the wave-function to identify critical points in the total electronic charge density distribution ρ(r) by analyzing the gradient vector field rρ(r). These critical points can further

114

6. Theory developments and applications of next-generation QTAIM

be divided into four types of topologically stable critical points according to the set of ordered eigenvalues λ1 < λ2 < λ3, with corresponding eigenvectors e1, e2, e3 of the Hessian matrix. The Hessian of the total electronic charge density ρ(r) is defined as the matrix of partial second derivatives with respect to the spatial coordinates. These critical points are labeled using the notation (R, ω), where R is the rank of the Hessian matrix, the number of distinct non-zero eigenvalues and ω is the signature (the algebraic sum of the signs of the eigenvalues); the (3,3) [nuclear critical point (NCP), a local maximum generally corresponding to a nuclear location], (3,1) and (3,1) [saddle points, called bond critical points (BCP) and ring critical points (RCP), respectively] and (3,3) [the cage critical points (CCP)]. In the limit that the forces on the nuclei become vanishingly small, an atomic interaction line [34] becomes a bond-path, although not necessarily a chemical bond [35] with the complete set of critical points and the bond-paths of a molecule being referred to as the molecular graph. The total local energy density H(rb) [33,36]. Hðrb Þ ¼ Gðrb Þ + V ðrb Þ,

(6.1)

where G(rb) and V(rb) in Eq. (6.1) are the local kinetic and potential energy densities at a BCP, defines a degree of covalent character: A negative value for H(rb) < 0 for the closed-shell interaction, a value of the Laplacian r2ρ(rb) > 0, indicates a BCP with a degree of covalent character and conversely a positive value of H(rb) > 0 reveals a lack of covalent character for the closed-shell BCP. A shared-shell BCP always possesses both Laplacian r2ρ(rb) < 0 and H(rb) < 0. The interpretation of the eigenvalues is different between QTAIM and the stress tensor: in QTAIM the most “easy” preferred direction is simply the shallowest direction based on the readiness of the electronic charge density to accumulate or move. For the stress tensor however, the most preferred “easy” direction is determined as the most compressible, i.e., the least tensile. The eigenvalues are ordered λ1σ < λ2σ < λ3σ for the stress tensor with λ3σ being the purely tensile and λ1σ being the most compressive. For QTAIM, the ordering is λ1 < λ2 < λ3 with λ2 being shallower and more changeable than λ1, enabling us to understand that λ2 is comparable most compressible λ1σ stress tensor eigenvalue. Consequently, the stress tensor eigenvectors e1σ and e2σ frequently do not coincide with the QTAIM e1 and e2 eigenvectors, respectively, particularly for symmetrical bonds such as the central CdC bond in biphenyl that links the two phenyl rings. Using QTAIM and the stress tensor, a new understanding of bond torsion was revealed; the electronic charge density around the twisted bond was found not to rotate in accordance with the nuclei of the rotated –CH2 methylene group. To describe the torsion β of a bond-path within QTAIM subject to a molecular torsion α, we define the bond-path framework as the set of orthogonal e1, e2 and e3 eigenvectors and the basin path sets of the two bonded nuclei that comprise the torsional bond-path. We define the basin path set using a plane defined using either a single plane normal vector or a pair of noncollinear vectors in the plane passing through a specified point in space. The basin path set is a set of trajectories of a vector function, e.g., the gradient of the charge density, seeded at equidistant points around the circumference of a circle of small radius centered on the specified point and lying in the specified plane. The trajectories terminate where the charge density falls below 0.001 a.u.; they are not restricted to lie in the original seeding plane but rather go where the local direction of the vector function dictates. With the QTAIM definition of the

2. Origins of the NG-QTAIM bond-path framework set B

115

bond-path framework in place there is no difficulty in describing a bond-path framework as being twisted. We will use the term “bond-path torsion” or similar to mean bond-path framework torsion from now on. For the BCP of a twisted bond-path with non-negligible ellipticity ε, we can always define a non-zero bond-path torsion. The orientation of the eigenvector e3 associated with the bondpath direction does not change during the applied rotation (twist) α, but information about the response β of the bond-path to the twist α can be found from the e1 and e2 eigenvectors that are always perpendicular to the e3 eigenvector. The reason is that the relative orientation of the e1 and e2 eigenvector framework will rotate about the fixed e3 eigenvector with an applied rotation α and hence the orientations of the e1 and e2 eigenvector vary uniquely, whilst still always being orthogonal to each other and to the e3 eigenvector, see Scheme 6.1. The nature of the bond-path torsion can be seen more in detail investigating how the different regions of the charge density respond to the bond torsion for a same state. The interatomic surface paths and the associated basin path sets in the form of the e1-e2-e3 bond-path framework set for the S0 state are presented in the left and right panels, respectively, of Scheme 6.2. The left panel of Scheme 6.2 presents the calculated basin path set starting the trajectories in the plane formed by e2 and the C1dC6 bond path and the right panel of Scheme 6.2 in the plane formed by the C1dC6 bond path and e1. It can be seen that the continuous surface of the basin path set defined by the plane containing e1 (right panel of Scheme 6.2) and the torsional C1dC6 bond-path has a much larger surface area than that formed from the e2 eigenvector (left panel of Scheme 6.2) and the torsional C1dC6 bond-path. We relate this result to the different topological stabilities of the charge density along the e1 and e2 directions, where e2 is the direction of preferred electronic accumulation. The charge density has the most stable morphology along the e2 direction and therefore is less resistant to the applied twist. This is reflected in the small surface area of the basin path sets. Conversely, e1 is the direction of least facile electronic accumulation, and this results in a larger surface area for the basin path sets, i.e., higher resistance to the twist. Notice also that the direction of torsion for the basin path sets defined by the plane including the e2 eigenvector and torsional bond-path rotates in the opposite sense to that of the e1 eigenvector according to the orientation presented. Therefore, we can say that bond-path torsion was also characterized within QTAIM by twisted C1 and C6 atomic basin path sets as well as the e1-e2-e3 bond-path

SCHEME 6.1 An aerial view down the torsional C1dC6 bond-path axis of the fulvene molecule and the orientation of the Cartesian coordinate system. The angle of mechanical twist α of the methylene group and the response β (QTAIM) and βσ (stress tensor).

116

6. Theory developments and applications of next-generation QTAIM

SCHEME 6.2 Views of the gradient vector field distribution —ρ(r) down the Z-axis of the atomic basin paths sets of the C6 and C1 nuclei, where the e1 and e2 interatomic surface eigenvector paths defined at the C1dC6 BCP, are highlighted in red. The atomic basin paths sets of the C6 and C1 nuclei are defined by the planes of the CH2 and five membered ring.

framework set of the Hessian of the density being rotated such that it was no longer contiguous with either the five carbon ring or the methyl group of the fulvene molecular graph. 2.1.1 The construction of NG-QTAIM 3-D bond-path framework set B ¼ {p,q,r} We refer to the next-generation QTAIM interpretation of the chemical bond as the bondpath framework set, denoted by B, where B ¼ {p,q,r}, with the consequence that for a given electronic state a bond is comprised of three “linkages”; p, q and r associated with the e1, e2 and e3 eigenvectors, respectively. Here the p and q are 3-D paths constructed from the values of the least (e1) and most (e2) preferred directions of electronic charge density accumulation ρ(r) along the bond-path, referred to as (r). The orbital-like packet shapes that the pair of q- and q0 -paths form along the BCP are referred to as a {q,q0 } path-packet. Extremely long {q,q0 } path-packets indicate the imminent rupture caused by the coalescence of a BCP with the associated RCP. Larger {q,q0 } and {qσ,qσ0 } path-packets in the vicinity of a BCP signify an easier passage of the BCP as opposed to smaller {q,q0 } and {qσ,qσ0 }-path packets. With n scaled eigenvector e2 tip path points qi on the q-path, where εi ¼ ellipticity at the ith bond-path point ri on the bond-path r and the analogous expression for the e1 tip path points on the p-path: qi ¼ r i + εi e2,I ,pi ¼ r i + εi e1,I

(6.2)

The bond-path is associated with the λ3 eigenvalues of the e3 eigenvector and does not take into account differences in the λ1 and λ2 eigenvalues of the e1 and e2 eigenvectors. We see for shared-shell BCPs, in the limit of the ellipticity ε  0, i.e., corresponding to single bonds, we then have pi ¼ qi ¼ ri and hence the value of the lengths H⁎ and H attain their lowest possible limit; the bond-path length (r) BPL. Conversely, higher values of.

2. Origins of the NG-QTAIM bond-path framework set B

117

the ellipticity ε, e.g., corresponding to double bonds will always result in values of H⁎ and H > BPL. We referred to the next-generation QTAIM interpretation of the chemical bond as the bondpath framework set, denoted by B, where B ¼ {p,q,r} with the consequence that for the S0 state a bond is comprised of three “linkages”; p, q and r associated with the e1, e2 and e3 eigenvectors, respectively. The p and q parameters define eigenvector-following paths with lengths H⁎ and H, see Scheme 6.3: n1 (6.3) H∗ ¼ Σn1 i1 pi+1  pi ,H ¼ Σi1 qi+1  qi The lengths of the eigenvector-following paths H⁎ or H refers to the fact that the tips of the scaled e1 or e2 eigenvectors sweep out along the extent of the bond-path, defined by the e3 eigenvector, between two bonded nuclei connected by a bond-path. In the limit of vanishing ellipticity ε ¼ 0, for all steps i along the bond-path then H ¼ BPL. Expressions for the QTAIM bond-path framework sets for the ground state S0, first excited state S1 and second excited state S2 at the S0 geometry (r0i): B0 ¼ ({q0,q00 },{p0,p00 },r0) B1 ¼ ({q1,q10 },{p1,p10 },r0) B2 ¼ ({q2,q20 },{p2,p20 },r0 The natural transition orbitals (NTOs) were applied to the S0 ! S1 and S0 ! S2 electronic transitions of fulvene S01 and S02 corresponding to the S0 ! S1 and S0 ! S2 electronic transitions also at the S0 geometry (r0i): B01 ¼ ({q01,q010 },{p01,p010 },r0) B02 ¼ ({q02,q020 },{p02,p020 },r0) With the corresponding p and q path-packet expressions: p0i ¼ r0i + ε0ie10,i, p1i ¼ r0i + ε1ie11,i, p2i ¼ r0i + ε2ie12,i, p01i ¼ r0i + ε01ie101,i, p02i ¼ r0i + ε02ie102,i q0i ¼ r0i + ε0ie20,i, q1i ¼ r0i + ε1ie21,i, q2i ¼ r0i + ε2ie22,i, q01i ¼ r0i + ε01ie201,i, q02i ¼ r0i + ε02ie202,i SCHEME 6.3 The pale-blue line (left panel) represents the path with length H⁎, swept out by the tips of the scaled e1 eigenvectors, shown in magenta, defined by Eq. (6.3). The red path (right panel) corresponds to the H, constructed from the path swept out by the tips of the scaled e2 eigenvectors, shown in mid-blue and is also defined by Eq. (6.3). The pale and mid-blue arrows representing the e1 and e2 eigenvectors are scaled by the ellipticity ε, respectively, where the vertical scales are exaggerated for visualization purposes. The green sphere indicates the BCP.

118

6. Theory developments and applications of next-generation QTAIM

SCHEME 6.4 The H2O, (H2O)2, (H2O)4 and (H2O)6 QTAIM molecular graphs with atom numbering scheme used throughout where the undecorated green, red and blue spheres representing the bond critical points (BCPs), ring critical points (RCPs) and cage critical point (CCP), respectively.

2.2 Applications of the bond-path framework set B: Bonding environments and structural preferences 2.2.1 The excited state photochemical reaction path from benzene to benzvalene The creation of the additional CdC bonds in benzvalene compared with benzene along the formation pathway, was explained in terms of an increasing the favorability of the reaction process by maximizing the bonding density [24]. The topological instability of the benzvalene structure was determined using vector-based measures to explain the short chemical half-life of benzvalene in the form of the competition between the formation of these two unstable new CdC bonding that also destabilizes nearest neighbor CdC bonds. The explosive character of benzvalene is demonstrated, for the first time, by the tendency of the short strong CdC bonds to rupture as easily as weak bonding. These short strong CdC bonds, were as topologically unstable as classically weak bonds, possessed twisted 3-D bonding descriptors. 2.2.2 The role of the natural transition orbital density in the S0 ! S1 and S0 ! S2 transitions of fulvene For the first time, the S0 ! S1 (S01) and S0 ! S2 (S02) natural transition orbital (NTO) densities for fulvene were presented, using NG-QTAIM that visualized and quantified the rearrangement of the charge density in response to the applied bond torsion [37]. A characteristic of the presence of a conical intersection (CI) for the S0 ! S1 (S01) transition was found in the symmetrization of the position of the bond critical point (BCP) of the torsional BCP along the containing bond-path. 2.2.3 The excited state deactivation reaction of fulvene The complete 3-D bond-path framework set B ¼ {(p0,p1), (q0,q1), (r0,r1)} from the quantum theory of atoms in molecules (QTAIM) and BσH ¼ {(pσH0,pσH1), (qσH0,qσH1), (r0,r1)} and

2. Origins of the NG-QTAIM bond-path framework set B

119

Bσ ¼ {(pσ0,pσ1), (qσ0,qσ1), (r0,r1)} for the ground and first excited states of fulvene from the stress tensor within the QTAIM partitioning are presented [14]. Both the QTAIM bond-path framework sets B ¼ {(p0,p1), (q0,q1), (r0,r1)} and the stress tensor Bσ ¼ {(pσ0,pσ1), (qσ0,qσ1), (r0,r1)} provide a quantitative 3-D rendering of the bonding consistent with interpretation of the bonding provided by the Lewis structures. 2.2.4 Halogen and hydrogen bonding in halogenabenzene/NH3 complexes compared The bond-path framework set B was used in an investigation into the competition between hydrogen bonding and halogen bonding for the recently proposed (Y ¼ Br, I, At): halogenabenzene: NH3 complex [25]. Differences between using the SR-ZORA Hamiltonian and effective core potentials (ECPs) were found to account for relativistic effects with increased atomic mass. NG-QTAIM was found to be much more responsive than conventional QTAIM. Subtle details of the competition between halogen bonding and hydrogen bonding were indicated a mixed chemical character shown in the 3-D paths constructed from the bond-path framework set B. In addition, the use of SR-ZORA reduced or removed entirely spurious features of B on the site of the halogen atoms. Computational details. To obtain the wavefunctions using ECPs, the complexes of NH3 interacting with the halogenabenzene structures with Y ¼ Cl, Br, I and At were optimized using the mPW2-PLYP double hybrid density functional [38] and the aug-cc-pVTZ basis set [39,40] for all atoms except Y ¼ I, At; For the I and At atoms, the aug-cc-pVTZ-PP basis set [41,42] was used, which includes relativistic ECPs. Double hybrid functionals have been identified as the most accurate density functionals for ground-state thermochemistry [43]. A method comparison focal study on iodobenzene•••H2O in previous work [44] showed that mPW2-PLYP/aug-cc-pVTZ(-PP) gives interaction energies and halogen-bond distances in excellent agreement with DLPNO-CCSD(T)/ma-def2-QZVP results, evidencing that this level of theory is accurate for studying halogen bonds. All geometry optimizations were performed with Gaussian09 Rev. E.01 [45] and used Gaussian’s “ultrafine” integration grid and “very tight” (RMS force 0, the e2F for these bonds changed direction relative to each other in the region between the covalent and hydrogen bond BCPs. Therefore, an absence of coupling between the hydrogen bonding and covalent (sigma) bonding corresponded to {q,q0 } path-packets of the hydrogen bond and covalent (sigma) bond were twisted relative to each other. The Ehrenfest Force F(r) is successful, compared with QTAIM and the stress tensor σ(r), is that F(r) provides an understanding of the forces on the electrons, e.g., F(r) ¼ 0 at the Ehrenfest Force F(r) BCP (Table 6.3). The direction of the e2F eigenvectors demonstrated by coplanar {q,q0 } path-packets, for instances of coupling, in the region between the covalent (sigma) and hydrogen bond BCPs was explained by the relative ease of electron momentum transfer from the hydrogen atom of a sigma bond to the hydrogen bond (Fig. 6.4–6.6). 2.3.2 3-D bond-paths of QTAIM and the stress tensor in neutral lithium clusters, Lim (m ¼ 2–5), presented on the Ehrenfest force molecular graph We pursued the stress tensor σ(r) to see if any indication of NNA character in neutral lithium clusters, can be determined without using the dependency on the QTAIM partitioning, since previously σ(r) was only calculated within the QTAIM partitioning [26]. The stress tensor σ(r) lacks an associated scalar- or vector-field unlike the QTAIM and Ehrenfest F(r) partitioning schemes, therefore, a stress tensor σ(r) partitioning scheme cannot be constructed. Therefore, in this investigation, we used the Ehrenfest Force F(r) molecular graph to display the stress tensor σ(r) bond-path framework set Bσ and BσH due to the matching of the positions of the Ehrenfest Force F(r) and the stress tensor σ(r) BCPs. We will also calculated the QTAIM bond-path framework set and display on the Ehrenfest Force F(r) molecular graph to compare the QTAIM and Ehrenfest Force F(r) bond-path framework set properties. We classified the degree of NNA character in the absence of NNAs, where a much higher degree of NNA character was found to be present for the stress tensor σ(r) 3-D bond-paths than for the corresponding QTAIM or Ehrenfest Force F(r) 3-D bond-paths. The stabilizing effect of the NNA was found by subjecting the Li2 bond-path to sufficient compression and stretching distortions to result in the annihilation of the NNA.

128

6. Theory developments and applications of next-generation QTAIM

TABLE 6.3 Total local energy H(rbF) and the Ehrenfest e1F and e2F eigenvector components at the Ehrenfest BCPs for the H2O, (H2O)2, (H2O)4 and (H2O)6 molecular graphs, as well as the vector dot product between the hydrogen-bond BCP e1F and e2F eigenvectors and those of its collinear sigma BCP. BCP

H(rbF)

e1F(HdO) e1F(OdH)

e1F

e2F(HdO) e2F(OdH)

e2F

H2O O1-H2

0.5524

0.0000

0.6391

0.7692

O1-H3

0.5524

0.0000

0.6391

0.7692

H2–O6

0.0012

0.0321

0.0578

0.9978

O3-H2

0.5353

0.2336

0.0233

0.9720

O3-H1

0.5537

0.8594

0.0274

0.5106

O6-H5

0.5513

0.2670

0.6584

0.7037

H4– O1

0.0030

0.7208

0.6932

0.0039

O5-H4

0.4990

0.8204

0.5718

0.0027

O1-H3

0.5536

0.0150

0.7523

0.6587

H11– O4

0.0031

0.2556

0.9648

0.0621

O10H11

0.4988

0.4855

0.8742

0.0023

H17– O13

0.0013

0.2945

0.9445

0.1458

O16H17

0.5074

0.1155

0.8997

0.4209

H9– O1

0.0001

0.7969

0.5140

0.3174

O7-H9

0.5152

0.7090

0.6525

0.2676

H2– O16

0.0089

0.3517

0.9313

0.0947

O1-H2

0.4653

0.4436

0.8819

0.1596

N.A.

1.0000

0.0000

0.0000

1.0000

0.0000

0.0000

0.0024

0.9983

0.0579

0.0052

0.9997

0.0252

0.0434

0.9989

0.0195

0.8726

0.1447

0.4665

0.0019

0.0037

1.0000

0.0071

0.0150

0.9999

N.A.

0.9979

0.0525

0.0372

0.9677

0.0041

0.0653

0.9979

0.0016

0.0018

1.0000

0.9555

0.2889

0.0587

0.9896

0.1407

0.0294

0.5493

0.8352

0.0267

0.6776

0.7355

0.0021

0.0540

0.1212

0.9912

0.0734

0.2132

0.9743

N.A.

(H2O)2 0.9787

N.A.

0.9995

N.A.

(H2O)4 0.9877

0.9999

N.A.

(H2O)6

0.8224

0.8155

0.9622

0.9980

0.9879

0.9864

0.9359

Hydrogen-bond BCPs with a degree of covalent character, H(rbF) < 0 are highlighted in bold font. The remaining sigma-bonds that do not share a hydrogen atom with hydrogen-bond with associated BCPs are presented in an italic font, with e1F(HdO)e1F(OdH) and e2F(HdO)e2F(OdH) values shown as N.A.

2. Origins of the NG-QTAIM bond-path framework set B

129

FIG. 6.4 The QTAIM bond-path framework set B ¼ ({q,q0 },{p,p0 }, r) and QTAIM on the Ehrenfest F(r) molecular graph bond-path framework set BρF ¼ ({qρF,qρF0 }, {pρF,pρF0 }, rF) for the H2O, (H2O)2, (H2O)4 and (H2O)6 are presented in pairs in the left and right panels of sub-figures (A–D), respectively. The “ρF” and “rF” refers to using QTAIM eigenvectors on the Ehrenfest Force F(r) molecular graph, with q, qρF (dark-magenta), q0 , qρF0 (light-magenta), p, pρF (dark-blue) and p0 , p0 ρF (light-blue), see Scheme 6.4.

2.4 Applications of the NG-QTAIM bond-path framework set B: Molecular devices, switches, wires and ring-opening reactions 2.4.1 Predicting competitive and noncompetitive torquoselectivity of thermal ringopening reactions We provide a new criterion to determine whether the five reactions are competitive or non-competitive. The selection of five competitive and non-competitive reactions include; methyl-cyclobutene, ethyl-methyl-cyclobutene, isopropyl-methyl-cyclobutene, ter-butylmethyl-cyclobutene and phenyl-methyl-cyclobutene [60]. We find three reactions are competitive and two are non-competitive reactions in contrast to the results from the activation energies, calling into question the reliability of activation energies. The bond-path frame-work set B ¼ {p,q,r}, where p, q and r represent three paths with corresponding eigenvector-

130

6. Theory developments and applications of next-generation QTAIM

FIG. 6.5 The stress tensor σ(r) on the QTAIM molecular graph bond-path framework set Bσρ ¼ ({qσρ,qσρ0 }, {pσρ,pσρ0 }, r) and stress tensor σ(r) on the Ehrenfest F(r) molecular graph bond-path framework set BσF ¼ ({qσF,qσF0 }, {pσF,pσF0 }, rF) with a magnification factor of 5 for the H2O, (H2O)2, (H2O)4 and (H2O)6 are presented in pairs in the left and right panels of sub-figures (A–D), respectively. The stress tensor σ(r) ellipticity εσ ¼ jλ2σj/jλ1σj  1 is used, see the caption of Fig. 6.4 for further details.

following paths with lengths H⁎, H and the bond-path length from the quantum theory of atoms in molecules (QTAIM). Longer path lengths H of the ring-opening bonds predict the preference for the transition state inward (TSIC) or transition state outward (TSOC) of thermal ring opening reactions in agreement with experiment for all five reactions. 2.4.2 The design of quinone-based switches with hydrogen tautomerization Investigation of the hydrogen transfer tautomerization process yielded metallic hydrogen bonds in the benzoquinone-like core of the switch [21]. The QTAIM and stress tensor σ(r) bond-path framework sets B and Bσ were visualized and revealed the destabilizing effects on the hydrogen bond of the presence of an Fe atom, the lengths of B and Bσ quantified this effect and the dependence on the position of a fluorine substituent.

2. Origins of the NG-QTAIM bond-path framework set B

131

FIG. 6.6 The {qFB,qFB0 } (left panel) and {pFB,pFB0 } (right panel) path-packets are calculated on the Ehrenfest Force F(r) molecular graphs for the H2O, (H2O)2, (H2O)4 and (H2O)6 are presented in sub-figure (A–D). The Ehrenfest Force F(r) ellipticity εFB ¼ (jλ2Fj/jλ1Fj)  1 is used, see the caption of Fig. 6.4 for further details.

2.4.3 Scoring molecular wires in electric-fields for molecular electronic devices We used low enough E-field to not measurably alter the molecular geometry, using conventional QTAIM, the shifting of the CdC and CdH bond critical points (BCPs) demonstrates polarization via an interchange in the size of the atoms involved in a bond, since a BCP is located on the boundary between a pair of bonded atoms [28]. The polarization effect was demonstrated in the form of significant asymmetries with bond-path framework set B that was more direct with a change in morphology of the 3-D envelope around the BCP. When the E-field was directed along the CdC bond, the BCP moved in response and was accompanied by the envelope constructed from p and q path-packets. The response displayed a polarization effect that increased with increasing magnitude of the Ex-field parallel and antiparallel to the CdC bond. Computational details. The molecular structure was optimized with Gaussian 09 [52] using DFT at the PBE0 [61] /cc-pVTZ [39] level of theory using Grimme’s empirical 3-center dispersion correction with Becke-Johnson damping [62,63]. Gaussian’s “ultrafine” DFT integration grid was employed for all calculations. These settings were retained and the converged

132

6. Theory developments and applications of next-generation QTAIM

atomic geometry (nuclear positions) were kept fixed in all subsequent calculations. Singlepoint SCF calculations were then performed with the DFT settings and basis set with additional stricter convergence criteria; Ez, see Fig. 6.7. The decrease in the relative energy ΔE is proportional to the dipole moment and polarizability of the ethene molecule in that direction. The orientations of the Ex, Ey and Ez-fields coincide with the e3, e1 and e2 eigenvectors of the Hessian matrix that correspond to the bond-path, and least and most preferred directions of charge density accumulation, respectively. This is consistent with the variation in the ellipticity ε for the C1-C2 BCP with associated bond-path aligned with Ex-field is ordered in terms of decreasing response: Ex > Ez > Ey, see the left panel of Fig. 6.7B. The Ex and Ey-fields comprise components along the bond-paths of the CdH BCPs and so the magnitude of the ellipticity ε response is Ex > Ey > Ez, see the right panel of Fig. 6.7B. There is a decrease in the ellipticity ε  2% for the C1-C2 BCP subjected to the Exfield ¼ 200  105 au. The variation in H(rb) for the C1-C2 BCP in order of decreasing strength Ez > Ex > Ey is explainable by the alignment with the bond-path (e3) and the most (e2) and least (e1) preferred directions of charge density accumulation, respectively, see the left panel of Fig. TABLE 6.4 The values of the bond-path lengths (BPL) (in a.u.) for the applied E-field (in a.u.); Ex, Ey and Ez (up to 200  105 a.u.) are presented, see Fig. 6.7 for further details. Ex

Ey

Ez

E-field

C1-H4

C1-H4

C1-H4

200

2.0160

2.0159

2.0174

100

2.0167

2.0167

2.0174

0

2.0174

2.0174

2.0174

+100

2.0182

2.0183

2.0174

+200

2.0189

2.0189

2.0174

The BPL of the C2-C1 BCP bond-path ¼ 2.5006 a.u. in all cases.

2. Origins of the NG-QTAIM bond-path framework set B

133

FIG. 6.7 The variation in the relative energy ΔE for varying values of the E-field up to [200  105 a.u. (60.91 kJ/mol)], where Ex, Ey and Ez correspond to the directions x, y, z shown in the inset of (A). Here a positive sign of Ex corresponds to the C1 ! C2 direction and a negative sign to C1 C2. The numbering scheme used throughout for the molecular graph of the ethene molecule is shown in the inset of (A). The black lines indicate the bond-path (r) with the green spheres denoting the bond critical points (BCPs). The variations of the ellipticity ε and the total local energy density H(rb) with the applied E-field are presented in sub-figures (B and C).

134

6. Theory developments and applications of next-generation QTAIM

6.7C. The response of the H(rb) values of the CdH BCPs is almost linear, see the right panel of Fig. 6.7C. The response of the bond-path lengths (BPL) to the applied E-field can be seen in Table 6.4. The BPL for the bond-path of the C1-C2 BCP is insensitive to the magnitude and direction of the applied E-field. The BPL of the C1-H4 BCP responds in terms of bond shortening/lengthening with an increase in the /+ values of the applied Ex and Ey directions equally. The movement of the C1-C2 BCP, toward to the C1 NCP, which occurs for -Ex (5 C1 C2), indicates that the electronic charge density is shifting from the C1 NCP toward the C1 NCP across the atomic boundary defined by the C1-C2 BCP, see Fig. 6.7. This is seen from the fact that the position of the BCP indicates the boundary of the two atoms (NCPs) that are defined to be bonded, indicating that the C2 NCP is increasing in size at the expense of the C1 NCP. For the application of the +Ex (5 C1 ! C2), the converse is true. The BPL of the C1-C2 BCP bond-path is consistently 2.501 a.u. and is therefore insensitive to either the magnitude or direction of the applied E-field. The partial bond-path lengths (BPL) demonstrates how the C2-BCP and BCP-C1 distances vary with the applied E-field, as presented in Table 6.4, see also Fig. 6.7. The significant variations with the magnitude for the Ex direction for the partial C1-C2 BCP bond-lengths are explained by to the motion of the BCP being much greater than that of the bonded atoms. Therefore, the position of a BCP is a sensitive indicator of the effect of the applied E-field unlike the BPL. Significant and slight dependencies are also observed for the Ey and Ez directions, respectively, for the C1-H4 BCP, see Table 6.5. A linear dependency of the observed C1-C2 BCP and symmetry inequivalent CdH BCP shifts for the Ex and  Ey directions is apparent with larger magnitude of the Ex and  Ey-fields resulting in larger BCP shifts, see Fig. 6.8A and B, respectively. The largest BCP shift occurs for bond-path of the C1-C2 BCP that can be explained by its orientation and is directed along the +Ex direction, the C1-C2 BCP moves away from the direction of the applied Ex-field. This results in the C1-C2 BCP being located closer to the C1 NCP than the C2 BCP for Ex-field, the reverse is true for +Ex-field compared with the case of no applied E-field, where the C1-C2 BCP is positioned midway between the C1 NCP and C2 NCP. The C1-C2 BCP shifts are insignificant in both the Ey and  Ez direction. The application of the applied E-field in the Ez direction produces CdH BCP shifts of an order of magnitude lower than for the CdH BCP with a harmonic variation, see Fig. 6.8C. TABLE 6.5 The values of the partial bond-path lengths (BPL) (in a.u.) for the applied E-field; Ex, Ey and Ez (up to 200  105 a.u.) are presented, see Figs. 6.7 and 6.8 for further details. Ex

Ey

Ez

E-field

C1-BCP, C2-BCP

C1-BCP, BCP-H4

C1-BCP, C2-BCP,

C1-BCP, BCP-H4

C1-BCP, C2-BCP

C1-BCP, BCP-H4

200

1.2199, 1.2807

1.3370, 0.6790

1.2503, 1.2503

1.3383, 0.6776

1.2503, 1.2503

1.3142, 0.7032

100

1.2352, 1.2654

1.3249, 0.6918

1.2503, 1.2503

1.3256, 0.6911

1.2503, 1.2503

1.3132, 0.7042

0

1.2503, 1.2503

1.3129, 0.7045

1.2503, 1.2503

1.3129, 0.7045

1.2503, 1.2503

1.3129, 0.7045

+100

1.2654, 1.2352

1.3012, 0.7170

1.2503, 1.2503

1.3003, 0.7180

1.2503, 1.2503

1.3132, 0.7042

+200

1.2807, 1.2199

1.2898, 0.7291

1.2503, 1.2503

1.2876, 0.7313

1.2503, 1.2503

1.3142, 0.7032

2. Origins of the NG-QTAIM bond-path framework set B

135

FIG. 6.8 The variation in the BCP shift along the associated bond-path with the E-field; Ex, Ey and Ez are presented in sub-figures (A–C), respectively. The corresponding path-packet shifts are presented in sub-figures (D–F), respectively, see Fig. 6.7 and Table 6.4 for further details.

The motion of the C1-C2 BCP {q,q0 } path-packet subjected to the Ex-field tracks the C1-C2 BCP along the associated bond-path, where the C1-C2 BCP {q,q0 } path-packet shifts almost match the C1-C2 BCP shifts, compare Fig. 6.8A with Fig. 6.8B. This explainable in terms of the C1-C2 BCP {q,q0 } path-packet not spreading out as demonstrated by the lack of variability of the C1-C2 BCP {q,q0 } path-packet area, see the black plot line in Fig. 6.9A. The CdH BCP {q,q0 } path-packets however, do spread out as seen from the variability in the path-packet areas, see Fig. 6.9B–D and the lack of matching between the CdH BCP shifts and CdH BCP {q,q0 } pathpacket shifts, compare Fig. 6.8A–C and Fig. 6.8D–F. Significant C1-C2 BCP (left panel) and C1-H4 BCP shifts (right panel) are be observed for the applied E-field corresponding to the Ex and  Ey directions in terms of the bond-path ellipticity ε profiles, see Fig. 6.10A and B. The C1-C2 BCP bond-path ellipticity profile ε is symmetrical and the C1-C2 BCP is located at the mid-point of the bond-path of the C1-C2 BCP in the absence of any applied E-field, see Fig. 6.11B. The application of the E-field corresponding to the -Ex direction removes this symmetry resulting in the C1-C2 BCP shifting closer to the C2 NCP, see Fig. 6.11A. For increases in magnitude along the +Ex direction, the converse is true, where the C1-C2 BCP shifts away from the C2 NCP and closer to the C1 NCP, see Fig. 6.11C and the left panel of Fig. 6.10A. The height of the bond-path ellipticity profile ε for the C1-C2 BCP increases relative to the absence of the applied Ex-field in addition to the symmetry breaking caused by the Ex-field.

136

6. Theory developments and applications of next-generation QTAIM

FIG. 6.9 The variations of the {q,q0 } path-packet areas with the applied E-field (in a.u.), see Fig. 6.7 for further details. Note, the same y-axis range of 0.05 a.u. is used for comparison in sub-figures (A–D).

Dependencies on the ellipticity profile ε and the E-field are apparent as increases/decreases in the maximum peak in the ellipticity profile ε corresponding to more negative/positive values of the Ex-field, see the right panel of Fig. 6.10A. We provide a summary of the effect of the bond-path ellipticity profiles ε by providing the {q,q0 } path-packets, for E ¼ 0, 20  105 au that are superimposed for each of the Ex, Ey and  Ez directions, see the middle panels of Fig. 6.10. Examination of the {p,p0 } and {q,q0 } path-packets subject to E-field demonstrates differences for the application of the Ex and Ey-fields, see Fig. 6.10A and B, respectively. The application of the Ex-field demonstrates the largest changes compared with ethene with E ¼ 0, see Fig. 6.8A. The change in chemical character of the C1-C2 BCP relative to E ¼ 0 is seen, in addition to the BCP shift, in a symmetry breaking of the {q,q0 } path-packet, whereby the blunt end of the profile of the {q,q0 } path-packet faces the direction of the applied Ex-field. The pointed end of the prfile of the {q,q0 } path-packet shifts along with the C1-C2 BCP, in the direction of the Exfield such that the pointed end of the {q,q0 } path-packet indicates the direction of the C1-C2 BCP and applied Ex-field (Fig. 6.12).

2. Origins of the NG-QTAIM bond-path framework set B

137

FIG. 6.10 Variations of the bond-path ellipticity ε profile (in a.u.) with the applied E-field (in a.u.); Ex, Ey and Ez are presented for the C1-C2 BCP (left panel), the superimposed {p,p0 } and {q,q0 } path-packets (blue and magenta insets in the middle panel) and the bond-path ellipticity ε profile C1-H4 BCP (right panel) in sub-figures (A–C), respectively. See also Figs. 6.11 and 6.12.

138

6. Theory developments and applications of next-generation QTAIM

FIG. 6.11 The ethene bond-path framework set B displaying two views of the {q,q0 } path-packets for values of the external electric field, Ex ¼ 200  104 a.u. (pale-magenta), Ex ¼ 0 (red) and Ex ¼ +200  104 a.u. (dark-magenta), are presented in sub-figures (A–C), respectively, see the middle panel of Fig. 6.10 for the superimposed plots of Ex. For a complete set of the bond-path framework set B.

The C1-H4 BCP {q,q0 } path-packet and C1-H4 BCP bond-path are not in alignment with the Ex or Ey-fields, but share some component along both directions that results in the resultant C1-H4 BCP {q,q0 } path-packet responding according to the direction of the applied Ex or Eyfield, see Fig. 6.10A and B, respectively. There is no significant response of the C1-C2 BCP {q,q0 } or CdH BCPs {q,q0 } path-packets o to the Ez-field due to the planar nature of the ethene molecule, see Fig. 6.10C. 2.4.5 Summary NG-QTAIM has quantified the effect of a varying E-field on the electronic charge density of the ethene molecule, that does not sigificantly alter the molecular geometry. We find that for

2. Origins of the NG-QTAIM bond-path framework set B

139

FIG. 6.12 The ethene bond-path framework set B displaying two views of the {q,q0 } path-packets for values of the external electric field, Ey ¼ 200  104 a.u. (pale-magenta), Ey ¼ 0 (red) and Ey ¼ +200  104 a.u. (dark-magenta), are presented in sub-figures (A–C), respectively, also see the middle panel of Fig. 6.10 for the superimposed plots of Ey.

the C1-C2 BCP bond-path that is parallel to the Ex-field the associated {q,q0 } path-packet shifts along with the C1-C2 BCP parallel to the direction of the Ex-field due very little spreading of the {q,q0 } path-packet. The area of the {q,q0 } path-packet of the C1-C2 BCP subject to the Ex-field remains independent of the magnitude of Ex, where the height increased with Ex-field indicating a polarization effect that increases with applied Ex-field. Therefore, the decrease in the relative energy ΔE is proportional to the polarizability of ethene parallel to the Ex-field direction. There is modest  2% decrease in the C1-C2 BCP ellipticity ε when the Ex-field is parallel/anti-parallel to the C1-C2 BCP bond-path.

140

6. Theory developments and applications of next-generation QTAIM

The response of the bond-path and BCP ellipticity ε is asymmetrical for the CdH BCPs that are not aligned parallel/anti-parallel to any of the Ex, Ey and Ez orientations of the applied E-field and consequently the {q,q0 } path-packets spread out by almost a factor of three. QTAIM scalar measures such as bond-path lengths were found invariant to the application of the E-field, therefore, the sensitivity of NG-QTAIM to the application of modest E-fields demonstrates the ability to consider more subtle details than changes to the molecular geometries. This is the first approach to use 3-D next-generation analysis to determine the response of molecular wires in molecular electronic devices to an applied E-field. 2.4.6 The dihydrocostunolide (DHCL) photochemical ring-opening reaction The factors underlying two possible pathways for the Dihydrocostunolide (DHCL) photochemical ring-opening reaction were investigated with NG-QTAIM [19]. The first reaction pathway returned to the ring-closed conformation of the reactant and the second reaction pathway progressed to the ring-opened product. The first pathway comprised oscillations in the chemical character of the fissile bond both before and after the conical intersection that pulled the reaction back to reactant. The second pathway however, led directly forward to the product. 2.4.7 The cyclohexadiene (CHD) photochemical ring-opening reaction of cyclohexadiene (CHD) The factors underlying the branching ratio (70:30), observed from experiment of the (1,3cyclohexadiene) CHD ! HT (1,3,5-hexatriene) photochemical ring-opening reaction were investigated [18]. The NG-QTAIM analysis suggests in both S1 and S0 electronic states an attractive interaction exists between the ends of the fissile σ-bond of CHD that steers the ringopening reaction predominantly in the direction of restoration of the ring. We suggested that opening of the ring and formation of the reaction product (HT) can only be achieved with sufficient persistent nuclear momentum in the stretching direction of the fissile bond. The rarity during the dynamics of this orientation of the nuclear momentum vector provides an explanation of the observed low quantum yield of the ring-opening reaction.

3. The NG-QTAIM bond-path precession K 3.1 Theory background: The NG-QTAIM bond-path precession K We provide the example of benzene to explain the bond-path precessions. The presence of bond-path flexing corresponds to the presence of non-zero differences in the values of the bond-path length (BPL) and geometric bond length (GBL) associated with a given bond critical point (BCP) bond-path. The anharmonic response of the CdC BCP and CdH BCP to the four IR-active modes is quantified by the degree of relative BCP sliding along the containing bond-paths. This is explained that fact that a changing BCP shift in real space relative to the bond-path implies a change in chemical character due to the change in the charge density ρ(r) distribution [32]. To determine the presence of bond-path torsion, we need to quantify the degree of torsion of the {p,p0 } path-packet, the bond-path torsion may be defined as occurring

3. The NG-QTAIM bond-path precession K

141

about the BCP or an extended region around the BCP, either side of the BCP, along the containing bond-path. Bond rupture occurs if the BCP and ring critical point (RCP) coalesce, this process occurs most readily when the e2 eigenvector, that indicates the most facile direction, when the e2 eigenvector aligns along the BCP→RCP path. The maximum tendency to avoid BCP and RCP coalescence occurs when the e1 eigenvector, indicating the least facile, most resistant direction, is parallel/anti-parallel to the BCP→RCP path, denoted by the gray line in Scheme 6.5. The extent to which the {p,p0 } path-packet, constructed from the e1 eigenvector, precesses about the shared-shell C5-C6 BCP bond-path and the shared-shell C6-H12 BCP bond-path is defined by the precession K, see the left and right panels of Scheme 6.5, respectively. For the {p,p0 } path-packet, defined by the e1 eigenvector, we wish to follow the extent to which the {p,p0 } path-packet precesses about the bond-path by defining the precession K for IR-responsivity: K ¼ 1  cos 2 α, where cos α ¼ e1 ∙u and 0  K  1

(6.4)

Considering the extremes, K ¼ 0, where α is defined by Eq. (6.4), we have the maximum alignment of the BCP!RCP path with the e1 eigenvector, the least facile direction and for K ¼ 1 we have the maximum degree of alignment with the e2 eigenvector, the most facile direction. Values of K ¼ 0 and K ¼ 1 can be used to indicate bond-paths with the lowest and highest tendencies toward IR-responsivity respectively. The precession K is determined relative to the BCP, in either direction along the bond-path toward the nuclei at either ends of the bond-path from an arbitrarily small spacing of e1 eigenvectors. If we chose the precession of the {p,p0 } path-packet about the bond-path when the e1 eigenvector is parallel to u, that defines the BCP!RCP path, the BCP will have minimum facile character, i.e., IR-responsivity see Scheme 6.5. By following the variation in the precession K, we can quantify the degree of facile character of a BCP along an entire bond-path. For the precession of the {q,q0 }

SCHEME 6.5 The {p,p0 } precession K construction (left panel) and {q,q0 } path-packet precession K0 (right panel) for the C5-C6 BCP and C6-H12 BCP, where the u is a unit vector (red arrow) along the BCP→RCP path are denoted by the gray line. The vector v is a unit vector (black arrow) defined to be aligned parallel to the normal vector of the plane of the relaxed benzene molecule. The pale magenta line indicates the interatomic surface paths (IAS) that originate at the BCP. The e1 eigenvector (blue arrow) and e2 eigenvector (magenta arrow), the  signs of e1 and e2 are chosen to form the right handed orthogonal set {e1, e2, e3}. The undecorated red and green spheres indicate the locations of the bond critical points (BCPs) and ring critical points (RCPs), respectively.

142

6. Theory developments and applications of next-generation QTAIM

path-packet, defined by the e2 eigenvector, about the bond-path, β ¼ (π/2  α) and α is defined by Eq. (6.5), we can write an expression K0 for the bond-path IR-non-responsivity: K0 ¼ 1  cos 2 β, where cos β ¼ e2 ∙u,β ¼ ðπ=2  αÞ and 0  K0  1

(6.5)

Note, for the general case the e3 eigenvector defined along the bond-path is not perpendicular to the reference direction u. For K0 ¼ 0, we have a maximum degree of facile character and for K ¼ 1 we have the minimum degree of facile character and values of K0 ¼ 0 and K0 ¼ 1 therefore indicate bond-paths with the lowest and highest tendencies toward IR-non-responsivity, respectively. We include K0 because for the CdH BCP bond-path may move out of the plane of the benzene molecule, then the K and K0 will not necessarily be converses of each other.

3.2 Applications of the bond-path precession K: Excited states 3.2.1 Bond-path-rigidity and bond-path-flexibility of the excited state deactivation reaction Fulvene We determined the precessions K and K0 of the {p,p0 } and {q,q0 } path-packets for the excited state deactivation reaction of fulvene that includes the tendencies toward bond-path-rigidity and bond-path-flexibility [30]. Uniquely, the S1 state at torsion θ ¼ 0.0° was found to possess the maximum possible bond-path-rigidity, along the entire bond-path but then dropped lower than the S0 state at the conical intersection indicating photo-excitation facilitated the torsion.

3.3 Applications of the bond-path precession K: Normal modes 3.3.1 Bond flexing, twisting, anharmonicity and responsivity for the IR-active modes of benzene We used NG-QTAIM to fully quantity the response to the four IR-active modes of all the bonding in benzene in terms of the tendencies toward IR-responsivity and IR-nonresponsivity [31]. Bond-anharmonicity were absent for the CdC bonds of the lowest frequency mode (721.57 cm1) determined as the absence of relative sliding of the bond critical point (BCP) along the containing bond-path. Bond-flexing and bond-anharmonicity were absent for this mode from the variation in the wrapping (torsion) of the {p,p0 } path-packet, referred to as the Precession K, along the bond-path. The remaining three IR-active mode possessed step-like variations in the K profiles and displayed anharmonic character. Nonnuclear attractors were detected for the IR-active mode with frequency 1573.93 cm-1 along with CdC K profiles that most closely resembled those of the relaxed benzene. 3.3.2 Results and discussions An absence of bond-path flexing is indicated by relative bond-path lengths (ΔBPL) that are not significantly larger than the relative interatomic separations, referred to as the relative geometric bond lengths (ΔGBL) for the CdC BCP bond-paths, see Table 6.6. The CdH BCP bond-paths and bond-path flexing shows the converse is evident for the differences in the ΔBPL and ΔGBL values in Table 6.6. Due to the facile nature of the total charge density distribution ρ(rb) associated with the H nuclear critical point (NCP) there are larger

143

3. The NG-QTAIM bond-path precession K

TABLE 6.6 The symmetry inequivalent BCP values of the relative partial benzene bond-path lengths Δ(CBCP/NNA) ¼ ΔA, Δ(NNA/BCP-C/H) ¼ ΔB with the Δ(BPL), ΔGBL and the relative position along the bondpath that precession K jumps from K ¼ minimum to K ¼ 1 (ΔQ and + ΔQ) in a.u. for the amplitudes ¼ 1.0 and + 1.0 of the infrared (IR) benzene each of the four IR-active modes are presented. 21.0 BCP

ΔBPL (ΔGBL)

(ΔA, ΔB) a

+1.0 (+ΔQ, 2ΔQ)

(ΔA, ΔB)

ΔBPL (ΔGBL)

(+ΔQ, 2ΔQ)

1

IR-active mode 5 , ν ¼ 721.57 cm C1-C2

(0.000, 0.000)

0.000(0.000)

(0.037, 0.037)

(0.000, 0.000)

0.000(0.000)

(0.037, 0.037)

C1-H7

(0.116, 0.047)

0.169(0.163)

(--, --)

(0.116, 0.047)

0.169(0.163)

(--, --)

1

IR-active mode 14, ν ¼ 1097.69 cm C1-C2

(0.151, 0.151)

0.302 (0.302)

(0.071, 0.071)

(0.152, 0.152)

0.303(0.303)

(0.175, 0.175)

C2-C3

(0.085, 0.101)

0.185(0.184)

(0.075, 0.075)

(0.085, 0.101)

0.185(0.184)

(0.050, 0.150)

C3-C4

(0.065, 0.090)

0.154 (0.154)

(0.150, 0.050)

(0.065,-0.090)

0.154 (0.154)

(0.075, 0.075)

C4-C5

(0.152, 0.152)

0.303(0.303)

(0.175, 0.175)

(0.151,-0.151)

0.302 (0.302)

(0.071, 0.071)

C1-H7

(0.002, 0.030)

0.034(0.031)

(--, --)

(0.071, 0.004)

0.069(0.067)

(--, --)

C3-H9

(0.134, 0.052)

0.195(0.185)

(--, --)

(0.134, 0.052)

0.195(0.185)

(--, --)

C5-H11

(0.027, 0.006)

0.069(0.067)

(--, --)

(0.002, 0.030)

0.034(0.031)

(--, --)

b

1

IR-active mode 21 , ν ¼ 1573.93 cm C1-C2

(0.208, 0.208)

0.416(0.416)

(0.000, 0.000)

(0.208, 0.208)NNA

0.415 (0.415)

(0.028, 0.028)

C2-C3

(0.187, 0.026)

0.215(0.213)

(0.150, 0.175)

(0.177, 0.012)

0.164 (0.165)

(0.150, 0.200)

C3-C4

(0.012, 0.177)

0.164 (0.165)

(0.200, 0.150)

(0.026, 0.187)

0.215(0.213)

(0.175, 0.150)

C4-C5

(0.208, 0.208)NNA

0.415 (0.415)

(0.028, 0.028)

(0.208, 0.208)

0.416(0.416)

(0.000, 0.000)

C1-H7

(0.112, 0.027)

0.145(0.138)

(--, --)

(0.048, 0.021)

0.074(0.067)

(--, --)

C3-H9

(0.286, 0.102)

0.405(0.381)

(--, --)

(0.286, 0.102)

0.405(0.381)

(--, --)

C5-H11

(0.009, 0.021)

0.074(0.067)

(--, --)

(0.112, 0.027)

0.145(0.138)

(--, --) Continued

144

6. Theory developments and applications of next-generation QTAIM

TABLE 6.6 The symmetry inequivalent BCP values of the relative partial benzene bond-path lengths Δ(CBCP/NNA) ¼ ΔA, Δ(NNA/BCP-C/H) ¼ ΔB with the Δ(BPL), ΔGBL and the relative position along the bondpath that precession K jumps from K ¼ minimum to K ¼ 1 (ΔQ and + ΔQ) in a.u. for the amplitudes ¼ 1.0 and + 1.0 of the infrared (IR) benzene each of the four IR-active modes are presented—cont’d 21.0 BCP

(ΔA, ΔB) c

ΔBPL (ΔGBL)

+1.0 (+ΔQ, 2ΔQ)

(ΔA, ΔB)

ΔBPL (ΔGBL)

(+ΔQ, 2ΔQ)

1

IR-active mode 28 , ν ¼ 3298.32 cm C1-C2

(0.037, 0.037)

0.074 (0.075)

(U, U)

(0.038, 0.038)

0.078(0.076)

(0.225, 0.225)

C2-C3

(0.014, 0.059)

0.044 (0.045)

(0.050, U)

(0.027, 0.021)

0.049(0.048)

(0.000, 0.225)

C3-C4

(0.021, 0.027)

0.049(0.048)

(0.225, 0.000)

(0.059, 0.014)

0.044 (0.045)

(U, 0.050)

C4-C5

(0.152, 0.152)

0.303(0.303)

(0.225, 0.225)

(0.019, 0.019)

0.038(0.038)

(U, U)

C1-H7

(0.639, 0.386)

1.047(1.024)

(--, --)

(0.648,-0.377)

1.042 (1.024)

(--, --)

C3-H9

(0.009, 0.008)

0.000(0.000)

(--, --)

(0.009, 0.008)

0.000(0.000)

(--, --)

C5-H11

(0.648, 0.377)

1.042 (1.024)

(--, --)

(0.639, 0.386)

1.047(1.024)

(--, --)

IR-active mode 5: the values of ΔQ were estimated as half way up the slope of the non-step-like variation in the K profile. IR-active mode 21: “NNA” indicates the presence of a non-nuclear attractor (NNA). IR-active mode 28: the CdC BCP bond-paths that are unresponsive are indicated by “U.” The Δ(C-BCP/NNA) and Δ(BCP/NNA-C/H) were calculated by subtracting off the relaxed benzene values. For the relaxed benzene, Q and + Q values are 0.700 and 0.700, respectively. a

b c

differences in the BPL and GBL apparent for the CdH BCP bond-path. We note additionally that values of CdH BPL < GBL occur due to the position of the H NCP not coinciding with the geometric center of the H NCP. The much larger range of BCP sliding values, quantified by the relative shifts of the CdC BCPs and CdH BCPs contrasts with the insignificant differences between the BPL and GBL values for the CdC BCP bond-paths, see Table 6.6. The CdC BCP sliding values provide a measure of bond-anharmonicity from Δ(C-BCP) and Δ(BCP-C) and there exists an increasingly anharmonic response of the CdC BCP bond-paths: IR-active mode 5 < IR-active mode 28 < IR-active mode 14 < IR-active mode 21. The CdC BCP bondpaths of IR-active mode 5 possess insignificant bond-anharmonicity since the Δ(C-BCP) and Δ(BCP-C) values are 0.000. For the CdC BCP bond-paths, the K and K0 are the converse of one another so we only provide the K plots. The values of the relative position along the bond-path that the values of the precession K jumps from K ¼ minimum to K ¼ 1 (ΔQ and + ΔQ) for the CdC BCP bond-paths are not correlated with the BCP shifts, e.g., see the entries for the C1-C2 BCP for the IR-active mode 14 presented in Table 6.6. For amplitudes 1.0, the K (and K0 ) profiles

3. The NG-QTAIM bond-path precession K

145

FIG. 6.13 The variation in the precession K along the CdC BCP bond-paths of the benzene IR-active mode 5 for the amplitudes of vibration 1.0 (black), 0.0 (red) and + 1.0 (blue). The {p,p0 } path-packets are superimposed onto the molecular graph and shown on the insets for amplitude 1.0 (left) and + 1.0 (right). Bond critical points (BCPs) are located at a distance ¼ 0.0 a.u. along the bond-path, the ring critical point (RCP) is visible as the green sphere located outside of the plane of the benzene ring, see the caption of Table 6.6 for further details.

of the CdC BCP bond-paths of IR-active mode 5 possess a harmonic-like morphology, explainable by the non-planar molecular graph of IR-active mode 5, where the RCP is visible the side view, see Fig. 6.13. Conversely, the relaxed benzene and the other three IR-active modes possess step-like K (and K0 ) profiles for the CdC BCP bond-paths and planar molecular graphs, see Figs. 6.14–6.16. Mode 14 is the only mode where the values of jΔQj are the same for the amplitudes 1.0, although the precession K (and K0 ) profiles are not symmetrical, see Fig. 6.14. For mode 21, the presence of the non-nuclear attractors (NNAs) [67,68] for both the C1-C2 BCP bond-path (amplitude ¼ +1.0) and C4-C5 BCP (amplitude ¼ 1.0) bond-path is noted and results in values of ΔQ comparable to the relaxed benzene and symmetrical precession K (and K0 ) profiles, see Fig. 6.15. Conversely, for mode 21 the bond-paths lacking an NNA possess a greater range of the values of ΔQ and asymmetrical precession K (and K0 ) profiles. Lower/higher values of Q determined for the precession K indicate that the associated CdC BCP bond-path is more/less IR-responsive. A complete lack of IR-responsivity is seen, e.g., for the C1-C2 BCP bond-path of mode 28 with amplitude ¼ 1.0 and is indicated by the initial U (non-responsive), see Table 6.6 and Fig. 6.16. The converse relation between K and K0 does not hold for the CdH BCP bond-paths, see Figs. 6.17–6.20 and the theory section. Values of the Precession K ¼ 0 and K ¼ 1 indicate bondpaths with the lowest and highest tendencies toward IR-responsivity and this explains why the K values for facile CdH BCP bond-path remain close to K ¼ 1, exactly 1.0 for the CdH BCP bond-path of the relaxed benzene with some exception for IR-active mode 21, see Figs. 6.17–6.20. The CdH BCP bond-paths of IR-active mode 28 contain the lowest (C3-H9 BCP) and highest (C1-H7 BCP) degree of bond-anharmonicity as well as bond-flexing. 3.3.3 Summary NG-QTAIM was used to complete the quantification, of the changes that occur in 3-D along a bond-path, for the four IR-active modes of benzene in terms of the bond-flexing, bondtorsion and bond-anharmonicity that includes the tendencies toward IR-responsivity and

146

6. Theory developments and applications of next-generation QTAIM

FIG. 6.14 The variation in the precession K along the CdC BCP bond-paths of the benzene IR-active mode 14, see the caption of Fig. 6.13 for further details.

IR-non-responsivity. The bond-anharmonicity is determined from the relative BCP shifts that quantify the sliding of at the BCP along the containing bond-path defined by e3. The lack of dependency of the relative positions along the CdC BCP bond-path referred to as ΔQ of the jumps in K and the BCP shifts is explained by the orthogonality of the {e1, e2, e3} explains that e3 will move with the BCP independently of the {p,p0 } and {q,q0 } path-packets constructed from the e1 and e2, respectively. BCP shift is demonstrated to be a good indicator of bond-anharmonicity from the presence of the harmonic-shaped K profile for mode 5 along with the zero values for the relative CdC BCP shifts. This is because non-zero BCP shifts correspond to a change in chemical character due to the change in the charge density ρ(r) distribution [32], therefore, for mode 5 we have no change in chemical character. This finding is consistent with mode 5 being the only mode for

3. The NG-QTAIM bond-path precession K

147

FIG. 6.15 The variation in the precession K along the CdC BCP bond-paths of the benzene IR-active mode 21, see the caption of Fig. 6.13 for further details. Note the presence of non-nuclear attractors (NNAs) for the C1-C2 BCP bondpath (blue spheres) and C4-C5 BCP bond-path (black spheres).

which the benzene molecular graph is non-planar since the CdH BCP bond-paths flex out the plane of the benzene ring that results in the ring critical point (RCP) being located outside of the plane of the C6 ring. Unlike the three other IR-active modes the minimum value of the CdC bond-path K profile of mode 5 is non-zero, that indicates lower shared-shell character, lower IR-responsivity and a more pliable, less stiff response of the bond-path. The measurement of the relative CdC BCP shifts in mode 21 is disrupted due to the presence of non-nuclear attractors (NNAs) that results in two BCPs and an NNA located along the CdC bond-path that renders the topology non-equivalent to that of the CdC bond-path of

148

6. Theory developments and applications of next-generation QTAIM

FIG. 6.16 The variation in the precession K along the CdC BCP bond-paths of the benzene IR-active mode 28, see the caption of Fig. 6.13 for further details.

FIG. 6.17 The precession K and K0 values for the CdH BCP bond-paths of the benzene IR-active mode 5 for the amplitudes of vibration 1.0 (black), 0.0 (red) and + 1.0 (blue). The variation in the precession K with inset {p,p0 } pathpackets (left panel) and K0 with inset {q,q0 } (right panel) along the CdC BCP bond-paths of the relaxed benzene. Bond critical points (BCPs) are located at a distance ¼ 0.0 a.u. along the bond-path.

3. The NG-QTAIM bond-path precession K

149

FIG. 6.18 The precession K and K0 values for the CdH BCP bond-paths of the benzene infrared IR-active mode 14, see the caption of Fig. 6.17 for further details.

FIG. 6.19 The precession K and K0 values for the CdH BCP bond-paths of the benzene IR-active mode 21, see the caption of Fig. 6.17 for further details.

FIG. 6.20 The precession K and K0 values for the CdH BCP bond-paths of the benzene IR-active mode 28, see the caption of Fig. 6.17 for further details.

152

6. Theory developments and applications of next-generation QTAIM

the relaxed benzene. The effect of the NNAs on the torsion of the bond-path can be determined however, by examination of the precession K profile and the relative positions along the bond-path ΔQ of the jumps in K. The K profiles corresponding to the presence of the NNAs and the values of ΔQ are the most similar to that of the relaxed benzene. High IR-responsivity values for the CdH BCP bond-path K profiles are consistently indicated on the basis of the K profiles values being non-zero and much closer to K ¼ 1.0 than is the case for the CdC BCP bond-path K profiles. Therefore, the CdH BCP bond-paths are more flexible and pliable than the CdC BCP bond-paths. There are large deviations apparent from the high/high IR-responsivity/non-responsivity K ¼ 1.0/K0 ¼ 1.0 of the CdH BCP bondpaths to mode 28. Conversely, the K profiles of the CdC bond-paths of mode 28 maintain K ¼ 0.0, referred to as unresponsive (U) and therefore possess the stiffest CdC bonds. Future work could include investigations on IR-inactive modes using NG-QTAIM to determine the degree of anharmonicity present, comparisons could then be made with the experimentally observed vibrational corrections [69].

3.4 Applications of the bond-path precession K: Switches 3.4.1 Fatigue and fatigue resistance in S1 excited state Diarylethenes in electric fields Directional electric-field effects on the bonding of the undoped and sulfur doped diarylethene (DTE) switch molecule were investigated using NG-QTAIM [29]. Chemical bonding concepts were introduced in the form of the least and most preferred directions of charge density accumulation relative to a bond-path, namely the precessions K and K0 that are demonstrated to be much more responsive to the electric-field than the conventional QTAIM Laplacian r2ρ(rb). Bond fatigue is presented as the tendency for a bond-path to rupture that provides directional versions of familiar bonding QTAIM concepts. Fatigue resistance and fatigue are included where the applied electric-field reduces and increases the tendency toward bond-path rupture, respectively. Applications of the precessions K and K0 including switches, ring opening reactions and molecular rotary motors in the presence of fields that cause a redistribution of ρ(r) are included.

4. The NG-QTAIM Uσ-space stress tensor trajectory Tσ(s) 4.1 Theory background: The NG-QTAIM Uσ-space stress tensor trajectory Tσ(s) We use Bader’s formulation of the stress tensor [11] within the QTAIM partitioning that is a standard option in the QTAIM AIMAll [66] suite. The Bond Critical Point (BCP) ellipticity, ε, quantifies the relative accumulation of ρ(rb) in the two directions perpendicular to the bondpath at a BCP with position rb. For values of the ellipticity ε > 0, the shortest and longest axes of the elliptical distribution of ρ(rb) are associated with the λ1 and λ2 eigenvalues, respectively, and the ellipticity ε is defined as ε ¼ jλ1j/jλ2j 2 1. Earlier, we demonstrated that the most preferred direction for bond motion, corresponding to most preferred direction of electronic charge density motion, is the e1σ eigenvector of the stress tensor [1]. The Tσ(s) is constructed

4. The NG-QTAIM Uσ-space stress tensor trajectory Tσ(s)

153

using the change in position of the BCP, referred to as dr, for all displacement steps dr of the calculation. Each finite BCP shift vector dr is mapped to a point {(e1σ∙dr), (e2σ∙dr), (e3σ∙dr)} in sequence, forming the Tσ(s), constructed from the vector dot products (the dot product is a projection or a measure of vectors being parallel to each other) of the stress tensor Tσ(s) eigenvector components evaluated at the BCP. The projections of dr are, respectively, associated with the bond torsion: e1σ.dr → bond-twist, e2σ.dr → bond-flexing and e3σ.dr → bondanharmonicity [9,17,21,32,70–72]. The e1σ corresponds to the most preferred direction of charge density ρ(r) accumulation and therefore the most facile direction in the plane perpendicular to the bond-path, where bond-torsion about the BCP does not involve structural distortion in the form of any increase in bond-path length from the straight-line bonded separation. A value of {e1σ.dr}max ¼ 0.0 corresponds to a constant orientation of the e1σ eigenvector in real space and therefore constant bond-path torsion (bond-twist). The subscript “max” corresponds to the difference between the minimum and maximum value of the projection of the BCP shift dr onto e1σ or e2σ or e3σ along the entire stress tensor trajectory Tσ(s). We will denote the maximum stress tensor projections Tσ(s)max ¼ {bond-twistmax, bond-flexingmax, bondanharmonicitymax}; these quantities therefore define the dimensions of a “bounding box” around each Tσ(s). The e2σ corresponds to the least preferred, i.e., the least readily distorted, direction of charge density ρ(r) accumulation and therefore least facile direction in the plane perpendicular to the bond-path. This is because bond-flexing requires a greater structural distortion than bond-twist (torsion) in the form of an increase in bond-path length from the straight-line bonded separation. A value {e2σ.dr}max ¼ 0.0 corresponds to constant bond-flexing in real space. A value of {e3σ.dr}max > 0.0 indicates the bond-anharmonicity corresponding to a changing BCP shift dr in real space relative to the bond-path, and therefore greater freedom for the BCP to slide along the bond-path. Conversely, {e3σ.dr}max ¼ 0.0 corresponds to a constant BCP shift dr in real space along the bond-path and therefore an absence of both helical motion about the BCP and bond-anharmonicity. Previously, we established the stress tensor trajectory Tσ(s) classifications of S and R stereoisomers based on the counterclockwise (CCW) vs. clockwise (CW) torsions for the e1σ.dr components of Tσ(s) for lactic acid and alanine [9]. The calculation of the stress tensor trajectory Tσ(s) for the torsional BCP is undertaken using the frame of reference defined by the mutually perpendicular stress tensor eigenvectors {e1σ,e2σ,e3σ} at the torsional BCP, corresponding to the minimum energy geometry; this frame that is referred to as the stress tensor trajectory space is also referred to as Uσ-space. This frame of reference is used to construct all subsequent points along the Tσ(s) for dihedral torsion angles in the range  180.0°  θ  +180.0°, where θ ¼ 0.0° corresponds to the minimum energy geometry. We adopt the convention that CW circular rotations correspond to the range 180.0°  θ  0.0° and CCW circular rotations to the range 0.0°  θ  +180.0°. To be consistent with optical experiments, we defined from the Tσ(s) that S (left-handed) character is dominant over R character (right-handed) for values of (CCW) > (CW) components of the Tσ(s). The gap that we observe for the Tσ(s) between the CCW and CW torsions at a torsion θ ¼ 0.0° is due to their BCP shift vectors dr being oppositely directed and of non-zero

154

6. Theory developments and applications of next-generation QTAIM

magnitude. For example, for a CCW torsion starting at θ ¼ 0.0° with a finite BCP shift dr 5 {+0.01, 0.00, 0.00}, the corresponding CW finite BCP shift dr  {20.01, 0.00, 0.00}, giving a Uσ-space separation of the Tσ(s) of the CCW and CW torsions at a torsion θ ¼ 0.0° of approximately 0.01-(0.01) ¼ 0.02. This mapping is sufficiently symmetry breaking to enable the Tσ(s) of S and R stereoisomers at the equilibrium configuration to be distinguished, in contrast to conventional scalar QTAIM. The Tσ(s) comprises a series of contiguous points as a 3-D vector path displaying the effect of the structural change and is analyzed here in terms of the CW and CCW directions of bond torsion. Computational details. The first step of the computational protocol is to perform a constrained scan of the potential energy surface (ΔE), see Scheme 6.6. The scan was performed with a constrained (Z-matrix) geometry optimization performed at all steps with all coordinates free to vary except for the torsion coordinate θ, where θ was defined by the dihedral angle C3-C1-C2-H8 (for the S stereoisomer) and C3-C1-C2-H7 (for the R stereoisomer), in the range 180.0°  θ  +180.0° with 1.0° intervals. These torsions correspond to the Cartesian clockwise (CW) and Cartesian counter-clockwise (CCW) directions of the torsion θ, applied to the S and R stereoisomers. For further analysis, we first focused on the physical quantities measured at the bond-critical point of the C1dC2 bond (hereafter referred to as the C1dC2 BCP). We chose the C1dC2 bond attached to the chiral center C1 because it is free to rotate: the other choices C1dC3 and C1dO4 are unsuitable due to the applied torsion θ being hindered by the H12dO4 bond, see Scheme 6.6. For a torsion performed about a nonchiral atom (e.g., C3), the variation in the relative energy ΔE for the S and R rotational isomers is distinguishable corresponding to the torsion about the C3dO11 BCP. The B3LYP DFT functional was used with the 6-311G(2d,3p) basis set and computations performed using Gaussian 09E01 [45]. In the calculations, the following tight convergence thresholds (stated here in atomic units) were used: SCF energy 106, density matrix 108, maximum force 4.5  104, RMS force 3  104. A built-in pruned “Ultrafine” DFT integration grid with 99 radial shells and 950 angular points/shell convergence criteria was also used. Subsequent single point energies for each step in the potential energy surface were evaluated using the same theory level, convergence criteria and integration grids. QTAIM and stress tensor analysis was performed with the AIMAll [66] suite on each wave function obtained in the previous step.

SCHEME 6.6

The molecular graphs of the S(left panel) and R-stereoisomers (right panel) of lactic acid with the torsional C1-C2 bond critical point (BCP) indicated by the undecorated green sphere, with the undecorated red sphere indicating the location of the ring critical point (RCP). The clockwise (CW) direction of torsion θ is indicated by dashed black lines surrounding the torsional C1-C2 BCP. The chiral center is located at the carbon atom C1.

4. The NG-QTAIM Uσ-space stress tensor trajectory Tσ(s)

155

All molecular graphs were additionally confirmed to be free of non-nuclear attractor critical points.

4.2 Applications of the stress tensor trajectory Tσ(s): Bonding environments and structural preferences 4.2.1 The Uσ-space trajectories of the four infrared active normal modes of benzene The four infrared active Uσ-space trajectories of the normal modes are found to be unique and the Uσ-space projections indicate mixed CdC and CdH bonding character [73]. 4.2.2 A stress tensor eigenvector projection space for the (H2O)5 potential energy surface We found a significant contribution of O-–O BCPs from the stress tensor trajectory length Lσ of the stress tensor trajectory Tσ(s) [72]. We demonstrated covalent coupling between covalent OdH and hydrogen bonds and dynamic coupling effects between pairs of covalent OdH and pairs of hydrogen bonds.

4.3 Applications of the stress tensor trajectory Tσ(s): Molecular devices: Switches and molecular rotary motors 4.3.1 Fatigue and photochromism S1 excited state reactivity of diarylethenes Improvements of the switching properties in complex diarylethene structures previously were attained on a “trial and error” basis through chemical substitutions aimed at tuning the chemical properties of the core of the diarylethene [70]. Therefore, we presented new guiding principles to analyze the first excited state reactivity of diarylethenes based on NG-QTAIM that provided consistent theoretical justification to partner the already successful symmetric substitution patterns obtained from experiments. The guiding principles provided indicated that more complex asymmetric patterns should be included for the systematic design of new technologically relevant functional compounds. The stress tensor trajectory Tσ(s) analysis was used to characterize the photochromism reaction as reusable and the fatigue reaction as irreversible and we discovered new candidate sites for alteration by future experiment. 4.3.2 The CdO ring-opening photo-reactions of oxirane with the Ehrenfest force F(r) trajectories The Ehrenfest Force F(r) trajectories were constructed for the CdO ring-opening photoreactions of oxirane in an eigenvector-space corresponding to bond-flexing, bond-twist and bond-anharmonicity associated with the least and most preferred directions of charge density accumulation and bond critical point (BCP) sliding, respectively [74]. The presence of the torsion of a CH2 group for one of the photo-reactions led to greater symmetry breaking and greater reaction pathway preference we have applied to the photochemical ring-opening reactions of oxirane.

156

6. Theory developments and applications of next-generation QTAIM

4.3.3 Quinone-based switches for candidate building blocks of molecular junctions The effect of an electric field on a recently proposed quinone-based molecular switch was investigated. The reversal of a homogenous external electric field was demonstrated to improve the “OFF” functioning of the switch via destabilization of the H atom participating in the tautomerization process along the hydrogen bond that defines the switch [71]. The “ON” functioning of the switch is also improved by the reversal of the homogenous external electric field: this result was previously inaccessible. Visualization of the “ON” and “OFF” functioning of the switch was undertaken in terms of the response of the most preferred directions of motion of the electronic charge density to the applied external field that provided consistent results for the factors affecting the “ON” and “OFF” switch performance. Future use for the design of molecular electronic devices was therefore indicated. 4.3.4 The dynamics trajectories of a light-driven rotary molecular motor NG-QTAIM was applied to analyze, along an entire bond-path, intramolecular interactions known to influence the photo-isomerization dynamics of a light-driven rotary molecular motor [75]. We undertook the first use of the stress tensor trajectory Tσ(s) analysis on selected non-adiabatic molecular dynamics trajectories that was found to be well suited to follow the dynamics trajectories that included the S0 and S1 electronic states through the conical intersection. The stress tensor trajectory Tσ(s) also provided a new measure to assess the degree of purity of the axial bond rotation for the design of rotary molecular motors.

4.4 Applications of the stress tensor trajectory Tσ(s): Isoenergetic phenomena: Stereoisomers 4.4.1 The chirality-helicity equivalence in the S and R stereoisomers of lactic acid and alanine We located the unknown chirality-helicity equivalence in lactic acid and alanine that possess a chiral center and as a consequence the degeneracy of the S and R stereoisomers of lactic acid was lifted [9]. Consistency with the naming schemes of S and R stereoisomers from optical experiments was found. This was enabled by the construction of the stress tensor trajectories Tσ(s) by the variation in the position of the torsional bond critical point upon a structural change, along the torsion angle, θ, involving a chiral carbon atom. This was undertaken by applying a torsion θ, 180.0°  θ  +180.0° corresponding to clockwise and counterclockwise directions. We explained why scalar measures can only partially lift the degeneracy of the S and R stereoisomers, as opposed to vector-based measures that can fully lift the degeneracy. The consequences for stereochemistry in terms of the ability to determine the chirality of industrially relevant reaction products are outlined. The same stress tensor trajectory Tσ(s) analysis was also performed on the S and R stereoisomers and rotational isomers of the chiral amino acid Alanine, where the magnitudes of the t1,max component of the S stereoisomer are larger (by 5.3%) for the CCWH direction than the CWH direction, also in agreement with the optical experiment results. We also performed the same analysis on the formally achiral amino acid glycine, where no helical

157

4. The NG-QTAIM Uσ-space stress tensor trajectory Tσ(s)

FIG. 6.21 The helical stress tensor trajectories Tσ(s) of the S and R stereoisomers (θ ¼ 0.0°) and rotational isomers of lactic acid for the Cartesian torsions CW (θ ¼ 30.0°, 60.0°, 90.0°, 120.0°, 150.0°, 180.0°) and CCW (θ ¼ 30.0°, 60.0°, 90.0°, 120.0°, 150.0°, 180.0°) of the torsional C1dC2 BCP for the most preferred (left panel) and least preferred (right panel) directions of charge density accumulation, respectively. The maximum projections Tσ(s)max ¼ {t1,max ¼ (e1σ∙dr)max, t2,max ¼ (e2σ∙dr)max, t3,max ¼ (e3σ∙dr)max}, where dr are the BCP shift vectors. The chirality assignments of the Tσ(s) helix screw axes: CWH and CCWH may not coincide with the Cartesian CW and CCW directions of torsion θ shown in Scheme 6.6. The subscript “H” indicates the left- or right-handed screw axes of the Tσ(s) helices.

geometry in the trajectory Tσ(s) was observed, due to the achiral nature of the molecule. This finding is consistent with recently published work on the formally achiral ethene molecule subjected to CW and CCW torsions where the no helical character was found to be present for the Tσ(s) [1] (Fig. 6.21). To summarize: in this investigation of stereochemistry, we located the unknown chiralityhelicity equivalence present in chiral molecules required to explain the chemical origins of differences in behaviors of the S and R stereoisomers of lactic acid. We find agreement with the prediction made by David Z. Wang [76] that the molecular electronic properties and not solely steric hindrance would be the source of the unknown chirality-helicity equivalence. This finding is consistent with the recent groundbreaking experimental work that uses the helical motion of bound electrons instead of the traditional magnetic field effects [77] (Table 6.7).

TABLE 6.7 The maximum stress tensor projections {t1max, t2max, t3max} for the S and R stereoisomers for the torsional C1-C2 BCP corresponding are shown highlighted in a bold font. {t1max, t2max, t3max} S

R

CWH

CCWH

CWH

CCWH

{1.900, 0.971, 0.880}

{1.978, 1.077, 0.883}

{1.976, 1.066, 0.884}

{1.899, 0.967, 0.881}

BCP C1-C2

The chirality assignments of the left- and right-handed screw axes of the Tσ (s) helices are denoted as CCWH and CWH, respectively, the subscript “H” refers these helices as opposed to the Cartesian torsion CCW and CW shown in Scheme 6.6.

158

6. Theory developments and applications of next-generation QTAIM

4.4.2 Comparison of QTAIM and the stress tensor for the chirality-helicity equivalence in S and R stereoisomers The QTAIM trajectories T(s) were calculated to locate the unknown chirality-helicity equivalence for the S and R stereoisomers of lactic acid and alanine to compare with the recently located using the stress tensor trajectories Tσ(s) [12]. Weak and good agreement with the naming schemes from optical experiment, respectively, were provided by the QTAIM trajectories T(s). The implications of the torsional directional preferences found are discussed and the T(s) analysis was applied to formally achiral glycine.

4.5 Applications of the stress tensor trajectory Tσ(s): Isoenergetic phenomena: Competitive ring-opening reactions, isotopomers and rotomers 4.5.1 Distinguishing and quantifying the torquoselectivity in competitive ringopening reactions The directional properties of the TSIC and TSOC ring-opening reactions of 3-fluoromethylcyclobutene and 1-cyano-1-methylcyclobutene were determined by constructing the stress tensor trajectory space Uσ with stress tensor trajectories Tσ(s) with length l in real space, where longer l correlated with greater relative energetic stability [78]. 4.5.2 A non-scalar approach to the intramolecular mode coupling of the isotopomers of water The bond-flexing, bond-twist and bond-anharmonicity were quantified during the bending (Q1), symmetric-stretch (Q2) and anti-symmetric-stretch (Q3) normal modes of vibration of the H2O/D2O/HDO isotopomers of water [32]. The presence of curved bonding was used to detect bond-flexing and bond-anharmonicity was detected by motion (sliding) of the bond critical point (BCP) relative to the oxygen. These 2-D scalar measures are insufficient for the description of the 3-D nature of the normal modes of vibration and therefore the susceptibility toward normal mode coupling or to fully distinguish the three isotopomers. Bond-twist was detected by the use of a vector-based measure in the form of the bond critical point (BCP) T(s) trajectory, constructed in terms of preferred directions of electronic motion, defined by the variation in the position of the BCP during the normal modes of vibration. The BCP T(s) trajectories described the coupling of the intramolecular bending and symmetric-stretch normal modes as well as distinguishing all three isotopomers within the harmonic approximation. The absence of bond-twist was indicated to be responsible for the coupling of the bending and symmetric-stretch normal modes that disrupted the coupling between sigma OdH bonds and hydrogen-bonding. Partial coupling was found to be present for the mixed isotopomer HDO. 4.5.3 The tunneling pathways of the flip rearrangements between permutationinversion isomers of (H2O)5 Tunneling pathways of the flip rearrangement between permutation-inversion isomers corresponding to the energetically degenerate global energy minima of (H2O)5 were analyzed [79]. The presence of the asymmetrical energy barrier enabled the scalar measures to distinguish the pairs of permutation-inversion isomers everywhere except at the transition state

5. Summary, future outlook and suggestions for further work

159

however, they cannot determine the most and least facile directions of the flip rearrangement. The vector or directional character of the two sides of the pathway is captured by the stress tensor trajectories were defined by the variation in the position of the bond critical point. The stress tensor trajectories enabled the quantification of the bond-flexing, bond-twist and bondanharmonicity of the flip rearrangement between permutation-inversion isomers to be quantified. The clockwise and counter-clockwise directions of the flip rearrangement are found to be the most and least facile, respectively, using the stress tensor trajectories.

5. Summary, future outlook and suggestions for further work 5.1 Summary Next-Generation QTAIM (NG-QTAIM) provides the first directional 3-D interpretation of chemical bonding, including bond-flexing, bond-twist and bond-anharmonicity. Unique developments of NG-QTAIM include: stereochemistry, distinguishing and quantifying isotopomers, tunneling pathways of the flip rearrangement between permutation-inversion isomers corresponding to the energetically degenerate global energy minima. NG-QTAIM has been used to distinguish properties relating to the S0 and S1 degenerate excited states at a conical intersection. The 3-D normal modes of vibration were presented in the 3-D Uσ-space. In addition, NG-QTAIM can be used, with or without nuclear motion present, to visualize the response to laser irradiation. The chemical origin of chirality, the missing chirality-helicity equivalence, was found using the stress tensor trajectory Tσ(s) in the space U(s). Agreement was found with the naming schemes of S and R stereoisomers from optical (laser) experiments. Essential insights into molecular switch functioning not available from the energy barrier or any scalar measures and a new measure to assess the degree of purity of the axial bond torsion for the design of rotary molecular motors are also provided.

5.2 Active control of molecular devices NG-QTAIM can fulfill the increasing need to find mechanistic theoretical models to interpret and understand how a shaped laser pulse, down to the sub-femtosecond timescale, perturbs and induces phenomena in the molecular device, to implement coherent control strategies. Next-generation QTAIM (NG-QTAIM) provides the directional and 3-D perspective needed to adapt to the needs of new molecular device environments. In practical terms, this would involve the implementation of techniques for extraction of the salient features of the induced electronic and nuclear dynamics from an ensemble of dynamics trajectories. An example of such a “smoothing” methods is “essential dynamics” [80–82]. One of the main challenges for materials science in actively controlling molecular devices using laser pulses is the ability to understand the time-scales and control the intramolecular energy and electron redistribution in molecules. In contrast, kinetic and thermodynamic control are both types of passive control that are unable to access molecular device mechanisms on a bond-by-bond basis. There is therefore an increasing need to find mechanistic theoretical models to interpret and understand how a shaped laser pulse, down to the sub-femtosecond timescale, perturbs and induces phenomena in the molecular device to implement coherent

160

6. Theory developments and applications of next-generation QTAIM

control strategies. The phenomena associated with switches and light-driven devices in response to external agents such as laser irradiation are inherently directional in nature. The current conventional computational treatments reduce the response of a given molecule to external agents to a set of scalar-based measures either using conventional QTAIM, orbital or orbital-free electronic structure methods.

5.3 Prediction of asymmetric synthetic reactions The prediction of asymmetric synthetic reactions, including the highly stereospecific SN2 reactions, is one of the most fundamental problems in industrial organic and catalytic chemistry. The ability to quantify the degree of chirality of reactants and any intermediate species of SN2 and other reactions will enable knowledge to be gained on the bias toward the stereocontrol of chiral reaction products. The Cahn-Ingold-Prelog (CIP) classifications are inaccessible in two circumstances: firstly for chiral reactions with intermediate structures where the chiral carbon of the reactant gains one or more additional bonds and secondly for formally achiral reactions and achiral species. This will be undertaken by the quantification of the chirality along an entire reaction pathway whereby achiral reaction species may also be considered.

Acknowledgments The National Natural Science Foundation of China is acknowledged, project approval number: 21673071. The One Hundred Talents Foundation of Hunan Province is gratefully acknowledged for the support of S.J. and S.R.K.

References [1] J.H. Li, W.J. Huang, T. Xu, S.R. Kirk, S. Jenkins, Stress tensor eigenvector following with next-generation quantum theory of atoms in molecules, Int. J. Quantum Chem. 119 (2018), e25847. [2] H. Nakatsuji, Common nature of the electron cloud of a system undergoing change in nuclear configuration, J. Am. Chem. Soc. 96 (1974) 24–30. [3] R.G.A. Bone, R.F.W. Bader, Identifying and analyzing intermolecular bonding interactions in van Der Waals molecules, J. Phys. Chem. 100 (1996) 10892–10911. [4] S. Jenkins, M.I. Heggie, Quantitative analysis of bonding in 90o partial dislocation in diamond, J. Phys. Condens. Matter 12 (2000) 10325–10333. [5] P.W. Ayers, S. Jenkins, An electron-preceding perspective on the deformation of materials, J. Chem. Phys. 130 (2009) 154104–154111. [6] S. Jenkins, S.R. Kirk, A.S. Cote, D.K. Ross, I. Morrison, Dependence of the normal modes on the electronic structure of various phases of ice as calculated by ab initio methods, Can. J. Phys. 81 (2003) 225–231 (7). [7] S. Jenkins, I. Morrison, The dependence on structure of the projected vibrational density of states of various phases of ice as calculated by ab initio methods, J. Phys. Condens. Matter 13 (2001) 9207–9229. [8] R.F.W. Bader, Atoms in Molecules: A Quantum Theory; International Series of Monographs on Chemistry, Oxford University Press, USA: New York, 1994, ISBN: 0-19-855865-1. [9] T. Xu, J.H. Li, R. Momen, W.J. Huang, S.R. Kirk, Y. Shigeta, S. Jenkins, Chirality-helicity equivalence in the S and R stereoisomers: a theoretical insight, J. Am. Chem. Soc. 141 (2019) 5497–5503. [10] K. Finzel, How does the ambiguity of the electronic stress tensor influence its ability to serve as bonding indicator, Int. J. Quantum Chem. 114 (2014) 568–576. [11] R.F.W. Bader, Quantum topology of molecular charge distributions. III. The mechanics of an atom in a molecule, J. Chem. Phys. 73 (1980) 2871–2883.

References

161

[12] T. Xu, S.R. Kirk, S. Jenkins, A comparison of QTAIM and the stress tensor for chirality-helicity equivalence in S and R stereoisomers, Chem. Phys. Lett. 738 (2020) 136907. [13] W.J. Huang, T. Xu, S.R. Kirk, S. Jenkins, The 3-D bonding morphology of the infra-red active normal modes of benzene, Chem. Phys. Lett. 710 (2018) 31–38. [14] W.J. Huang, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, QTAIM and stress tensor bond-path framework sets for the ground and excited states of fulvene, Chem. Phys. Lett. 713 (2018) 125–131. [15] W.J. Huang, R. Momen, A. Azizi, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited states of fulvene, Int. J. Quantum Chem. 118 (2018), e25768. [16] X. Bin, T. Xu, S.R. Kirk, S. Jenkins, The directional bonding of [1.1.1] propellane with next generation QTAIM, Chem. Phys. Lett. 730 (2019) 506–512. [17] T. Xu, R. Momen, A. Azizi, T. van Mourik, H. Fr€ uchtl, S.R. Kirk, S. Jenkins, The destabilization of hydrogen bonds in an external E-field for improved switch performance, J. Comput. Chem. 40 (2019) 1881–1891. [18] T. Tian, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited state of the ring-opening of cyclohexadiene (CHD), Int. J. Quantum Chem. 119 (2019), e25862. [19] T. Tian, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the ground and excited state of DHCL, Chem. Phys. Lett. 717 (2019) 91–98. [20] X. Bin, A. Azizi, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the photochemical ring-opening reactions of oxirane, Int. J. Quantum Chem. 119 (2019), e25957. [21] T. Tian, T. Xu, T. van Mourik, H. Fr€ uchtl, S.R. Kirk, S. Jenkins, Next generation QTAIM for the design of quinonebased switches, Chem. Phys. Lett. 722 (2019) 110–118. [22] L. Wang, A. Azizi, T. Xu, S.R. Kirk, S. Jenkins, Explanation of the role of hydrogen bonding in the structural preferences of small molecule conformers, Chem. Phys. Lett. 730 (2019) 206–212. [23] X. Bin, R. Momen, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, A 3-D bonding perspective of the factors influencing the relative stability of the S1/S0 conical intersections of the penta-2,4-dieniminium cation (PSB3), Int. J. Quantum Chem. 119 (2019), e25903. [24] T. Malcomson, A. Azizi, R. Momen, T. Xu, S.R. Kirk, M.J. Paterson, S. Jenkins, Stress tensor eigenvector following with next-generation quantum theory of atoms in molecules: excited state photochemical reaction path from benzene to benzvalene, J. Phys. Chem. A 123 (2019) 8254–8264. [25] S. Li, T. Xu, T. van Mourik, H. Fr€ uchtl, S.R. Kirk, S. Jenkins, Halogen and hydrogen bonding in halogenabenzene/NH3 complexes compared using next-generation QTAIM, Molecules 24 (2019) 2875. [26] A. Azizi, R. Momen, S.R. Kirk, S. Jenkins, 3-D bond-paths of QTAIM and the stress tensor in neutral lithium clusters, Li m (m ¼ 2-5), presented on the Ehrenfest force molecular graph, Phys. Chem. Chem. Phys. 22 (2019) 864–877. [27] S. Li, A. Azizi, S.R. Kirk, S. Jenkins, An explanation of the unusual strength of the hydrogen bond in small water clusters, Int. J. Quantum Chem. 120 (2020), e26361. [28] A. Azizi, R. Momen, H. Fr€ uchtl, T. van Mourik, S.R. Kirk, S. Jenkins, Next-generation QTAIM for scoring molecular wires in E-fields for molecular electronic devices, J. Comput. Chem. 41 (2020) 913–921. [29] X. Nie, Y. Yang, T. Xu, S.R. Kirk, S. Jenkins, Fatigue and fatigue resistance in S1 excited state diarylethenes in electric fields, Int. J. Quantum Chem. 121 (2021), e26527. [30] B. Mahara, A. Azizi, Y. Yang, M. Filatov, S.R. Kirk, S. Jenkins, Bond-path-rigidity and bond-path-flexibility of the ground state and first excited state of fulvene, Chem. Phys. Lett. 766 (2021) 138339. [31] Y. Yang, T. Xu, S.R. Kirk, S. Jenkins, Bond flexing, twisting, anharmonicity and responsivity for the infraredactive modes of benzene, Int. J. Quantum Chem. 121 (2021), e26584. [32] T. Tian, T. Xu, S.R. Kirk, I.T. Rongde, Y.B. Tan, S. Manzhos, Y. Shigeta, S. Jenkins, Intramolecular mode coupling of the isotopomers of water: a non-scalar charge density-derived perspective, Phys. Chem. Chem. Phys. 22 (2020) 2509–2520. [33] S. Jenkins, L. Blancafort, S.R. Kirk, M.J. Bearpark, The response of the electronic structure to electronic excitation and double bond torsion in fulvene: a combined QTAIM, stress tensor and MO perspective, Phys. Chem. Chem. Phys. 16 (2014) 7115–7126. [34] R.F.W. Bader, A bond path: a universal indicator of bonded interactions, J. Phys. Chem. A 102 (1998) 7314–7323. [35] R.F.W. Bader, Bond paths are not chemical bonds, J. Phys. Chem. A 113 (2009) 10391–10396. [36] E. Kraka, D. Cremer, Description of chemical reactions in terms of the properties of the electron density, J. Mol. Struct. THEOCHEM 255 (1992) 189–206.

162

6. Theory developments and applications of next-generation QTAIM

[37] L. Wang, A. Azizi, T. Xu, M. Filatov, S.R. Kirk, M.J. Paterson, S. Jenkins, The role of the natural transition orbital density in the S0 ! S1 and S0 ! S2 transitions of fulvene with next generation QTAIM, Chem. Phys. Lett. 751 (2020) 137556. [38] T. Schwabe, S. Grimme, Towards chemical accuracy for the thermodynamics of large molecules: new hybrid density functionals including non-local correlation effects, Phys. Chem. Chem. Phys. 8 (2006) 4398–4401. [39] T.H. Dunning, Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen, J. Chem. Phys. 90 (1989) 1007–1023. [40] R.A. Kendall, T.H. Dunning, R.J. Harrison, Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions, J. Chem. Phys. 96 (1992) 6796–6806. [41] K.A. Peterson, D. Figgen, E. Goll, H. Stoll, M. Dolg, Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements, J. Chem. Phys. 119 (2003) 11113–11123. [42] K.A. Peterson, B.C. Shepler, D. Figgen, H. Stoll, On the spectroscopic and thermochemical properties of ClO, BrO, IO, and their anions, J. Phys. Chem. A 110 (2006) 13877–13883. [43] L. Goerigk, N. Mehta, A trip to the density functional theory zoo: warnings and recommendations for the user*, Aust. J. Chem. 72 (2019) 563–573. [44] E.L. Cates, T. van Mourik, Halogen bonding with the halogenabenzene bird structure, halobenzene, and halocyclopentadiene, J. Comput. Chem. (2019). [45] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G.A. Petersson, et al., Gaussian 09, Revision E.01, Gaussian, Inc, Wallingford, CT, 2009. [46] F. Neese, The ORCA program system, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2 (2012) 73–78. [47] J. Zheng, X. Xu, D.G. Truhlar, Minimally augmented Karlsruhe basis sets, Theor. Chem. Accounts 128 (2011) 295–305. [48] D.A. Pantazis, F. Neese, All-electron scalar relativistic basis sets for the 6p elements, Theor. Chem. Accounts 131 (2012) 1292. [49] D. Aravena, F. Neese, D.A. Pantazis, Improved segmented all-electron relativistically contracted basis sets for the lanthanides, J. Chem. Theory Comput. 12 (2016) 1148–1156. [50] T.A. Keith, AIMAll, Revision 17.01.25, TK Gristmill Software, Overland Park KS, USA, 2017. [51] P. Ramachandran, G. Varoquaux, Mayavi: 3D visualization of scientific data, Comput. Sci. Eng. 13 (2011) 40–51. [52] M. Frisch, G. Trucks, H. Schlegel, G. Scuseria, M. Robb, J. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. Petersson, et al., Gaussian 09, Revision B.01, Gaussian, Inc, Wallingford, CT, 2009. [53] A.D. Becke, Density-functional thermochemistry. III. The role of exact exchange, J. Chem. Phys. 98 (1993) 5648–5652. [54] W. Kohn, A.D. Becke, R.G. Parr, Density functional theory of electronic structure, J. Phys. Chem. 100 (1996) 12974–12980. ˚ . Malmqvist, B.O. Roos, Density matrix averaged atomic natural orbital (ANO) basis sets for [55] P.-O. Widmark, P.-A correlated molecular wave functions, Theor. Chim. Acta 77 (1990) 291–306. [56] B.O. Roos, V. Veryazov, P.-O. Widmark, Relativistic atomic natural orbital type basis sets for the alkaline and alkaline-earth atoms applied to the ground-state potentials for the corresponding dimers, Theor. Chem. Accounts 111 (2004) 345–351. ˚ . Malmqvist, V. Veryazov, P.-O. Widmark, New relativistic ANO basis sets for transi[57] B.O. Roos, R. Lindh, P.-A tion metal atoms, J. Phys. Chem. A 109 (2005) 6575–6579. [58] D. Feller, The role of databases in support of computational chemistry calculations, J. Comput. Chem. 17 (1996) 1571–1586. [59] K.L. Schuchardt, B.T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J. Chase, J. Li, T.L. Windus, Basis set exchange: a community database for computational sciences, J. Chem. Inf. Model. 47 (2007) 1045–1052. [60] A. Azizi, R. Momen, A. Morales-Bayuelo, T. Xu, S.R. Kirk, S. Jenkins, A vector-based representation of the chemical bond for predicting competitive and noncompetitive torquoselectivity of thermal ring-opening reactions, Int. J. Quantum Chem. 118 (2018), e25707. [61] C. Adamo, V. Barone, Toward reliable density functional methods without adjustable parameters: the PBE0 model, J. Chem. Phys. 110 (1999) 6158–6170. [62] S. Grimme, J. Antony, S. Ehrlich, H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, J. Chem. Phys. 132 (2010) 154104.

References

163

[63] S. Grimme, S. Ehrlich, L. Goerigk, Effect of the damping function in dispersion corrected density functional theory, J. Comput. Chem. 32 (2011) 1456–1465. [64] Field j Gaussian.Com, Available online: https://gaussian.com/field/. (Accessed 2 December 2019). [65] S. Sowlati-Hashjin, C.F. Matta, The chemical bond in external electric fields: energies, geometries, and vibrational stark shifts of diatomic molecules, J. Chem. Phys. 139 (2013) 144101. [66] T.A. Keith, AIMAll, Revision 19.10.12, TK Gristmill Software, Overland Park KS, USA, 2019. [67] A. Azizi, R. Momen, S.R. Kirk, S. Jenkins, 3-D bond-paths of QTAIM and the stress tensor in neutral lithium clusters, Lim (m ¼ 2–5), presented on the Ehrenfest force molecular graph, Phys. Chem. Chem. Phys. 22 (2020) 864–877. [68] A. Azizi, R. Momen, T. Xu, S.R. Kirk, S. Jenkins, Non-nuclear attractors in small charged lithium clusters, Limq (m ¼ 2–5, q ¼ 1), with QTAIM and the Ehrenfest force partitioning, Phys. Chem. Chem. Phys. 20 (2018) 24695–24707. [69] M. Piccardo, E. Penocchio, C. Puzzarini, M. Biczysko, V. Barone, Semi-experimental equilibrium structure determinations by employing B3LYP/SNSD anharmonic force fields: validation and application to semirigid organic molecules, J. Phys. Chem. A 119 (2015) 2058–2082. [70] P. Yang, T. Xu, R. Momen, A. Azizi, S.R. Kirk, S. Jenkins, Fatigue and photochromism S1 excited state reactivity of diarylethenes from QTAIM and the stress tensor, Int. J. Quantum Chem. 118 (2018), e25565. [71] T. Xu, L. Wang, Y. Ping, T. van Mourik, H. Fr€ uchtl, S.R. Kirk, S. Jenkins, Quinone-based switches for candidate building blocks of molecular junctions with QTAIM and the stress tensor, Int. J. Quantum Chem. 118 (2018), e25676. [72] T. Xu, J. Farrell, R. Momen, A. Azizi, S.R. Kirk, S. Jenkins, D.J. Wales, A stress tensor eigenvector projection space for the (H2O)5 potential energy surface, Chem. Phys. Lett. 667 (2017) 25–31. [73] M.X. Hu, T. Xu, R. Momen, A. Azizi, S.R. Kirk, S. Jenkins, The normal modes of vibration of benzene from the trajectories of stress tensor eigenvector projection space, Chem. Phys. Lett. 677 (2017) 156–161. [74] X. Nie, M. Filatov, S.R. Kirk, S. Jenkins, Photochemical ring-opening reactions of oxirane with the Ehrenfest force topology, Chem. Phys. Lett. (2021) 138432. [75] L. Wang, A. Azizi, R. Momen, T. Xu, S.R. Kirk, M. Filatov, S. Jenkins, Next-generation quantum theory of atoms in molecules for the S1/S0 conical intersections in dynamics trajectories of a light-driven rotary molecular motor, Int. J. Quantum Chem. 120 (2020), e26062. [76] D. Zhigang Wang, A Helix theory for molecular chirality and chiral interaction, Mendeleev Commun. 14 (2004) 244–247. [77] S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G.A. Garcia, R. Geneaux, A.G. Harvey, F. Legare, Z. Masˇ´ın, L. Nahon, et al., Photoexcitation circular dichroism in chiral molecules, Nat. Phys. 14 (2018) 484–489. [78] H. Guo, A. Morales-Bayuelo, T. Xu, R. Momen, L. Wang, P. Yang, S.R. Kirk, S. Jenkins, Distinguishing and quantifying the torquoselectivity in competitive ring-opening reactions using the stress tensor and QTAIM, J. Comput. Chem. 37 (2016) 2722–2733. [79] T. Xu, X. Bin, S.R. Kirk, D.J. Wales, S. Jenkins, Flip rearrangement in the water pentamer: analysis of electronic structure, Int. J. Quantum Chem. 120 (2020), e26124. [80] A. Amadei, A.B.M. Linssen, H.J.C. Berendsen, Essential dynamics of proteins, Proteins: Struct. Funct. Bioinf. 17 (1993) 412–425. [81] M. Orozco, A. Perez, A. Noy, F.J. Luque, Theoretical methods for the simulation of nucleic acids, Chem. Soc. Rev. 32 (2003) 350–364. [82] I. Daidone, A. Amadei, Essential dynamics: foundation and applications, WIREs Comput. Mol. Sci. 2 (2012) 762–770.

C H A P T E R

7 Real-space description of molecular processes in electronic excited states Jesu´s Jara-Cort
esa,* and Jesu´s Herna´ndez-Trujillob,* a

Unidad Acad
emica de Ciencias Ba´sicas e Ingenierı´as, Universidad Auto´noma de Nayarit, Tepic, M
exico bDepartamento de Fı´sica y Quı´mica Teo´rica, Facultad de Quı´mica, UNAM, M
exico City, M
exico *Corresponding author.

1. Introduction Given their relevance in a diversity of fields spanning from physics to biology, and due to their technological role, over the past decade there has been a growing interest in the characterization of processes taking place in excited electronic states (EEs). The theoretical study of EEs provides information complementary to the experimental one which can be used for the interpretation or rationalization of experimental results, because it allows to propose microscopic interaction models for the understanding of the molecular structure and reaction mechanisms involved. The use of scalar and vector fields to understand the electronic structure of atoms and molecules was pioneered by Prof. Richard Bader, who investigated the properties of the electron density, ρ(r), the central object that led to the development of the quantum theory of atoms in molecules (QTAIM) [1]. The field has been extended to what nowadays is known as quantum chemical topology (QCT) [2]. The goal of this chapter is to present a brief review on the use of QCT in the study of molecules in EEs. It is illustrated how QCT provides interpretative tools for the study of the charge and energy redistribution taking place in real space upon electronic transitions and chemical transformations in excited states. Understanding the physical basis of these types of processes is of relevance in photophysical or photochemical applications. It has to be pointed out that the number of studies reported in the literature to date using QCT for EEs is relatively small. This is in part due to the complexity and computational requirements of the electronic structure methods necessary to carry out this type of studies. The computational methods should be able to correctly incorporate static correlation in order to account for avoided crossings of states, Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00008-6

165

Copyright # 2023 Elsevier Inc. All rights reserved.

166

7. Real-space description of molecular processes in electronic excited states

conical intersections, and Rydberg states, among other features of the potential energy surfaces (PES) frequently involved in these processes. Whereas CASSCF and RASSCF have often been used in QCT in order to account for these effects, the quantitative description provided by multiconfigurational methods has been less frequently reported. In addition, being less computationally demanding, time-dependent density functional theory (TD-DFT) has often been used, although in general this approach is less suitable for the qualitative description of the PES for excited states than multiconfigurational or multireference methods. Many applications of QCT to excited states resort to the QTAIM as presented in Ref. [1]. They include the analysis of the electron density, ρ(r), related scalar fields at the bond critical points (BCPs), and atomic charges, for example. The electron sharing between topological atoms has also been studied in terms of the delocalization index [3, 4] for EEs systems. Beyond the QTAIM, the electron localization function (ELF), the interacting quantum atoms (IQA) method, the stress tensor and, to a lesser extent, nonstationary electron densities have also been reported [5–9]. The use of this wealth of real-space defined tools is illustrated in Section 2 by some examples including the excitation of molecules, excimer formation, and excited-state intramolecular proton transfer (ESIPT), among others. The aromaticity of organic molecules in EEs in terms of electron delocalization descriptors is presented in Section 3. Computational implementations of IQA for electronic structure methods that account for dynamic correlation (EOM-CCISD, MRCI-SD, and CASPT2) useful for the study of EEs have recently been reported in the literature; they are briefly described in Section 4. Finally, some concluding remarks and perspectives for future work for the study of excited states using QCT are presented in Section 5.

2. Charge distribution of electronic excited states This section presents some examples of the changes taking place on the electron distribution upon electronic excitation and how they affect the nature of covalent and noncovalent interactions. The use of molecular multipole moments and their partition into atomic contributions using the QTAIM is also illustrated.

2.1 Electron density differences The first step of any photochemical process of a molecular system is the electronic transition to an EEs. Electronic excitations are usually interpreted in terms of orbitals. For example, the UV-visible spectra of benzene and polycyclic aromatic hydrocarbons (PAHs) are assigned regarding energy levels and configurations, so that the transitions are of the type π ! π ? (see for example figure 4 of Ref. [10]). The QCT provides other means to analyze this type of processes in terms of real-space properties that in many cases provide complementary information to that obtained from orbital models. Nevertheless, as molecular orbitals correspond to a ground-state monodeterminantal description, QCT-based descriptors are often more suitable for the study of excited states [11]. For example, the local changes on the electron density upon a vertical electronic excitation, ΔρðrÞ ¼ ρexcited ðrÞ  ρground ðrÞ,

(7.1)

2. Charge distribution of electronic excited states

167

in which the nuclear framework remains unchanged, provides useful details on the electronic rearrangements that take place in a Franck-Condon transition. In addition, the analysis of the local charge increment or depletion can be complemented by the study of the properties at the BCPs, atomic charges, and the electric moments, as is shown in the following examples. 2.1.1 Formaldehyde in the S1 state The n ! π ? excitation of formaldehyde is an illustrative example of the use of QCT for the description of EEs. Ferro-Costa et al. used orbital invariant quantities (electron densities, IQA energy components, atomic charges, among others) to propose an explanation for the process [11]. In terms of orbitals, the structural changes involved (C–O bond elongation and pyramidalization of the C atom) are the consequence of populating a π ? antibonding C–O molecular orbital and an increase of the corresponding σ bond character followed by sp3 hybridization of the C atom. In terms of ρ(r), these authors analyzed the process in three steps, namely, the vertical excitation followed by lengthening of the C–O bond and C atom pyramidalization in the first singlet excited state (S1). An electron density reorganization takes place during the S0 ! S1 vertical transition of formaldehyde that consists of ρ(r) removal from the vicinity of the O atom toward both the C and O atoms and the internuclear region. This can be associated to an increase in population of a C–O π ∗ orbital. As a result, there is a decrease in the value of ρ(r) at the BCP, ρb, for the C–O bond and a charge transfer of 0.489 e to the C atom from the rest of the molecule, mostly from the O atom. Concerning the geometric relaxation in the S1 state, ρb decreases because of the C–O bond elongation accompanied by an additional depletion of the O atomic population. In addition, the electronic reorganization is negligible in the pyramidalization of the C atom, the last step of the process. Moreover, according to the Laplacian of the electron density and the ELF, the n ! π* excitation can be interpreted as a rotation of the O atom lone pairs to a charge concentration associated to a π lone pair in the pyramidalized structure accompanied by the weakening of the C–O double bond. 2.1.2 Excimer formation Excimers are complexes in electronic excited states formed between atoms or molecules of the same type. The mechanism of excimer formation involves the electronic excitation of a monomer, M !M*, and its interaction with another monomer in the ground state, M +M*! (MM)*. The nature of the interactions stabilizing the excimers has been studied using QCT. For example, the ρb values at the intermolecular BCPs and the atomic charges, among other descriptors, allow to identify specific contacts and the degree of charge transfer taking place on excimer formation. Intermolecular interactions in the ðC6 H6 Þ*2 and ðC10 H8 Þ*2 excimers

The excimers of benzene and naphthalene in the S1 state have been studied at the CASPT2 level [12]. They are formed by the following processes: C6 H6 + ðC6 H6 Þ* C10 H8 + ðC10 H8 Þ*

! ðC6 H6 Þ*2 ! ðC10 H8 Þ*2

168

7. Real-space description of molecular processes in electronic excited states

˚ ) at the minima of the PES and TABLE 7.1 Intermolecular separations (in A binding energies (in eV) for the complexes of benzene and naphthalene. System

RS0

RS 1

ΔES0

ΔES1

(C6H6)2

3.60

3.00

 0.12

 0.53

(C10H8)2

3.70

3.10

 0.24

 0.82

Values taken from J. Jara-Cort
es, T. Rocha-Rinza, J. Herna´ndez-Trujillo, Electron density analysis of aromatic complexes in excited electronic states: the benzene and naphthalene excimers, Comput. Theor. Chem. 1053 (2015) 220–228, ISSN 2210-271X, https://doi.org/10.1016/j.comptc.2014.09.031, (special Issue: Understanding structure and reactivity from topology and beyond).

Both excimers are arranged in stacked parallel configurations. Table 7.1 displays the corresponding intermolecular distances at the minima of the PES and the binding energies, defined as the difference of energy between the complex and the monomers at the dissociation limit. These values show that the intermolecular interactions in the excimers are greater than in their ground-state counterparts. The analysis of the mechanism of excimer formation begins comparing the electron densities of the monomers in the ground and excited states, their Δρ ¼ ρS1  ρS0 envelopes are displayed in Fig. 7.1A and B. In the case of benzene, there is an increase of ρ(r) above and below the aromatic ring in the regions of the C–H bonds, away from the z-axis (the symmetry

(A)

(B)

(C)

(D)

FIG. 7.1 Δρ ¼ 0.001 au envelopes for the S0 !S1 vertical excitation of (A) benzene and (B) naphthalene at their ground state geometries, and the (C) benzene and (D) naphthalene excimers at their corresponding equilibrium geometries. Positive envelopes are shown in light gray and negative in dark gray, respectively. Taken with permission from J. Jara-Cort
es, T. Rocha-Rinza, J. Herna´ndez-Trujillo, Electron density analysis of aromatic complexes in excited electronic states: the benzene and naphthalene excimers, Comput. Theor. Chem. 1053 (2015) 220–228, ISSN 2210-271X, https://doi.org/10.1016/j. comptc.2014.09.031 (special Issue: Understanding structure and reactivity from topology and beyond).

169

2. Charge distribution of electronic excited states

axis perpendicular to the molecular plane); for naphthalene, although the changes also take place above and below the molecular plane the removal of ρ(r) toward outer regions is not as clearly defined as for benzene. These changes can be quantified by the zz component of the electron density contribution to the traceless quadrupole moment tensor: Z   1  2   2  1 ρðrÞð3z2  r2 Þdr ¼ z2  x + y (7.2) Qzz ¼  2 2 where r2 ¼ x2 + y2 + z2; the minus sign before the integral takes into account the negative sign of the electronic charge. The terms within brackets,  2 x ¼

Z ρðrÞx2 dr ,

 2 y ¼

Z ρðrÞy2 dr ,

 2 z ¼

Z ρðrÞz2 dr ,

(7.3)

are the second moments of the electron distribution. Qzz accounts for the spatial extent of ρ(r) along the z-axis with respect to the xy-plane, as its expression in terms of the second moments in Eq. (7.2) indicates [13]. The Qzz values for the ground-state benzene and naphthalene molecules are  6.006 and  9.586 au, respectively; the corresponding values after a vertical excitation are  5.145 and  9.640 au. These changes of Qzz are relatively small, in keeping with the small electron density shifts observed. After the geometry relaxation in the excited state, they are further modified to  5.571 and  10.028 au, respectively. Concerning the S0 !S1 excitation of the dimers, Fig. 7.1C and D shows that there is an electron density accumulation in the intermolecular region because of the enhanced π–π interaction in the excited state. In addition, the atomic charges do not indicate any charge transfer provoked by the electronic excitation and the properties of ρ(r) at the noncovalent BCPs do not vary significantly compared to the ground-state dimers. Moreover, the exact electrostatic energiesa of the ground- and excited-state complexes were also calculated and did not support any dominance of this long-range contribution to the interaction energy in the excited state; the larger values observed for the excimers are due to the proximity of the monomers in the excited state. Furthermore, as the molecular dipole polarizabilities do not have important changes in the excitation process (see Ref. [12] for further details), it is likely that increases in the dispersion or induction interactions are not responsible for the increased stability of the excited-state complexes. The role of electron delocalization in the stabilization of these excimers was also explored. According to the QTAIM, the number of electrons shared between two regions of threedimensional (3D) space is obtained by the integration of the exchange-correlation density over the corresponding basins, thus involving an integration of the pair density. The standard M€ uller approximation can be used to define a delocalization index [15]: X 1=2 1=2 δðA,BÞ ¼ 2 ηi ηj Sij ðAÞSij ðBÞ (7.4) ij

where ηi is the occupation number of the ith natural orbital and Sij(B) and Sij(B) are the overlap integrals of the natural orbitals i and j over the atomic basins A and B. This approximation has a

In this context, the electrostatic energy is defined by the classical interaction between the molecular charge distributions [14]. This should not be confused with the electrostatic energy defined in the context of the IQA approach to be discussed in Section 4.

170

7. Real-space description of molecular processes in electronic excited states

been successfully applied to excited-state molecules at the EOM-CCSD level [16]. It has to be emphasized that the atomic basins do not have to share a bond path in order to have a delocalization index value. In the case of the benzene and naphthalene excimers, the δ(A, B) values for the intermolecular contacts remain nearly the same as in the ground-state complex. It is also possible to obtain the number of electrons shared between the monomers M and M0 by means of the P sum of pair contributions of the atoms belonging in different monomers, δðM,M0 Þ ¼ AM,BM0 δðA,BÞ. Accordingly, the number of electrons shared between the monomers in the benzene dimer increases from 0.161 to 0.406 electrons in the vertical excitation; in the case of the naphthalene complex, there is also an increase from 0.667 to 0.861 electrons. In addition, there is an increase in electron sharing in the global M +M*! (MM)* process in both cases, in line with the excimers being more stable than the ground-state counterparts. It can be concluded that whereas the electrostatic interaction energy is not the main contribution stabilizing the excimers compared to the ground-state complexes, the increased electron delocalization suggest its participation on their extra stability. Bonding properties of excimers of He2 in singlet electronic states

Jara-Cort
es et al. studied the charge redistribution accompanying the formation of the lowest-energy He2 excimers [17]. In this case, the process begins with the excitation of one He atom to the lowest triplet (13S1) or first singlet (21S0) excited states. Excimers A1Σ+u and C1Σ+g are obtained by the interaction of a ground-state He atom (11S0) with another in the 21S0.bTable 7.2 shows that the internuclear separations in the excimers are substantially shorter than for the ground-state complex and that the interaction energies become three orders of magnitude larger upon excitation. In addition, there is an important increase in the number of electrons shared between the He atoms, from nearly 0 in the ground state to 1.0 (A1Σ+u) and 0.96 (C1Σ+g ) electron pairs, signaling a change from an extremely weak noncovalent interaction to a covalent one stabilizing the lowest singlet excited-state complexes [18]. ˚ ), TABLE 7.2 ΘRR values (au), internuclear distances (A interaction energies (kcal/mol), and delocalization index (electrons) for the ground and lowest singlet excites states of the He2 complex. X1Σ+g Reqa ΔE 0

δ(He, He ) ΘRRa a

b

A1Σ+u

C1Σ+g

3.015

1.048

1.100

 0.036

 54.316

 23.937

0.0037  4.857

1.00  37.572

0.96  58.040

Ref. [17]

Another possibility is He(11S0) + He(13S1), which gives rise to two triplets, which are not to be analyzed here. See Ref. [17] for further details.

2. Charge distribution of electronic excited states

171

FIG. 7.2 Contour plots of ρ(r) and Δρ ¼ 0.005 for the lowest singlet excited states of He2 at an internuclear dis˚ . The  marks in C1 Σ+ denote positions of nonnuclear attractors. Taken from J. Jara-Cort
es, J.M. Guevaratance of 1 A g ´ Vela, A.M. Penda´s, J. Herna´ndez-Trujillo, Chemical bonding in excited states: energy transfer and charge redistribution from a real space perspective, J. Comput. Chem. 38 (13) (2017) 957–970.

Further information concerning the spatial extent of ρ(r) is provided by the trace of the matrix of electric second moments[19], ΘRR ¼ hx2 i+hy2 i+hz2 i,

(7.5)

which were defined in Eq. (7.3). The ΘRR values for the He atom in the 11S0, 21S0, and 13S1 states are:  2.433,  36.548, and  24.850 au, respectively; their absolute values indicate an important increase in the spatial extent of the atom upon excitation. Fig. 7.2 shows that the electron distributions of the lowest singlet He2 excimers are of a diffuse nature, in agreement with the large jΘRRj values, shown in Table 7.2, compared to the ground-state dimer. Moreover, the Δρ values show that ρ(r) increases in the bonding and nonbonding regions of the excimers at expense of the nuclear positions.

2.2 Atomic multipole moments The charge redistribution in molecular systems accompanying electronic excitation has also been addressed in terms of the atomic contributions to multipole moments [20–22]. Examples of dipole and quadrupole moments are presented next.c In the QTAIM, the

c

Although not oriented to excited states, see Ref. [23] for the use of atomic multipole expansions of the electrostatic energy in the design of classical potentials for molecular dynamics simulations.

172

7. Real-space description of molecular processes in electronic excited states !

molecular dipole moment is conveniently partitioned in polarization, μ pol, and charge trans! fer, μ CT , terms [1]: X! X ! ! ! ! μ ðAÞ+ qðAÞRA (7.6) μ ¼ μ pol + μ CT ¼ A !

A

! RA

In this expression, μ ðAÞ, q(A), and are the atomic polarization, atomic charge, and nuclear position of the basin A, respectively. According to the physicist’s convention, the dipole moment is oriented from the negative to the positive end, and is the one followed here. 2.2.1 Multipole moments of CO As an example, consider the dipole moment of carbon monoxide. In the ground-state !

!

(11Σ+), μ is oriented from the carbon to the oxygen atom, a situation represented as CO. This charge distribution is considered anomalous on electronegativity grounds. In terms of Eq. (7.6), the small dipole moment of 0.11 D is a consequence of opposing and nearly canceling atomic polarization and charge transfer contributions [1]. In addition, the dipole moment of CO changes to  1.37 and  0.34 D in the lowest singlet (11Π) and triplet excited states (13Π), respectively; its orientation is a reversed to CO upon excitation, a situation that is denoted by the minus sign on the corresponding values. Terrabuio et al. carried out a real-space analysis of the electron density reorganization of the molecule in order to explain these trends [21]. For 11Σ+, μpol ¼ +6.63 and μCT ¼ 6.52 D are the origin of the small dipole moment in the ground state. The corresponding values in 13Π (11Π) are: μpol ¼ +4.14 (+4.18) D and μCT ¼ 5.73 (4.67) D. In both cases, charge transfer is opposed to atomic polarization and is larger in ! magnitude. Therefore, μ CT is responsible for the inversion of polarity of the molecule upon excitation. The zz component of the quadrupole moment tensor, Eq. (7.2), of CO was also decomposed in atomic contributions (in this case z is the molecular axis) to obtain additional information concerning the charge redistribution in the molecule. The atomic Qzz values for the 11Σ+, 13Π, and 11Π states are: 0.048, 2.176, and 2.743 au for the C atom; and 0.027, 0.621, and 0.812 au for the O atom. These values are positive and larger for the excited states. They indicate a larger spread of electron density along the z-direction compared to the σ h symmetry plane, once the excited states are reached, and show that electronic excitation has a larger effect on the C than on the O atom for the 11Π than for the 13Π state. This trend can be interpreted as resulting from a larger σ to π* population transfer in the former atom. 2.2.2 Photochromism in the water excimer Another illustrative application of atomic moments is the study of the solvatochromic 00 effects in the transitions of the water dimer to the first (11A ) and second (21A0 ) singlet states. A small red shift and a larger blue shift relative to a single water molecule were reported for 00 11A and 21A0 , which were assigned as local excitations centered in the hydrogen bond donor and acceptor molecules, respectively [24]. Ferna´ndez-Alarco´n et al. carried out an EOMCCSD QCT study of these excitation processes [22]. Concerning the isolated water molecule, in the ground state it has a dipole moment of 0.6 D oriented along the C2 symmetry axis in

3. Excited-state aromaticity

173

FIG. 7.3 (A) Dipole moment of the water molecule in the 11A1 and 11B2 electronic states. (B) Charge transfer in the water dimer: δq ¼ 0.018, 0.021, and 0.091 electrons in the ground, 00 21A0 and 11A states, respectively.

(A)

(B)

the manner shown in Fig. 7.3. As in the case of CO, the 11A1 ! 11B2 electronic excitation yields a reversal in polarity of the water molecule, which acquires a dipole moment of  0.7 D. Concerning the water dimer, based on the analysis of the atomic charges, these authors 00 found that the red shift in the 11A corresponds to photoexcitation of the hydrogen bond do1 0 nor and the blue shift in 2 A is a result of excitation of the hydrogen bond acceptor. This statement is also supported by the trends of dipole moment magnitudes of the monomers in the excimers as a function of the O⋯ O separation compared to the (H2O)2 ground-state complex. They also calculated the dipole-dipole electrostatic energy using the dipole moments of the water molecules in the dimers.d It turns out that photoexcitation alters the interactions in the excimers. The dipole-dipole interaction energy is repulsive for all O⋯ O distances in 21A0 with the accompanying hypsochromic effect; on the other hand, the bathochromic excitation 00 energy shift upon excitation to 11A agrees with the dipole-dipole interaction becoming attractive for close proximity of the water molecules.

3. Excited-state aromaticity 3.1 Aromaticity of polycyclic arenes The properties of aromatic molecules in electronic excited states is an important field of research in organic chemistry. According to N.C. Baird, the rules for aromaticity are reversed in the triplet with respect to the ground state: 4n rings become aromatic, whereas 4n + 2 become antiaromatic [26]. Baird’s rule has been extended to excited states of other multiplicities [27]. Being aromaticity a multidimensional concept, a wide variety of indicators have been defined using criteria based on electronic, geometric, or magnetic properties [28]. A number of aromaticity indexes have been defined in terms of electron delocalization [29]. Some of them have been used to analyze the aromaticity of excited-state molecules [30–33]. The vertical excitation energies of benzene and several PAHs were computed by Est
evezFregoso et al. at the CASPT2 level and the trends observed were analyzed in terms of the changes in the electron distribution [32]. Fig. 7.4 shows that there is a decreasing trend on the excitation energy values with the system size, where the angular molecules show deviations with respect to the acenes. Because of the π ! π ? nature of the electronic transitions, d

An analysis of electrostatic energy calculated with the multipole moments of the topological atoms in the environment of cation-aromatic complexes can be found in Ref. [25].

174

7. Real-space description of molecular processes in electronic excited states

FIG. 7.4 Lowest vertical singlet excitation energies of the PAHs studied. Numbering scheme: 1 benzene, 2 naphthalene, 3 anthracene, 3b phenanthrene, 4 tetracene, 4b pyrene and 5 pentacene. Available experimental (+) and other theoretical () values are also displayed. In the case of the first and second excitations, the systems joined by a solid line belong in the p and β bands, respectively, and form separate families according to Clar’s classification [34] of the electronic transitions. Taken from M. Estevez-Fregoso, J. Hernandez-Trujillo, Electron delocalization and electron density of small polycyclic aromatic hydrocarbons in singlet excited states, Phys. Chem. Chem. Phys. 18 (17) (2016) 11792–11799, https://doi.org/10.1039/C5CP06993A. Published by the PCCP Owner Societies.

illustrated by the Δρ(r) envelopes in Fig. 7.1A and B, the local properties of ρ(r) at the BCPs, which are located at the molecular plane, are not good to characterize the trends of excitation energies. Being the delocalization index a global descriptor, in the sense of it resulting from a double integration in three-dimensional (3D) space, it provides a better understanding of the excitation process in PAHs. A similarity index between the S0 and the Sn excited states, based on the delocalization index between bonded C atoms in the molecules, was defined as X Dδ ¼ ð1=NÞ dj (7.7) in terms of the Euclidean distance dj ¼

hP

N k¼1

j



δkj,Sn  δkj,S0

2 i1=2 , in the space of delocalization

indexes. In Eq. (7.7), N is the number of bonded interactions in a given PAH and δkj, Sn is the delocalization index between atoms i and j in the state Sn with a similar definition for δkj, S0. In this manner, the changes in the delocalization pattern in a molecule are quantified for the S0 !Sn excitation. Using this descriptor, linear correlations with the excitation energies were obtained for the p and β bands of the linear and angular acenes (with a correlation coefficient of 0.855 for the latter). Although more data to improve the correlations are desired, the computational cost to carry out the CASPT2 calculations of larger systems turns out to be prohibitive nowadays.

175

3. Excited-state aromaticity

FIG. 7.5 θ0 aromaticity indexes for benzene, naphthalene, anthracene, and phenanthrene in the S0 and S1 states. Adapted from M. Estevez-Fregoso, J. Hernandez-Trujillo, Electron delocalization and electron density of small polycyclic aromatic hydrocarbons in singlet excited states, Phys. Chem. Chem. Phys. 18 (17) (2016) 11792–11799, https://doi.org/10.1039/ C5CP06993A, with permission from the PCCP Owner Societies.

The following aromaticity index defined in terms of electron delocalization was used [35]: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X6 0 ðδ  δ0 Þ2 θ ¼ (7.8) i¼1 i where δ0 is the total delocalization of a C atom with the others C atoms in benzene, and δi is the total electron delocalization of a C atom in a given ring in a PAH; this definition follows closely that reported by Matta et al. [36]. The largest the value of θ0 , the less aromatic a given ring is. The aromaticity indexes for the smallest PAHs reported are shown in Fig. 7.5. In general, the aromaticity index for the rings in a PAH decreases upon excitation, in agreement with Baird’s aromaticity rules. Benzene is considered as an antiaromatic molecule in the singlet excited states. It is the molecule of the series with the largest decrease of aromaticity, which cannot be attributed to bond length alternation (the molecule retains the D6h symmetry) but to a decrease in δ(C,C0 ) upon excitation. A similar finding was also reported for benzene at the CASSCF level [30]. As an additional example, the aromaticity of the rings in anthracene is reversed, and the inner ring of phenanthrene becomes more aromatic at the expense of the outer rings. Aromaticity of the PAHs studied decreases upon excitation to their singlet excited states.

3.2 Excited-state intramolecular proton transfer of salicylideneaniline The aromatic character of a molecule using electron delocalization indexes has also been studied in photophysical and photochemical processes. In particular, Guti
errez-Arzaluz et al. carried out a study of the ESIPT tautomerism of the Schiff base salicylideneaniline (2(phenyliminomethyl)phenol) [31]. In this process, electronic excitation of the enol form is followed by tautomerization in the S1 state to yield a keto species, which further evolves into other conformers. These authors analyzed the evolution of the atomic energy contributions along the PES and other QCT descriptors. In particular, the θ0 index defined in Eq. (7.8) was used. Accordingly, the aromaticity of the phenol moiety decreases along the tautomerization process in both the ground and excited states. In addition, aromaticity is even lower in the excites state because of a decrease of θ’ in the S0 !S1 transition. Therefore, the

176

7. Real-space description of molecular processes in electronic excited states

(A)

(B)

FIG. 7.6 (A) Potential energy curves for the S0 and S1 states of hypoxanthine along the ϕ ¼ ∠C3N6C4N9 dihedral angle (in degrees). (B) Iring aromaticity index for the rings in the molecule.

difference in the reactivity for the proton transfer in the ground an excited states is associated to the decrease in the aromaticity of the molecule upon excitation.

3.3 Conical intersection of hypoxanthine In another study, Guit
errez-Arzaluz et al. analyzed the π ! π* excitation and subsequent deactivation of hypoxanthine, which can be considered as a parent model for guanine [33]. These authors used TD-DFT to analyze the evolution of ρ(r) and related properties for the photophysical process. The potential energy curves for the S0 and S1 states in terms of the distortion of the six-membered ring are shown in Fig. 7.6A. After electronic excitation to the S1 state, the deactivation takes place through a conical intersection with the ground state. Among other aromaticity descriptors, they considered the multicenter Iring index expressed in terms of closed-shell single determinant molecular orbitals [37]: X I ring ¼ 2m Si1 i2 ðΩ1 ÞSi2 i3 ðΩ2 Þ…Sim i1 ðΩm Þ (7.9) i1 …iN

where Si1 i2 ðA1 Þ , etc., are overlap integrals of molecular orbitals over the atomic basins Ω1 ,…,Ωm . For this index, larger values of Iring correspond to more aromatic rings. As in the case of the PAHs and the ESIPT of salicylideneaniline, the aromaticity of hypoxanthine decreases in the S0 !S1 excitation (see Fig. 7.6). In addition, in the approach to the conical intersection, there is loss of aromaticity for both rings in the molecule, which is more pronounced for the ground-state molecule.

4. Interacting quantum atoms The nonrelativistic electronic energy (E) of a molecular system can be expressed as E ¼ T + V en + J + V nn + V xc

(7.10)

where the terms on the right-hand side correspond to the kinetic and to the electron-nucleus (Ven), the classical electron-electron repulsion (J), nucleus-nucleus (Vnn), and exchangecorrelation (Vxc) potential energies. For a particular electronic structure method, these

4. Interacting quantum atoms

177

contributions can be evaluated from the information contained in the reduced first-order ρðr, r 0 Þ and the exchange-correlation probability distributions, where the latter canPbe obtained from the pair density as ρxc ðr 1 , r 2 Þ ¼ ρðr 1 Þρðr 2 Þ  ρ2 ðr 1 , r 2 Þ. Except for Vnn ¼ A, BZAZB/RA,B, where ZA is the nuclear charge of the atom A and RAB the internuclear distance between A and B, the other energies can be obtained from Z X Z ρðr 1 ÞZA 1 r2 ρðr , r 0 Þjr0 !r dr Ven ¼  dr 1 T ¼ 2 r1A A (7.11) Z Z Z Z 1 ρðr 1 Þρðr 2 Þ 1 ρxc ðr 1 , r 2 Þ J ¼ dr 1 r 2 Vxc ¼  dr 1 r 2 2 r12 2 r12 The starting point of the IQA method of Martı´n-Penda´s et al. consists on decomposing the above expressions in mono- and bicentric terms [7, 38]. To carry out the above procedure, two of the main results of the QTAIM theory are used. First, the expectation value of a monoelectronic operator is written as a sum of contributions evaluated on the atomic basins; and second, for a bielectronic operator the addition involves all the basin pairs. Take as an example the evaluation of T and J: X 1Z X T ¼  r2 ρðr , r 0 Þjr0 !r dr ¼ TA 2 rΩA A A Z Z X1Z X 1Z ρðr 1 Þρðr 2 Þ ρðr 1 Þρðr 2 Þ J ¼ dr 1 dr 2 + dr 1 dr 2 2 r 2 r12 12 r1 ΩA r2 ΩA r1 ΩA r 2 ΩB A A6¼B ¼

X A

JA +

1 X A,B J 2 A6¼B

Note that if the double integration of the electronic coordinates is carried out on the same basin, the term is classified as monocentric (written JA instead of JA,A). Applying the same procedure to each energy component of Eq. (7.10), as well as grouping the sums into one and twocenter terms, the electronic energy is expressed as X 1 X A,B E¼ EA E (7.12) self + 2 A6¼B int A The first one (EA self ) is called self-energy of atom A and comprises kinetic, nucleus-electron, and electron-electron contributions A A A A EA self ¼ T + V en + J + V xc

whereas the interaction (EA,B int ) between atoms A and A involves the electrostatic and exchange-correlation parts EA,B int

A,B A,B A,B ¼ V A,B + V A,B nn + V en + V ne + J xc |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} V A,B ele A,B ¼ V A,B ele + V xc

178

7. Real-space description of molecular processes in electronic excited states

A,B The V A,B ele and V xc energies can be respectively associated with the ionic (classical) and covalent contributions of the interaction. Further elaboration about the method can be found in the literature [38]. Among the main advantages of IQA is that the analysis can be performed hierarchically, starting with atoms and functional groups and ending with molecules or the total system; in other words, it is applicable to intra- and intermolecular cases. Also, as opposed to approaches based on the atomic virial theorem, the IQA partition can be applied at any point on the PES, so it can be used to study processes involving structures other than equilibrium geometries. Importantly, it does not require the introduction of arbitrary reference states to perform the analysis. These features make the use of IQA appealing for the study of excited-state processes with respect to other energy partitioning schemes such as SAPT or EDA [39, 40]. For example, in the case of SAPT it is not possible to describe in detail more than three fragments while there is no computational implementation to deal with excited states. On the other hand, defining reference states for excited-state dissociation processes, other than intermolecular complexes, can be problematic with EDA. Eqs. (7.10) and (7.11) make it clear that in order to perform IQA partitioning with a particular method, it is necessary to obtain ρðr, r 0 Þ and ρ2 ðr 1 , r 2 Þ. In terms of the molecular orbital basis (ϕi) which constitutes the Fock space of the system, these latter quantities can be expressed as X ρðr, r 0 Þ ¼ Dpq ϕp ðr Þϕq ðr 0 Þ pq

ρ2 ðr 1 , r 2 Þ ¼

X

dpqrs ϕp ðr 1 Þϕq ðr 1 Þϕr ðr 2 Þϕs ðr 2 Þ

pqrs

where Dpq and dpqrs are the first- and second-order density matrices. The electron density ρðr Þ is obtained from ρðr, r 0 Þ through r0 !r. In other words, if we can obtain the density matrices it is possible to recover ρðr Þ and ρ2 ðr 1 , r 2 Þ. Except for monodeterminantal and small CI expansions, obtaining Dpq and dpqrs is not a trivial task, it represents the major drawback in order to be able to apply the IQA partitioning scheme with a general electronic structure method. The Appendix presents a brief sketch of how to obtain these density matrices for some selected methods suitable to describe molecular processes in electronic excited states.

4.1 Applications The theoretical framework provided by IQA is useful to rationalize many photochemical or photophysical processes. To mention a few instances, it allows to analyze how the molecular energy is redistributed after an electronic transition as well as to describe which are the energetic contributions that favor the relaxation of an excited molecule. For example, using the IQA partition, Ferna´ndez-Alarco´n et al. studied the changes in the atomic and interaction terms of small molecules following light absorption [41]. In addition, Jara-Cort
es et al. performed a study for some representative excited systems, with the purpose of describing bond breaking, excimer formation, charge transfer complexes, as well as the energetic redistribution in the neighborhood of a conical intersection [17]. Other studies have analyzed the batho- and hypsochromism in the water dimer, the excited-state relaxation of H2CO, and the photodissociation of C4H8 [11, 22, 42].

179

4. Interacting quantum atoms

TABLE 7.3

IQA energy changes (eV) for the S0 !S1 electronic transitions of C2H4, CH4, H2O, and CO.

Transition C2H4 11Ag ! 11B1u 11Ag ! 11B3u Transition CH4 11A1 ! 11T2 Transition H2O 11A1 ! 11B2 Transition CO

11Σ+ ! 11Π

2ΔECself

4ΔEH self

ΔEC,C int

4ΔECH int

4ΔEC,H int

2ΔEH,H int

1.59 2.97 4.35 6.17 ΔEC self 0.64 1.18 ΔEO self 0.24 1.25 ΔEC self 1.88 1.31

1.81 2.16 1.96 1.93 4ΔEH self 5.91 6.09 2ΔEH self 0.62 0.72 ΔEO self 3.04 3.00

1.47 0.72 0.64 0.42 4ΔEC,H int 1.74 0.85 2ΔEO,H int 8.49 5.92 ΔEC,O int 13.59 13.01

1.20 0.56 0.30 0.11 4ΔV C,H xc 3.63 3.32 2ΔV O,H xc 0.74 1.56 ΔV C,O xc 1.39 1.47

0.65 0.58 0.05 0.18 4ΔV C,H ele 1.88 2.47 2ΔV O,H ele 7.75 4.36 C,O ΔV ele 14.98 14.48

0.23 0.35 0.34 0.30 P H,H0 ΔEint 1.98 2.40 ΔEH,H int 1.04 0.52

0

0

00

2ΔEH,H int

,H ΔEH int

0.09 0.04 0.17 0.10

0.02 0.12 0.14 0.10

Eabs 6.84 7.33 7.85 8.01 Eabs 10.28 10.53 Eabs 7.07 7.37 Eabs 8.67 8.70

For C2H4, the transition to the 1ππ? state (11B3u) is also included; for the other molecules, the decomposition of the interaction energy in electrostatic and correlation-exchange components is performed. The data in the first and second rows correspond to the MRCI-SD/Sadlej-pVTZ and EOMCCSD/Sadlej-pVTZ levels of theory, respectively. The EOM-CCSD values were taken from A. Ferna´ndez-Alarco´n, J.L. Casals-Sainz, J.M. Guevara-Vela, A. Costales, E. Francisco, A.M. Penda´s, T. Rocha-Rinza, Partition of electronic excitation energies: the IQA/EOM-CCSD method, Phys. Chem. Chem. Phys. 21 (25) (2019) 13428–13439, ´ . M. Penda´s, https://doi.org/10.1039/C9CP00530G and others were evaluated using the procedure indicated in J. Jara-Cort
es, J.M. Guevara-Vela, A J. Herna´ndez-Trujillo, Chemical bonding in excited states: energy transfer and charge redistribution from a real space perspective, J. Comput. Chem. 38 (13) (2017) 957–970.

4.1.1 Electronic absorption The IQA approach allows to analyze how the molecular energy is redistributed internally after photon absorption. Table 7.3 presents the changes in the self-energies and interaction terms following the first electronic transition of a group of small molecules including CO, H2O, CH4, and C2H4. As an example, from the data for CO it can be seen that following the 11Σ+ ! 11Π process a decrease in the self-energies of C and O takes place, while ΔEC,O int is positive. Simultaneously, the atomic charge of C decreases by 0.30 e, going from 1.17 to 0.87 e, with opposite changes for oxygen. These changes result in an increase in the magnitude of V Cen , thus explaining the negative values in ΔECself ; on the other hand, for oxygen the electron-electron repulsion decreases, accounting for EO self < 0. Moreover, the positive changes C,O in ΔEC,O int and ΔV ele have their origin in the decrease of the interatomic nucleus-electron attraction. Another important aspect that can be inferred from the data is that there are certain regularities according to the nature of the transition. For example, for methane it can be seen that the electronic excitation has a greater effect on the hydrogen contributions, mainly in ΔEH self H,H0 and ΣΔEint . These results agree qualitatively with what is expected in terms of the MO analysis, given that the valence (σ) and Rydberg orbitals (R3s) have very large contributions from hydrogen atoms and this transition is ascribed as σ !R3s. The same trend appears for the 11Ag ! 11B1u transition in ethylene, i.e., the changes in hydrogen contributions are comparable in magnitude to those of the carbons, in agreement with the Rydberg 3s nature of the excited state. This is in contrast to 11Ag ! 11B1u for the same molecule, in which the largest changes

180

7. Real-space description of molecular processes in electronic excited states

FIG. 7.7 Vertical electronic transitions in H2O and (H2O)2 molecules, as well as their decomposition into group and interaction contributions. The energies are from EOM-CCSD/6-311G++(2d,2p) calculations, and the sketch was made using the data from the supplementary material of reference [22].

take place in the self-energies of the carbons, in line with the valence nature ascribed to the π ! π ? transition. The batho- and hypsochromic double shift for the excitation of (H2O)2 was explained in Section 2.2 in terms of the classical electrostatic interactions between the monomers in the complex. The IQA decomposition has also been used to rationalize the process [22]. Briefly, 00 the two 11B2 equivalent states of H2O +H2O (in the dissociation limit) are split into the 11A and 21A0 levels of (H2O)2 upon approach of the water moieties, with a consequent shift of the excitation energy from 7.38 to 7.09 and 7.54 eV, respectively. Fig. 7.7 shows a summary of the 00 main changes in the group and interaction contributions. The trends for 11A0 ! 11A and 11A0 ! 21A0 are very similar, involving positive water deformation energies and a negative 00 water-water interaction term. However, the major changes in the 11A state take place on 1 the hydrogen bonding donor water molecule (EW def ¼ 8:29 eV) with minor changes on the 1 0 2 HB acceptor (EW def ¼ 0:62 eV), whereas the contributions in the 2 A state are reversed, show-

W2 1 ing values of EW def ¼ 0:44 eV and Edef ¼ 7:71 eV. A quick comparison of these data with those obtained for the 11A1 ! 11B2 transition for H2O (Table 7.3) suggests that the IQA partition allows to identify locally the fragment where the excitation occurs. Moreover, the analysis of the electrostatic and covalent intermolecular contributions as a function of the O–O dis1 ,W2 tance shows that the EW component is the one that modulates the solvatochromic shift. ele

4.1.2 Bond formation in excited states The H2 molecule can be taken as an archetype system in order to study the chemical bond formation in ground and excited states [17]. Fig. 7.8A shows the potential energy curves of the X1Σ+g , B1Σ+u, E,F1Σ+g , and b3Σ+u states of H2 as a function of internuclear distance (R); Fig. 7.8B–D shows the evolution of the respective IQA energy components. The deformation energy (Fig. H H 7.8B), defined as EH def ðRÞ ¼ Eself ðRÞ  Eself ðR ¼ ∞Þ, accounts for the atomic redistribution that takes place upon approach of the fragments. The information presented in Fig. 7.8 allows to quantify the energetic components that favor the formation of the molecule for the different states. For example, the bond formation in the X1Σ+g state originates mainly from the V H,H’ xc contribution, which is counterbalanced by the increase on the self-energies; as for the

181

4. Interacting quantum atoms

(A)

(B)

(C)

(D)

FIG. 7.8 (A) Potential energy curves for the ground and the lowest excited states of H2. IQA electronic energy partition for the X1Σ+g, B1Σ+u, E,F1Σ+g, and b3Σ+u states of H2. (B) Deformation energies, (C) electrostatic and exchange-correlation 0 contributions to EH,H int , and (D) components of the atomic energy. The data are from MRCI-SD/d-aug-cc-pVDZ calculations ´ .M. Penda´s, J. Herna´ndez-Trujillo, Chemical bonding in excited states: and were taken from J. Jara-Cort
es, J.M. Guevara-Vela, A energy transfer and charge redistribution from a real space perspective, J. Comput. Chem. 38 (13) (2017) 957–970.

formation of any typical covalent bond, it does not involve a charge transfer. On the other H hand, for b3Σ+u the increase in EH def (mainly due to T ) is not compensated by a stabilization in the interaction part, thus explaining the repulsive curve. Moreover, the analysis for B1Σ+u and E,F1Σ+g shows that in these cases the interaction is long range in nature. For example, ˚ for the singlet excited states have the same value as those of note how the Edef values at R ¼ 6 A 1 + X Σg at the minima of the PES. Also, the V H,H’ contributions are more negative in the excited xc state for almost the entire range of internuclear distances, which can be explained in terms of ˚ , delocalization indices increased electronic delocalization. In fact, between 2.5 and 3.5 A greater than 1.5 are observed, which are atypical values for two-electron two-center bonds and can be explained in terms of the H+ , H and H, H+ resonant structures by analyzing the electron distribution functions [18]. Further quantitative information can be obtained by analyzing the correlation diagram for the molecule in terms of the data of Fig. 7.8, making possible to relate the H+H dissociation processes and the electronic states of H2 with the united atom limit of He. The X1Σ+g and b3Σ+u states connect the 1s2 and 1s2s configurations of He with the dissociation into two hydrogens H(1s)+H(1s), whereas B1Σ+u and E,F1Σ+g dissociate to H(1s)+H(2l), so that the analysis is more elaborated in the last case. The correlation diagram shows the connection between B1Σ+u, the 1s2pσ configuration of He, and the dissociation in H+ +H. Similarly, the E state should link He-1s2s to H(1s)+H(2l), and the F state He-2p2 to H+ +H. However, all the involved states are

182

7. Real-space description of molecular processes in electronic excited states

(A)

(B)

(C)

FIG. 7.9 (A) Potential energy curves for the 11Σ+ and 21Σ+ states of LiF. Evolution of the (B) interaction components and the (C) atomic self-energies. The data were from MRCI-SD/d-aug-cc-pVDZ calculations and were taken from J. Jara´ .M. Penda´s, J. Herna´ndez-Trujillo, Chemical bonding in excited states: energy transfer and charge Cort
es, J.M. Guevara-Vela, A redistribution from a real space perspective, J. Comput. Chem. 38 (13) (2017) 957–970.

Σ singlets, and also, the energy of H+ +H is between H(1s)+H(4l) and H(1s)+H(5l), so due to the noncrossing rule the dissociation in the ionic state does not take place. In other words, a change in the nature of B and F must occur along the potential energy curve, although nearequilibrium geometries both B and F must show ionic character. For example, the electron˚ in the excited singlets coincides with the value of electron repulsion (Vee ¼ J + Vxc) at R ¼ 3 A  Vee  192.6 kcal/mol for H , which provides evidence that in that region the states are clearly ionic. This information agrees with changes on ΘRR, Eq. (7.5), ranging from  13.29 in the outer minimum (F) to  52.07 in the inner one (E), evidencing a transition from an ionic to a Rydberg state. Also, note how the change in the electronic nature is reflected in the variation 1 + of V H,H xc , and how this component explains the appearance of the double well for E,F Σg . 4.1.3 Avoided crossing of states in LiF The formation of the LiF molecule is a model for charge transfer processes and for the description of the harpoon mechanism. During the approach of the atoms, there is a narrow ˚ , where a sudden charge transfer internuclear distance interval centered at ca. Rc ¼ 6.64 A of nearly one electron from the Li to the F atom takes place because of an avoided crossing of the two lowest singlet Σ electronic states [17, 43]. Near the ground-state equilibrium geometry, the 11Σ+ and 21Σ+ states are named ionic and covalent states, whereas at the dissociation limit the trends are inverted. The potential energy curves as well as the respective changes of the IQA energetic components are shown in Fig. 7.9. Note how the self-energies of the Li and F atoms remain relatively constant in the approach, and how the energetic trends reverse from Li–F to Li+F in the avoided crossing region. For instance, if a narrow region around RC was removed, the neutral Li-F or ionic Li+F species would still be perceived as noninteracting, except for their V Li,F ele interaction term. The evolution of electrostatic contribution for 11Σ+ evidences the drastic change in the nature of the electronic state, because V Li,F ele changes from 0, expected for a neutral compound, to nearly 50 kcal/mol for an ionic com˚ . An interesting aspect is the fact that all the curves for the plex in a short interval of  0.3A IQA energy components cross, while those for total energy do not. 4.1.4 Conical intersections in C2H4 The IQA scheme has also been used to analyze the energy redistribution that takes place in the neighborhood of conical intersections, regions on the PES of high transition probabilities

183

4. Interacting quantum atoms

(A)

(D)

(B)

(E)

(C)

(F)

FIG. 7.10 IQA energy contributions for the S0 and S1 states of C2H4 in the neighborhood of the conical intersection of the twisted-pyramidalized geometry. (A) and (B): total and kinetic energies; (C) and (D): self-energies of the ! ! ! ! methylene fragments along h and g ; and (E) and (F) interfragment interaction energies along h and g . The data were ´ from CASSCF/d-aug-cc-pVDZ calculations and were taken from J. Jara-Cort
es, J.M. Guevara-Vela, A.M. Penda´s, J. Herna´ndezTrujillo, Chemical bonding in excited states: energy transfer and charge redistribution from a real space perspective, J. Comput. Chem. 38 (13) (2017) 957–970.

between electronic states of vital importance in photochemistry. Particularly, the analysis has been performed on the g-h plane of the crossing point associated with the twistedpyramidalized geometry of ethylene [17]. It is known that subsequent to the π ! π ? vertical transition of C2H4, the minimum energy path in the excited state leads directly from the Franck-Condon region to the S1/S0 seam space. Therefore, the energetic description in the neighborhood of the surface crossing is relevant to understand the energetic changes after the internal conversion process. Fig. 7.10 shows the changes in!E, kinetic energies, and ! ! IQA contributions of the methylene fragments along the! g and h directions, where g is the energy gradient difference between the two states and h is the nonadiabatic coupling vec! ! tor. Displacements along g and h are dominated by the RCC distance and the twisting angle α measured with respect to the nonpyramidalized methylene fragment F2 at the conical intersection point, respectively. By labeling the contributions in terms of the electronic states (S0 or S1), notice how all the energies are discontinuous at the crossing because the ground and excited states change in nature after passing the conical intersection along the RCC and α coordinates. Following the contributions in terms of the nature of the state (π 2 or ! ? ππ ), it can be clearly observed that along h all the energetic contributions (EF1 , EF2 , and ! EF1 ,F2 ) favor the surface crossing, whereas along g only the energies of the F1 methylene fragment modulate the approach of ES0 and ES1. In other words, it is shown that the IQA analysis allows the identification of fragment energies and interactions that favor surface crossing.

184

7. Real-space description of molecular processes in electronic excited states

4.1.5 Photodissociation of cyclobutane The C4H8 ! 2C2H4 decomposition is the classic example of a [2+2] cycloreversion reaction according to the Woodward-Hoffmann rules. Unlike the Diels-Alder [4+2] cycloreversion, the concerted path is thermally forbidden but photochemically allowed. The IQA analysis of the concerted mechanism in ground and excited states provides information concerning why the energy barrier is so large in S0, while the process proceeds favorably (small barrier) in S1. Fig. 7.11A and B shows the intrinsic reaction coordinate for the ground and excited concerted dissociation of C4H8. An energy barrier of 4.08 eV is observed for S0, of comparable magnitude to the absorption energy of a small conjugated molecule. Going from C4H8 to 2 C2H4, there is the breaking of two C–C single bonds, the formation of two double bonds with a formal change in hybridization from sp3 to sp2 in the four carbon atoms (Fig. 7.11B). In addition, Fig. 7.11C shows that the energy required for the concerted breaking of C–C single bonds exceeds the stabilization associated with the formation of two double bonds as well as the change in hybridization, thus explaining the appearance of the barrier. The energy landscape changes considerably in the first excited state. Following the S0 !S1 electronic transition (10.08 eV), the minimum energy path (MEP) proceeds until a minimum is reached; subsequently, a small barrier must be overcome in order to access the surface crossing with the ground state. In the region near the transition state, a change in nature of the electronic state occurs, passing from Rydberg-like to ππ ?. In the middle part of the MEP the analysis is very similar to that of the S0, i.e., the stabilization gained by the formation

(A)

(B)

(C)

(D)

FIG. 7.11 (A) Intrinsic reaction coordinate (IRC) for the ground-state concerted dissociation of cyclobutane. (B) Energies of S0 and S1 along the minimum energy path of S1. Changes on the self-energies and interaction energies (C) along the IRC for S0 and (D) the MEP for S1. The labels “carbons” and “hydrogens” comprise the sum of the deformation energies of the C or H atoms, respectively; “Remainder Eint” is the sum of the C–H and H–H interaction energies. The data were calculated at the CASPT2/def2-SVP level of theory. The data were taken from J. Jara-Cort
es, E. LealSa´nchez, E. Francisco, J.A. P
erez-Pimienta, A. Martı´n Penda´s, J. Herna´ndez-Trujillo, Implementation of the interacting quantum atom energy decomposition using the CASPT2 method, Phys. Chem. Chem. Phys. 23 (48) (2021) 27508–27519, https://doi. org/10.1039/D1CP02837E.

Appendix. One- and two-electron densities matrices

185

of C¼C bonds is less than the energy required to break two C–C bonds. However, near the transition state, a considerable decrease in hydrogen self-energies and from the C–H interactions is observed, so that these contributions drive the MEP of S1 toward the surface crossing, and therefore promote the cycloreversion of the cyclobutane moiety. In other words, IQA analysis reveals that the energy contributions “Hydrogens” and “Remaining Eint” are relevant to explain the deactivation of the excited state. This kind of information is not recovered from the analysis of the correlation diagram, which only focuses on the carbon-centered molecular orbitals and completely ignores the contributions of the hydrogens.

5. Conclusions The study and characterization of molecular processes in electronic excited states has become an area of growing interest, given its importance in a variety of fields ranging from materials design to the study of biological systems. Due to the complex nature of these processes, it is essential to have a set of interpretative tools to complement the information obtained from the PES calculations. The above combination makes it possible to establish a connection between the molecular information and chemical concepts based on bonding models, which ultimately allow for the information obtained to be rationalized or systematized in the case of excited-state processes. The QCT toolbox involves the analysis of scalar fields, their integration over atomic basins, or energy decomposition schemes along photophysical/photochemical deactivation paths. It was illustrated how the visualization of the one-electron density differences and the partition of multipole moments into atomic contributions enables to analyze the charge redistribution that takes place after an electronic transition. For example, the Δρ(r) isosurfaces allow the identification of the spatial regions where the spatial redistribution of charge density occurs, while multipole moments quantify the extent of charge density within atomic regions. Electron delocalization indexes provide additional information from the exchange-correlation density. In addition, the IQA energy partition complements this type of analysis by tracking the energy redistribution involved. It was illustrated how these tools provide information that complements the traditional analysis in terms of molecular orbitals. There is scarce information on other QCT molecular descriptors that could provide valuable information for the understanding of excited-state processes. Among them, further work on nonstationary electron densities, the ELF, or the Ehrenfest forces should be carried out. In addition, methods to extract information from TD-DFT could trigger the use of QCT in the study of excited states. This contribution can serve to encourage further exploration of these and other scalar and vector fields for the understanding of molecular processes in electronic excited states.

Appendix. One- and two-electron densities matrices Obtaining the one- and two-electron density matrices represents the main shortcoming to be able to apply the IQA partitioning scheme with a general electronic structure method. The following subsections present a brief summary of how to obtain these quantities in selected approaches suitable to describe molecular processes in electronic excited states.

186

7. Real-space description of molecular processes in electronic excited states

A.1 Configuration interaction methods The configuration interaction methods express the system wave function (Ψ) as a linear combination of Slater determinants (DET), configurations (CFG), or configuration state functions (CSF) X jΨi ¼ ci jϕi i i

The variational parameters are the {ci} coefficients, and in some cases, the coefficients of the expansion of the basis functions that compose the molecular orbitals. At convergence, the electronic energy of a molecular system is given by X 1X ^ E ¼ hΨjHjΨi ¼ Dpq hpq + dpqrs gpqrs (A.1) 2 pqrs pq where hpq and gpqrs are the mono and bielectronic integrals in the MO basis, and Dpq and dpqrs are the elements of the density matrices. The latter can be obtained from the expectation values of the one and two particle excitation operators X X ^ pq jΨi ¼ Dpq ¼ hΨjE ci cj hψ j ja{pσ aqσ jψ i i (A.2) σ¼fα,βg i,j

dpqrs ¼ hΨj^epqrs jΨi ¼

X

^ pq E ^ rs  δqr E ^ ps jψ i i ci cj hψ j jE

(A.3)

i,j

which involve the product of the CI vector with the one- and two-electron coupling coefficients. X ji γ pq ¼ hψ j ja{pσ aqσ jψ i i σ¼fα,βg ji Γpqrs

^ pq E ^ rs  δqr E ^ ps jψ i i ¼ hψ j jE

Since most of the programs provide the information of the coefficients ci at the end of the calji ji culation, the difficult part corresponds to acquiring γ pq and Γpqrs. They are implicit in the electronic structure programs, but in most cases, they are not easily accessible to the user in a useful form. The usual path is recalculating them, and subsequently, perform the products involved in Eqs. (A.2) and (A.3) to obtain the elements of Dpq and dpqrs. Explicit formulas for their evaluation have been presented elsewhere [44–46]. An efficient procedure relies on an indexing scheme for the determinants (CSF or CFG), the decomposition of the deterji ji minants in alpha and beta chains, and the obtention of Γpqrs from γ pq through the introduction of the resolution to the identity.

A.2 MRCI-SD The density matrices for the MRCI-SD method can be evaluated directly from Eqs. (A.2) and (A.3). However, in order to decrease the number of variational parameters, in most of

Appendix. One- and two-electron densities matrices

187

programs the wave function is expressed using a contracted basis. For example, the following expansion is used for jΨi in the MRCI-SD implementation of molpro [47]: X XX X X X ijp jΨi ¼ cI jΨI i+ cSa jΨaS i+ cab jΨab ijp i I

S

a

ij

p

ab

where jΨIi and jΨaS i are CSFs spanning the internal space generated by single excitations, and jΨab ijp i are contracted bielectronic functions defined from ^ ij,kl E jΨab ijp i

^ ij E ^ kl  δjk E ^ ij ¼E

 1 ^ ^ bi,aj jΨ0 i ¼ Eai,bj + pE 2

in which p ¼ {1,1} and jΨ0i is the zero-order wave function (e.g., CASSCF); in turn, jΨ0i is written as a linear combination of CSF. Other implementations of MRCI are fully contracted, and therefore, use only bielectronic functions instead CSF for the internal and single external ji ji spaces [48]. The main problem for obtaining γ pq and Γpqrs is that the jΨab ijp i functions are not orthonormal 1 ðpÞ cd hΨab ijp jΨklq i ¼ δpq ðδac δbd + pδad δbc ÞSij,kl 2 ðpÞ ^ ik,jl + pE ^ il,jk jΨ0 i Sij,kl ¼ hΨ0 jE Therefore, the evaluation of coupling coefficients on this basis becomes relatively complicated. The evaluation of the density matrices can be facilitated by first expressing the wave function in terms of Slater determinants. In such way, it is possible to use conventional CI techniques without presenting any major difficulties beyond the size of the expansion. To accomplish this, it is necessary to transform the basis functions of the jΨIi, jΨaS i, and jΨ0i spaces from CSF to DET, for example, using the genealogical coupling scheme, in which the transformation coefficients are expressed as products of the Clebsch-Gordan coefficients. To uncontract jΨab ijp i, the procedure involves multiplying the coefficients of normalization and ijp cab by those of the determinants in jΨ0i, and finally, a summation is performed over unique determinants [47]. The evaluation of the fully uncontracted jΨi, as well as the as subsequent obtaining of Dpq and dpqrs from Eqs. (A.2) and (A.3) has been carried out previously in order to perform the IQA decomposition [17].

A.3 CASPT2 One of the main problems with perturbative approaches is that although it is possible to obtain Dpq and dpqrs from Eqs. (A.2) and (A.3) using the expansion jΨi  jΨ0 i+jΨ1 i+⋯, the kinetic and potential components obtained through them do not allow to recover the energy obtained from the electronic structure calculation. One way to avoid the above problem is to rewrite the energy expression of the method in the same form as Eq. (A.1), and from this, define effective density matrices. This procedure was first used to perform IQA partitioning with the MP2 and CCSD methods [49, 50], an approach that has been extended to CASPT2 [42].

188

7. Real-space description of molecular processes in electronic excited states

The CASPT2 energy for a specific electronic state of an atomic or molecular system is given by [51] ^ 0 i+hΨ0 jHjΨ ^ 1 i+hΨ1 jHjΨ ^ 0 i+hΨ1 jH ^ 0  E0 jΨ1 i E ¼ hΨ0 jHjΨ

(A.4)

where jΨ i and jΨ i, respectively, correspond to the CASSCF reference and the first-order ^ 0 jΨ0 i, and H ^ 0 is an effecperturbative correction to the wave function. In addition, E0 ¼ hΨ0 jH tive one-electron operator 0

1

^ f^Q ^ ^ 0 ¼ P^f^P^ + Q H X ^ pq f^ ¼ f pq E pq

f pq

¼ hpq +

X rs

D0,0 pq ðgpqrs  gprqs Þ

^ ¼ 1  P^ are projectors onto the CASSCF wave function and its orwhereas P^ ¼ j0ih0j and Q thogonal complement, respectively. Note that the CASSCF one-electron density matrix (D0,0 pq ) ^ 0 and H, ^ taking into account the peris involved in fpq. Introducing the explicit definitions of H mutational symmetry of gpqrs and grouping the result in mono and bielectronic contributions, it is possible to rewrite Eq. (A.4) as X 1X 0 D0pq hpq + d g E¼ 2 pqrs pqrs pqrs pq Let M and N be 0 or 1 so that jΨMi or jΨNi correspond to the reference function or to the firstorder correction. Using the notation ^ pq jΨN i DM,N ¼ hΨM jE pq M ¼ hΨ j^epqrs jΨN i dM,N pqrs D0pq and d0pqrs are obtained as 1,1 1,0 0,1 D0pq ¼ ð1  hΨ1 jΨ1 iÞD0,0 pq + Dpq + Dpq + Dpq 0,1 1,0 0,0 1,1 d0pqrs ¼ d0,0 pqrs + dpqrs + dpqrs + 2Drs Dpq 1,1 1 1 0,0 0,0 1 1 0,0 0,0 D0,0 qs Dpr  2hΨ jΨ iDrs Dpq + hΨ jΨ iDqs Dpr

As in the case of MRCI, the CASPT2 perturbative correction is usually expressed using contracted bielectronic functions, but can be expressed in terms of DETs in analogy to the MRCI procedure (see the Section A.2).

A.4 EOM-CCSD The Coupled cluster wave function can be written as [52] ^

jΨCC i ¼ eT j0i

189

Acknowledgments

where j0i is the Hatree-Fock determinant, and at the single and doubles truncation level (CCSD), the T^ operator is given by c c1 + T T^  T X 2 X w { { { ¼ ti aw ai + tw,x i,j aw ai ax aj w,i

w,x,i,j

in which the i, j labels correspond to the reference space (occupied in j0i) and w, x to the virtual space. Collectively, the t coefficients are referred to as amplitudes. Once the t’s are known from the solution of the CCSD equations hw,x i,j j

^ N eT j0i ¼ 0 eT H

^N H

^  h0jHj0i ^ ¼H

^

^

{ { where hw,x i,j j ¼ h0jaw ai ax aj , the correlation energy can be obtained from ^

^

T T ^ ECCSD corr ¼ h0je H N e j0i

The equation of motion is an extension of CC in order to calculate excited states. The EOMCCSD wave function for the kth excited state is expressed as ^ jΨCC i jki ¼ R Xk { ^ k  r0 + R rw i aw ai i,w

^

^ j0i ¼ eT R Xk { { + rw,x i,j aw ai ax aj wn Þdxi>n , n B

(8.3)

Qn 0 0 with 1nA 1A n ¼ i¼1 ωA ðxi ÞωA ðxi Þ, xin ¼ x1 …xn and xi>n ¼ xn+1 …xN . The subsystem A is now necessarily a mixed system with N + 1 possible values of its number of electrons. Each sector contains its own density matrix ρnA, which can be diagonalized to obtain its spectrum. For every n, ρnA is normalized to pA(n), the probability of finding n electrons in A and N  n in B [36], Z Z n n 0 TrρA ¼ ρA ðxin ; xin Þjx0 !xi dxin ¼ ρnA ðxin Þdxin ¼ pA ðnÞ: (8.4) i

1

Hence, we can define normalizedPdensity operators for subsystem A as  ρnA ¼ ½pA ðnÞ ρnA A n that satisfy Tr ρnA ¼ 1 and ρA ¼ N n¼0 p ðnÞ ρA . It is now time to examine how these objects can be actually computed. We start by examining SDWs in order to introduce a set of concepts that will be further used in more general multideterminant situations.

2.1 Sector density operators for SDW Let us assume an SDW jΨi written in terms of orthonormal spin-orbitals ui:    u1 ðx1 Þ … uN ðx1 Þ    ⋱ ⋮ : jΨi ¼ ðN!Þ1=2  ⋮  u1 ðxN Þ … uN ðxN Þ  Using the Laplace expansion, jΨi can be expanded in terms of its first n rows

(8.5)

196

8. Open quantum systems, electron distribution functions

jΨi ¼ ðN!Þ1=2

X detjuk1 ðx1 Þ…ukn ðxn Þj  detðnjkÞ,

(8.6)

k

where k denotes the ordered set k1 < k2 < … < kn, det(njk) that depends only on the coordinates xn+1, …, xN, is the determinant obtained from detju1 ðx1 Þ…uN ðxN Þj by deleting the first n rows and columns k1, k2, …, kn. With this Ψ, the expression for ρnA (Eq. 8.3) can be obtained by using Eq. (47) of Ref. [40]. The result is X ee ρnA ¼ jUk i hUl j detjSB ðkj lÞj, where (8.7) k,l

jUk i ¼ ðn!Þ1=2 juk1 ðx1 Þ…ukn ðxn Þj,

(8.8)

hUl j ¼ ðn!Þ1=2 jul1 ðx01 Þ…uln ðx0n Þj,

(8.9)

we have assumed real spin-orbitals, S is the (N  N) atomic overlap matrix (AOM) in B, deee lÞ the (N  n)  (N  n) array obtained from SB by selecting the fined as SBij ¼ hui juj iB, and SB ðkj e rows kand the columns e l, complementary of k and l, respectively. Each jUki is normalized to 1 B

in R3 , hUk jUk iR3 ¼ 1. However, the normalization in A is hUkjUkiA ¼ detjSA(kjk)j. It is then convenient to define jψ k i ¼ jUk idetjSA ðkjkÞj1=2 ,

such that hψ kjψ kiA ¼ 1. Then, Eq. (8.7) becomes X jψ k i hψ l j Pkl , ρnA ¼

(8.10)

(8.11)

k,l

where ee Pkl ¼ detjSA ðkjkÞj1=2 detjSA ðljlÞj1=2 detjSB ðkj lÞj:

(8.12)

2.2 Sector density operators for multideterminant wave functions To generalize Eq. (8.7) to multideterminant wave functions (MDW), we first expand the size of the basis of spin-orbitals ui to a size m > N. Then, jΨi ¼

XM r¼1

Cr ðN!Þ1=2 jψ r i

(8.13)

where jψ r i ¼ detjur1 …urN j, and ður1 …urN Þ denotes a N-elements subset of the ui’s. Then, we can write X   ρnA ðxin ; x0in Þ ¼ Cr C*s ρnA rs (8.14) r,s

X  n ee ρA rs ¼ jUrk i hUsl j detjSBrs ðkj lÞj, k,l

with

(8.15)

2. Open quantum systems

197

jUrk i ¼ ðn!Þ1=2 jurk1 ðx1 Þ…urkn ðxn Þj,

(8.16)

hUsl j ¼ ðn!Þ1=2 jusl1 ðx01 Þ…usln ðx0n Þj:

(8.17)

In the above equations, k and l are n-elements subsets of ðr1 …rN Þ and ðs1 …sN Þ, respectively, ke ee lÞ the AOM in and e l their corresponding (N  n)-elements complementary subsets, and SBrs ðkj r B between the spin-orbitals not contained in jUk i and the spin-orbitals not contained in hUrl j.

2.3 Using orthogonal spin-orbitals in A, B, and R3 Up to now, we have assumed that the ui’s are orthonormal in R3, hui juj iR3 ¼ δij, but not in A B or B, i.e., SA ij ¼ hui juj iA 6¼ 0 and Sij 6¼ 0 for i6¼j. However, it is possible to subject them to a unitary transformation, by diagonalizing SA (or SB), such that the transformed spin-orbitals are orthonormal in R3 , and orthogonal in A and B [41]. Let us consider that our initial wave function is initially expressed in an orthonormal basis ϕi (i ¼ 1, m), hϕi jϕj iR3 ¼ δij. Let SA the (m  m) Hermitian matrix defined by SA ij ¼ hϕi jϕj iA or, in A matrix form, S ¼ hϕjϕiA. Now, we define u ¼ ϕU, where U is the unitary matrix that diagonalizes SA, SAU ¼ Us. Defining  S ¼ hujuiA , we have e A ¼ hujui ¼ U{ hϕjϕi U ¼ U{ SA U ¼ diagðsA Þ: S A A

(8.18)

Given that SB ¼ hϕjϕiB ¼ diag(I) SA, U also diagonalizes SB with diag(sB) ¼ diag(I sA) being their eigenvalues. The basis u is thus the same no matter one diagonalizes SA or SB, and it is simultaneously orthonormal in R3 and orthogonal in A and B. Now, let Urj be a (N  N) matrix (N  m) formed by rows r1 …rN ¼ r and columns j1 …jN ¼ j of U, and jΨi an N-electron wave function given by XM jΨi ¼ C ðN!Þ1=2 jψ r i, (8.19) r¼1 r where jψ r i ¼ detjϕr1 ð1Þϕr2 ð2Þ…ϕrN ðNÞj. Using ϕ ¼ uU† and elementary properties of the determinants, it can be shown that X jψ r i ¼ detðUrj Þ detjuj1 ð1Þ…ujN ðNÞj: (8.20) j

where j runs over the m!/[N!(m  N)!] possible ordered subsets that can be extracted from the basis u. Using Eq. (8.20) in Eq. (8.19) we have X jΨi ¼ Dj ðN!Þ1=2 detjuj1 ð1Þ…ujN ðNÞj: (8.21) j

where Dj ¼

XM r¼1

Cr detðUrj Þ:

(8.22)

An interesting point related to Eq. (8.20) is the following. In general, Urj is not unitary, so det(Urj)6¼1. However, for an SDW m ¼ N, j ¼ r ¼ ð1,2,…,NÞ, and Urj coincides with U, which is unitary. Consequently, in this case, det(Urj) ¼ 1 and ψ r ¼ detju1 ð1Þ…uN ðNÞj. This shows the

198

8. Open quantum systems, electron distribution functions

well-known invariance of an Slater determinant under an unitary transformation of all of its spin-orbitals. Thus, without loss of generality, we can assume that the ui’s actually satisfy [41] SA ij ¼ hui juj iA ¼ si δij , SBij ¼ hui juj iB ¼ ð1  si Þδij

(8.23) and

B SA ij + Sij ¼ hui juj iR3 ¼ δij :

(8.24) (8.25)

ee Assuming this, each matrix SB ðkj lÞin Eq. (8.7) is diagonal. Even more, if any li is different from B ee ee lÞj will be zero. On the ki, S ðkjlÞ will contain one or more zeros in the diagonal and detjSB ðkj B ee e e contrary, when k ¼ l, detjS ðkjlÞjwill be equal to the product of N  n AOM elements in B, 1  si, with i ¼ ke1 ,…,keNn . 2.3.1 Single-determinant wave functions

For an SDW, ρnA is given by Eq. (8.7) or (8.11). However, in this case, ke ¼ e l ,k ¼ l, and ke 6¼ e l n ,k6¼l. Taking into account the comments above, ρA for an SDW becomes ρnA ¼

X

jUk ihUk j

k

0 Y

ð1  si Þ,

(8.26)

i

Q where the prime (’) in means that only terms with i ¼ ke1 ,…,keNn have to be included in the product. The physical meaning of this product is the following. Assuming a single determie Qi0(1  si) represents the probability of finding in B nant built with the spin-orbitals given by k, the N  n electrons described with u~k1 , …, u~kNn If one prefers to write ρnA as in Eq. (8.11), we have X jψ k i hψ k j Pk : (8.27) ρnA ¼ k

Q e For each k, Pk is the product of N factors Pk ¼ N i pi, where pi ¼ si if i  k and pi ¼ 1  si if i  k. 3 For instance, for a system with N ¼ 4 and n ¼ 3, ρA takes the form ρ3A ¼ jψ 123 i hψ 123 j s1 s2 s3 ð1  s4 Þ+jψ 124 i hψ 124 j s1 s2 s4 ð1  s3 Þ + jψ 134 i hψ 134 j s1 s3 s4 ð1  s2 Þ+jψ 234 i hψ 234 j s2 s3 s4 ð1  s1 Þ,

(8.28) (8.29)

where jψ ijki is the normalized (in A) wave function given by 1 jψ ijk i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jui ðx1 Þuj ðx2 Þuk ðx2 Þj: 3!si sj sk P Since Trjψ ki hψ kj ¼ 1, we have TrρnA ¼ k Pk , and [4] X k

N Y pi ¼ pA ðnÞ,

^ Pk ¼ S

i

(8.30)

(8.31)

199

2. Open quantum systems

^ is a symmetrizing operator that takes into account electron indistinguishability. where S Q Moreover, when Eqs. (8.23)–(8.25) are valid, SA(kjk) is diagonal and detjSA(kjk)j ¼ iksi. In this case, apart from jψ ki, defined in Eq. (8.10), we can define spin-orbitals ψ i as 1=2

ψ i ¼ si

ui ,

(8.32)

which form an orthonormal set in A, hψ ijψ jiA ¼ δij. In terms of the ψ i’s, each normalized (in A) wave function jψ ki can be written in the standard way: 1 jψ k i ¼ pffiffiffiffi jψ k1 ðx1 Þψ k2 ðx2 Þ…ψ kn ðxn Þj n!

(8.33)

Due also to Eqs. (8.23)–(8.25) the property hψ kjψ liA ¼ δkl is satisfied, and consequently X ρnA ρnA ¼ jψ k i hψ k j P2k : (8.34) k

Tre ρAn

Since Pkk  1 and ¼ 1, we have Tr½e ρAn ρeAn   1. Let us analyze some particular cases of ρnA, starting with n ¼Q N. In this case, there is a unique term in Eq. (8.26) (k ¼ 1, k2 ¼ 2, …, kn ¼ N), the product i0 (1  si) is absent, and ρN A ¼ Ψ* ðx01 ,…,x0N ÞΨðx1 ,…,xN Þ. Using again the properties (8.23)–(8.25) Z

TrρN A ¼

jΨðx1 ,…,xN Þj2 dxiN ¼ pA ðNÞ ¼ A

Y si :

(8.35)

i 1=2

e A ¼ ½pA ðNÞ e A can be defined as Ψ Ψ From this equation, a normalized wave function in A, Ψ e A ihΨ e A j ¼ 1. When the ui’s do not meet the conditions (8.23)–(8.25), the first equalthat fulfills TrjΨ ity in Eq. (8.35) is still valid. In this case, pA ðNÞ ¼ detjSA ij j. When n ¼ N  1, the sum in Eq. (8.26) contains N terms, corresponding to the N possibilities of eliminating a single ui in constructing Uk. For instance, for N ¼ 3, the diagonal element of ρN1 results A or ρ2A ðx1 ,x2 Þ ¼ jUx1 ,x2 j2 ð1  s3 Þ+jUx1 ,x3 j2 ð1  s2 Þ+jUx2 ,x3 j2 ð1  s1 Þ   X  Ui ðx1 Þ Uj ðx1 Þ 2 2   ρA ðx1 , x2 Þ ¼ with k 6¼ i ^ k 6¼ j:  Ui ðx2 Þ Uj ðx2 Þ  ð1  sk Þ,

(8.36) (8.37)

im !xi>m dxi>m , (8.43) ρnA,m ðxim ; x0im Þ ¼ ðn  mÞ! R with the spinless mth order RDM given by ρnA,m ðr im ; r 0im Þ ¼ ρnA,m ðxim ; x0im Þjσ0 !σi dσ im . i Using Eq. (8.3), ρA,m can be put in the form n Z 0 m ρnA,m ðxim ; x0im Þ ¼ 1mA 1A Λ ρðx; x0 Þdxi>m , (8.44) m N,n D

where x ¼ x1 …xN , Λm N,n ¼ N!=½ðN  nÞ!ðn  mÞ!, D is a domain such that electrons m + 1 to n are integrated over A, and electrons n + 1 to N over B. The expression (8.44) is related to the so-called coarse-grained density matrices (CGDM) [42]. Let us imagine that n1 electrons of the system are integrated or condensed over Ω1, the following n2 electrons over Ω2, …, and the last np over Ωp, [with n1 + n2 + ⋯+np ¼ N ð NÞ], leaving the remaining m ¼ N  N electrons free or not condensed. The CGDM of this condensation is defined as Z ρm ðxim ; x0im Þ½C ¼ I C W C ρðx; x0 Þdxi>m , (8.45) where ½C≡½n1 ,n2 ,…np , I C ¼ N!=½n1 !n2 !…np !, and W C ¼ W C ðxm+1 …xN Þ is the product of N ωk weight functions such that electrons (m + 1) to (m + n1) are weighted through ωΩ1 ðxm+1 Þ, …, ωΩ1 ðxm+n1 Þ, electrons (m + Pn1 + 1) to (m + n1 + n2) through ωΩ2 ðxm+n1 +1 Þ, …, ωΩ2 ðxm+n1 +n2 Þ, and so on. Using the property k ωΩk ðxÞ ¼ 1 [43], one has 1¼

X ½C

N! ðN  mÞ! X WC ¼ IC W C , n1 !n2 !…np ! N! ½C

(8.46)

201

2. Open quantum systems

where the last summation runs over all possible condensations [C] of the N electrons. Any ordinary mth order RDM (mRDM) of the full system, ρm ðxim ; x0im Þ, defined (in the McWeeny normalization [44]) as Z N! ρðx; x0 Þjx0 i>m !xi>m dxi>m , ρm ðxim ; x0 im Þ ¼ (8.47) ðN  mÞ! may then be recovered as ρm ðxim ; x0im Þ ¼

X ρm ðxim ; x0im Þ½C:

(8.48)

½C

When the system in made up of only two fragments A and B, and take n1 ¼ n  m, n2 ¼ N  n, n1 + n2 ¼ N ¼ N  m, so that [C] ¼ (n  m, N  n), one has I C ¼ Λm N,n. In this case, the integration over domain D in Eq. (8.44) is equivalent to the one in Eq. (8.45), and the sum over [C] in Eq. P (8.48) can be replaced by N n¼0 . Then, Z XN 0 m Λ ρðx; x0 Þdxi>m : (8.49) ρm ðxim ; xim Þ ¼ n¼0 N,n D

As a consequence, the sum of the mRDMs of all sectors n of domain A is given by X 0 ρA,m ¼ ρnA,m ¼ 1mA 1A (8.50) m ρm : n

ρeAn’s

n If the are used in the RHS integral of Eq. (8.43), one obtains ρeA,m the normalized mRDMs P A n of sector n. Then, ρA,m ¼ n p ðnÞe ρA,m . In the following two subsections, we will consider separately the RDMs of OQSs for SDWs and MDWs.

2.4.1 Single-determinant wave functions We will derive now expressions for the first- and second-order RDM. From their definition in Eq. (8.43), we have Z n 0 ρA,1 ðx1 ; x1 Þ ¼ n ρnA ðxin ; x0in Þjx0 !xi>1 dxi>1 (8.51) i>1

ρnA,2 ðx1 x2 ; x01 x02 Þ ¼ nðn  1Þ

Z

ρnA ðxin ; x0in Þjx0

Considering Eq. (8.44), we can write ρnA,1 ðx1 ; x01 Þ as Z 0 n 0 A A ρA,1 ðx1 ; x1 Þ ¼ 11 11  n

i>2

!xi>2 dxi>2 :

ρðx; x0 Þdxi>1 :

(8.52)

(8.53)

D

In terms of the spin-orbitals ui’s satisfying Eqs. (8.23)–(8.25), ρnA,1 ðx1 ; x01 Þ, for a SDW, is given 0 as (we will suppress the factor 1mA 1A m with m ¼ 1 or m ¼ 2 in the following equations): XN ρnA,1 ðx1 ; x01 Þ ¼ u* ðx01 Þuj ðx1 Þ pA (8.54) j ðn  1Þ, j¼1 j

202

8. Open quantum systems, electron distribution functions

where pA j ðn  1Þ represents the probability of having n  1 electrons in A and N  n electrons in B for a hypothetical (N  1)-electron SDW built with all ui spin-orbitals except uj. For the above example (N ¼ 4, n ¼ 3), Eq. (8.54) results ρ3A,1 ðx1 ; x01 Þ ¼ u*1 ðx01 Þu1 ðx1 Þ½s2 s3 ð1  s4 Þ+s2 s4 ð1  s3 Þ+s3 s4 ð1  s2 Þ

(8.55)

+ u*2 ðx01 Þu2 ðx1 Þ½s1 s3 ð1  s4 Þ+s1 s4 ð1  s3 Þ+s3 s4 ð1  s1 Þ

(8.56)

+ u*3 ðx01 Þu3 ðx1 Þ½s1 s2 ð1  s4 Þ+s1 s4 ð1  s2 Þ+s2 s4 ð1  s1 Þ

(8.57)

(8.58) + u*4 ðx01 Þu4 ðx1 Þ½s1 s2 ð1  s3 Þ+s1 s3 ð1  s2 Þ+s2 s3 ð1  s1 Þ: N1 A Each pj ðn  1Þ consists of n1 terms, each of them given by the product N  1 factors. In each factor, the first n  1 numbers are AOM integrals in A, sm (m6¼j), and the last N  n numbers are AOM integrals in B, 1  sm (m6¼j). The expression (8.54) is normalized to 3pA(3) or to npA(n) in a general SDW case. If we have used ρeAn instead of ρnA , we had obtain n

ρ A,1 ðx1 ;x01 Þ ¼

N X j¼1

u∗j ðx01 Þuj ðx1 Þ

pA j ðn  1Þ pA ðnÞ

,

(8.59)

n that integrates to n. As we can see, ρeA,1 ðx1 ; x01 Þ is diagonal in the basis of ui spin-orbitals, with eigenvalues that depend on n. The ui’s coincide with Ponec spin-orbitals which, in turn, are those that diagonalize the domain average Fermi Hole (DAFH) in A. Regarding ρnA,2 ðx1 ,x2 Þ ≡ ρnA,2 ðx1 x2 ; x1 x2 Þ, we obtain X0 1 uj ðx1 Þuk ðx2 Þu∗j ðx1 Þu∗k ðx2 ÞpA ρnA,2 ðx1 , x2 Þ ¼ A12 A012 (8.60) jk ðn  2Þ, 2 j, k

where A12 ¼ 1  p^12 is an operator that antisymmetrizes with respect to variables in the unstarred spin-orbitals, and A012 which are likewise for the starred spin-orbitals, and pA jk ðn  2Þ represents the probability of having n  2 electrons in A and N  n electrons in B for a hypothetical (N  2)-electron SDW built with all ui spin-orbitals except uj and uk. Assuming real spin-orbitals, ρnA,2 ðx1 ,x2 Þ can also be written as 2 N  X   uj ðx1 Þ uk ðx1 Þ  pA ðn  2Þ, ρnA,2 ðx1 ,x2 Þ ¼ (8.61)  uj ðx2 Þ uk ðx2 Þ  jk j > E. The first operator on the left-side of the equality is the scalar relativistic contribution and the second term is the spin-orbit contribution. Also notice that if EV replaces V in the Hamiltonian and Langrangian density in the ZORA argument that the zero-flux condition for large component Dirac equation with zero-vector potential is   ð ð

{ 

{  1 ħ2 c2 — F F ¼0 drς F , F ¼ dr —  (10.50) 2mc2 + E  V 2 Ω Ω that leaves the definition of the zero-flux condition for one particle unaltered in practical cases just as in the SR-ZORA case.

5. Relativistic effects on atoms in molecules The previous sections have motivated and justifiedusing the zero-flux condition to compute atomic properties with relativity included. Remarkably with reasonable assumptions the zeroflux condition is unchanged. The only change is using the relativistically corrected electron density and computing atomic properties in the same way as the nonrelativistic case. As expected there is little change in the charges of molecules only containing light atoms as seen in the investigation by Anderson et al. while those with heavier atoms have a larger change at the SR-ZORA level of theory [50]. They found a change in the charge of about 0–1% for molecules only containing light atoms. When comparing the change in charge when relativistic effects are included is 16% for gold in Au4, and 27% for uranium in UF6. Furthermore, it was determined that the electron density and its Laplacian evaluated at bond critical points for Au4 differ by 8% and 13%, respectively. For UF6, the differences in the electron density and its Laplacian evaluated at the bond critical point are about 2% and 19%, respectively. In the investigation by Wang et al. [51] on the all-metal [Sb3Au3Sb3]3 sandwich complex, they found that when evaluating the stress tensor analysis a topological instability in the SbdAu BCPs and SbdSb BCPs exists when using a pseudopotential approach. It is expected that there will be two negative eigenvalues and one positive eigenvalue for the stress tensor when evaluated at a bond critical point, but the pseudopotential method yielded three negative eigenvalues. However, when the SR-ZORA method is used the instability is removed. This highlights that relativistic corrections are important even at bond critical points even though they are far away from the nuclei where relativistic corrections would be expected to be most important.

5. Relativistic effects on atoms in molecules

261

A surprising result by Anderson et al. [48] is that the number of critical points can be altered when relativistic corrections are included. They computed the QTAIM topology for a family of gold clusters at the nonrelativistic level of theory and at the SR-ZORA level of theory. Using different basis sets and different density functionals while keeping the geometry fixed they found that at the SR-ZORA level of theory that the same critical points appear but also additional ring and bond critical points also appear when compared with the nonrelativistic level of theory with the same functional and basis set as seen in Fig. 10.1.

FIG. 10.1

QTAIM topology of benzene dithiol gold cluster calculated at the unrestricted PBE QZ4P singlet level of theory for (A) nonrelativistic and (B) SR-ZORA levels of theory at the same geometry. The unlabelled red dots, dark gray in the print version, that lie on lines, bond paths, denote the bond critical points. The unlabelled green dots, light gray in the print version, that do not lie on lines denote the ring critical points. Reproduced with permission from J.S.M. Anderson, et al., Molecular QTAIM topology is sensitive to relativistic corrections, Chem. Eur. J. 25 (10) (2019) 2538–2544.

262

10. Relativistic QTAIM

The Pointcare-Hopf relationship is satisfied in all cases (this relationship is discussed in Chapter 1) [1,48]. The bond path(s) [115] that form when relativity is included closes a ring. These are weak bond paths since the bond paths deviate from a straight-line (curved), however their appearance changes the interpretation the system, with the inclusion of relativistic effects two nonbonded atoms now have a bonded interaction. These bond paths are classified as non-covalent interactions since the Laplacian of the density evaluated at the bond critical point of each of these new bond paths is around 0.03. To the best of our knowledge this is the first time it has been observed that the inclusion of relativistic effects has changed the number of critical points and bond paths at the same geometry. This is not to be expected in the majority of calculations, but it is important to point out that this happens and to ensure that the level of theory is sufficiently high so to make the proper interpretation. Pilme [53,54] explored several At-X diatomics where X ¼ H, F, Cl, Br, and I at the 2c-B3LYP/ aug-cc-PTZ-PP-2c level of theory. They found that spin-orbit effects changed the charge by more than 4.6% in AtF and by more than 52.6% in the case of AtI. In absolute terms, the change in charge was smallest in AtH by 0.01 and by as much as 0.1 in AtI. In all cases, the charge of At was positive with and without spin-orbit coupling and in all cases the charge increased (more positive) with the the inclusion of spin-orbit coupling. They found that spin-orbit coupling decreased the electronegativity in the At molecules. An interesting result is that in the molecular dipole. In the molecule AtH, the dipole moment inverts (changes sign) due to spin-orbit effects and the magnitude is primarily due to the spin-orbit effects. From the scalar relativistic effects, the dipole moment is 0.05 D and the spin-orbit effect changes this value by 0.26 D resulting in 0.21 D. Given that intuition suggests that relativistic effects occur near the nuclei it should be expected that relativistic effects will impact the integrated properties. In the other molecules, the value of the dipole changes by as little as 21% and as much as 60%. In all cases, spin-orbit effects increase the magnitude of the dipole moment and in direction away from At. The Laplacian of the density evaluated at bond critical points has a lot of information regarding bonding. The spin-orbit coupling correction in AtCH3 at the bond critical point changes from a 0.10 e bohr5 to 0.01 e bohr5. This changes the sign of the Laplacian albeit the value is essentially zero. Spin-orbit effects were also found to impact the delocalization index. The impact is 18% in At2, 11% in AtI and AtBr, 17% in AtCl, and 14% in AtF of the final magnitude. In all cases, the delocalization is reduced. There is less impact in the At triatomic halogenic anions on the Laplacian of the density at the bond critical point as well as the delocalization index. The Laplacian changes in the third decimal place increasing the value. The delocalization has a larger impact. It has 9% of the magnitude on IAtI between At and I, 10% of the magnitude on BrAtBr between At and Br, and 1% of the magnitude in IAtBr- between At and I but negligible change in the delocalization between At and Br.

6. Conclusion The zero-flux condition remains unchanged when relativistic effects are included at different levels of theory when using the Bader’s application of the Schwinger stationary action principle to a volume (an atom) when a reasonable Lagrangian is taken. Even at the Dirac-

References

263

Coulomb level of theory the zero-flux condition can be recaptured albeit its inclusion requires a somewhat ad hoc assumption. This justifies the use of QTAIM and the zero-flux condition above that of a pragmatic choice. Though there is some dispute regarding this approach for forming an open system the work of Bader, coworkers, and users of theory have seen its value for providing insight into the behavior of molecules, clusters, and other chemical systems. Despite the definition of QTAIM remaining unchanged in the Hamiltonians examined here including relativistic effects into the calculation of the density can change the properties, and even the topology. Properties evaluated over an atomic basin or at a critical point can be altered substantially. Properties investigated while including relativistic effects within the QTAIM context when evaluated over a basin include charge, delocalization, and dipole moment. At the bond critical point, properties investigated include the density and the Laplacian of the density. These properties have been found to sometimes have substantial changes and sometimes negligible when compared to the nonrelativistic treatment. It is also found that sometimes changes are more pronounced in systems with light and heavy atoms than those with only heavy atoms, for example the QTAIM properties of AtH are often more sensitive to relativistic effects than for AtF, AtCl, AtBr, and AtI. The inclusion of relativity can even change the number of critical points altering the interpretation. Though relativistic effects are dominated at the nuclei and are more pronounced with the heavier nuclei the inclusion of relativity can have effects away from the nuclei including on the number and properties of critical points, and the properties of atoms that contain a light nucleus. Therefore, when wanting to obtain an accurate QTAIM description even though the definition of QTAIM is robust enough to remain unchanged for practical purposes at different levels of theory it is important to select a relativistic level of theory high enough to capture all the important properties.

References [1] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon, Oxford, 1990. [2] R.F.W. Bader, T.T. Nguyen-Dang, Quantum-theory of atoms in molecules—Dalton revisited, Adv. Quantum Chem. 14 (1981) 63–124. [3] R.F.W. Bader, Principle of stationary action and the definition of a proper open system, Phys. Rev. B 49 (1994) 13348–13356. [4] P.F. Zou, R.F.W. Bader, Variational principle and path-integrals for atoms in molecules, Int. J. Quantum Chem. 43 (1992) 677–699. [5] R.F.W. Bader, Why define atoms in real-space, Int. J. Quantum Chem. 49 (3) (1994) 299–308. [6] R.F.W. Bader, The zero-flux surface and the topological and quantum definitions of an atom in a molecule, Theor. Chem. Accounts 105 (2001) 276–283. [7] R.F.W. Bader, H. Essen, The characterization of atomic interactions, J. Chem. Phys. 80 (1984) 1943–1960. [8] R.F.W. Bader, et al., Quantum topology: theory of molecular structure and its change, Isr. J. Chem. 19 (1980) 8–29. [9] R.F.W. Bader, S. Srebrenik, T.T. Nguyen-Dang, Subspace quantum dynamics and quantum action principle, J. Chem. Phys. 68 (8) (1978) 3680–3691. [10] S. Srebrenik, R.F.W. Bader, Towards the development of the quantum mechanics of a subspace, J. Chem. Phys. 63 (9) (1975) 3945–3961. [11] R.F.W. Bader, Everyman’s derivation of the theory of atoms in molecules, J. Phys. Chem. A 111 (2007) 7966–7972. [12] S. Srebrenik, R.F.W. Bader, T.T. Nguyen-Dang, Subspace quantum-mechanics and variational principle, J. Chem. Phys. 68 (8) (1978) 3667–3679.

264

10. Relativistic QTAIM

[13] R.F.W. Bader, The Lagrangian approach to chemistry, in: C.F. Matta, R.J. Boyd (Eds.), The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007, pp. 35–59. [14] C.F. Matta, R.J. Boyd, An introduction to the quantum theory of atoms in molecules, in: C.F. Matta, R.J. Boyd (Eds.), The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007, pp. 1–34. [15] S. Shahbazian, Extending the topological analysis and seeking the real-space subsystems in non-coulombic systems with homogeneous potential energy functions, in: R. Chauvin, et al. (Eds.), Applications of Topological Methods in Molecular Chemistry, Springer International Publishing, Cham, 2016, pp. 89–100. [16] H. Joypazadeh, S. Shahbazian, What does shape a topological atom? Found. Chem. 16 (1) (2014) 63–75. [17] P. Nasertayoob, M. Goli, S. Shahbazian, Toward a regional quantum description of the positronic systems: primary considerations, Int. J. Quantum Chem. 111 (9) (2011) 1970–1981. [18] M. Goli, S. Shahbazian, Atoms in molecules: beyond born-oppenheimer paradigm, Theor. Chem. Accounts 129 (2) (2011) 235–245. [19] S. Shahbazian, Beyond the orthodox QTAIM: motivations, current status, prospects and challenges, Found. Chem. 15 (3) (2013) 287–302. [20] M. Goli, S. Shahbazian, The two-component quantum theory of atoms in molecules (TC-QTAIM): foundations, Theor. Chem. Accounts 131 (5) (2012). [21] M. Goli, S. Shahbazian, The two-component quantum theory of atoms in molecules (TC-QTAIM): the unified theory of localization/delocalization of electrons, nuclei, and exotic elementary particles, Theor. Chem. Accounts 132 (12) (2013). [22] M. Goli, S. Shahbazian, The two-component quantum theory of atoms in molecules (TC-QTAIM): tensor formulation and its implications, Theor. Chem. Accounts 132 (6) (2013). [23] T.E. Jones, M.E. Eberhart, The bond bundle in open systems, Int. J. Quantum Chem. 110 (8) (2010) 1500–1505. [24] M. Eberhart, A quantum description of the chemical bond, Philos. Mag. B 81 (8) (2001) 721–729. [25] F.H. Zadeh, S. Shahbazian, Toward a fuzzy atom view within the context of the quantum theory of atoms in molecules: quasi-atoms, Theor. Chem. Accounts 128 (2) (2011) 175–181. [26] F. Heidarzadeh, S. Shahbazian, The quantum divided basins: a new class of quantum subsystems, Int. J. Quantum Chem. 111 (12) (2011) 2788–2801. [27] J.R. Maza, et al., The Ehrenfest force topology: a physically intuitive approach for analyzing chemical interactions, Phys. Chem. Chem. Phys. 15 (41) (2013) 17823–17836. [28] A. Azizi, et al., Non-nuclear attractors in small charged lithium clusters, Limq (m ¼ 2–5, q ¼ 1), with QTAIM and the Ehrenfest force partitioning, Phys. Chem. Chem. Phys. 20 (38) (2018) 24695–24707. [29] D.X. Zhao, et al., A model of atoms in molecules based on potential acting on one electron in a molecule: I. partition and atomic charges obtained from ab initio calculations, Int. J. Quantum Chem. 118 (15) (2018). [30] J. Dillen, The topology of the Ehrenfest force density revisited. A different perspective based on slater-type orbitals, J. Comput. Chem. 36 (12) (2015) 883–890. [31] A.M. Pendas, J. Hernandez-Trujillo, The Ehrenfest force field: topology and consequences for the definition of an atom in a molecule, J. Chem. Phys. 137 (13) (2012). [32] P. Salvador, E. Ramos-Cordoba, Communication: an approximation to Bader’s topological atom, J. Chem. Phys. 139 (7) (2013), 071103. [33] B. Rousseau, A. Peeters, C. Van Alsenoy, Atomic charges from modified Voronoi polyhedra, J. Mol. Struct. Theochem 538 (1) (2001) 235–238. [34] J.I. Rodriguez, et al., An efficient grid-based scheme to compute QTAIM atomic properties without explicit calculation of zero-flux surfaces, J. Comput. Chem. 30 (2009) 1082–1092. [35] A. Morgenstern, T.R. Wilson, M.E. Eberhart, Predicting chemical reactivity from the charge density through gradient bundle analysis: moving beyond Fukui functions, J. Phys. Chem. A 121 (22) (2017) 4341–4351. [36] A. Morgenstern, et al., The influence of zero-flux surface motion on chemical reactivity, Phys. Chem. Chem. Phys. 18 (7) (2016) 5638–5646. [37] A. Morgenstern, et al., In search of an intrinsic chemical bond, Comput. Theor. Chem. 1053 (2015) 31–37. [38] J.S.M. Anderson, P.W. Ayers, J.I.R. Hernandez, How ambiguous is the local kinetic energy? J. Phys. Chem. A 114 (2010) 8884–8895. [39] P. Cassam-Chenai, D. Jayatilaka, Some fundamental problems with zero flux partitioning of electron densities, Theor. Chem. Accounts 105 (2001) 213–218.

References

265

[40] R.F.W. Bader, A comment on "some fundamental problems with zero-flux partitioning of electron densities", Theor. Chem. Accounts 107 (2002) 381–382. [41] P. Cassam-Chenai, D. Jayatilaka, A complement to "some fundamental problems with zero flux partitioning of electron densities", Theor. Chem. Accounts 107 (6) (2002) 383–384. [42] P. Cassam-Chenai, Frequently asked questions on "some fundamental problems with zero flux partitioning of electron densities", J. Math. Chem. 31 (2002) 145–153. [43] E.S. Kryachko, Comments on "some fundamental problem with zero flux partitioning of electron densities", Theor. Chem. Accounts 107 (2002) 375–377. [44] J.S.M. Anderson, P.W. Ayers, Quantum theory of atoms in molecules: results for the SR-ZORA Hamiltonian, J. Phys. Chem. A 115 (45) (2011) 13001–13006. [45] S.W.R. Hamilton, On a General Method in Dynamics, Richard Taylor, 1834. [46] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980. [47] J. Schwinger, The theory of quantized fields. I, Phys. Rev. 82 (6) (1951) 914–927. [48] J.S.M. Anderson, et al., Molecular QTAIM topology is sensitive to relativistic corrections, Chem. Eur. J. 25 (10) (2019) 2538–2544. [49] J.S.M. Anderson, P.W. Ayers, The general setting for the zero-flux condition: the Lagrangian and zero-flux conditions that give the Heisenberg equation of motion, J. Comput. Chem. (2018). [50] J.S.M. Anderson, et al., Relativistic (SR-ZORA) quantum theory of atoms in molecules properties, J. Comput. Chem. 38 (2) (2017) 81–86. [51] L.L. Wang, et al., Insights into the all-metal Sb3Au3Sb3 (3) sandwich complex from a QTAIM and stress tensor analysis, Chem. Phys. Lett. 685 (2017) 127–132. [52] B. Ni, et al., Protonolysis of the HgC bond of chloromethylmercury and dimethylmercury. A DFT and QTAIM study, J. Phys. Chem. A 110 (30) (2006) 9451–9458. [53] C. Gomez Pech, et al., Quantum chemical topology at the spin-orbit configuration interaction level: application to astatine compounds, J. Comput. Chem. 41 (23) (2020) 2055–2065. [54] J. Pilme, et al., QTAIM analysis in the context of quasirelativistic quantum calculations, J. Chem. Theory Comput. 10 (11) (2014) 4830–4841. [55] S. Sadjadi, et al., Relativistic-consistent electron densities of the coinage metal clusters M-2, M-4, M-4(2-), and M4Na2 (M ¼ Cu, Ag, Au): A QTAIM study, J. Phys. Chem. A 115 (45) (2011) 13024–13035. [56] T.A. Keith, M.J. Frisch, Subshell fitting of relativistic atomic core electron densities for use in QTAIM analyses of ECP-based wave functions, J. Phys. Chem. A 115 (45) (2011) 12879–12894. [57] W. Zou, et al., An open library of relativistic core electron density function for the QTAIM analysis with pseudopotentials, J. Comput. Chem. 39 (22) (2018) 1697–1706. [58] J. Cioslowski, J. Karwowski, Quantum-mechanical theory of atoms in molecules: a relativistic formulation, in: Fundamentals of Molecular Similarity, Kluwer, Dordrecht, 2001, p. 101. [59] L. Bucinsky´, D. Jayatilaka, S. Grabowsky, Importance of relativistic effects and electron correlation in structure factors and electron density of diphenyl mercury and triphenyl bismuth, J. Phys. Chem. A 120 (33) (2016) 6650–6669. [60] L. Bucinsky´, et al., Picture change error in quasirelativistic electron/spin density, Laplacian and bond critical points, Chem. Phys. 438 (2014) 37–47. [61] P. Pyykko, Relativistic effects in structural chemistry, Chem. Rev. 88 (3) (1988) 563–594. [62] G. Eickerling, et al., Relativistic effects on the topology of the electron density, J. Chem. Theory Comput. 3 (6) (2007) 2182–2197. [63] N. Hebben, et al., The electronic structure of the tris(ethylene) complexes M(C2H4)(3) (M¼Ni, Pd, and Pt): a combined experimental and theoretical study, Chem Eur J 13 (36) (2007) 10078–10087. [64] G. Eickerling, M. Reiher, The shell structure of atoms, J. Chem. Theory Comput. 4 (2008) 286–296. [65] K. Batke, G. Eickerling, Topology of the electron density of d(0) transition metal compounds at subatomic resolution, J. Phys. Chem. A 117 (45) (2013) 11566–11579. [66] A. Fischer, et al., Experimental and theoretical charge density studies at subatomic resolution, J. Phys. Chem. A 115 (45) (2011) 13061–13071. [67] S. Fux, M. Reiher, Electron density in quantum theory, in: D. Stalke (Ed.), Electron Density and Chemical Bonding II: Theoretical Charge Density Studies, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012, pp. 99–142.

266

10. Relativistic QTAIM

[68] P.A.M. Dirac, Quantum mechanics of many-electron systems, Proc. R. Soc. London Ser. A 123 (1929) 714–733. ˚ gren, Relativistic theory of atoms and molecules. A bibliography 1916–1985. By P. Pyykk€ [69] H. A o, Lecture Notes in Chemistry, Vol. 41, Edited by G. Berthier, et al., Springer-Verlag, Berlin, 1986, Wiley Online Library, 1988. [70] P. Pyykk€ o, Relativistic Theory of Atoms and Molecules II: A Bibliography 1986–1992, first ed., Lecture Notes in Chemistry, vol. VIII, Springer Berlin, Berlin, 1993, p. 479. Heidelberg. [71] P. Pyykk€ o, Relativistic Theory of Atoms and Molecules III: A Bibliography 1993–1999, vol. 76, Springer Science & Business Media, 2013. [72] P. Pyykk€ o, Relativistic effects on periodic trends, in: S. Wilson, I.P. Grant, B.L. Gyorffy (Eds.), The Effects of Relativity in Atoms, Molecules, and the Solid State, Springer US, Boston, MA, 1991, pp. 1–13. [73] K.G. Dyall, in: K. Faegri (Ed.), Introduction to Relativistic Quantum Chemistry, Oxford University Press, New York, 2007. [74] M. Reiher, in: A.D. Wolf (Ed.), Relativistic Quantum Chemistry : The Fundamental Theory of Molecular Science, Wiley-VCH, Weinheim, 2009. [75] J.P. Desclaux, Relativistic Dirac-Fock expectation values for atoms with Z ¼ 1 to Z ¼ 120, At. Data Nucl. Data Tables 12 (4) (1973) 311–406. [76] J. Almlofl, O. Gropen, Relativistic effects in chemistry, in: Reviews in Computational Chemistry, Volume 8, vol. 8, 2009, p. 203. [77] T. Saue, Relativistic Hamiltonians for chemistry: a primer, ChemPhysChem 12 (17) (2011) 3077–3094. [78] I.P. Grant, Relativistic atomic structure calculations, in: S. Wilson (Ed.), Methods in Computational Chemistry: Volume 2 Relativistic Effects in Atoms and Molecules, Springer US, Boston, MA, 1988, pp. 1–71. [79] C.T. Chantler, et al., Convergence of the Breit interaction in self-consistent and configuration-interaction approaches, Phys. Rev. A 90 (6) (2014), 062504. [80] D. Cremer, W. Zou, M. Filatov, Dirac-exact relativistic methods: the normalized elimination of the small component method, WIREs Comput. Mol. Sci. 4 (5) (2014) 436–467. [81] R.E. Stanton, S. Havriliak, Kinetic balance: a partial solution to the problem of variational safety in Dirac calculations, J. Chem. Phys. 81 (4) (1984) 1910–1918. [82] A. McLean, Y. Lee, in: R. Carbo´ (Ed.), Current Aspects of Quantum Chemistry 1981, Elsevier, Amsterdam, 1982. [83] Y.S. Lee, A.D. McLean, Relativistic effects on Re and De in AgH and AuH from all-electron Dirac-Hartree-Fock calculations, J. Chem. Phys. 76 (1) (1982) 735–736. [84] I.P. Grant, Conditions for convergence of variational solutions of Dirac’s equation in a finite basis, Phys. Rev. A 25 (2) (1982) 1230–1232. [85] P. Pyykko, J.P. Desclaux, Relativity and the periodic system of elements, Acc. Chem. Res. 12 (8) (1979) 276–281. [86] R. Ahuja, et al., Relativity and the lead-acid battery, Phys. Rev. Lett. 106 (1) (2011), 018301. [87] J. Autschbach, Perspective: relativistic effects, J. Chem. Phys. 136 (15) (2012). [88] W. Kolos, K. Szalewicz, H.J. Monkhorst, New Born-Oppenheimer potential-energy curve and vibrational energies for the electronic ground-state of the hydrogen molecule, J. Chem. Phys. 84 (6) (1986) 3278–3283. [89] W. Kołos, L. Wolniewicz, Accurate adiabatic treatment of the ground state of the hydrogen molecule, J. Chem. Phys. 41 (12) (1964) 3663–3673. [90] W. Kołos, L. Wolniewicz, Potential-energy curves for the X 1Σg+, b3Σu+, and C 1Πu states of the hydrogen molecule, J. Chem. Phys. 43 (7) (1965) 2429–2441. [91] W. Kolos, L. Wolniewicz, Improved theoretical ground-state energy of the hydrogen molecule, J. Chem. Phys. 49 (1) (1968) 404–410. [92] C. Michauk, J. Gauss, Perturbative treatment of scalar-relativistic effects in coupled-cluster calculations of equilibrium geometries and harmonic vibrational frequencies using analytic second-derivative techniques, J. Chem. Phys. 127 (4) (2007), 044106. [93] Y. Nomura, Y. Takeuchi, N. Nakagawa, Substituent effects in aromatic proton nmr spectra. III substituent effects caused by halogens, Tetrahedron Lett. 10 (8) (1969) 639–642. [94] C.M. Marian, Spin-orbit coupling in molecules, Rev. Comput. Chem. 17 (2001) 99. [95] A. Einstein, On the special theory of relativity, Ann. Phys. 17 (1905) 891–921. [96] P. Schwerdtfeger, U. Kaldor, S. Wilson (Eds.), Relativistic pseudopotentials, in: Theoretical Chemistry and Physics of Heavy and Superheavy Elements, Springer Netherlands, Dordrecht, 2003, pp. 399–438. [97] M. Dolg, X. Cao, Relativistic pseudopotentials: their development and scope of applications, Chem. Rev. 112 (1) (2012) 403–480.

References

267

[98] L. Seijo, Z. Barandiara´n, The Ab initio model potential method: a common strategy for effective core potential and embedded cluster calculations, in: Computational Chemistry: Reviews of Current Trends, World Scientific, 1999, pp. 55–152. [99] M. Klobukowski, S. Huzinaga, Y. Sakai, Model core potentials: theory and applications, in: Computational Chemistry: Reviews of Current Trends, World Scientific, 1999, pp. 49–74. [100] C. Chang, M. Pelissier, P. Durand, Regular 2-component Pauli-like effective-Hamiltonians in Dirac theory, Phys. Scr. 34 (5) (1986) 394–404. [101] J.L. Heully, et al., Diagonalization of the Dirac Hamiltonian as a basis for a relativistic many-body procedure, J. Phys. B 19 (1986) 2799–2815. [102] E. van Lenthe, E.J. Baerends, J.G. Snijders, Relativistic regular 2-component Hamiltonians, J. Chem. Phys. 99 (6) (1993) 4597–4610. [103] E. van Lenthe, E.J. Baerends, J.G. Snijders, Relativistic total-energy using regular approximations, J. Chem. Phys. 101 (11) (1994) 9783–9792. [104] E. van Lenthe, The ZORA Equation, 1996. [105] M. Filatov, D. Cremer, On the physical meaning of the ZORA Hamiltonian, Mol. Phys. 101 (14) (2003) 2295–2302. [106] J.G. Snijders, A.J. Sadlej, Perturbation versus variation treatment of regular relativistic Hamiltonians, Chem. Phys. Lett. 252 (1) (1996) 51–61. [107] E.J. Baerends, T. Ziegler, A.J. Atkins, J. Autschbach, O. Baseggio, D. Bashford, A. B
erces, F.M. Bickelhaupt, C. Bo, P.M. Boerrigter, L. Cavallo, C. Daul, D.P. Chong, D.V. Chulhai, L. Deng, R.M. Dickson, J.M. Dieterich, D.E. Ellis, M. van Faassen, L. Fan, T.H. Fischer, C.F. Guerra, M. Franchini, A. Ghysels, A. Giammona, S.J.A. van Gisbergen, A. Goez, A.W. G€ otz, J.A. Groeneveld, O.V. Gritsenko, M. Gr€ uning, S. Gusarov, F.E. Harris, P. van den Hoek, Z. Hu, C.R. Jacob, H. Jacobsen, L. Jensen, L. Joubert, J.W. Kaminski, G. van Kessel, C. K€ onig, F. Kootstra, A. Kovalenko, M.V. Krykunov, E. van Lenthe, D.A. McCormack, A. Michalak, M. Mitoraj, S.M. Morton, J. Neugebauer, V.P. Nicu, L. Noodleman, V.P. Osinga, S. Patchkovskii, M. Pavanello, C.A. Peeples, P.H.T. Philipsen, D. Post, C.C. Pye, H. Ramanantoanina, P. Ramos, W. Ravenek, J.I. Rodrı´guez, P. Ros, R. R€ uger, P.R.T. Schipper, D. Schl€ uns, H. van Schoot, G. Schreckenbach, J.S. Seldenthuis, M. Seth, J.G. Snijders, M. Sola`, M. Stener, M. Swart, D. Swerhone, V. Tognetti, G. te Velde, P. Vernooijs, L. Versluis, L. Visscher, O. Visser, F. Wang, T.A. Wesolowski, E.M. van Wezenbeek, G. Wiesenekker, S.K. Wolff, T.K. Woo, A.L. Yakovlev, ADF, in: SCM: Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 2017. [108] F. Neese, The ORCA program system, Wiley Interdiscip. Rev. Comput. Mol. Sci. 2 (1) (2012) 73–78. [109] S. Faas, J.G. Snijders, J.H. van Lenthe, Ab-initio ZORA calculations, in: A. HernandezLaguna, et al. (Eds.), Quantum Systems in Chemistry and Physics, Vol 1—Basic Problems and Model Systems, 2000, pp. 251–261. [110] S. Faas, et al., An ab initio two-component relativistic method including spin-orbit coupling using the regular approximation, J. Chem. Phys. 113 (2000) 4052–4059. [111] S. Faas, J.H. Van Lenthe, J.G. Snijders, Regular approximated scalar relativistic correlated ab initio schemes: applications to rare gas dimers, Mol. Phys. 98 (2000) 1467–1472. [112] W. Klopper, J.H. van Lenthe, A.C. Hennum, An improved ab initio relativistic zeroth-order regular approximation correct to order 1/c(2), J. Chem. Phys. 113 (2000) 9957–9965. [113] I.N. Levine, Quantum Chemistry, Prentice Hall, Englewood Cliffs, 1999. [114] D.J. Griffiths, Introduction to Quantum Mechanics, Cambridge University Press, 2016. [115] Bond paths are introduced in Chapter 1, but briefly summarized as the gradient paths that join maxima to bond critical points.

C H A P T E R

11 Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry Carlo Gattia,b and Giovanna Brunoc a

b

CNR-SCITEC, Istituto di Scienze e Tecnologie Chimiche “Giulio Natta”, Milano, Italy Istituto Lombardo, Accademia di Scienze e Lettere, Milano, Italy cDipartimento di Chimica, Universita` degli Studi di Milano, Milano, Italy

1. Introduction This chapter reviews the Source Function (SF) approach [1–7], as applied to the study of three scalar fields, the electron density, the electron spin density and the molecular electrostatic potential, all of them carrying relevant physical and chemical information. Differently from the Quantum Theory of Atoms in Molecules (QTAIM) [8] and from all other quantum chemical topological (QCT) methods [9], that have the topological study of the scalar fields mentioned above or of other fields as their main focus, the SF approach is aimed at interpreting the local nature of a field in terms of a remote cause and of a local effect description [2,3,10]. Indeed, within such an approach the local value of a scalar is viewed as arising from concurring or opposing sources from the remaining points of a system. Such a view applies to any of the position points of a system and therefore the SF approach belongs only in a quite broad sense to the QCT methods. The use this approach makes of topology, either as a natural way of partitioning in convenient subunits the space of a system or for selecting the system points of particular interest, is essentially instrumental and surely not its main scope [11]. However, when the point sources are integrated over meaningful chemical objects, like the atoms in a molecule of the QTAIM, precious chemical insight on chemical bonding [2,3,11–23], or on spin delocalization and polarization mechanisms [4,5,24,25], or on the effect of remote or nearby chemical groups to the negative or positive regions of the electrostatic potential [6,7,26,27], is obtained.

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00003-7

269

Copyright # 2023 Elsevier Inc. All rights reserved.

270

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

The SF approach, as applied to the electron density has been comprehensively presented and critically reviewed in Ref. [2], while concise or narrative accounts can be found in Refs. [28] and [3], respectively. Other precious reviews, characterized by privileged foci on some aspects of the SF approach, form an important part of Refs. [5, 11, 15, 16]. The application of the SF to electron spin density is reviewed in detail in Refs. [5,24], while no specific reviews have thus far appeared as for the SF application to the molecular electrostatic potential. The present book chapter is organized as follows. First, the basic and common tenets of the SF approach (Section 2), along with the specific ways these tenets impact on the considered scalar fields (section 3) are illustrated. Then for each of the named fields, some relevant and didactic applications of the SF approach are reviewed (Sections 4–6).

2. The basic tenets of the SF approach Chapter I, entitled Types of Fields, of the famous Morse & Feshbach book on Methods of Theoretical Physics [29], includes a brief paragraph on “A solution of Poisson’s Equation” which acted as a seed for the proposal of the SF approach back in 1998 [1]. Consider the Poisson equation, r2 φðrÞ ¼ qðrÞ

(11.1)

where q is a finite function of r which goes to zero at infinity and where the only requirement on the solution φ is that it also vanishes at infinity. Under these conditions, a unique solution of Eq. (11.1) is given by. ð qðr0 Þ dr0 (11.2) φð r Þ ¼ 3 4π |r2r0 | R while, for other boundary conditions, the correct solution is φ plus some solution ψ of the Laplace equation r2ψ 5 0 such that φ + ψ satisfy these conditions [29]. If we write Eq. (11.1) as an identity, by taking φ equal to the electron density (ED) ϱ and q as r2ϱ, we get Eq. (11.3), ð r2 ϱðr0 Þ 0 ϱðrÞ ¼ dr (11.3) 0 R3 4π |r2r | which may be conveniently rewritten as ϱðrÞ ¼

ð R3

LSρ ðr, r0 Þdr0

(11.4)

r 2 ϱðr0 Þ 4π|r2r0 |

(11.5)

with LSρ(r, r0 ) given by. LSρ ðr, r0 Þ ¼ 

The ED may be so envisaged as determined by contributions from a local source LSρ(r, r0 ) operating at all other points of the space r’ and where the local source is given by the negative of the Laplacian of the function ρ to be reconstructed, weighted by a term proportional to the

271

2. The basic tenets of the SF approach

inverse of the distance between the local source and the reconstruction point [2]. The term (4π | r 2 r0 |)1 is a Green’s function or an influence function [10], expressing the influence or the effectiveness of r2ρ(r’)dr’ in contributing to cause the effect ρ(r). For the sake of chemical interpretation, the operation of the local source over the whole space (Eq. 11.4) may be recast [1,2] in terms of separate LS integrations over conveniently selected atomic basins Ω (Eq. 11.6): ð Xð X ϱðrÞ ¼ LSρ ðr, r0 Þdr0 ¼ LSρ ðr, r0 Þdr0 ¼ SFρ ðr, ΩÞ (11.6) R3

Ω

Ω

Ω

Though these latter may be chosen arbitrarily, as of any conceivable mutually exclusive partitioning of space or as of one of the many existing fuzzy boundary partitioning schemes [2], adoption of the QTAIM recipe has a clear advantage [1,2]. It ensures association of the atomic basin source contributions SFρ(r, Ω) to atoms or groups of atoms rigorously defined through quantum mechanics [8]. Eq. (11.6) shows that ρ(r) may be envisaged as determined by a sum of atomic contributions SFρ(r, Ω), each of them being called the Source Function from the atom Ω to the ED ρ at the reference point (hereinafter, rp) r [1,2]. Thus, the SF is an interpretive tool intimately connected to the very important operative notion of chemistry that any local property and chemical behavior of a system is, in some way, always affected by the other system’s moieties. Whether quite limited or very significant such an influence is, it is rigorously quantified through the SF. Even more importantly, the SF tool is not simply telling us that the ED is only apparently a local quantity. This kind of information is already firmly known from the Density Functional Theory (DFT), according to which the ground state ED of a system is at any point uniquely determined by the spatial location and the nuclear charge of all of its nuclei (and by any externally applied field, if present). However, the SF approach goes one step further [2,3]. Not only it rigorously quantifies such ED nonlocality but it also decodes the nonlocality into a chemical language, as each SFρ(r, Ω) value represents a measure of how an atom Ω, or group of atoms Ω, contributes to determine the ED at r, relative to the contributions from other atoms or group of atoms in the system. The SF approach may be straightforwardly extended to the reconstruction of the electron spin density (ESD) s(r), the difference between the α- and β-electrons densities, s(r) ¼ ρα(r)ρβ(r) [4,5]. By replacing ρ(r) by s(r) in both the local cause, r2ρ(r’) ! r2s(r’), and the effect, ρ(r) ! s(r), definitions, Eqs. (11.5) and (11.6) become   r2 ρα ðr0 Þ  ρβ ðr0 Þ r2 sðr0 Þ 0 ¼ (11.7) LSs ðr, r Þ ¼  4π|r2r0 | 4π|r2r0 | ð Xð X 0 0 sðrÞ ¼ LSs ðr, r Þdr ¼ LSs ðr, r0 Þdr0 ¼ SFs ðr, ΩÞ (11.8) R3

Ω

Ω

Ω

where SFs (r, Ω) is the SF contribution from atom Ω to s(r) and LSs is the local source for the ESD. In Eq. (11.7), the Green function, (4π | r 2 r0 |)1, remains unchanged relative to Eq. (11.5), being a purely geometrical (effectiveness) factor, totally independent from the scalar function to be reconstructed by the SF approach [10]. Customarily, the ρ ! s scalar replacement has not been applied also to the definition of the atomic basins. These latter have been retained as

272

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

those defined through the rρ(r) zero-flux QTAIM recipe [8] in order to associate atomic ESD source contributions to the rigorously defined quantum atoms or group of atoms of Bader’s theory. However, in a recent study on the topology of the spin density [30], both the usual QTAIM definition and that based on rs(r) zero flux ESD surface boundaries have been explored. Even when the QTAIM basin definition is maintained for both fields, quite different atomic sources reconstruct ρ and s at a given rp [4,5,24]. This is not surprising as these two scalar fields were found, in molecules, to concentrate [r2u(r’) < 0, u ¼ s or ρ] or dilute [r2u(r’) > 0] themselves in a quite distinct manner [4,5,24]. Their different behavior reflects and implies distinct mechanisms for the transmission of the ED and of the ESD information throughout a molecule [4,5]. Moreover, since s(r) is given by a difference of densities, the mechanisms of transmission of s(r) are dictated by the relative, rather than the absolute, concentration or dilution of the α- and β-electron density components (see Section 3.2.1). Let’s now consider the SF reconstruction of the molecular electrostatic potential (MEP) V(r) [6,7]. The electronic contribution, Vel(r), to V(r) has an expression, Eq. (11.9), ð ϱðr0 Þ dr 0 (11.9) V el ðrÞ ¼  0 R3 jr  r j formally resembling that for the reconstruction of ρ(r) in terms of its Laplacian distribution (Eq. 11.3) at all other points of the space. This is not surprising since both Vel(r) and ρ(r) are solutions (Eq. 11.2) of the Poisson’s equation (Eq. 11.1), where φ and q are either Vel and ρ or ρ and r2ρ, respectively. The perfect formal analogy between Eqs. (11.3) and (11.9) is then made evident by exploiting the well-known electrostatic relationship (also a Poisson’s equation), r2 V el ðrÞ ¼ 4πρðrÞ

(11.10)

that enables us to substitute ρ(r) with (4π)1r2Vel(r) in Eq. (11.9). (Note that the rhs of Eq. (11.10) carries a positive rather than the customary negative sign, because ρ(r), as in all other equations of this chapter, is taken as a positive function despite being associated with a negative charge distribution). Interestingly, a comparison of Eqs. (11.3) and (11.9) reveals that the ED ρ(r) may be envisaged as the potential generated by its own Laplacian distribution, r2ρ(r’) [1]. Based on the illustrated analogies, the local source for Vel has the following expression: LSV el ðr, r0 Þ ¼ 

r2 V el ðr0 Þ ϱðr0 Þ ¼  4π|r2r0 | |r2r0 |

(11.11)

and, similarly to the case of the ED and of the ESD distributions, Vel(r) can be expressed as a sum of atomic contributions, Vel(r, Ω), ð Xð X X V el ðrÞ ¼ LSV el ðr, r0 Þdr0 ¼ LSVel ðr, r0 Þdr0 ¼ V el ðrÞðr, ΩÞ ¼ SFVel ðr, ΩÞ (11.12) R3

Ω

Ω

Ω

Ω

each of which can be termed as the Source Function contribution SFVel(r, Ω) from atom Ω to Vel(r). The MEP at r, V(r), is given by the sum of its electronic, Vel(r), and nuclear, Vnuc(r), contributions, which may be both decomposed into a sum of corresponding atomic contributions Vel(r, Ω) and Vnuc(r, Ω),

2. The basic tenets of the SF approach

V ðrÞ ¼ V el ðrÞ + V nuc ðrÞ  ð X  ð ϱðr0 Þ ϱðr0 Þ 0 X ZA ZΩ 0 ¼ dr + dr + 5 Ω  0 0 A,nuclei |r2R | |r2RΩ | A R3 |r2r | Ω |r2r | X X X V ðrΩÞ + V nuc ðrΩÞ ¼ V ðrΩÞ ¼ SFV ðrΩÞ ¼ Ω el Ω

273

(11.13)

Ω

yielding MEP atomic contributions V(r, Ω) termed as the Source Function contribution SFV(r, Ω) from atom Ω to V(r) [6,7]. Although it is evident that the V(r) SF reconstruction given by Eq. (11.13) is quantitatively analogous to evaluating V(r) as a sum of electrostatic potential contributions from all the atomic basins of a molecule [31], it is nevertheless of some interest to analyze also this function in terms of the same SF concept introduced for the ED and the ESD scalar fields [6,7,26,27]. As in the conventional SF for the ED, QTAIM atomic basins Ω have been customarily adopted in the application of Eq. (11.13) [6,7,26,27]. The association of SF contributions to rigorously defined quantum objects implies the possibility of valuable chemical interpretations. Through this link, the variations of the MEP values at V(r) maxima or minima on a given ED isosurface, caused by chemical substitution or change of conformation, are led to a rigorous interpretation in terms of concurring or opposing V(r, Ω) contributions from the various atoms or group of atoms of a molecule (see Section 6). Being based on the ED, the ESD or the MEP, and requiring just the knowledge of their associated Laplacian fields, the SF approach is also experimentally accessible. Namely, either for SFρ and SFV from crystal X-ray diffraction or, for SFs, from crystal X-ray plus polarized neutron diffraction (PND) data [2,3,12,15,16,24,25,28]. In both cases, the diffraction intensity data are refined through appropriate multipole models (MMs), yielding analytical expressions for “experimental” in crystal estimates of the ρ(r), s(r) and V(r) fields and of their Laplacian distributions [15,32–34]. Overall, the SF tool provides a privileged bridge between theory and experiment, as it allows to compare their outcomes on the same grounds [2,3,15,16,28,35]. In fact, for given ED, ESD or MEP distributions and for a given space partitioning scheme, the SFρ, SFs and SFV values are fully independent from the specific ways these scalar fields are obtained and expressed. The SF values would not change whether these fields had been calculated through ab-initio calculations or obtained from diffraction experiments and whether given numerically on a grid or expressed through an atomic nuclearcentered basis set or through a MM expansion. Regardless of the scalar field they refer to, SFx(r, Ω) contributions (with x being in our case either ρ, s or V) may be conveniently discussed either in term of their values or in terms of percentage contributions, SFx % (r, Ω) [2,4,6], SFx %ðr, ΩÞ5

SFx ðr, ΩÞ  100,x ¼ ρ,s or V xð r Þ

(11.14)

The SFx % (r, Ω) value expresses the relative capability of an atom or group of atoms Ω to determine the field x at r. The SFx % (r, Ω) values are usually portrayed using a ball and stick molecular model where each “atomic” ball volume is taken proportional to the corresponding SF percentage contribution and where the location of the adopted rp is signaled by a dot. Although any point r ER3 may be selected as a rp, some guiding criteria need to be adopted [2,4,6,7]. The

274

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

criteria depend both on the kind of study and of chemical questions one is addressing and on the specific scalar field one is investigating. A number of illustrating examples for each of the examined fields are reported in several paragraphs of the following Sections 3–6. SFx(r, Ω) values may be in general either positive or negative. While this appears as quite obvious for those scalar fields, like s(r) and V(r), that have both positive and negative regions, negative SFρ(r, Ω) values may look at a first sight surprising since ρ(r) is everywhere positive. However, such perplexity is not motivated [2,3,13]. Whatever the field, the SFx(r, Ω) values may differ in sign from that of the x value they contribute to reconstruct [4–7]. As we have discussed earlier in this chapter, SFx(r, Ω) values should be just envisaged as contributions to determine x(r) in a cause and effect relationship [2,3,10,23]. If the cause, rather than operating over R3, it does so just in one of its portions, then the produced effect may in some instances just oppose to the cumulative effect the cause determines by operating over the whole space. Only in the limit of the isolated atom, SFx(r, Ω) needs to have the same sign of the SF reconstructed x(r) value. In such a case, a single atomic basin, alone, will determine x(r). For an atom in a polyatomic system, the situation may often be different and, for instance, the sign and magnitude of the SF contribution from the H atom involved in a hydrogen bond (HB) to the ED at the HB bond critical point (bcp) turns out to be distinctly related to the HB nature [2,3,13]. The SFρ(HBbcp, H) is largely negative for the weak, mostly electrostatic in nature, HBs, while it is largely positive for the strong and largely covalent, charge assisted HBs. This behavior, along with its clear chemical rationale, is illustrated in detail in the Section 4.1. Regardless of the SFx(r, Ω) sign, SFx % (r, Ω) will be always positive when the source from Ω has the same sign of the x(r) value and favors its reconstruction, while it will be invariably negative when it has a different sign from the x(r) value and, thus, opposes its reconstruction [2]. The relative accuracy by which the field x(r) is obtained as a sum of atomic sources SFx(r, Ω) is given by the quantity ER%(r),   X xðrÞ  SFx ðr, ΩÞ    Ω   100 (11.15) ER%ðrÞ ¼   xð r Þ     ER%(r) is typically lower or much lower than one when jx(r)j is larger than 102 au, but for one order of magnitude lower jx(r)j values may rise up even to 3–5 [2,4,6,7]. In regions where jx(r)j < 104 au, a careful SF reconstruction of x(r) may become challenging. This accuracy issue implicitly involves a minimum jx(r)j value constraint (roughly 103 au) on the choice of rps. For more details on this problem and on how it may be partly circumvented, see Refs. [2,14].

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions 3.1 The case of the ED Several uses of the SF approach for the ED field have been explored in molecules and crystals [2,3]. They include, to cite just the most relevant applications: (i) the study of the chemical bond nature [2,3,11–21,36–112] and, in general, of any potential chemical interaction existing

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

275

in a system; (ii) the detection of features associable to electron localization/delocalization mechanisms [2,3,5,28,43,113–120]; (iii) the assessment of the extent of chemical transferability [2,3,28], if any, in series of chemically related systems. Besides that, the SF has been largely employed also as a valuable tool to compare calculated and experimentally derived electron densities (and their Laplacians) and to judge the adequacy of the adopted multipolar model expansions [2,3,12,15,16,28,35–37,39,40,43,58]. The general features inherent to each of the various ED SF applications are illustrated below. 3.1.1 ED SF studies of chemical bonding: General aspects Within QTAIM, and despite many controversial debates (Ref. [121] and references therein), the occurrence of a bond critical point (bcp) and of a bond path linking two nuclei in a system at an equilibrium geometry is assumed as a relevant (if not a necessary) indicator for the existence of a chemical bond or in general of a chemical interaction between their atoms [8,19,122]. Not unexpectedly, bcps have therefore been assumed as the most representative rps to discuss chemical bonding issues using the ED SF approach [2,3]. The pattern of the atomic SF contributions, as represented through a ball and stick model, provides [2,3,13,14] a direct representation of the more or less delocalized nature of the interaction (Fig. 11.1). If the SF approach is not simply a mathematical identity, providing a “trivial tautological reconstruction of the ED” [2,3,23], but rather a useful descriptor capable of chemical insight, it is expected that the more covalently will two atoms be bonded to one another, the higher will be their SFρ % (r, Ω) contributions to the value of the ED at their associated bcp [2,3]. This behavior is neatly revealed by the SF ED reconstruction at the CdC bcp in the series ethane, ethene, and ethine (Fig. 11.1). The SF contribution from the C atom dominates the bcp ED value and increases along the series from 0.092 au in ethane, to 0.148 au in ethene and to 0.182 au in ethine. More importantly, in parallel to the formal CC bond order augmentation along the series, also the source function percentage contribution from the two C atoms, SFρ % (rCC, bcp, C + C), keeps increasing from an initial value of 79% in ethane, to 84% in ethene and a final value of 96% in ethine [14]. The increasing dominance of the C atom sources along the series, implies a consequent decrease in the SF percentage contribution from the whole set of H atoms, SFρ % (rCC, bcp, all H). This trend holds true also for the SF contribution

FIG. 11.1

From left to right: Source Function percentage (SF%) contributions to the electron density (values in au) at the CC bond critical point (bcp) in ethane, ethene, ethine and at the B-Hbridge bcp in diborane. The position of the bcp is denoted by an azur dot. SF% contributions are displayed also as spheres having a volume proportional to their value. Adjusted from Fig. 3 and reproduced with permission from C. Gatti, D. Lasi, Source function description of metal-metal bonding in d-block organometallic compounds, Faraday Discuss., 135 (2007), 55–78.

276

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

of each H atom, SFρ % (rCC, bcp, H), Fig. 11.1, due to the parallel increase in the ED value at the bcp. Indeed, the absolute contribution from H, SFρ(rCC, bcp, H), is found to remain almost constant throughout the series (0.0084, 0.0086 and 0.0080 au). The simple, localized picture of standard CC covalent bonds is largely altered when the SF approach is applied to the less conventional and supposedly more delocalized B-Hbridge-B 3c-2e bonds in diborane (Fig. 11.1). While the terminal B-H bond has SF percentage contributions at bcp from its B and H atoms typical of a conventional covalent bond (82%), the SFρ % (rBHbridge,bcp, B + Hbridge) is as low as only 55% of the B-Hbridge bcp ED value, with the hydridic H atom contributing about 2/3 of this value. The residual 45% contribution to the B-Hbridge bcp density arises, for its largest part (about 80%), from the two terminal H bonded to the B atom and from the other Hbridge atom, all of these atoms having almost equal sources of about 11%–12%, a value just half that of the B atom directly participating to the B-Hbridge bond. The SFρ% contribution to the B-Hbridge bcp density from the other B atom is surprisingly low (3%) and even smaller than that from its bonded terminal H atoms (8% in total). Overall, the resulting pattern of sources looks fairly less localized than for the CC bond series (Fig. 11.1). An interesting comparison between the SF description and that provided by the QTAIM delocalization indices in diborane and in the CC bond series is reported in Ref. [14]. The SF ED analysis has been applied to several classes of bonding types, including metal-metal and metal-ligand bonds in organometallic complexes [2,3,11,12,14,15,18,35–65], bonds between main groups elements in inorganic [2,3,11–13,66–74] and organic [2,3,5,75–79] systems, hydrogen bonds of diverse nature [2,3,13,16,18,22,80–96], intra- and intermolecular noncovalent interactions of various kinds [16,97–110], bonds in biomolecules [92–96,111,112]. Examples of applications are discussed in Sections 4.1 and 4.2. A number of interesting features and advantages of the application of the SF to the study of chemical bonding deserve to be mentioned [2,3]. From a strictly topological point of view, a QTAIM bond path (or atomic interaction line) is associated only to the pair of atoms it connects, which implies that it is intrinsically incapable to graphically represent an interaction involving more than two atoms, while it is commonly acknowledged that a chemical bond may in some instances be multi-centered, or fluxional or better described in terms of competing bonding schemes. Nevertheless, both the bond path shape and the profile of the ED values along it, including the bcp ED value, all depend on the whole set of physical interactions existing between the particles (nuclei and electrons) in a system, as are accounted for by its Hamiltonian operator (or, in the language of DFT, all of them would depend on the external potential). Such an apparent QTAIM inconsistency is nicely resolved by the SF approach, since both the ED at the bcp and at any point along the bond path turn out to be determined not only from the pair of linked atoms but also to some extent from the sources of all other system’s atoms. The resulting SF patterns for the ED reconstruction at bcp will so reflect and disclose any possible nonlocal role to bonding. As shown later in this chapter, the profiles of the SF contributions along the bond path represent a further interesting tool to reveal how the local and not local contributions to bonding evolve along the path [2]. Moreover, similarly to other bonding descriptors giving access to nonlocal information on bonding, such as the QTAIM delocalisation index DI [123] or the synaptic order [124] of an Electron Localization Function (ELF) [125,126] valence basin, also the SF approach does not necessarily require a bond path to reveal the existence of a chemical interaction [2,3]. As said earlier in this chapter, any rp choice is potentially suitable. Evaluating the ED SF contributions

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

277

along an internuclear axis linking the nuclei of two atoms Ω and Ω’, characterized by a significant delocalization index δ(Ω,Ω’) and yet by the lack of a bond path joining them, it is a perfectly valid strategy and one that has been followed in some quite interesting cases [2,3,14,36,46]. Examples to this regard will be discussed in Section 4.2. Relative to the powerful interpretive tools based on the knowledge of the electron pair density or at least of the first order density matrix [2,3,11,14,15], the SF approach retains the distinctive advantage to be directly amenable to experimental determination. This fact places the SF in a privileged position for extracting comparative bond information from experimentally and theoretically derived ED distributions, which is of relevant importance, especially for the nonconventional bonding situations [2,3]. Dismissing, in some cases, the privileged role of the bond path as a universal indicator of bonding should not particularly worry the reader. Indeed, clear evidence of bonding in terms, besides other evidences, of a moderate or even large electron sharing between two atoms Ω and Ω’, also occurs when a bond path is lacking [2,3,14,36,127]. The typical situation is that of two distinct pairs of atoms entering in competition for a bond path. In such a case, an abrupt change in the molecular or crystal graph may occur following a negligible change of geometry and a hardly detectable variation of the delocalization indexes associated with these atomic pairs [14,36,121]. While the ED topology is sharply discontinuous in such an event, the description in terms of electron sharing within the two distinct pair of atoms remains perfectly smooth and continuous regardless a bond path is linking one or the other atom pair. In a similar smooth way are found to behave the various ED SF contributions through the structural catastrophe [14,36]. By using the Interacting Quantum Atom (IQA) approach [128], Pendas et al. could indeed show that the bond paths act as privileged electron exchange channels [121]. If two pairs of atoms compete for a bond path, the winning pair, i.e. that linked through a bond path, is regularly that having the larger exchange energy contribution. Therefore, even minimal changes of these contributions may lead to a switch of the bond path between the two competing pairs and to a structural catastrophe [36]. Yet, in the structural neighborhoods of the catastrophe point, the physics beneath is that of the occurrence of two almost equivalent interactions. Chemical bonding studies using the ED SF contributions are often complemented by the analysis of the local source for the ED, LSρ(r, r0 ), and of its energetic components. This analysis, providing further insights on the chemical bonding nature, is illustrated in the next paragraph. ED SF studies of chemical bonding: The local source for the ED, its energetic components and its profiles

By exploiting the virial theorem in its local expression [8], Eq. (11.16), 1 2 r ϱðrÞ52  GðrÞ + VEðrÞ 4

(11.16)

where G is the positively defined kinetic energy density and VE is the electronic potential (or virial) energy density, LSρ is given [2,22], Eq. (11.17), in terms of a sum of two local source contributions associated with such energy densities,

278

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

1 2  GðrÞ + VEðrÞ 1 2  GðrÞ 1 VEðrÞ ¼    ¼ LGðr, r0 Þ + LVEðr, r0 Þ (11.17) LSρ ðr, r0 Þ ¼   π π jr  r0 j π jr  r0 j jr  r0 j Therefore, those regions of a system that are electronic charge concentrated (r2ϱ(r0 ) < 0) and where the potential energy dominates over the kinetic energy [8] behave as positive sources for ϱ(r),while those regions that are charge depleted (r2ϱ(r0 ) > 0) and where the kinetic energy is dominating, act as negative sources for ϱ(r) [2,3,22]. The extent by which ϱ(r0 ) contributes to determine, as a source or as a sink, the ED at another point r is proportional to the value of its charge concentration or depletion at r0 and with a weight equal to the inverse of the distance between the two points r and r’ multiplied by 1/4π (see Eq. 11.3). Eq. (11.17) has also a further interesting interpretation. It discloses that the local source to the ED arises because of the local failure of the global virial relationship that dictates the total kinetic energy to be twice the magnitude of the total virial energy for a system at equilibrium [2,3,22]. Such global condition, it is locally realized only where r2ϱ(r0 ), hence LSρ(r, r0 ), is null. LSρ(r, r0 ) and its energy density components LG (r, r0 ) and LVE(r, r0 ) may be conveniently studied in terms of their profiles along a line and as a function of a suitably selected rp r0 [2,14,22]. These local functions provide further detail, relative to the corresponding atomic integrated sources, such as SFρ(r, Ω) [the SFρG(r, Ω) and SFρVE(r, Ω) have not been investigated thus far, but they will be likely the subject of forthcoming studies]. When the selected line for the local source and its component profiles is a bond path, the bcp is the natural choice for the rp [2,14,22]. Those regions of the bonded atomic pair that along the bond path yield positive or negative LSρ(r, r0 ) contributions to the ED at the bcp are made distinctly visible, this way [14,22]. The profile of LSρ(r, r0 ) may be interestingly compared with the corresponding profile of r2ϱ(r0 ) in the molecule or to the atomic radial profiles of r2ϱ(r0 ) in the two atoms of the pair [2,14,22]. This comparison evidences the link between the positive or negative local sources with, respectively, regions of charge concentration and depletion, but also the relevant impor1 tance of the weighting factor jrr 0 j in accentuating or weakening the role of such regions in determining the ED at the rp. Several other applications of the LSρ(r, r0 ) profiles have been reported thus far [14,22]. For instance, comparison of such profiles in a set of chemically related systems, where chemical substitution or simply a bond length change induces the appearance or the disappearance of a bond path, may disclose whether significant differences or simply marginal ones emerge in the way the ED at the rp is determined in the two situations. This comparison is particularly appropriate and rigorous when, due to symmetry reasons, the bcp is located at the internuclear axis midpoint, that is thus used as a convenient rp in the two cases [14]. LSρ(r, r0 ) profiles for the same molecule and adopting the same rp may be also expediently compared when obtained through different model wavefunctions or diverse multipolar model expansions [14,35]. The comparison returns a deeper comprehension of the changes in the bonding description at the bcp owed to a given ED model [14,35]. Examples of such uses of the LSρ(r, r0 ) profiles are a comprehensive SF study of metal-metal bonding in a number of d-block organometallic compounds (Ref. [14], and Section 4.2 in this chapter) and a detailed SF analysis of the bias on the ED induced by a refinement of the theoretical structure factors through various multipole model expansions adopted in the X-ray studies of the binuclear carbonyl coordination complex Mn2(CO)10) [35]. Studies involving the energy density components of the LSρ(r, r0 ) profiles were found to add further insight [14,22]. They have, for instance, being carefully analyzed in the HX series (X ¼ Li through F), taking the bcp as the rp [22]. The LSρ(r, r0 ) profiles show an increasing asymmetry, relative to the bcp location, with increasing

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

279

electronegativity of X through the series. However, from HC to HF the local source asymmetry is essentially due to a corresponding LVE(r, r0 ) asymmetry, denoting a dominance of the potential energy distribution. On the contrary, in the HLi and HBe molecules it is the LG (r, r0 ) sink contribution related to the kinetic energy that almost alone determines their LSρ(r, r0 ) profiles in the region around the bcp [22]. Typical shared interactions [8], characterized by a pile up of the electronic charge along the bond, are dominated by the LVE(r, r0 ) component in regions around the bcp, hence determining positive sources around this point. Smaller and positive, rather than larger and negative, LSρ(r, r0 ) sources from regions around the bcp are instead observed [22] in the case of the typical not shared interactions [8]. They are dominated by the LG (r, r0 ) component and characterized by a removal of the electronic charge in the regions around the bcp and by its accumulation in the innermost part of either one or both of the two interacting atomic basins [22]. Although this picture nicely fits the usual classification [8,19,129] based on r2ϱ(bcp) for these two limiting cases of chemical interactions, the LSρ(r, r0 ) and its components profiles act as a magnifying glass for their characteristic bonding features [14,22]. The latter include the asymmetric local source contributions from the two interacting partners that result from their asymmetric ED sharing [22]. 1 Once again, it is the weighting factor jrr 0 j that produces the magnifying glass effect. An interesting observation is that the analysis of the LSρ(r, r0 ) and of its components profiles places the supposedly dominant role of the bcp in bond classification schemes within a more correct perspective [2,3,22]. Indeed, while these profiles take the bcp as their rps, they are also conveying bonding information at bcp from all remaining points along the bond, thereby emphasizing the relevant role of these other points too. LSρ(r, r0 ) and its components profiles have been also investigated in the case of the intermolecular hydrogen bonds (HBs) in crystals, taking the HB bcp as rp. The study might be at first sight surprising, since it is well acknowledged that both the deformation densities (crystal density minus Independent Atom Model, IAM, density) and the interaction densities (crystal density minus superposition of molecular densities) reach absolute minimum values in regions close to the HB critical point [22,130,131]. However, both these difference densities are instead much larger in magnitude in regions closer to the H nucleus or to the nuclei of the heavier atoms involved in the HB [22,130,131]. By plotting the changes ΔLSρ(bcpHB, r0 ), ΔLG(bcpHB, r0 ) and ΔLVE(bcpHB, r0 ) of the local source and of its components profiles, obtained by subtracting the local profiles calculated through the deformation or the interaction density from those calculated through the “exact” crystal ED, it is possible to get precious understandings [22]. It is found that the negligible variation of the HB bcp ED in the crystal, relative to the corresponding ED value in the deformation or interaction densities, it is the result of a compensation of large local source differences in regions far from the HB bcp [22,130,132]. Analysis of the ΔLSρ(bcpHB, r0 ), ΔLG(bcpHB, r0 ) and ΔLVE(bcpHB, r0 ) profiles along the juxtaposition of the DdH and H…A (D]N, A ¼ O) bond paths in the urea crystal clearly highlights such a behavior [22].

ED SF studies of chemical bonding: The SF profiles

Similarly to the study of the LSρ(r, r0 ) profiles along a generic line or a bond path, SFρ(r, Ω) and SFρ % (r, Ω) may also be both analyzed along this same line or bond path [14,22]. However, the SFρ(r, Ω) profile [and the related SFρ % (r, Ω) one] is distinctly different from the local

280

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

source profiles. In the former, the rp is the varying position along the line or the bond path, rather than a given fixed point like in the latter. In addition, global SF contributions from the various atoms or group of atoms of a system, rather than the local source contribution due to a system’s point, are profiled, as a function of the moving rp, by the SFρ(r, Ω) and SFρ % (r, Ω) profiles. Therefore, these latter provide information on how the total or percentage SF “contribution” from the atom or group of atoms Ω to ρ(r) evolves along the line or the bond path (at r ¼ bcp, the usual SF ED reconstruction for the ED at bcp is obviously retrieved). Along a bond path joining atoms Ω and Ω’, the SF contribution from Ω’ (Ω) will generally increase (decrease) moving from Ω to Ω’, while those of the remaining atoms will also change, but in a different way. Typically, those latter contributions are generally higher in regions around the bcp and diminish when r approaches either of the two linked nuclei. Relatively large or relative small SF contributions from atoms other than Ω or Ω’ in the region around the Ω-Ω’ bcp denote a low or, respectively, high covalent character of the Ω-Ω’ bonding [14,22]. 3.1.2 The ED SF and electron delocalization: General aspects Choice of rps different from bcps or from points located along a bond path (or an internuclear axis), has been adopted in many other interesting situations [46,113–120]. Several studies have, for instance, explored whether the SF ED approach can detect electron delocalization effects, such as those responsible of aromatic character, and whether this approach may do so in a powerful, complementary way relative to other existing descriptors [2,3,5,28,113–120]. Let’s put this simply: if some electron delocalization takes place between atoms A and B, does this have any influence on the way atom A determines the ED in the basin of atom B and vice versa? [2,3,113]. A few examples of positive answers to this question, both in molecules and in crystals [28,113–120], and using either experimental or theoretically derived densities are illustrated in Section 4.3. In force of their conveying the extent of electron sharing between any pair of atoms in a system, the delocalization indices (DIs) are intimately related to the electron delocalization mechanism and are thus used as fundamental ingredients in a number of well-known global aromaticity descriptors [113,133–135]. However, differently from the SF approach, DIs require the knowledge of the electron pair density, generally unavailable at the experimental level [2,3,15]. Moreover, through the choice of different rps, the SF approach may provide further detail, being in this way able to sample the effects of electron delocalization in the various molecular regions and arising from “ensembles” of differently correlated electrons [2,3,28,113]. For instance, using as rps the points along a line perpendicular to a CC bond path and directed above and below the molecular σ-plane in a polycyclic aromatic hydrocarbon, a progressive increase in the SFρ% contributions from the C atoms other than those linked by the studied bond path is awaited as the rp is gradually displaced from the molecular plane [5,28,113]. In fact, the relative contribution from π-orbitals to the total ED is in parallel becoming larger and larger. These changes in the SFρ% contributions with the rp displacement from the molecular plane provide valuable insight into the π-electron conjugation mechanisms and allow to characterize the extent of conjugation in a system, without requiring or imposing any σ/π-electron separation constraint [5,28,113]. Once more, this fact is of paramount importance when electron conjugation effects using theoretically or experimentally derived EDs are compared, since for the experimental EDs the σ/π-electron separation is unfeasible, even for planar conjugated systems [28]. Several examples to this regard are reported in Section 4.3.

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

281

3.1.3 Further ED SF studies using reference points other than the bcps Another interesting example of a choice of rps other than bcps, concerns the comparison of the SF ED reconstruction at the ED non-nuclear attractor (NNA) maxima [136,137] and at the nuclear positions maxima to explore [13] whether the SF approach is able to distinguish the presumably quite different nature of their associated basins. Indeed, while the SF contribution from an ordinary basin at its own ED maximum hardly differs from 100%, that from a NNA basin may largely deviate from this value [13]. In a study on Li clusters, this internal source was as low as 74% in all investigated systems, the remaining 26% contribution coming from the closest linked Li basin(s) and from the other NNA basins, if present [13]. The fact that the NNA basins do not exist in isolation has, as a physical consequence, that even the ED at their maximum, that is at their innermost basin locations, is determined by significant external contributions [13]. Local charge concentrations or depletions [8], as detected by the ED Laplacian minima and maxima, respectively, may also be selected as convenient rps for some studies [4,5,24,25]. For instance, this rp choice has been explored [4,5,24,25] to investigate the distinct ways the ED and the ESD are reconstructed by the SF approach at these special locations (see Section 5.1). 3.1.4 ED SF and chemical transferability Looking back over, one of the major aims that initially stimulated Richard Bader and myself to apply the SF concept in chemistry was that of reaching a deeper understanding of chemical transferability and how it is realized [1]. This cornerstone of chemical thinking through the past two centuries [8] had always been like a compass needle for Richard Bader’s research and had also been at the basis of one of the most compelling “experimental” proofs of QTAIM [8,138,139]. In fact, a large evidence had been gathered through years that the topological definition of an atom or of a group of atoms allows to quantitatively recover the atoms and the atomic groups of chemistry, together with their experimentally measured transferable properties [8,140,141]. The degree of chemical transferability of an atom or group of atoms in a molecule is measured in terms of how a number of its characteristic properties are affected when the atom or the group of atoms is placed in different chemical environments. A perfect transferability occurs when the ED of this piece of matter is indeed fully transferable [8], while only partial transferability is realized whenever, due to mechanisms of compensatory transferability [140,142], only some of its properties remain constant. For instance, a constant atomic group electron population, resulting from compensating charge transfers within the group, or a constant atomic electron population achieved at the expense of a significant charge polarization within the atomic basin, are two typical examples of compensatory transferability. The electron population of the atomic group or even of the individual atom does not change, but their ED is no longer fully transferable in such a case. Perfect and compensatory transferability may be neatly distinguished and quantified using the SF ED approach [1–3,5,13,28], which also provides a detailed description on how perfect or compensatory transferability is realized in a given situation [13]. For instance, by expressing ϱ(r) in Eq. (11.6) in terms of a sum of a SF internal contribution from the atomic basin or atomic group Ω hosting the rp and an external SF contribution from the remaining atomic basins Ω’ in the system, Eq. (11.18), X ϱðrÞ ¼ SFρ ðr, ΩÞ + SFρ ðr, Ω0 Þ ¼ SFρ,int ðrÞ + SFρ,ext ðrÞ (11.18) Ω0 6¼Ω

282

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

an interesting constraint on the occurrence of perfect transferability of an atomic group property is obtained [13]. Constancy in a group property, if expressible in terms of the ED of the group, is only achieved when at any point r of the basin of the group remains constant the sum of SF contributions to ϱ(r) from the remaining atoms in the system, SFρ,ext(r), besides the contribution of the group itself, SFρ,int(r) (the constancy of this latter being automatically ensured by the requirement of a fixed ED of the group for a perfect transferability). The SF approach has been systematically applied to transferability issues especially in the first SF publications [1,13]. Due to the lack of space these studies are not reported among the SF ED applications (Section 4), yet two paradigmatic examples are concisely summarized below for the sake of completion. They represent two limiting cases, one where the SF approach confirms that a supposed almost perfect transferability is realized and another one where, differently from the established belief, a compensatory transferability mechanism takes place. Later on, the SF approach has been used to compare the ED and the ESD transferability at the CdH bcp of the terminal CH2• group of a series of n-alkyl radicals [5] and for assessing the transferability of electron conjugation effects in chemically related aromatic systems, both in the vacuo and in the crystalline phase [28]. Few results about this specific use of the SF tool are discussed in Section 4.3. The first paradigmatic case concerns the terminal methyl group in n-alkanes. Past ethane, this group is characterized by very transferable atomic properties, regardless of the length of the alkyl chain. Transferability does not apply only to the integral properties of the methyl group, such as its energy, electron population, volume and spectroscopic behavior [8], but it so good that it is found to cover also many local properties such as the value of the ED at the bcp of the unique CdH bond of the terminal methyl group. By applying the ED SF reconstruction to the ethane, propane, butane and pentane series, it is found that the internal SF contribution at this bcp ED from the atoms in the methyl group remains invariably constant throughout the series and that the external SF contribution hardly changes, irrespective of the length of the chain [1]. Constancy of the SFρ, ext value is achieved through a fall-off of the methylene groups and of the other terminal methyl group SFρ contributions with the chainlength increase. The value of the SFρ contribution for each succeeding increment is essentially predetermined by the condition of being equal to the SFρ contribution of the hydrogen atom it replaces. In summary, the terminal methyl group in n-alkanes is found to neatly fulfill the requirements for perfect transferability, as precisely stated through the SF approach (see previous discussion of Eq. 11.18). The second paradigmatic example addresses the transferability of Li atom in the Li-X (X ¼ F, O, N, Cl, H) diatomics [13]. These molecules played a prominent role in the historical development of QTAIM for a number of important reasons. The Li atom exhibits almost constant and transferable integral properties, including its net charge, through this series of molecules [8,138]. Such constancy of properties represented a neat example of the fundamental observation behind the concept of a functional group, according to which “atoms or linked groupings can exhibit characteristics forms and properties in spite of gross changes in their immediate neighbours” [140]. Moreover, the charge distributions of Li in Li-X (X ¼ H, O and F) had been selected back in 1972 by Bader to “illustrate the all important observation that the transferability of atom’s density is accompanied by a paralleling transferability in its kinetic energy density” [138]. Such a result had been reckoned as crucial to establish QTAIM in tandem with the empirical proof [139] that the virial theorem is satisfied not only by a system as a whole but also by

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

283

any of the system’s subspaces provided these are bounded by a —ρ zero-flux surface. Despite all this, a question remained to be answered. Was the Li atom transferability a case of a perfect transferability or simply of a compensatory one? The SF could provide the right answer to this important question [13]. In the limit of perfect transferability, the SF contribution from the Li atom to the bcp density, SFρ(bcp, Li), should be constant through the Li-X series, irrespective of the ED value at the bcp. Any change in this value should be ascribed only to SFρ(bcp, X).Yet, rather than remaining constant, SFρ(bcp, Li) is found to decrease with the diminishing of the X electronegativity through the series, being halved in LiH relative to LiF, the final and the first members of the series, respectively. The decrease in SFρ(bcp, Li) follows a corresponding decrease in the ED bcp value along the series, while both the Li electron population and energy are hardly changed. Interestingly, rather than SFρ(bcp, Li), it is the percentage source contribution from Li, SFρ % (bcp, Li), that remains almost constant to 40% through the series. Thus, the SF approach enables us to disclose the occurrence of a compensatory transferability mechanism [13]. This latter ensures a hardly changing Li energy and electron population through the series, realized by a constant percentage share between X and Li of the total SF contribution to the ED bcp rather than through a constant SF contribution from Li. The discovered mechanism has a clear chemical rationale [13]. The Li atom is behaving essentially as a cation through the series, a fact which implies a smaller percentage contribution to the bcp density relative to that of the anion and, quite reasonably, their fixed ratio. Yet, how is this fixed ratio achieved? It is found that with decreasing electronegativity of the X atom, the Li atom basin significantly expands toward the X basin. The ensuing displacement of the bcp position toward the X nucleus is the main responsible of the observed trends in the SFρ(bcp, Li) values. Indeed, the SFρ(r, Li) profiles, with the rp r moving along the bond path from the Li nucleus to the bcp, hardly change with X for the r interval common to all systems (Fig. 11.2). However, through the series, the position of the bcp displaces progressively toward the X nucleus in such a measure as to achieve a constant SFρ % (bcp, Li) and a constant Li electron population [13]. Though here limited to a brief outline of two simple cases, the SF represents an interesting tool to “measure” chemical transferability. One may envisage its potential use in pharmacological, biological and materials science investigations since the SF is capable to measure how a more or less geometrically distant chemical substitution or how any other kind of perturbation (for instance the placement of a molecule in a corresponding molecular crystal) may affect the extent of chemical transferability, hence of the properties, of a specific molecular moiety [3]. 3.1.5 ED SF software and computational details MULTIWFN [143] and AIMALL [144] general purpose packages for molecular systems include the ED SF analysis as one of their various options. The XD-2006 [34] and CRITIC2 [145] (linked to Quantum Expresso) packages perform ED SF analysis for crystalline systems, using X-ray experimentally derived and ab initio EDs [146], respectively. Early versions of the TOPOND code [147], that implements QTAIM for periodic systems are able to perform ED SF analysis for molecules, polymers, slabs and crystals. However, this capability has not yet been coded in the last TOPOND version [148] that it is fully embedded into the CRYSTAL 17 code [149], rather than been simply interfaced to this series of codes as it was for all the previous TOPOND versions. A complete in vacuo SF analysis for the ED, including the evaluation of the LSρ(r, r0 ), LG (r, r0 ) and LVE(r, r0 ) profiles and their differences for two different

284

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

FIG. 11.2 Source Function contribution of Li atom to the electron density at r, SFρ(r,Li), along the Li-X (X ¼ F, O, N, Cl, H) internuclear axis. The Li nucleus is at r¼ 0. The SFρ(r, Li) value is hardly dependent from X up to r equal to about 1.1 au, where the LiF bcp, denoted by a black bar in the inset, is located. The changes in SFρ(bcp,Li), as a function of X, are thus due to the shift of the bcp location on passing from X ¼ F to X ¼ H. Reproduced from Fig. 3 of C. Gatti, F. Cargnoni, L. Bertini, Chemical information from the source function, J. Comput. Chem. 24 (2003), 422–436. with kind permission from Springer Science + Business Media B.V.

model wavefunctions, is performed by a set of codes [150], originating from the Richard Bader’s AIMPAC code [151] and written as development versions by C. Gatti (CG) for his everyday work. This set of codes is available upon request to the author. More technical details about this CG SF software may be found in Refs. [2,13] and, for the local source and its components profiles, in Ref. [22].

3.2 The case of the ESD The ESD distribution is the result of the interactions taking place among the system’s electron spins and thus plays a fundamental role for understanding magnetic phenomena [152–156]. A detailed knowledge of the molecular ESD distribution sheds light on how spin polarization propagates in molecular complexes, enabling us to rationalize the magnetic

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

285

interactions among the assembled molecules, as a function of molecular orientation and packing [154,157,158]. Though these interactions are comparable in energy to the thermal energy, kBT, they are yet of utmost relevance for designing magnetic materials with important technological applications [153]. Obtaining accurate enough ESDs is, however, not a trivial task, not to talk about their interpretation. Experimentally, the s(r) distributions are obtained from polarized neutron diffraction (PND) experiments on magnetic crystals, which call for a neutron source and a sophisticated setup [159,160]. The very small number of reflections obtainable from PND experiment (only 10% of all the reflections in the reciprocal space) and the restricted access to neutron beamtime are the most important factors that hamper the reconstruction of good experimental ESD distributions [161], though the recent proposal of a spin-split (ρα and ρβ) version of the Hansen & Coppens multipolar model [32] for a joined fitting of X-ray and PND data, is a significant progress forward [161]. On the other hand, ab-initio electron ESDs can be easily calculated, but they are usually far from being reliable. Systematic studies demonstrated that DFT is often unable to treat open-shell systems properly, leading to nonaccurate ESDs [162–164]. Ab-initio electron correlation methods or density-matrix renormalization group approaches have to be called for [164]. Yet, they are both often computationally too demanding and unsuited for large systems. Besides all these mentioned difficulties, extracting chemical information from ESDs is not as an easy and extensively experienced task as it is for the ED, even for very simple molecular systems [4]. This has been one of the main reasons that motivated the extension of the SF concept to the ESD distributions and its use to study the magnetic patterns in some organic systems [5] and transition metal complexes [24,25]. Analogously to the standard ED SF, the ESD SF provides quantitative insights into the relative capability of different atoms or groups of atoms in a system to determine the ESD at given system’s locations [4,5,24,25]. The ESD SF does not only show, within a cause-effect view, how spin information propagates from the paramagnetic to nonmagnetic centers, but also whether the nonmagnetic centers are in turn capable to back influence the ESD distribution of the paramagnetic ones [4,5,24,25]. If that occurs, the ESD SF enables us also to quantify whether an atom or a group of atoms is facilitating or opposing the action of the paramagnetic centers in determining spin polarization at a given rp and whether they do so in relevant or negligible measure. The SF SD serves also as a useful tool for discussing the SD accuracy and for revealing the origin of ESD discrepancies when approaches of increasing quality are used [4,24,25]. A number of representative examples are presented in Section 5.1. 3.2.1 The origin of the positive and negative ESD local source contribution LSS Analogously to the ED, the local source for the ESD at a point r’ is directly proportional to the Laplacian of the scalar function it reconstructs (Eq. 11.16). However, as we anticipated in Section 2, important differences emerge relative to the case of the ED, due to the fact that s(r) is given by the difference rather than by the sum of the α- and β-electron density components. Concentration or dilution of both density components does not automatically ensure, respectively, a positive and a negative local source contribution LSs. What it is decisive is the relative concentration or dilution of the two components [4]. The list of all possible situations, including those where one component is concentrated while the other is diluted, is reported in Table 11.1 (from Ref. [4]). So, if both density components are concentrated LSs turns out to

286

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

TABLE 11.1 The sign and the α or β effect of the ESD local source function contribution LSS as a function of the local concentration (r2ρx(r0 ) < 0; x ¼ α, β) or local dilution (r2ρx(r0 ) > 0; x ¼ α, β) of the α- and β-electron density components. Sign [—2ρα(r’)]

Sign [—2ρβ(r’)]

Relative magnitudes

—2s(r)

LSS (r,r’)

Effect on s(r)

>0

>0

r ρα > r ρβ r2ρα < r2ρβ

>0 0

0

jr2ρα(r0 )j. If both density components are diluted, then the reverse will be true. LSs is positive when r2ρβ(r0 ) > r2ρα(r0 ) and negative when r2ρα(r0 ) > r2ρβ(r0 ). If the density components are one concentrated and the other diluted, the LSs will be positive when the α-density is concentrated and negative when it is the β-density that is concentrated, irrespective of the concentration or dilution magnitude of the two component densities (Table 11.1). A positive local source LSs, r2s(r’) < 0, will cause the α-component of the total ED to increase at a given rp r. Such an α-spin polarization increase is named an α-effect, the β-effect being, instead, the β-spin polarization increase at r due to a negative local source LSs, r2s(r’) < 0 (Table 11.1). The nature of the effect, α or β, depends only on the source point location r’. Yet, the effect magnitude is proportional to jr2s(r0 )j and inversely proportional to the distance between r’ and the rp. 3.2.2 Reference points for the ESD SF studies In Sections 3.1.1, “ED SF studies of chemical bonding: the local source for the ED, its energetic components and its profiles” and “ED SF studies of chemical bonding: The SF profiles”, a detailed discussion on how to select the most suited rps for the SF ED studies was reported. The analogous problem for the SF ESD studies is addressed below. If bcps look as the most representative and least biased choice for rp’s associated with the ED chemical bonding studies, this same choice becomes hardly a significant one in the case of the ESD. Indeed, the ESD if found to have quite often very low values at bcps. At and around these locations, especially for covalent-like bonds, the pairing of electrons is maximized and the ED is barely spin polarized, even for spin polarized systems [4,5]. Therefore, different rps choices have been customarily adopted. Representative rps for the ESD were found to be the critical points (maxima and saddles) of r2ρ(r ) in regions mostly associated with the unpaired electron(s) distribution and, for the sake of comparison, also those (3,-3) r2ρ(r ) critical points (CPs) associated with lone pair electrons [4,24,25]. A recent, first study on the ESD topology [30] used the CPs of s(r) as rps, a practice that is clearly to be recommended and further explored.

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

287

3.2.3 Easing the interpretation of the atomic ESD SF contributions to s(r): The magnetic and the relaxation components Although not so many papers have thus far appeared on the ESD SF approach, one observation seems already well established. The interpretation of the SFs(r, Ω) patterns is, in general, far from being trivial but it can be greatly facilitated [4,5,24,25] by decomposing the ESD s(r) in two physically meaningful components, the magnetic, sma(r) and the reaction or relaxation sre(r) electron spin densities, sðrÞ ¼ sma ðrÞ + sre ðrÞ

(11.19)

The former component is that spin density distribution due to the fully unpaired α-electrons, while the reaction (or relaxation) component accounts for the spin density distribution arising from the remaining α- and β-electrons in a system [4,5,24,25]. The two composing spin densities differ also in their integrated properties over the whole space. By definition, the integral of sma(r) yields the total number of unpaired electrons in the system as does the integral of s(r), while that of sre(r) needs to be equal to zero, though locally sre(r) is, in general, different from zero. The sre(r) distribution is named reaction or relaxation spin density as the constraint on s(r) of Ðan equal number of integrated electrons for its composing α- and β-electrons distriÐ butions [ R3ρα(r)dr ≡ R3ρβ(r)dr] is locally relaxed (ρα(r) 6¼ ρβ(r)) in reaction to the concomitant presence of the α-electrons fully unpaired ED [4]. Analogously to Eq. (11.19), also the LSs may be expressed as a sum of a magnetic, LSs,ma, and a relaxation, LSs,re, local source component. Likewise LSS,re, LSS,ma may be either positive or negative and be so the cause of either α or β effect, despite being due to α-electrons only (Table 11.1). In addition, LSs,re may either concur or oppose to the α or β effect due to the magnetic density. From a computational point of view, a simple recipe may be followed to obtain the magnetic and relaxation densities [4,24]. After a diagonalization of the first order density matrix, only those natural orbitals having occupation numbers equal to or very close to one are picked up to evaluate sma(r), the remaining natural orbitals being those associated with sre(r). More technical details on such ESD decomposition are given in the main text and SI of Ref. [4], and Refs. [24, 25]. As anticipated, the magnetic component is found to largely dominate the regions of high ESD and, similarly do some, if not all, of the magnetic SF contributions, SFs,ma(r, Ω), reconstructing the ESD in such regions. A clear example to this regard is given in Fig. 11.3 that shows [24] two quite opposite situations taking place in a centrosymmetric azido-bridged dicopper molecular complex. The first case is related to the SF ESD reconstruction at a (3,+3) r2ρ(r ) charge depletion CP in the valence shell charge concentration (VSCC) of the Cu atom, pointing toward the N1 atom ligand and with very high (0.278 au) ESD value. The second case adopts as the rp the copper (3,3) r2ρ(r) charge concentration CP pointing toward the other Cu atom and with an ESD value about 25 times as low. The first rp is roughly associated with an ED dominated by the contribution of a singly occupied natural orbital, hence leading to high ESD, while the second rp lies in a ED region dominated by double occupancy, hence with very low ESD. In the former case, the SFs,ma % (r, Ω) are marginally different from the total SF contributions, SFs(r, Ω), while those due to the relaxation spin density, SFs,re % (r, Ω), are very small and comparatively almost negligible. In the second scenario, handling the case of SF reconstruction in low ESD regions, SFs,ma % (r, Ω) and SFs,re % (r, Ω) display instead values of the same order of magnitude. It has also been found that while the

FIG. 11.3 End-on centrosymmetric ferromagnetic azido-bridged di-copper molecular complex. Top (from left to right): Molecular structure and atomic labelling (black and white balls denote C and H atoms, respectively); contour plots of the CASSCF(6,6) Electron Spin Density (total, s(r), magnetic, sma(r), and reaction or relaxation, sre(r), components) in the least squares plane of the four N ligand atoms around each Cu (red positive and, dotted blue, negative contour values; contour maps are drawn at interval of (2,4,8)10n, 4  n  0 au). Bottom (from left to right): Cu atom 3d electron asphericity and its CASSCF(6,6) electron density negative Laplacian [L(r) ¼ -r2(ρ)] representation in the plane of the N atoms. Locations of the (3,3) L(r) Charge Concentration (CC) and (3,+3) L(r) Charge Depletion (CD) used as reference points (rps) for the SF reconstructions in the panels at the right are indicated, along with their associated s(r) values; CASSCF(6,6) SF percentage contributions to the Electron Spin Density (total, SFs % (r, Ω), magnetic, SFs,ma % (r, Ω), and relaxation, SFs,re % (r, Ω), spin density components) at the CC and CD rps indicated in the left panel. Green (red) atomic balls denote an α (β) effect on the spin density or on its components at the rp. SF percentage contributions are positive or negative whether they concur or oppose to reconstruct the corresponding spin density or spin density component value at the rp. All panels of the figure, except those of the SF reconstructions that are fully original, are adapted from Figs. 2, 3 and 7 of C. Gatti, G. Macetti, L. Lo Presti, Insights on spin delocalization and spin polarization mechanisms in crystals of azido Cu (II) dinuclear complexes through the Electron spin density source function, Acta Crystallogr. B 73 (2017), 565–583 with permission of the International Union of Crystallography.

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

289

SFs,ma(r, Ω) values are in general quite stable against the quality of the theoretical model adopted for evaluating the ESD, those associated with the relaxation densities, SFs,re(r, Ω), are very much model dependent [4,24,25]. Needless to say, the same holds true for sma(r) relative to sre(r). The decomposition of the SF ESD contributions in magnetic and relaxation components has been shown to be a very useful tool to discuss the general accuracy of ESD distributions since it detects and rationalizes the changes brought in the ESD distribution by wavefunctions of increasing quality [4,24,25]. It is also a valid help when comparing experimentally and theoretically derived ESD distributions, as it allows to clarify which are the fundamental features of a given, accurate ESD, that have to be retrieved by either approach and which are instead those features that are very model dependent and less decisive [4,24,25,163,164]. Further precious insight is provided by the partial Source Function ESD reconstructions which will be introduced in Section 3.4. 3.2.4 ESD SF software and computational details The set of codes [150] described in Section 3.1.5, originating from the Richard Bader’s AIMPAC code and written as development versions by C. Gatti (C G) for his everyday work, include separate modules to evaluate the atomic SF ESD contributions and the ESD local source profiles. There are also modules for calculating the ESD and its Laplacian (along with their α and β components) at the ED CPs and as (density) profiles along a line or as isocontours on a selected plane. More technical details about this ESD SF software may be found in the Sections 1 and 2 of the Supplementary Information of Ref. [4]. The developed codes are available upon request to the author.

3.3 The case of the MEP SF 3.3.1 Preliminary MEP SF studies: Their motivation and the choice of the reference points The decomposition of V(r) as a sum of electrostatic potential contributions from all the atomic basins of a molecule (Eq. 11.13) has been explored long time ago [31]. The scope of this earlier investigation was to compare the exact QTAIM atomic contributions to V(r) with those calculated through QTAIM electrostatic multipole moments based on spherical tensors in order to assess and prove the fast convergence of such expansions. The proposal of viewing the atomic decompositions of V(r) in terms of the SF approach is instead quite recent and so are the applications derived thereof [6,7,26,27]. The motivation of such studies was that of providing a powerful interpretive tool for the stereospecific recognition of chiral molecules in analytical enantioseparations using High Performance Liquid Chromatography (HPLC) [27]. These separations are driven by the formation of transient diastereomeric selector-selectand complexes in a chiral environment generated by a chiral selector. Diverse short-range directional interactions, including HBs, π-π, dipole-dipole, and van der Waals interactions, underlie the formation of the selector-selectand complexes and promote the enantioseparation [27]. Quite recently, also σ- and π-hole interactions have been shown to play an important role to this regard [6,27,165]. Since the σ- and π-holes are generally associated with regions of relative ED depletion, MEP maxima on properly selected ED isosurfaces (typically 0.002 au) have been adopted as one of their most relevant and useful

290

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

signatures [27]. Still, none is quantitatively, and often even qualitatively known about the role the various moieties of a molecule play in producing such holes [6,7]. However, by using the MEP SF approach [6,7,26,27] the impact of chemical substitution and chemical conformation on the σ- and π-holes MEP maxima values can be dissected in separate contributions from chemically meaningful molecular moieties. Irrespective of their being close to or far from the holes, these moieties can either contribute (and even overdetermine in some cases) or oppose to the observed MEP maxima (see next paragraph). The SF MEP studies thus provide a clear analysis and a rationale of why and how this influence is realized and may so offer precious clues for designing changes in the σ- and π-holes molecular regions, with the precise scope of affecting their potential involvement in noncovalent interactions in a desired way [6]. Applications of the SF MEP tool to a series of 4,40 -bipyridine derivatives containing atoms from Groups VI (S, Se) and VII (F, Cl, Br) (yielding σ-holes) and the pentafluorophenyl group acting as a π-hole have been recently reported [6,7,26]. A number of exemplar cases is illustrated in Section 6. 3.3.2 Evaluating meaningful SF contributions to the MEP The reconstruction of V(r) in terms of atomic SF contributions SFV(r, Ω) leads in general to quite useless results, if the SFV(r, Ω) are not gathered into suitable atomic group contributions [6,7,26]. Indeed, Fig. 11.4 shows that the atomic reconstruction of V (on the 0.002 au ED isosurface) at the σ-hole maximum of the starred chlorine atom in a atropoisomeric chlorine substituted 4,40 -bipyridine is given by a sum of opposing positive and negative percentage contributions from all atoms in the molecule, and even from the more distant ones from the chosen rp. Moreover, the atomic contributions are all of comparable magnitude to the V(r) value they concur to reconstruct. Indeed, each SFV(r, Ω) value results from the balance FIG. 11.4 Reconstruction of the Molecular Electrostatic Potential, V(r), in terms of atomic SF percentage contributions,SFV % (r, Ω), at the σ-hole maximum (on the 0.002 au electron density isosurface) of the starred Cl atom in a atropoisomeric chlorine substituted 4,40 -bipyridine. SFV % (r, Ω) contributions are displayed as numerical values (the size of the atoms in this figure are not related to the source contribution magnitude). The atomic reconstruction of V is given by a sum of opposing positive and negative percentage contributions from all atoms in the molecule. The atomic contributions are all of comparable magnitude of the V(r) value they concur to reconstruct. Interpretation of V(r) in terms of atomic SF contributions is hardly useful (see text) if these atomic sources are not gathered into suitable atomic group contributions [6,7,26].

3. Specific features and general applications of the ED, ESD and MEP SF reconstructions

291

of a positive term due to the Ω nucleus and a negative one due to the ED in the basin Ω, with each of the two terms being typically four orders in magnitude larger than their sum. The sign and the magnitude of each SFV (r, Ω) contribution are extremely dependent on the rp position and interpreting the V(r) values in terms of their atomic SFV(r, Ω) contributions is apparently a nightmare. However, when the single atom contributions are gathered into those of chemically meaningful molecular moieties, very clear and useful pictures emerge. This step seems a mandatory one for a meaningful SF MEP analysis. A few exemplar cases are discussed in Section 6. 3.3.3 MEP SF software and computational details Application of the SF MEP approach requires as a first step the location of representative rps. In all papers published thus far, extrema of MEP have been located on ED isovalue surfaces (typically 0.002 au) using the Multiwfn code [143] and the module thereof performing quantitative analyses of molecular surfaces [166] for a given isovalue surface and a mapped property thereon (ED and V(r), respectively, in the present case). The set of MEP extrema is then analyzed by a few lines code, VEXTLOC [167], that based on the MEP value of each extremum and on its distance from the various nuclei of the system, allows for selecting the subset of extrema of interest for the current SF MEP analysis. The set of codes [150] described in Section 3.1.5, originating from the Richard Bader’s AIMPAC code and written as development versions by C. Gatti (CG) have been modified so as to perform both the ED and the MEP source function reconstructions at the same set of rps. A final code, named ANASFR_MEP (Analyze SF Results for the MEP field) [167], extracts the SFV (r, Ω) data for the various considered MEP extrema and gathers them according to all combinations of chemically meaningful atomic groups that have been defined in input. All these in-house developed codes are available upon request to the author (CG).

3.4 Partial SF reconstructed densities or potentials As we discussed earlier in this chapter, the choice of the most suitable rps for a SF study is a challenging one, regardless of the specific scalar field it applies. Any choice has, to some extent, its degree of arbitrariness. As a way to partly obviate this problem, the concept of Partial Source Function reconstructed density (PSFRD, for ϱ and s) or Partial Source Function reconstructed potential (PSFRP, for V) has been introduced [18,24,25,89]. The PSFRD (or PSFRP) are defined as follows, X X ð xΩsubset ðrÞ 5 LSx ðrr0 Þdr0 5 SFx ðrΩÞ where x ¼ ρ,s, V (11.20) Ωsubset Ω

Ωsubset

and they represent hypothetical density (or potential) distributions determined only by a given subset of the atoms in a system. PSFRD (or PSFRP) may be calculated along a line, on a 2D surface or in a 3D volume, using a suitable mesh. When calculated at a point or at a line, they correspond to a standard SF reconstruction at a given rp or to a SF profile, but limited to a subset of system’s atoms. When PSFRD (or PSFRP) are evaluated on a 2D or a 3D grid and then compared on these grids against the full density (or potential) the major or minor role a given subset of atoms plays in determining the density or the potential in the various analyzed system’s regions may be highlighted [18,24,25,89]. The 2D plots of the PSFRD or PSFRP are particularly revealing (Fig. 11.5). For instance, they make manifest

FIG. 11.5

Electron Density (ED) SF analysis in the Adenine:Thymine (AT) DNA base pair complex: (left) Ball and stick representation of the ED SF percentage atomic contributions SFρ% at the bcp ED of the three O…HC, NH…N and O…HN hydrogen-bond linkages. The volume of each atom is proportional to the magnitude of the associated SFρ% value and the bcp location is indicated by an azur dot. Positive and negative SFρ% values are distinguished by the blue and yellow colored atoms, respectively; (central and right panels) Partial Source Function Reconstruction Electron Density (PSFRD), in the least squares molecular plane of the AT complex, either using only the SFρ contributions from the ring 3 (R3) and the ring 4 (R4) atoms (top right panel), or only those from the ring R3 (bottom central panel) or only those from the ring R4 atoms (bottom right panel). For the sake of comparison, the top central panel displays the ED SF reconstruction in terms of all the system’s atom that yields, within the numerical integration errors, the total ED. Adapted with permission from Figs. 3 and 5 of C. Gatti, G. Macetti, R. J. Boyd, C. F. Matta, An electron density source-function study of DNA bases pairs in their neutral and ionized ground states, J. Comput. Chem., 39 (2018), 1112–1128. Copyright 2018 Wiley-VCH Verlag 15,530 GmbH&Co. KGaA.

4. Applications of the ED SF

293

whether the SF contribution from a given atom (or group of atoms) accounts or does not account for the most relevant features of the x scalar distribution in its (their) own basin(s) (Fig. 11.5). This behavior is clearly related to how much local or not local are the causes for the SF effects in those basins. In addition, the 2D plots of the PSFRD or PSFRP, when compared with the corresponding plots for the full density or potential, unveil whether the (percentage) SF reconstructions from a given subset of atoms remain or do not remain stable against rp displacements in a given region of space [18,24,25,89]. This analysis enables us to judge whether the choice of a specific rp in that region of space is a representative one or if it is to some large or small extent arbitrary. Another interesting aspect needs to be mentioned. At variance with standard partial densities, calculated by zeroing all basis set contributions of atoms not included in the considered subset of system’s atoms, the SF partial densities (or potential) represent a rigorous cause-effect decomposition of a scalar x that it is fully independent from the specific set of functions used to compute its spatial distribution. Fig. 11.5 illustrates the conventional ball and stick representation of the ED SF percentage atomic contributions SFρ% at the bcp ED of the three O⋯HC, NH⋯N and O⋯HN hydrogenbond linkages in the Adenine:Thymine (AT) DNA base pair complex and also the electron PSFRD in the least squares molecular plane of the complex, using various atomic subsets of the complex [89]. Relative to the NH⋯N and O⋯HN hydrogen bonds that have relatively localized sources, the longest and weakest O⋯HC HB has very delocalized sources and distinctly different SFρ% values, with largely negative SFρ% contributions from the H and the O atoms involved in the HB. This behavior is in agreement with the ED SF description of the HBs that is discussed later in this chapter (Section 4.1), as a function of the donor to acceptor distance. Yet, the peculiarity of the O⋯HC HB clearly emerges also from the four 2D PSFRD contour maps shown in Fig. 11.5. One of these maps (top central panel), include the SFρ contributions of all the atoms of the complex and therefore coincides with the total ED, within the numerical integration errors. This contour map serves for the sake of comparison with the other three that include either the SFρ contributions of only the ring 3 (R3) and the ring 4 (R4) atoms (top right panel) or only those from the ring R3 (bottom central panel) or only those from the ring R4 atoms (bottom right panel). While the central NH⋯N HB is reasonably well described by the SFρ contributions of either the R3 or the R4 atoms only and the O⋯HN HB by those of the R4 atoms only, the longest and weakest O⋯HC HB calls for the atomic sources from the remaining rings also in order to be adequately represented and to eventually reach a positive ED at bcp [89].

4. Applications of the ED SF 4.1 ED SF studies of the hydrogen bonds The ED SF has been largely applied to the study of hydrogen bonded systems using either calculated or X-ray diffraction derived EDs [2,3,13,16,18,22,80–96]. Reference 2, published in 2012, provides a thoughtful and highly detailed review on the subject, though it does not obviously include the results of the more recent studies. Here, for didactic reasons, only few results on a number of simple, prototypical cases are illustrated.

294

11. Chemical insights from the Source Function reconstruction of scalar fields relevant to chemistry

The extraordinary variety of the hydrogen bonds (HBs) geometries, strengths and dominating energetic contributions as a function of the nature of the H-donor and of the H-acceptor HB, has been generally found to be mirrored by dramatic changes in the SFρ % (HBbcp, Ω) contributions at the HB bcp ED from the atoms Ω of the hydrogen bonded molecular complexes or crystals [2,13,89]. By selecting a simple model system made by two water molecules that approach each other within the linear dimer Cs constraint, it is possible to explore how the atomic sources behave when the HO and OO distances vary from the values typical of the long and weak isolated HBs, to those characteristic of the very short and strong HBs (Fig. 11.6A). Typically, the former HBs have energies around 1–4 Kcal/mol and may be classified as weak electrostatic or van der Waals interactions, while the latter have much larger HB energies (15–50 Kcal/mol) and are almost covalent or, in the case of the so called chargeassisted HBs, strongly ionic in nature [168]. As shown in Fig. 11.6A, the atomic SF% contributions to the ED at the HB bcp radically change along the approach of the two water molecules, despite both the H-donor and the H-acceptor molecules always maintain a similar and hardly affected total source function contribution [13]. Three representative SF patterns ˚ is the OO distance at equilibrium, while are shown in Fig. 11.6A, where ROO ¼ 3.020 A

FIG. 11.6 Electron Density (ED) Source Function (SF) studies of hydrogen-bonded systems: (A) Evolution of Source Function percentage contributions SFρ% at the hydrogen bond bcp along the reaction path for the approach of two water molecules. The volume of each atom is proportional to the magnitude of the associated SFρ% value and the bcp location is indicated by an azur dot. Positive (negative) sources in blue (yellow); (B) L(r) ¼ r2ϱ(r) isocontour levels in the plane containing the H-donor molecule and the O acceptor atom for the same hydrogen-bonded systems shown in the first row (red contours, negative L(r) values; blue dotted contours, positive L(r) values); (C) SFρ% values at the hydrogen bond bcp (azure dot) in a number of prototypical hydrogen-bonded complexes [(H2O⋯H⋯OH2)+, 1; the open form of the formic acid-formate anion complex, 2; malonaldeyde, in its Cs equilibrium form, 3, and in its C2v transition state, TS, 4; cyclic water trimer, 5; water dimer at equilibrium geometry, 6]. Adapted from Fig. 4 of C. Gatti, The source function descriptor as a tool to extract chemical information from theoretical and experimental electron densities, Struct. Bond. 147 (2012), 193–286. With kind permission from Springer Science+Business Media B.V.

4. Applications of the ED SF

295

additional patterns along the reaction path are displayed in Ref 13. The SF% contributions to the HB bcp ED from the triad of atoms (OdHO) mostly involved in the HB interaction increase (from 32% up to 94%), while that of the remaining atoms correspondingly decrease ˚ ) and increasing energy, covalency and local with decreasing OO distance (from 3.25 to 2.0 A character of the HB. In particular, it is manifest that the SF% contribution from the H atom directly involved in the HB, SFρ % (HBbcp, H), represents a characteristic marker of the HB nature along the investigated reaction path. Such contribution is large in magnitude and negative (72%) at the equilibrium distance, then it increases to reach a slightly negative value (12%) at the OO distance typical of resonance assisted HBs (RAHBs) and becomes eventually positive (3%) only at the very short OO distance typical of charge assisted HBs, (CAHBs) [13]. Also interesting, is the evolution of the sum of the SF% contributions, SFρ % (HBbcp, H + Oacceptor) from the H atom and its acceptor O atom. This sum has a negative and large magnitude value at equilibrium distance, it becomes close to zero at distances ˚ ) typical of the long chains of OdHO bonds in water and alcohols and it (OO ¼ 2.750 A ˚ , OO distance. Note that reaches a positive value of about 50%, at the very short, 2.250 A a 50% SF contribution is still quite far from that observed for the covalent bonds that typically amounts to about 80%–90% (Refs. [2,13] and Section 3.1.1). Percentage sources of a compara˚ and ble value to those of standard covalent case occur only for OO distances below 2.5 A provided the source contribution from the donor O atom, SFρ % (HBbcp, Odonor), is also included in the sum. This observation gives support to the fact that the HB retains at least a three-center nature, even at such short OO separations [169,170]. At the equilibrium distance, the other atoms of the water dimer concur for almost half of the ED at the HB critical point providing a fairly delocalized picture of sources at the Cs linear dimer energy minimum. The dramatic change of the SFρ % (HBbcp, H) contributions with varying OO separations and, even more, the large distance interval where these contributions are highly negative may be puzzling, if not disturbing, at a first sight. Yet, this behavior has a clear-cut interpretation. Fig. 11.6B shows the contour maps of the negative ED Laplacian L(r) overlaid by the intersections of the atomic boundaries in the symmetry plane containing the donor water molecule and the HB bond path. By focussing on the H atom involved in the HB, one may notice that at equilibrium distance its basin is quite asymmetric in shape and that the L(r) distribution within the basin is greatly polarized along the HB bond path. Being L(r) largely positive toward the Odonor-H bcp and largely negative toward the H-Oacceptor bcp, a usual H atom positive source (+36%) at the Odonor-H bcp is recovered and a source, as largely negative as 72%, results for the ED at the HB bcp [13]. More precisely, the former positive H source contribution arises by the dominant contribution of the small region of high charge concentration (L(r) >0) close to and surrounding the OdH bcp, while the negative H source at the HB bcp ED is due to this bcp being located in a wide region of charge depletion (L(r) 0). The form of the Laplacian of the charge density provides a more complete resolution of the shell structure of s and p atoms than the radial density function [10]. However, r2ρ(r) cannot resolve each expected atomic shell due to the last two shells of the elements after the fourth-row collapse [11]. The mapping of the Laplacian of the conditional pair density (LCPD) for same-spin electrons shows that every charge concentration can be associated with a corresponding maximum in LCPD, denoting a partial condensation of the pair density toward individually localized electron pairs [12]. The topology of r2ρ(r) of an atom in a molecule is a measure of the distortion of the electron density from the spherical distribution of an isolated atom where charge concentration and depletion regions appear due to the perturbations by the presence of the rest of the atoms. During the formation of a covalent bond, two isolated atoms approach their electron density to polarize in two charge concentration regions, bonding and nonbonding, as shown in Fig. 15.1 [13]. Within the valence shell, a charge concentration (CC) indicates a Lewis base or a nucleophilic site. In contrast, a charge depletion (CD) denotes a Lewis acid or an electrophilic site. Several chemically meaningful concepts have been linked with CCs and CDs. For instance, CCs have been related to the essential chemical idea of electron-pair domains [14], which are regions of real space where there is a high probability of finding two electrons of opposite spin as a result of the Pauli antisymmetry principle.

r2 ρðrÞ ¼

FIG. 15.1

∂2 ρðrÞ ∂2 ρðrÞ ∂2 ρðrÞ + + ∂x2 ∂y2 ∂z2

Evolution of the Laplacian of electron density as two nitrogen atoms approach.

(15.1)

4. Atomic graph

391

FIG. 15.2 Laplacian envelope maps of CrCO6 (left) and chromium atom in complex (right).

3. Valence shell As we noted before, in real space, it is not possible to define the valence in terms of molecular orbitals. Instead, r2ρ(r) locates the valence shell at a distance from the nucleus where the last charge concentration and depletion shells are placed. It is possible to estimate the valence region from the shell structure of isolated atoms using Eq. (15.2), where the valence distance (γ υs) is a function of the atomic number (Z) and the principal quantum number (n). γ υs ¼ αeβZ

α ¼ 0:106e1:78n

β ¼ 0:058n  0:352

(15.2)

r ρ(r) is useful to identify the polarization of the core and the valence shells [1,2,15], Bader used this property to support the valence-shell electron-pair repulsion model or VSEPR, developed by Gillespie. VSEPR predicts molecular geometries based on the expected behavior of the electron-pair domains around an atom, assuming that the valence density is spatially localized in regions whose arrangement maximizes the separation among them [14]. In this way, the VSEPR is reduced to one postulate: the most stable molecular geometry of a molecule AXn corresponds to maximizing the separations between the local maxima in the valence shell of charge concentration of the atom A defined by r2ρ(r) [3]. Bader, Gillespie, and Martin proposed that a heavy central atom of a metal complex may be susceptible to ligand-induced polarization of the outer shell. These authors further concluded that r2ρ(r) is useful to localize the signatures of the local polarization of the central atom [1,2,16]. Fig. 15.2 shows the valence shell polarization of the chromium in the CrCO6 complex. It is possible to observe that the metal valence shell presents concentration and depletion sites. The carbon’s charge concentration of the ligands is directed to the charge depletion of the metal [17]. 2

4. Atomic graph The information about the atomic polarization described by r2ρ(r) is condensed in one topological object called atomic graph (AG), which is the set of critical points (CP, a point where the gradient of a scalar field vanishes) and the gradient paths of r2ρ(r), in a polyhedron shape around the atomic core [18]. CP allows identifying the valence shell (VS) of an atom in a

392

15. Spin polarization of the atomic valence shell

FIG. 15.3 The atomic graph is specified by a polyhedron, where the charge concentrations, (3,+3) CPs, define the vertices (V). The edges (E) are associated with the pair of flux lines originating at intervening (3,+1) CPs. The faces (F) represent depletions, (3,1) CPs, of the electron charge.

molecule, which is composed of local charge concentrations (CC) and depletions (CD). From a topological point of view, the CCs in the VS are (3,+3) CPs, and they define the vertices (V) of the polyhedron. The edges (E) are set by the pair of flux lines originating at intervening (3,+1) CPs. Every face (F) of this polyhedron has a (3,1) CP in the center, which represents a depletion of charge in the VS. An AG is denoted by the specific set [V, E, F], and it must satisfy the Euler’s formula for a polyhedron VE + F ¼ 2, Fig. 15.3 [19,20]. Popelier performed extensive work to determine the entire r2ρ(r) topology and its changes, identifying four regions of an atom in a molecule: the core-shell charge concentration (CSCC), the core-shell charge depletion (CSCD), the valence shell charge concentration (VSCC) and the valence shell charge depletion (VSCD). Each region has a set of critical points associated with a graph. At the ground state, the atomic graph has been used to define functional groups [21], metal-ligand donor-acceptor [17], metal-metal interactions [22], and conformational changes [23]. Also, the AG can explain the shielding or deshielding of agostic protons and the description of reaction paths observing the expansions or contractions of the AG in every individual step of reaction [24]. AG’s evolution at excited states gives insight into the valence shell’s polarization as a mechanism to compensate for the atomic destabilization produced by an electronic excitation [25]. The AGs can be obtained by accurate experimental electron density distribution, determined through X-ray diffraction at low temperatures and multipolar refinements [26]. Our group found a relationship between the number (and spin states) of electrons in d orbitals and the set of critical points that constitute the atomic graph of a metal center in a set of selected hexaaquo complexes with octahedral symmetry. The calculations were performed at the PBE0/def2-TZVPD level of theory [27]. This led us to classify the atomic graphs of the metals analyzed with different combinations of charge and multiplicity in only five types (I–V) represented in Table 15.1. Some types of atomic graphs present the same set of critical points (even adding the connectivity information of CC critical points). However, the orientation of the critical points and the directionality with respect to the ligands allows such classification. For example, Type I and III atomic graphs have the set of critical points 6, 8, 12 but in Type I the ligands point to (3,+1) critical points while in Type III the ligands interact with (3,1) critical points.

393

4. Atomic graph

TABLE 15.1 Atomic graphs (AG) of metal center in complexes [M(H2O)6]m+, where M is first-row d-block metals with oxidation states m and multiplicity 2 s + 1. AG

Type I

II

III

IV

[V(C),E,F] [8(3),12,6]

[6(4),12,8]

[8(3),12,6]

[4(4),8,6]

Mm+

2s+1

Electron configuration

2

[Ar]4s 3d

1,0

Fe2+

5

[Ar]4s03d6

5,1

Ti2+ V3+

3

[Ar]4s03d2

2,0

Co2+ Ni3+

4

[Ar]4s03d7

5,2

V2+ Cr3+ Mn4+

4

[Ar]4s03d3

3,0

Ni2+

3

[Ar]4s03d8

5,3

Cr2+ Mn3+

5

[Ar]4s03d4

4,0

Cu2+

2

[Ar]4s03d9

5,4

Sc Ti3+ V4+

0

α,β

1

2+

Continued

394

15. Spin polarization of the atomic valence shell

TABLE 15.1 Atomic graphs (AG) of metal center in complexes [M(H2O)6]m+, where M is first-row d-block metals with oxidation states m and multiplicity 2 s + 1—cont’d AG

Type V

[V(C),E,F] [6(4),8,6]

Mm+ Sc

3+ 2+

2s+1

Electron configuration 0

0,0

0

5

[Ar]4s 3d

1

α,β

0

Mn Fe3+

6

[Ar]4s 3d

5,0

Zn2+

1

[Ar]4s03d10

5,5

(3,+3) CPs in purple, (3,+1) CPs in red and (3,1) CPs in yellow. The atomic graphs are described as a polyhedron by the number of its vertices, V (charge concentration and its connectivity, C), its edges, E, and the faces, F, [V(C),E,F]. α and β are the spin electron occupation in the valence shell.

5. Atomic spin graphs and catastrophe process With the possibility to split P the electron density into symmetry or spin components following an additive scheme, ρ ¼ i ρi, the systematic behavior of the atomic graphs of the previous sections and even the structure and reactivity of a metal center can be described from the effects of each component of the electron density. McWeeny showed in the electron density the explicit spin dependence of density matrices, where the nonvanishing spin components of ρ1(r1; r10 ) and ρ2(r1; r2) for a spin state are shown in Eq. (15.3) [28]. The analysis of the spin density (ρs ¼ ρα  ρβ) is more common than the separate analysis of the α and β spin components of electron density (ρ ¼ ρα + ρβ).      ββ  0 0 ρ1 r1 ; r01 ¼ ραα 1 r1 ; r1 + ρ 1 r1 ; r1 (15.3) αβ βα ββ ρ2 ðr1 ; r2 Þ ¼ ραα 2 ðr1 ; r2 Þ + ρ2 ðr1 ; r2 Þ + ρ2 ðr1 ; r2 Þ + ρ2 ðr1 ; r2 Þ On the other hand, it is also possible to apply the Laplacian operator to each spin component of the electron density of its components. r2ρ(r) or its spin version is a consequence of each molecular orbital contribution and not just the valence orbitals or a specific set of orbitals. X X  2 r2 jϕiα ðrÞj2 + r2 ϕiβ ðrÞ (15.4) r2 ρðrÞ ¼ r2 ρα ðrÞ + r2 ρβ ðrÞ ¼ i

i

Therefore, an atomic graph can be associated with each spin component and the polarization of the valence shell in an atom is a consequence of the interaction between both spin shells. To exemplify this point, Fig. 15.4 represents the fictitious situation where the Laplacian of the β electron density is progressively added to the Laplacian of α electron density, r2ρ0 ¼ r2ρα + λr2ρβ, where λ is the control parameter that allows r2ρ0 ¼ r2ρ when λ ¼ 1.

5. Atomic spin graphs and catastrophe process

395

FIG. 15.4 Evolution of r2ρ0 ¼ r2ρα + λr2ρβ for different λ values: (A) 0.02, (B) 0.06, (C) 0.10, (D) 0.20, (E) 0.60, (F) 1.00.

The result of this process depends on the magnitude of the curvatures in opposite directions. Equal magnitudes result in the annihilation of both critical points but with unequal magnitudes, the topological contribution of one critical point remains. In the case of metal atoms, Fig. 15.5 presents a series of examples for the different types of atomic spin graphs and their overlapping in order to analyze the interaction between spin shells. The simplest case to analyze is the Type III atomic graph for vanadium and nickel complexes, Fig. 15.5A. The main difference between these two examples is the number of electrons in the d orbitals (or valence shell). V2+ has three α d electrons while Ni2+ has five α d electrons and three β d electrons. It is important to distinguish that the same classification in five types of atomic graphs can be done in the spin components of the electron density, however, the difference in electron configurations have a direct consequence in the types of atomic spin graphs. The α spin component shows a Type III atomic graph in the case of V2+, but we observe this same distribution of critical points in the β spin component for Ni2+, while the other spin component in each case presents a spherical distribution. This behavior is observed in all the Types of atomic graphs of metal complexes with high spin electronic state,

396

15. Spin polarization of the atomic valence shell

(A) α and β atomic graphs of complexes [V(H2O)6]2+ and [Ni(H2O)6]2+ are displayed together as they are overlapping. (B) Comparison between α and β atomic graphs of the five different atomic graph types of aquo complexes with high spin electron configuration. Type I: Fe2+, Type II: Ni3+, Type III: Ni2+, Type IV: Cu2+, Type V: Zn2+.

FIG. 15.5

Fig. 15.5B. So that the classification of valence shell polarization patterns depends on the incomplete spin shell, meanwhile the complete spin shell always has a spherical distribution. If we analyze the interaction between spin shells by focusing on the critical points of the atomic spin graphs, the result is a series of catastrophe processes between critical points in specific regions of the valence shell of a metal. A catastrophe is discontinuous and abrupt changes of the atomic graph due to changes in the number or type of critical points, giving a new atomic graph. Bader introduced catastrophe theory to the analysis of the topology of ρ(r) to define molecular change [29] and Popelier extensively studied the changes of the topological features of r2ρ(r) during the conformational change from tetrahedral to planar NH3 [23]. In this sense, Fig. 15.5A shows how critical points of different nature coincide in specific regions, for example, CC and CD regions. The result of this interaction is the dominance of one spin shell over another to form the atomic graph of the total electron density through constructive processes or the annihilation of critical points.

6. Polarization of the metal valence shell in metal-ligand interaction

397

6. Polarization of the metal valence shell in metal-ligand interaction Up to this point, the polarization pattern in the valence shell of a metal has been analyzed at the equilibrium geometry of the complex. The next step is to describe the effect of the position of the ligands in the coordination sphere and their interaction with the valence shell of the metal. Modeling the gradual approach of the six water molecules in the hexaaquo complexes it is possible to distinguish this phenomenon (using MP2/def2-TZVPD theoretical level). Fig. 15.6 shows the polarization of the Laplacian of electron density and its spin components in the metal-ligand interaction in the complex [Ni(H2O)6]2+. ˚ distance there is no interaction between the metal center and the ligands and all At the 5.5 A the spin density is concentrated in the metal. An important characteristic is that at this distance the ligands are pointed in the direction of CC regions in the metal (for the total electron density and its spin components). However, as the metal-ligand distances begin to decrease, there is a characteristic distance where the metal-ligand interaction begins and there is a charge transfer from the ligand to the metal, Table 15.2. Consequently, the spin electron density appears in the water molecules and it is slightly lost in the metal. We denominated this distance as the spin transfer distance (STD). It is an interesting value because it determines the communication of the spin information between two chemical species. Furthermore, an important change at this point is the reorientation or polarization of the valence shell of metal center in a way that favors the metal-ligand interaction; that is CC of the ligands points to CD of the metal. As Ni is the metal center in Fig. 15.6, we observe these changes in the β electron density while the α electron density distribution remains spherical at all distances.

FIG. 15.6 Envelope maps of the Laplacian of electron density and its spin components during the changes of distance metal-ligands in the complex [Ni(H2O)6]2+. Metal center on the left side and oxygen atom on the right side of each contour.

398

15. Spin polarization of the atomic valence shell

˚ ) to the TABLE 15.2 Changes in electron population from a distance without metal-ligand interaction (5.5 A 2+ ˚ equilibrium distance in complexes [M(H2O)6] . ΔN ¼ NeqN5.5A.

Complex

Atom(s)

[Sc(H2O)6]

2+

Sc 6(H2O)

[Ti(H2O)6]

2+

Ti 6(H2O)

[V(H2O)6]

2+

V 6(H2O)

[Mn(H2O)6]

2+

Mn 6(H2O)

2+

[Fe(H2O)6]

Fe 6(H2O)

[Co(H2O)6]

2+

Co 6(H2O)

[Ni(H2O)6]

2+

Ni 6(H2O)

[Zn(H2O)6]

2+

Zn 6(H2O)

ΔN

ΔNs

ΔN

ΔN

0.29

0.16

0.06

0.23

0.29

0.16

0.06

0.23

0.27

0.16

0.06

0.21

0.27

0.16

0.06

0.21

0.32

0.18

0.07

0.25

0.32

0.18

0.07

0.25

0.29

0.21

0.04

0.25

0.29

0.21

0.04

0.25

0.34

0.16

0.09

0.25

0.34

0.16

0.09

0.25

0.38

0.13

0.12

0.26

0.38

0.13

0.12

0.26

0.42

0.12

0.15

0.27

0.42

0.12

0.15

0.27

0.40

0

0.20

0.20

0.40

0

0.20

0.20

˚) STD (A 4.5–5.0

4.5–5.0

4.5–5.0

4.0–4.5

4.0–4.5

4.0–4.5

4.0–4.5

–

7. Atomic graphs in the excited states The atomic graphs can be used to trace the origin of the geometrical changes of metal complexes after an electronic excitation. Photophysical properties of copper (I) complexes (CIC) with bis-diimine ligands, such as 1,10-phenanthroline (phen) and their 2,9-disubstituted derivatives, have been of great interest due to their drastical geometrical change observed after excitation and relaxation [30]. CIC has a D2d tetrahedral geometry at ground state, where the ligands show a perpendicular conformation. Chemical oxidation of the metal center provokes a geometrical change from tetrahedral to a planar square geometry with a lower D2 symmetry, which is the preferred arrangement by the copper (II) compounds [31–34]. During a

7. Atomic graphs in the excited states

399

FIG. 15.7 Photo-induced CIC flattening. The dihedral angle between phenanthroline rings scanned from tetrahedral to quasiplanar conformation.

vertical photoexcitation of CIC, a reversible and partial charge transfer from the metal center to the aromatic ligands is observed [35] and a latter structural relaxation within the excited state potential surfaces produces a flattening of the geometry that closely resembles the structure of copper (II) complexes [36–40], as shown in Fig. 15.7. The driving force of the photoinduced structural change of [Cu(phen)2]+ and its derivatives is usually explained in terms of the molecular orbital approach to the Jahn-Teller effect, which is a geometric distortion that reduces the symmetry and energy of a metal complex. This effect is observed after a charge transfer from the d orbitals of the metal to the lowest unoccupied molecular orbital (LUMO) of the ligands, which produces a degenerate electronic state and thus a structural change. However, the origin of the photoinduced geometrical change cannot be fully understood based on the molecular orbital changes. When the molecular orbitals of the whole metal-ligand system are considered, the electronic state of the excited state is actually not degenerate [31] and the molecular orbital change is similar for both the ground state and the first excited state, however, in S0, it is associated with a barrier, whereas in S1, a minimum is observed. After the photoexcitation the metal-to-ligand charge transfer produces a stabilization of the ligand and destabilization of the metal originated by the reduction of the attractive potential energy within the copper basin. To counterbalance the atomic destabilization the copper center polarizes its valence shell during the deactivation path that provokes the flattening of the structure to obtain the geometry with the maximum interaction between the charge depletion in the metal and the charge concentrations of the ligand. To understand this process, an analysis of the polarization of its valence shell was performed. Table 15.3 displays the contour maps and the atomic graphs of the metal center in four different states. A system without polarization presents a spherical distribution of the valence shell as an isolated atom or a transition metal with a full d shell; the polarization appears with the interactions with the ligands or in an electronic excitation. Cu(I) at the ground state, a d10 metal center, presents an almost spherical VSCC (the difference between the average of r2ρCC(r) and r2ρCD(r) is 12.65 a.u.) with the ligands in a tetrahedral configuration and the CC of the metal in the direction of the ligands. In this case, the metal-ligand interaction is mainly ionic, without a bond directionality, and with a tetrahedral

400

15. Spin polarization of the atomic valence shell

TABLE 15.3 Atomic graph, contours plot, and average values of CC and CD of r2ρ(r) after vertical excitation and relaxation. The set of critical points of the atomic graphs are shown as vertices (V), edges (E), faces (F) and symmetry (S). Electronic state

S0

S1

T1

S0

Torsion angle [degrees]

90

90

30

30

[4, 6, 4, Td]

[4, 8, 6, D4h]

[4, 8, 6, D4h]

[6, 8, 4, C1]

r2 ρðrÞCC

70.60

87.50

105.15

69.28

r2 ρðrÞCD

57.96

42.85

29.72

59.07

AG

[V, E, F, S] r ρ(r) 2

geometry—the best configuration of four atoms around a spherical distribution, according to Gillespie’s rules. In this case, the atomic graph is characterized by the critical point set [4,6] with a Td symmetry. The vertical excitation provokes a polarization of the VSCC, which causes a change in the atomic graph, from [4, 6, 4, Td] to [4, 8, 6, D4h], and the formation of different CC and CD regions. These changes in the atomic graphs can be explained by a catastrophe process. In this process, the difference between the charge concentrations and depletions at VSCC, increases during the deactivation process. This difference is around 12.65 a.u., 44.66 a.u. and 75.42 a.u. at S0 (90 degrees), S1 (90 degrees) and T1 (30 degrees), respectively. It is interesting that, regardless of the state, the sum of the values of charge concentrations and depletions at the VSCC remains almost constant, with an average value of 130.8 a.u. The polarization of the VSCC causes the formation of CD regions on some faces of the octahedron that can be used to represent the atomic graph of the S1 (90 degrees) excited state, shown in Fig. 15.8. A process of geometric relaxation takes place immediately after the initial polarization of the VSCC. This process leads to the flattening of the molecule to maximize the equatorial interaction between the charge depletion regions of the valence of the metal center and the nonbonded charge concentration regions of nitrogen atoms of the ligands. After the geometric relaxation on the potential energy surface of the S1 state, the atomic graph of the metal center could be described as the cube-like polyhedron shown in Fig. 15.8, where the CD regions are on the metal face and the nonbonded CC region are on the ligands.

8. Relationship between the atomic graph and the atomic polarization

401

FIG. 15.8 Contour plots and atomic graph of the metal center in [Cu(4,7-dmp)2] + (4,7-dmp ¼ 4,7-dimethyl-1,10phenanthroline) in different electronic states. Dihedral angle between ligands enclosed in parenthesis at the top.

A similar change is observed at the minimum of the T1 excited state shown in Table 15.3, where the charge concentrations and depletions reach their extreme values, 105.14 a.u. and 29.72 a.u. respectively, where the contour maps show the maximum polarization with the largest metal-ligand interaction with a directionality like that observed in the Cu(II) complexes. At this point, we found the shortest Cu-N distance and the ρbcp and δ(Cu, N) maxima [41]. The polarization is lost after deactivation to S0 state.

8. Relationship between the atomic graph and the atomic polarization To find a relationship between the atomic graph’s features with the atomic polarization we developed the “atomic graph descriptor” (AGD) as the difference between charge concentration and depletion values within an AG as shown by Eq. (15.5) and applied to a data set of 54 copper complexes. The data set contains complexes with 20 different ligands that link the central copper atom through nitrogen and oxygen atoms. All copper complexes were optimized at the M06-2/6-311++g(d,p) level of theory and a set of quantum topological descriptors and atomic graphs of the metal center were later calculated for Hexa-aqua copper(II) and all the complexes of the data set. X X r2 ρðrÞcd (15.5) AGD ¼ r2 ρðrÞcc  As expected from Tables 15.1 and 15.3, all the copper atoms in the data set have a Type IV atomic graph with 4 CC and 6 CD. With the numerical values of r2ρ(r) from the aforementioned CPs. The best model of the correlation is shown in Eq. (15.6) and the parameters are presented also in Table 15.4. The descriptors in the model are three copper quantum

402

15. Spin polarization of the atomic valence shell

TABLE 15.4

Examples of the complexes and its descriptors value. [Cu(H2O)6]2+ is the reference.

Reaction 2+

1

[Cu(H2O)6] + gly 2+

! [Cu(gly)(H2O)4]

1

[Cu(H2O)6] + 2 gly 2+

1

[Cu(H2O)6] + leu 2+

[Cu(H2O)6] + 2 leu 2+

1

[Cu(H2O)6] + pro 2+

1+

! cis-[Cu(leu)2(H2O)2]

! [Cu(pro)(H2O)4]

1

[Cu(H2O)6] + 2 pro 2+

! trans-[Cu(gly)2(H2O)2]

! [Cu(leu)(H2O)4]

1

1+

! trans-[Cu(pro)2(H2O)2]

4

[Cu(H2O)6] + EDTA

1+

! [Cu(EDTA)]

2

ΔQ1(Cu)

ΔNɑ(Cu)

ΔNβ(Cu)

0.2436

0.0237

0.0689

0.3806

0.0389

0.1021

0.2161

0.0213

0.0685

0.5207

0.0387

0.1064

0.2190

0.0221

0.0716

0.4192

0.0370

0.1079

0.0782

0.0436

0.1024

topological quantities: the change of α, β electronic populations and the quadrupole eigenvalue. The statistical parameters of the proposed model indicate that the three descriptors used are not statistically related and they can predict molecules not included in the data set. All the copper atoms from the data set gain an electronic population, both α and β electrons, if we compare with these same values at Hexa-aqua copper(II). In consequence, what sets apart one type of electronic population from the other is the sign of the coefficient: positive for the α contribution and negative for the β contribution. The increase in copper electronic population causes a change in its quadrupole moment. This descriptor measures the elongation (Q < 0) or depletion (Q > 0) of the electron density along an axis. In the data set, the AGD correlates with the quadrupole eigenvalue that is measured in the direction of the axial bonds, Fig. 15.9. In Table 15.4, we present some examples of the quadrupole values of complexes in the data set. The negative value of ΔQ1(Cu) in all the complexes indicates that

FIG. 15.9

(A) Representation of the change of copper (II) quadrupole moment from the metal aquo complex to any complex in the data set [CuL]. The atomic electron density elongates in the Q1(Cu) direction (height of the figure). (B) The Direction of Q1(Cu) inside the complexes (axial bond direction) is represented by a blue arrow.

9. Energy polarization within the valence shell

403

the copper electron density elongates further along the axial bond axis as the ligands change from water to other ligands, Fig. 15.9. This observation leads to the idea that the AGD can describe the electron density redistribution as the copper atom is being polarized by the ligands. AGD ¼ 234:113ΔN α ðCuÞ + 239:999ΔN β ðCuÞ + 11:653ΔQ1 ðCuÞ  142:693

(15.6)

9. Energy polarization within the valence shell As we mentioned, within the valence region can occur several electronic and energetic processes. Here we present the behavior of the Laplacian of the attractive and repulsive potential energies in the valence shell. Fig. 15.10 shows the isosurfaces of r2Vneα(A, A) and r2Vneβ(A, A) for Ni2+ and V2+ in hexaaquo complexes. As in Fig. 15.5, the spin component gives the same feature as r2ρ(r) for r2Vne(A, A). The spin is the origin of this behavior, where it is possible to relate a charge concentration with the maxima of the nuclear-electron interactions. It is important to emphasize that the incomplete spin shell determines the concentration regions in the attractive and repulsive potential energies. It is also possible to identify the valence shell in these energetic fields, where the polarization of the electron density is accompanied by an increase of the nuclear-electron attraction and the electron-electron repulsion in the same regions, i.e., when the electron density in the valence shell redistributes to regions where the nuclear-electron attraction is maximum, along with the corresponding increase in repulsive interactions between electrons. The mechanism

Contours of r2Vne(r) and r2Vee(r) for total electron density and its spin components in complexes [V(H2O)6] and [Ni(H2O)6]2+ at the equatorial plane with lines in gray representing the direction of the ligands. Positive values in blue and negative values in red.

FIG. 15.10 2+

404

15. Spin polarization of the atomic valence shell

Isosurfaces of the Laplacian of the attractive potential energy, r2Vne, during the photoinduced flattening of complex [Cu(phen)2]+. The isovalue for every envelope map is 1600 a.u.

FIG. 15.11

involved in the interaction between α and β shells are slightly different from that assumed by Linnett and Gillespie. Given a perturbation provoked by a ligand, in each spin shell the metal electron density redistributes to maximize the nuclear-electron interaction to compensate for the emergence of the electron-electron repulsion concentrations. Then, the spin shells interact by several catastrophe processes to produce the density and energetic features of the metal valence shell. In the case of excited states, we found stronger evidence that the origin of photoinduced geometrical distortion of copper (I) complexes is the change of the electron nuclear potential density. We found that the Laplacian of this component of the potential density, r2Vne, can also describe the polarization of the VSCC during the photophysical process and support our concept. Fig. 15.11 depicts the envelope maps of r2Vne around the copper core, in the four states. At S0, the concentration of Vne distributes as a spherical envelope with small depletions located opposite to the CudN bonds. After the photoexcitation and the first metal-ligand charge transfer (S1v), each depletion increases its size to include the closest CudN bond path. The geometrical relaxation (S1) places the bond paths in the center of each r2Vne depletions and the second electron transfer (T2) produces the axial depletions. The redistribution of the attractive potential density allows defining donor-acceptor interactions associated with the ligand-metal bonds based on electron-nuclear interactions. In the coordination sphere of an electron-deficient metal center, the configuration of the ligands depends on the location of the depletions of Vne on the metal VSCC, which are directed toward the concentration of Vne on the valence shell of the donor atoms of the ligands. In this way, it is possible to say that the r2Vne is homomorphic to r2ρ, at least where r2ρ < 0, due to the dominant role of potential energy density in those regions and thus the maximization of electronnuclear interactions is the driving force behind the polarization of electron density described by r2ρ.

References

405

10. Conclusions In this paper, we focused our attention on the properties of the atomic graph. This topological object summarizes the charge concentration, a depletion in the spin valence shells of an atom in a molecule or a metal within a complex. There are five basic atomic graphs within the octahedral symmetry, which shape is determined by the number of electrons into the valence shell. We found that an atomic graph is the result of catastrophe processes between the α and β components, where the local dominance of one spin shell over the other determines the final shape of the atomic graph and the disposition of the ligand in the coordination sphere. In this approach, it is possible to find the distance that determines the communication of the spin information between two chemical species. We also found that the separation between charge concentration in the valence shell of the metal, the shape of the atomic graph, is the product of the maximization of the nuclear-electron interactions in each spin shell to compensate for the emergence of the electron-electron repulsion concentrations. In the case of the excited states, the polarization of the valence shell causes the flattening of the structure to maximize the interaction between the charge depletion sites of the metal with the charge concentration regions of the ligand. The quantum topological origin of the Jahn-Teller effect is fork into two groups, axial and equatorial where the axial charge depletions increase its potential energy in a stronger way than the equatorial position.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

I. Bytheway, R.J. Gillespie, T.-H. Tang, R.F.W. Bader, Inorg. Chem. 34 (1995) 2407–2414. J. Muna´rriz, M. Calatayud, J. Contreras-Garcı´a, Chem. A Eur. J. 25 (2019) 10938–10945. R.F.W. Bader, R.J. Gillespie, P.J. MacDougall, J. Am. Chem. Soc. 110 (1988) 7329–7336. B. Silvi, A. Savin, Nature 371 (1994) 683–686. 3716499. M.T. Carroll, C. Chang, R.F.W. Bader, Mol. Phys. 63 (1988) 387–405. P.J. MacDougall, M.B. Hall, R.F.W. Bader, J.R. Cheeseman, Can. J. Chem. 67 (1989) 1842–1846. T.-H. Tang, L.R.F.W. Bader, P.J. MacDougall, Inorg. Chem. 24 (1985) 2047–2053. R.F.W. Bader, P.J. MacDougall, J. Am. Chem. Soc. 107 (1985) 6788–6795. R.F.W. Bader, P.J. MacDougall, C.D.H. Lau, J. Am. Chem. Soc. 106 (1984) 1594–1605. Z. Shi, R.J. Boyd, J. Chem. Phys. 88 (1988) 4375–4377. M. Kohout, A. Savin, H. Preuss, J. Chem. Phys. 95 (1991) 1928–1942. R.F.W. Bader, G.L. Heard, J. Chem. Phys. 111 (1999) 8789. J. Herna´ndez-Trujillo, R.F.W. Bader, J. Phys. Chem. A 104 (2000) 1779–1794. R.J. Gillespie, E.A. Robinson, Angew. Chem. Int. Ed. Engl. 35 (1996) 495–514. W. Scherer, P. Sirsch, D. Shorokhov, M. Tafipolsky, G.S. McGrady, E. Gullo, Chem. A Eur. J. 9 (2003) 6057–6070. R.F.W. Bader, R.J. Gillespie, F. Martı´n, Chem. Phys. Lett. 290 (1998) 488–494. F. Cortes-Guzma´n, R.F.W. Bader, Coord. Chem. Rev. 249 (2005) 633–662. P.L.A. Popelier, Coord. Chem. Rev. 197 (2000) 169–189. K. Collard, G.G. Hall, Int. J. Quantum Chem. 12 (1977) 623–637. N.O.J. Malcolm, P.L.A. Popelier, Faraday Discuss. 124 (2003) 353–363. discussion 393–403, 453–455. P.L.A. Popelier, J. Burke, N.O.J. Malcolm, Int. J. Quantum Chem. 92 (2003) 326–336. P. Macchi, A. Sironi, Coord. Chem. Rev. 238–239 (2003) 383–412. N.O.J. Malcolm, P.L.A. Popelier, J. Phys. Chem. A 105 (2001) 7638–7645. F. Cortes-Guzma´n, R.M. Go´mez, T. Rocha-Rinza, M.A. Sa´nchez-Obrego´n, J.M. Guevara-Vela, J. Phys. Chem. A 115 (2011) 12924–12932. [25] L. Gutierrez-Arzaluz, D.I. Ramı´rez-Palma, L.G. Ramı´rez-Palma, J.E. Barquera-Lozada, J. Peon, F. CortesGuzma´n, Chem. A Eur. J. 25 (2019) 775–784.

406 [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

15. Spin polarization of the atomic valence shell

P. Macchi, D.M. Proserpio, A. Sironi, J. Am. Chem. Soc. 120 (1998) 13429–13435. D.I. Ramı´rez-Palma, F. Cortes-Guzma´n, Phys. Chem. Chem. Phys. 22 (2020) 24201–24212. R. McWeeny, Y. Mizuno, Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 259 (1961) 554–577. R.F.W. Bader, Y. Tal, S.G. Anderson, T.T. Nguyen-Dang, Isr. J. Chem. 19 (1980) 8–29. A. Laviecambot, M. Cantuel, Y. Leydet, G. Jonusauskas, D. Bassani, N. McClenaghan, Coord. Chem. Rev. 252 (2008) 2572–2584. M. Iwamura, S. Takeuchi, T. Tahara, J. Am. Chem. Soc. 129 (2007) 5248–5256. M.W. Mara, K.A. Fransted, L.X. Chen, Coord. Chem. Rev. 282–283 (2015) 2–18. M. Ruthkosky, F.N. Castellano, G.J. Meyer, Inorg. Chem. 35 (1996) 6406–6412. M. Schmittel, C. Michel, S.-X. Liu, D. Schildbach, D. Fenske, Eur. J. Inorg. Chem. 2001 (2001) 1155–1166. N. Armaroli, G. Accorsi, F. Cardinali, A. Listorti, Top. Curr. Chem. 280 (2007) 69–115. L. Hua, M. Iwamura, S. Takeuchi, T. Tahara, Phys. Chem. Chem. Phys. 17 (2015) 2067–2077. M. Iwamura, S. Takeuchi, T. Tahara, Phys. Chem. Chem. Phys. 16 (2014) 4143–4154. J.J. Mcgarvey, S.E.J. Bell, K.C. Gordon, Inorg. Chem. 27 (1988) 4003–4006. J.J. McGarvey, S.E.J. Bell, J.N. Bechara, Inorg. Chem. 25 (1986) 4325–4327. M.T. Miller, P.K. Gantzel, T.B. Karpishin, Inorg. Chem. 37 (1998) 2285–2290. X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304–314.

C H A P T E R

16 A bond bundle case study of Diels-Alder catalysis using oriented electric fields Timothy R. Wilson and Mark E. Eberhart Department of Chemistry, Colorado School of Mines, Golden, CO, United States

1. Introduction—Atomic basins and bond bundles Bader observed that regions of electron charge density (ρ) bounded by zero-flux surfacesa possess unambiguous energies and energy-mediated properties [1]—useful for checking results against measured/predicted values, and for making contact with the broader physical sciences. For any system, there exists in ρ an infinite number of such regions, which are called gradient bundles [2]. This infinite set is elegantly reduced by appealing to the topology of ρ, which precisely defines a finite set of regions that correspond to atoms in molecules (atomic basins) [1,3]. In turn, ρ can be infinitesimally decomposed into differential gradient bundles (GBs), producing the gradient bundle condensed charge density (P). The topology of P defines a second set of regions corresponding to chemical bonds (bond bundles) or nonbonding regions (e.g., lone pairs) [2,4]. Partitioning ρ into either atomic basins or bond bundles produces a set of nonoverlapping regions that combine to fill space, and that can be integrated over to yield a set of condensed regional properties such as atomic (or bond) volume, energy, and electronic population (electron count). Comparing atomic condensed properties across chemical systems, one can inspect the property flow between atoms resulting from perturbations to the system. Likewise, comparing bond bundle condensed properties, one can inspect the property flow within and between bonds, and between the particular atomic regions (bond “wedges”) participating in a bond [5]. Taken together, the sets of atomic and bond condensed properties a Surfaces

through which the flux of electron density gradient is everywhere zero.

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00007-4

407

Copyright # 2023 Elsevier Inc. All rights reserved.

408

16. A bond bundle case study of Diels-Alder catalysis

provide a set of qualitative and quantitative tools for investigating molecular and condensed phase phenomena. Here, using the combined toolset of atomic basin/bond bundle decompositions, we partially reproduce the investigation by Meir et al. into Diels-Alder catalysis using oriented electric fields [6], which is a growing area of study especially as it pertains to enzymatic catalysis [7–10]. We will see how P and other property fields computable using gradient bundle decomposition enable the direct inspection of electron density, volume, and energy redistribution within an atomic basin, and how this can be used to reveal the subtle changes to structure that underly the catalytic effects of an applied electric field.

2. Background 2.1 Gradient bundle decomposition To summarize our earlier article [2], the field P is constructed by mapping electron density gradient (rρ) paths to points in P. The origin and terminus of a gradient path (G) are, respectively, a local minimum called a cage critical point (CP)—which may be at ∞—and a local maximum typically coincident with a nucleus and thus dubbed a nuclear CP. Each G is also parameterized by arclength (s). All Gs are radial as they approach their terminuses, allowing one to define a spherical polar coordinate system about each nuclear CP so that each G in the infinite set may be uniquely specified by its terminating nuclear CP index and the polar and azimuthal angles at which it terminates, i.e., Gi(θ, ϕ). Thus, the independent variables of ρ within an atomic basin i are transformed from a (x, y, z) to a (θ, ϕ, s) coordinate system. With this atomic gradient coordinate system, we can conceptually perform a gradient bun˚ ) centered at dle decomposition by first placing a sphere Si of radius dr (in practice, dr ≲ 0.2A nuclear CP i. We then partition the surface of Si into differential area elements dA ¼ dr2 dθ dϕ (see Fig. 16.1). The Gs intersecting Si interior to a particular area element constitute a differential volume element, bounded by local zero-flux surfaces and called a differential gradient

FIG. 16.1

A differential area element on a sphere.

409

2. Background

bundle, dGBi(θ, ϕ).b Importantly, the cross-sectional area of each dGBi itself varies with arclength, i.e., dA ¼ dA(s). The union of all dGBi recovers the atomic basin. Notably, dGBi are the smallest structures bounded by zero-flux surfaces, hence the smallest structures possessing well-defined energies and energy-mediated properties. The unambiguous regional electronic kinetic energy of zero-flux surface bounded regions applies only to the Laplacian family of kinetic energy operators. When operators outside this family are considered, there seems to be some ambiguity to regional kinetic energy regardless of how one partitions the system, so a “well-defined” kinetic energy may be unobtainable in an absolute sense [11].

2.2 The condensed charge density and bond bundles By mapping the integrated ρ of each dGBi(θ, ϕ) to its corresponding position at Si(θ, ϕ), we produce the gradient bundle condensed charge density, P i ðθ, ϕÞ, which has units of electrons per angular area, specifically electrons per steradian (sr). Additionally, for any atomic scalar property field, Fi, there exists a corresponding gradient bundle condensed scalar field, F i, that is a function of θ and ϕ and a functional of Fi, such that, Z F ½Fi ≡ F i ½ θ, ϕ, Fi ðθ, ϕ, sÞ  ¼ Fi ðsÞ dAðsÞ ds (16.1) Gi ðθϕÞ

Using ρ as the input function, a GB decomposition yields the condensed charge density ðF ½ρ ¼ P Þ, using the gradient or Laplacian forms of the kinetic energy density yields the condensed kinetic energy density ðF ½TG  ¼ F ½TL  ¼ T Þ, etc. If the constant function Fi(θ, ϕ, s) ¼ 1 is used as input, the gradient bundle condensed volume is produced (F ½1 ¼ VÞ,c which has been shown to be a functional of the Gaussian curvature of charge density isosurfaces within each gradient bundle [2,5]. An algorithmic GB decomposition looks quite similar, using a finite number of dGBi—at approximately 6000 per atom for this demonstration—bounded by Gs intersecting at and along the nodes and edges of a triangulated sphere mesh. Gaussian quadrature tetrahedral numerical integration is performed for each GB by decomposing it, first into convex polyhedra defined by the GB intersections with lower and upper ρ isosurfaces, and again into tetrahedra. The resulting integrals are mapped to the midpoint of each GB’s corresponding triangular sphere element and visualized as contours on the sphere mesh surface. By virtue of its construction, a point in P maps to a G in ρ, any arbitrary path in P maps to a zero-flux surface, and any closed path in P corresponds to a gradient bundle in ρ. Each sphere mapping of P (and other field variables) is called an atomic chart, and the collection of all atomic charts gives the system (molecular, crystalline, etc.) atlas. The critical points of P include maxima, minima, and saddle points. Fig. 16.2 shows P mapped onto nuclei-centered spheres in ethylene, along with representative gradient bundles

b Surfaces

through which the flux of r ρ is everywhere zero, i.e., r ρ is everywhere tangent to the surface. There also exist net zero flux surfaces whose

total, integrated gradient flux is zero. c In

open systems, a step function—defined to be one within some truncating isosurface (typically ρ ¼ 0.001 a.u.) and zero beyond—is used to calculate V.

410

16. A bond bundle case study of Diels-Alder catalysis

The condensed density P mapped as contours onto spheres centered at the C and H nuclei in ethylene. Gradient bundles are shown for a C atom corresponding to the maximum, minimum, and saddle CPs—intersecting the sphere along equiradial CP-centered circles. Contours of ρ are shown in the molecular plane k and the perpendicular, C]C bond axis coincident plane ?.

FIG. 16.2

corresponding to P CPs. Often, bond paths map to maxima, as in ethylene where the black bond paths intersect atomic charts at local maxima. Just as the union of all ρ gradient paths sharing a common terminal nuclear CP specifies a unique volume—the atomic basin—the union of all P gradient paths (G) sharing a common terminal maximum specifies a unique area—a maximum basin—corresponding to a unique gradient bundle called a bond wedge. When bond wedges on neighboring atoms share some portion of an interatomic surface, they combine to form a bond bundle. Fig. 16.3 depicts the bond wedges of a C atom in ethylene, delineated by red G linking saddle to minimum CPs on the sphere surface, and the corresponding ethylene bond bundle surfaces. Bond bundles possess many qualitative and quantitative properties. Early in bond bundle research, it was found that bond bundle valance electron population recovers bond order [12], total electron populations were found to correlate with the interfacial energy of a metalceramic interface [13], and the shapes of bond bundle surfaces were shown to be indicative of chemical functionality and delineate molecular regions prone to nucleophilic or electrophilic attack [14].d These examples employed a different method of bond bundle identification, although the topology of the gradient bundle condensed charge density recovers the same bond bundle surfaces as the previously used method [4] though with higher accuracy, and this method is generalizable to a broader range of chemical and condensed phase systems.

2.3 Gradient bundle condensed properties and charge density geometry In addition to enabling the identification of bond bundles, gradient bundle decomposition makes possible two additional types of charge density analysis: (i) analysis of condensed d

Previously, bond bundles were analyzed using a method of explicit bond bundle surface identification, rather than implicit identification using the

gradient bundle condensed charge density as at present.

2. Background

411

FIG. 16.3 The three carbon maximum basins in ethylene and their corresponding bond bundle surfaces.

deformation properties relative to a spherical atomic reference state, conceptually similar to the chemical deformation densities of Schwarz et al. [15], though here applicable beyond the charge density to any scalar field and (ii) local and global geometric analysis of the charge density [5]. Thus, there are three categories of gradient bundle condensed properties: condensed scalar fields from Eq. (16.1) (category A); property fields derived therefrom, such as condensed deformation properties (category B); and geometric charge density descriptors (category C), such as average gradient bundle curvature, net curvature, and torsion (κ,κnet ,and κτ). 2.3.1 Visual qualitative inspection of gradient bundle condensed properties Fig. 16.4 shows a selection of condensed property fields as contours mapped onto the atoms of ethylene. The top, middle, and bottom rows, respectively, show condensed properties from each category. Starting with the condensed scalar fields (category A; top row), in organic systems the condensed density (P) seems to correlate with the topology of the condensed kinetic energy (T ) and volume (V), that is, they typically have the same sets of often colocated maximum, minimum, and saddle CPs [2,4]. Thus, in many systems, bond bundles, defined according to P (see the red paths on the spheres in Fig. 16.4), wouldn’t differ dramatically if defined according to another condensed scalar property such as kinetic energy density. V, however, is also a measure of total geometric Gaussian isosurface curvature within gradient bundles [2,5], and can deviate from the behavior of P more so than T . In this case, note that on the carbon atoms, P and T each have four minimum CPs, one each above and below the molecular plane, and one each within the molecular plane on either side of the C]C bond. In V, however, there is an additional minimum CP on the “back” of the carbon atom at the HdCdH position on the sphere opposite the C]C bond path (indicated for P at top-left of Fig. 16.4). This character is mirrored in the geometric descriptors (category C; bottom row) in the average gradient bundle curvature ðκÞ and average net gradient bundle curvature ðκnet Þ that also have “extra” minimum CPs in the same region on the C atoms; evidence of the link between condensed volume and charge density geometric curvature. The average curvature-scaled gradient bundle torsione ðκτÞ shows immediately which gradient paths e

κ is computed as the gradient path integral of the angles between neighboring line segments along a discretized path, κ net is computed as the angle

between the originating and terminal ends of a gradient path, and κτ is computed as the gradient path integral of the of angles between planes defined by coincident pairs of line segments (three neighboring line segments, where the central segment is shared between the pairs) along the path, scaled at each line segment by the its value of path curvature. All three properties are computed for paths and then averaged over the gradient paths defining a particular gradient bundle to recover its values.

412

16. A bond bundle case study of Diels-Alder catalysis

FIG. 16.4 Gradient bundle condensed scalar properties (top row), derived properties (middle row), and geometric charge density descriptors (bottom row) shown as contours mapped onto spheres around the nuclei of ethylene. Values are everywhere positive and are shaded blue to white to indicate low to high values, except for ΔT sph , where blue and red indicate negative and positive values. Red lines on carbon atom spheres indicate the boundaries between maximum basins defined according to P.

within an atomic basin will bend and curve primarily within the same plane. Here, we see that gradient paths within the carbon atomic basins will have very low torsion if they are close to the molecular plane or the perpendicular plane—see the minimum CP representative gradient bundles in Fig. 16.2—and that those at approximately 45 degrees to both planes will achieve maximum torsion—see the saddle CP gradient bundle in the same figure. This is to be expected since both planes are symmetry planes and hence zero flux surfaces. The same general behavior is also observed on the H atoms, in the molecular plane and the perpendicular plane containing a CdH bond axis. Investigations into the chemical significance of these pure geometric descriptors are ongoing, but preliminary results show that the redistribution of charge density curvature plays an important role in such fundamental chemical processes as carbonyl bond activation [5]. Turning to the middle row of Fig. 16.4 where two derived condensed properties are shown. The first is average kinetic energy per electron (T =P ), which has been investigated by Morgenstern et al. using another type of gradient bundle decomposition applicable to systems with linear symmetry [16] (see Chapter 13). Unlike P and T , that typically achieve maxima at bond paths, for bonds considered covalent the kinetic energy per electron typically has minimum CPs at bond paths, reflecting the lower electronic kinetic energy content of valence density resulting from bonding MOs. Note also that the minimum basins of T =P are nearly identical to the bonding regions defined by P , providing an alternative way to define the boundaries between bonding regions [16].

413

2. Background

The second derived condensed property shown in Fig. 16.4 is the condensed deformation kinetic energy density (ΔT sph Þ. All condensed deformation properties are calculated in the same way, by subtracting from the condensed property field, the equivalent field in a spherical atomic reference state, ΔF sph ¼ F  F sph That is, comparing the value at a point to what the value would have been if the total atomic basin condensed value in the current system were spherically distributed. Each differential gradient bundle’s atomic reference state value is simply its share of the atomic basin condensed value as determined by its normalized solid angle (α), so the value of some condensed deformation property for differential gradient bundle i is ΔF sph,i ¼ F i  αi Fatom where Fatom is the value of the atomic basin condensed property. In Fig. 16.4, we see that kinetic energy accumulates at and near bond paths (positive ΔT sph values), and that there is a precise boundary separating these regions from those of kinetic energy depletion. Like all closed paths through gradient bundle condensed property fields, the ΔT sph ¼ 0 contour (separating blue from red in Fig. 16.4) specifies a precise zero-flux surface bounded region in ρ. Due to the virial theorem, we know that for systems where no forces are acting on the atoms, within regions bounded by zero-flux surfaces the total regional kinetic energy is equal and opposite to the total energy [17,18]. So, in this case, the regions in ρ specified by the ΔT sph ¼ 0 contour separates regions of total energetic stabilization from those of destabilization. We note that our calculations use the noninteracting kinetic energy density, which recovers more than 99% of the total kinetic energy [19] and is sufficient at this time. Condensed deformation properties may also be calculated by providing a reference value to use in place of Fatom. For example, if an accurate and comparable atomic or ionic energy is known, it may be used to define the spherical reference state. 2.3.2 Quantitative condensed property analysis Table 16.1 lists the property fields from Fig. 16.4, condensed over atomic basins, bond wedges, and bond bundles, where C bond wedges correspond to the regions delineated

TABLE 16.1

Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed properties in ethylene. ΔT2 sph [ΔHa]

ΔT+sph [ΔHa]

0.000

0.723

0.723

0.996

0.068

0.270

0.338

0.321

0.992

0.033

0.224

0.190

0.339

0.991

0.035

0.229

0.195

Region

ρ [e]

T [Ha]

V [a30]

k=V [rad/a30]

knet =V [rad/a30]

kτ/V [rad/a30]

T/ρ [Ha/e]

C1 AB

6.032

5.993

98.51

10.100

5.957

0.328

0.994

↳ BW

2.441

2.432

38.43

9.307

6.707

0.325

↳ BW

1.787

1.773

29.84

10.082

5.168

↳ BW

1.804

1.788

30.24

11.125

5.783

ΔTsph [ΔHa]

Continued

414

16. A bond bundle case study of Diels-Alder catalysis

TABLE 16.1 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed properties in ethylene—cont’d k=V knet =V kτ/V [rad/a30] [rad/a30] [rad/a30]

T/ρ [Ha/e]

ΔT+sph [ΔHa]

Region

ρ [e]

C2 AB

6.032

5.993

98.52

9.938

5.863

0.324

0.994

0.000

0.723

0.723

↳ BW

2.442

2.431

38.39

9.316

6.707

0.324

0.996

0.072

0.269

0.341

↳ BW

1.802

1.786

30.23

10.805

5.590

0.333

0.991

0.034

0.227

0.193

↳ BW

1.788

1.776

29.89

9.861

5.054

0.314

0.993

0.038

0.226

0.189

V

[a30]

ΔTsph [ΔHa]

ΔT2 sph [ΔHa]

T [Ha]

H1 AB

0.942

0.584

48.34

30.570

24.068

1.418

0.620

0.000

0.163

0.163

H2 AB

0.942

0.584

48.34

29.996

23.625

1.405

0.620

0.000

0.163

0.163

H3 AB

0.942

0.584

48.34

30.572

24.068

1.419

0.620

0.000

0.163

0.163

H4 AB

0.942

0.584

48.34

29.996

23.624

1.404

0.620

0.000

0.163

0.163

Total

15.830 14.322 390.38 20.056

14.794

0.864

0.905

0.000

22.097

2.097

C1dC2 BB

4.883

4.863

76.83

9.311

6.707

0.325

0.996

0.139

0.540

0.679

↳ C1 BW

2.441

2.432

38.43

9.307

6.707

0.325

0.996

0.068

0.270

0.338

↳ C2 BW

2.442

2.431

38.39

9.316

6.707

0.324

0.996

0.072

0.269

0.341

C1dH2 BB

2.729

2.357

78.18

22.395

16.580

0.991

0.864

0.033

0.386

0.353

↳ C1 BW

1.787

1.773

29.84

10.082

5.168

0.321

0.992

0.033

0.224

0.190

↳ H2 BW

0.942

0.584

48.34

29.996

23.625

1.405

0.620

0.000

0.163

0.163

C1dH3 BB

2.745

2.372

78.58

23.088

17.031

1.003

0.864

0.035

0.392

0.357

↳ C1 BW

1.804

1.788

30.24

11.125

5.783

0.339

0.991

0.035

0.229

0.195

↳ H3 BW

0.942

0.584

48.34

30.572

24.068

1.419

0.620

0.000

0.163

0.163

C2dH1 BB

2.744

2.370

78.57

22.966

16.959

1.000

0.864

0.034

0.390

0.356

↳ C2 BW

1.802

1.786

30.23

10.805

5.590

0.333

0.991

0.034

0.227

0.193

415

2. Background

TABLE 16.1 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed properties in ethylene—cont’d k=V [rad/a30]

knet =V [rad/a30]

kτ/V [rad/a30]

T/ρ [Ha/e]

ΔT+sph [ΔHa]

Region

ρ [e]

↳ H1 BW

0.942

0.584

48.34

30.570

24.068

1.418

0.620

0.000

0.163

0.163

C2dH4 BB

2.730

2.360

78.23

22.302

16.528

0.988

0.865

0.038

0.389

0.351

↳ C2 BW

1.788

1.776

29.89

9.861

5.054

0.314

0.993

0.038

0.226

0.189

↳ H4 BW

0.942

0.584

48.34

29.996

23.624

1.404

0.620

0.000

0.163

0.163

Total

15.830 14.322 390.38 20.056

14.794

0.864

0.905

0.000

22.097

2.097

V

[a30]

ΔTsph [ΔHa]

ΔT2 sph [ΔHa]

T [Ha]

All gradient bundles are truncated at the ρ ¼ 0.001 a.u. isosurface (98.9% of electrons recovered).

by red paths in Fig. 16.4. Note that it is known that the primary source of error in the current implementation arises from interpolation error from the exclusive use of regular volumetric property grids. Starting again with gradient bundle condensed scalar fields, the current gradient bundle decomposition method recovers accurate atomic basin populations and energies, with agreement between symmetry-degenerate atoms less than one part in a thousand, e.g.,  0.001 electrons or  0.001 Hartree ( 0.6 kcal/mol), when calculating P or T , respectively. Recall that bond wedges are defined in the condensed (θ, ϕ) space and coincide with a portion of an atomic nucleus, where electronic kinetic energies are highest. Thus, a slight error in the identification of bond wedge boundaries results in a misallocation of some core energy and density. Agreement between symmetry-degenerate bond wedges is thus around one part in a hundred; a respective hundredth of an electron or Hartree ( 6 kcal/mol). Because all gradient bundle condensed properties are additive, the atoms and regions known to be symmetry degenerate can be averaged, as done elsewhere in this work. When reporting regional condensed values for geometric descriptors, we find it useful to average these over some other regional property. Here, we have averaged over (divided by) volume, resulting in units of angle per volume for κ,κnet , and κτ that give a clear sense of one volumetric region being more or less curved (i.e., less or more spherical; “atomic”) or tortuous than another.

416

16. A bond bundle case study of Diels-Alder catalysis

Returning to ΔTsph which, like all deformation properties, integrates to zero over an atomic + basin. The signed components of the function, ΔT sph and ΔTsph, indicate the extent of intraatomic energy depletion and accumulation. ΔTsph is itself nonzero for bond bundles or any gradient bundle smaller than an atomic basin. Table 16.1 indicates that regional destabilization occurs exclusively within ethylene CdH bonds (lower kinetic energy thus higher total energy) and that the C]C bond solely experiences the offsetting regional stabilization. This captures one of the many important dualities at work in chemical bonding, where bonds are simultaneously a cause and result of system energy minimization. A negative ΔTsph value does not indicate that ethylene CdH bonds are “destabilizing,” contributing destabilization to the system, but rather stable enough to accept regional destabilization such that the C]C double bond may stabilize the system to a greater extent. So, ground state regional gradient bundle deformation properties indicate not the winners and losers, but rather the beneficiaries and benefactors. Furthermore, because atomic ΔTsph is always zero, the change in CdH ΔTsph is exclusively due to rearrangement within the C atom, which shifts  0.035 electrons each from its CdH bond wedges into its C]C bond wedge. The combined analysis of atomic basin, bond bundle, and individual bond wedge condensed properties can inform both atom-centric and bond-centric interpretations. Whether one interprets the numerical C atom ΔTsph values as indicative of an atomic response or as a bond response, they are indicative of both and hence both approaches will be correct, in that they will be discussing the well-defined properties of regions that correspond to atoms and bonds. Individual bond wedges can then be invoked to frame a particular bonding interaction in the context of the internal atomic rearrangement of its constituent atoms.

2.4 Electric field charge density response and catalysis Oriented electric fields are known to influence chemical reactions [7–9,20–24], and are thought to play a significant role in enzymatic catalysis [25–28]. Deliberate reaction rate and selectivity control in the biologically and industrially significant Diels-Alder reaction has been experimentally achieved [20,29,30] and theoretically predicted [6,31], and as the broader field of electrostatic catalysis progresses, Diels-Alder reactions continue to provide conceptual insight. In their theoretical investigation, Meir et al. modeled two Diels-Alder reactions with and without electric fields of varying orientation and magnitude [6]. The first reaction was the cycloaddition of ethylene and cis-butadiene to form cyclohexene (see Scheme 16.1) where

SCHEME 16.1 The prototypical nonpolar Diels–Alder reaction of butadiene and ethylene (left) and the orientation of the reactants (right; note the negation of the x-axis).

3. Computational methods

417

they found that an electric field oriented along the “reaction axis” pointing from the butadiene to the ethylene (the negative z direction; electric field direction points positive to negative) lowered the reaction barrier, but a field in the opposite direction did not raise the reaction barrier. Using this conventionally “simple” example of the charge density response to an electric field, we will examine the inter and intraatomic redistribution of electron charge density that underly the response of ρ through the reaction and from the applied fields.

3. Computational methods All DFT calculations, including those used to produce the energy and distance values in Scheme 16.2 were performed with the Amsterdam Modeling Suite [32–34] ab initio software using the Perdew-Burke-Ernzerhof (PBE) functional [35] and a triple-zeta with polarization (TZP) all-electron basis set. Implicit COSMO [36–38] solvation was applied to all simulations using Allinger solvent radii [39] and a dielectric constant of ε ¼ 8.93. All electric fields are of magnitude 0.0125 a.u. ( 64 MV/cm). Analysis was performed within the Tecplot 360 visualization package [40] using the Gradient Bundle Decomposition software of the in house Bondalyzer package by the Molecular Theory Group at Colorado School of Mines [41]. The reaction of cis-butadiene and ethylene was modeled by bringing the cis-butadiene down on top of the ethylene, as depicted in the right side of Scheme 16.1. The resulting reaction profile is shown in Scheme 16.2 for the reaction with no applied electric field (NEF). Optimized geometries were obtained for all four steps and used for single point calculations with the various oriented electric fields.

SCHEME 16.2 The reaction profile for the cycloaddition of cis-butadiene and ethylene into cyclohexene in kcal/ mol. R is the separate reactant molecules, R0 the reactant complex, TS the transition state, and P the final, relaxed product, that is, without the geometric constraints placed on R0 .

418

16. A bond bundle case study of Diels-Alder catalysis

4. Results and discussion 4.1 Ethylene electric field response Our analysis of the electric field response of the reaction starts with a lone reactant ethylene molecule (Scheme 16.3) and three 0.0125 a.u. applied external electric fields (EEFs) oriented orthogonally along the C]C bond axis (y, pointing from C2 to C1), in the molecular plane (x), and out of the molecular plane (z). Table 16.2 lists regional condensed electron counts for ethylene with no electric field (NEF) and for each of the oriented applied fields, along with their difference (ρEEF  ρNEF) and percent difference (ρEEF  ρNEF)/ρNEF  100. Starting with the atomic basins, overall there is an interatomic charge transfer of around a twentieth of an electron for the symmetry breaking x and y fields, and negligible transfer for the symmetry conserving z field. The direction of x and y charge transfer is as expected for a homogeneous electron gas; opposite that of the field.

SCHEME 16.3 Orientation of ethylene with respect to applied electric fields, and numbering of atoms.

TABLE 16.2 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed charge densities in ethylene with and without 0.0125 a.u. applied electric fields oriented in the x, y, and z directions. ρ [e] EEF x Δ

EEF y

EEF z

Δ

%Δ

5.988

0.044

0.723

6.033

0.001

0.013

0.003

6.076

0.045

0.741

6.032

0.001

0.011

0.050

5.337

0.987

0.046

4.838

0.941

0.001

0.088

0.891

0.050

5.346

0.895

0.047

4.956

0.941

0.001

0.060

0.942

0.992

0.050

5.316

0.895

0.047

4.950

0.941

0.001

0.091

H4 AB

0.942

0.991

0.050

5.298

0.987

0.045

4.820

0.941

0.001

0.061

Total

15.830

15.829

20.001

20.007

20.001

20.008

15.829

20.001

20.009

C1dC2 BB

4.883

4.883

0.000

0.003

0.024

0.489

4.859

0.023

0.475

Region

NEF

C1 AB

6.032

6.032

0.000

0.004

C2 AB

6.032

6.032

0.000

H1 AB

0.942

0.891

H2 AB

0.942

H3 AB

%Δ

15.829 4.906

Δ

%Δ

419

4. Results and discussion

TABLE 16.2 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed charge densities in ethylene with and without 0.0125 a.u. applied electric fields oriented in the x, y, and z directions—cont’d ρ [e] EEF x

EEF z

Δ

%Δ

Δ

%Δ

2.441

2.453

0.012

0.505

2.265

0.176

7.217

2.432

0.009

0.355

↳ C2 BW

2.442

2.429

0.012

0.500

2.642

0.200

8.194

2.427

0.015

0.594

2.729

2.770

0.041

1.506

2.741

0.013

0.466

2.735

0.007

0.243

↳ C1 BW

1.787

1.879

0.091

5.116

1.846

0.059

3.323

1.794

0.007

0.402

↳ H2 BW

0.942

0.891

0.050

5.346

0.895

0.047

4.956

0.941

0.001

0.060

2.745

2.691

0.054

1.965

2.772

0.027

0.968

2.747

0.001

0.050

↳ C1 BW

1.804

1.700

0.104

5.766

1.877

0.073

4.058

1.806

0.002

0.124

↳ H3 BW

0.942

0.992

0.050

5.316

0.895

0.047

4.950

0.941

0.001

0.091

2.744

2.783

0.039

1.429

2.714

0.029

1.063

2.746

0.002

0.081

↳ C2 BW

1.802

1.891

0.089

4.965

1.727

0.075

4.147

1.805

0.003

0.169

↳ H1 BW

0.942

0.891

0.050

5.337

0.987

0.046

4.838

0.941

0.001

0.088

2.730

2.702

0.028

1.010

2.695

0.035

1.291

2.741

0.012

0.424

↳ C2 BW

1.788

1.711

0.077

4.331

1.708

0.081

4.509

1.800

0.012

0.679

↳ H4 BW

0.942

0.991

0.050

5.298

0.987

0.045

4.820

0.941

0.001

0.061

15.830

15.829

20.001

20.007

20.001

20.008

15.829

20.001

20.009

C1dH2 BB

C1dH3 BB

C2dH1 BB

C2dH4 BB

Total

Δ

EEF y

↳ C1 BW

Region

NEF

%Δ

15.829

For the x field, charge density is “pushed” from H1 to H4 and from H2 to H3, leaving the C atoms unchanged, which makes sense as the CdH bonds are identical, so an equal and opposite response is not surprising. We see that each C atom donated and accepted the same amount of density to and from its bound H atoms, essentially leaving the C atoms unchanged by the field. The y field results pose something of an interpretative dilemma. Charge flows from H2 to H1, from H3 to H4, and from C1 to C2, all by nearly equal amounts. Alternatively, the charge flow could be pictured as occurring between CH2 groups; from the H2–C1–H3 group to the H1–C2–H4 group. But when the interactions between bonded atoms are considered, the picture is less clear; C1 loses 0.044 net electrons but gains 0.047 electrons from each hydrogen. To see how the charge density response to an electric field is distributed among CdH bonds and the C]C bond, we turn to analysis of interatomic bonding regions.

420

16. A bond bundle case study of Diels-Alder catalysis

Inspection of the ethylene maximum basin (bond wedge) values in Fig. 16.5 reveals a more intricate relationship between the charge redistribution to, from, and within C atoms. When considering the totality of charge within an atomic basin, the x-oriented field appears to push charge through C1 from H2 to H3, that is, C1 accepts charge from H2 and donates it to H3, with the charge moving in the  x direction. However, within the atomic basin C1 shifts charge in the opposite direction, from the C1dH3 bond wedge to the C1dH2 bond wedge, by an amount equal to that transferred to/from the H atoms;  0.05e gained from H2,  0.05e given to H3, and  0.05e shifted from the C1dH3 bond wedge to the C1dH2 bond wedge. The ambiguous case of the y-field response results in the same behavior. Within both C atoms, the intraatomic charge redistribution is in the positive y-direction, opposite the direction of interatomic redistribution and opposite that expected of an electron gas. We can also visually inspect the condensed charge density response to the electric fields by computing a difference condensed density, ΔP, similar in concept to the deformation kinetic energy density in Fig. 16.4, but now instead of a spherical atomic reference state, the difference is that between the no-electric-field system and those with applied fields, ΔP EEF ¼ P EEF  P NEF : Fig. 16.6 shows the difference densities corresponding to the ethylene C atomic response to the electric fields. There are clear regions of accumulation and depletion resulting from the fields, and no bonding region experiences exclusively one or the other. For

FIG. 16.5 P maximum basins with their values of and changes in regional charge density in neutral ethylene (NEF) in response to electric fields oriented along the x, y, and z directions. Basins are indicated by shading and separated by white borders. Red and gray shading indicate the C]C and CdH bonding regions, respectively.

4. Results and discussion

421

all three, the region directly above and below the C]C bond path intersection with the sphere—corresponding to the π-bond—appear to be the among the most responsive, making the C p-orbitals act like sails catching electronic wind. The compliance of the p-orbital region is also evident in the high charge density shifts to/from the C]C bond wedge regions resulting from the y-field of Fig. 16.5. The offsetting  signs of ΔP x and ΔP z about the C]C bond path in Fig. 16.6 show graphically why only ΔP y achieves a (high) nonzero regional value. Also of note: ΔP x shows a preference for accumulation on the + x (left) side of the sphere; ΔP y is primarily positive for C2, and negative regions are primarily in the CdH maximum basins, while the opposite applies to C1; and ΔP z shows that charge shifted opposite the field direction. The key finding here is that picturing the response of simple negative charges to an electric field appears to provide some intuition regarding the van der Waals regions, but the induced response of the inner valance molecular charge redistribution does not follow such a simplistic model. This interdependent behavior was not evident in the interatomic charge transfer, and only via the combined approach of atomic and bond regional property analysis have we arrived at a clear picture of ethylene’s response to electric fields.

4.2 Diels-Alder electric field catalysis Turning now to the charge density response accompanying the catalytic effect of a properly oriented electric field on the cycloaddition of ethylene and cis-butadiene. Recall that the effect was observed as a lowering of the TS energy—motivating our analysis of the atomic and bond bundle charge density response of the TS structure shown in Scheme 16.4. Meir et al. found that the electric field pointing in the negative z direction, from the butadiene to the ethylene, lowered the barrier while a field pointing in the positive z direction had no effect on the TS energy [6]. Our calculations indicate that both fields stabilize the TS relative to the reactants (R). However, the  z field had the stronger stabilizing effect, lowering the TS energy 5.9 kcal/mol compared with 2.2 kcal/mol from the +z field. Table 16.3 lists the atomic basin and bond bundle condensed valence electron density (ρv) values for the TS structure with/without the  z fields. The total valence density of an atom is calculated as its total density minus its number of core electrons, ρv ¼ ρ  ρcore. The gradient bundle condensed valence density is then calculated, like the condensed deformation energy of Fig. 16.4, by subtracting the spherical atomic reference amount of core density from each gradient bundle, P v ¼ P  P core,sph. Note that symmetry degenerate atomic basins have been averaged. Starting with the atomic basins, we see a response like that of the total charge density in ethylene. Interatomic valance charge transfer is never more than a twentieth of an electron, and the prevalent motion is opposite the field. Overall, charge shifts from the butadiene central C2,3 and H1,2 atoms to the ethylene C and H atoms, and the Δ values for every atom are nearly equal and opposite with respect to field direction. From a pairwise perspective, there is also a lesser transfer between the ethylene C atoms and the butadiene C1,4 atoms. Regarding

422

16. A bond bundle case study of Diels-Alder catalysis

FIG. 16.6 Difference condensed densities for the C2 atom in ethylene resulting from applied oriented electric fields. Maximum basin boundaries are overlaid as red lines. The columns show the same C2 atom, from angles 90 degrees apart, except for ΔP y where the right column shows the C1 atom.

423

4. Results and discussion

SCHEME 16.4

Numbering and orientation of atoms in TS structure.

TABLE 16.3 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) regional valence charge densities in the cis-butadiene + ethylene transition state with and without 0.0125 a.u. electric fields oriented along the  z directions. ρv [e] + z (ene to diene) Δ

2z (diene to ene) Δ

Region

NEF

C1 (C4) AB

4.074

4.064

0.010

0.234

4.087

0.013

0.324

C2 (C3) AB

4.036

3.995

0.042

1.028

4.082

0.046

1.133

C5 (C6) AB

4.050

4.081

0.031

0.777

4.022

0.028

0.680

H1 (H2) AB

0.953

0.923

0.030

3.138

0.980

0.027

2.851

H3 (H5) AB

0.944

0.941

0.003

0.325

0.944

0.000

0.006

H4 (H6) AB

0.954

0.943

0.011

1.150

0.963

0.009

0.985

H7 (H8) AB

0.948

0.983

0.035

3.641

0.911

0.037

3.882

H10 (H9) AB

0.948

0.975

0.027

2.840

0.918

0.030

3.190

Total

33.814

33.809

20.004

20.012

33.815

0.002

0.005

C1dC2 BB

2.981

2.931

0.050

1.682

2.937

0.044

1.487

↳ C1 BW

1.534

1.518

0.016

1.030

1.487

0.047

3.074

↳ C2 BW

1.448

1.413

0.034

2.373

1.450

0.003

0.195

0.312

0.356

0.044

14.004

0.299

0.013

4.278

↳ C1 BW

0.136

0.114

0.022

16.445

0.191

0.054

39.897

↳ C5 BW

0.176

0.242

0.066

37.645

0.108

0.068

38.574

2.161

2.160

0.001

0.062

2.167

0.006

0.265

1.217

1.219

0.002

0.142

1.223

0.006

0.476

C1dC5 BB

C1dH3 BB ↳ C1 BW

%Δ

%Δ

Continued

424

16. A bond bundle case study of Diels-Alder catalysis

TABLE 16.3 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) regional valence charge densities in the cis-butadiene + ethylene transition state with and without 0.0125 a.u. electric fields oriented along the z directions—cont’d ρv [e] 2z (diene to ene)

+z (ene to diene) Region ↳ H3 BW

Δ

NEF

Δ

%Δ

%Δ

0.944

0.941

0.003

0.325

0.944

0.000

0.006

2.140

2.156

0.016

0.748

2.134

0.005

0.250

↳ C1 BW

1.186

1.213

0.027

2.273

1.171

0.015

1.242

↳ H4 BW

0.954

0.943

0.011

1.150

0.963

0.009

0.985

2.785

2.672

0.113

4.045

2.817

0.032

1.150

↳ C2 BW

1.392

1.336

0.056

4.045

1.408

0.016

1.150

↳ C3 BW

1.392

1.336

0.056

4.045

1.408

0.016

1.150

2.137

2.155

0.019

0.868

2.157

0.020

0.939

↳ C2 BW

1.183

1.232

0.048

4.095

1.176

0.007

0.602

↳ H2 BW

0.953

0.923

0.030

3.138

0.980

0.027

2.851

2.970

2.992

0.023

0.763

2.865

0.105

3.545

↳ C5 BW

1.485

1.496

0.011

0.763

1.432

0.053

3.545

↳ C6 BW

1.485

1.496

0.011

0.763

1.432

0.053

3.545

2.142

2.137

0.005

0.226

2.174

0.032

1.504

↳ C5 BW

1.193

1.154

0.039

3.299

1.262

0.069

5.783

↳ H8 BW

0.948

0.979

0.031

3.319

0.914

0.034

3.586

2.136

2.132

0.004

0.175

2.118

0.018

0.850

↳ C5 BW

1.188

1.157

0.031

2.583

1.200

0.012

1.019

↳ H9 BW

0.948

0.975

0.027

2.840

0.918

0.030

3.190

33.772

33.718

20.054

20.159

33.652

20.120

20.354

C1dH4 BB

C2dC3 BB

C2dH2 BB

C5dC6 BB

C5dH8 BB

C5dH9 BB

Total

Symmetry-degenerate atoms have been combined as indicated parenthetically.

the charge density behavior responsible for the shift in the reaction barrier, it is difficult to say whether the apparent interatomic C1–C5 charge transfer should positively or negatively alter the barrier. Yet, when bond bundle properties are considered, this difficulty vanishes. To illustrate: in the conventional view and as shown in Scheme 16.1, the forward reaction direction decreases the ethylene and butadiene C]C bonds from double to single bond order, while increasing the butadiene C2dC3 bond from single to double. At the TS (Scheme 16.4),

425

4. Results and discussion

the ethylene and butadiene CdC bonds will have an intermediate bond order between 1 and 2, while the intermolecular C1dC5 and C4dC6 bonds will have a bond order between 0 and 1. Recalling that CdC bond bundle valance electron count has been found to correlate with bond order [12], shown in Table 16.4 are the regional valence electron counts for the NEF system in the reactant, transition, and product states. The data of Table 16.4 reveal that throughout the reaction the C]C bonds decrease by 1.1 and 1.3 valance electrons in butadiene and ethylene, respectively (34% and 40%), while the C2dC3 bond increases by 1 electron (42%). At the TS, these CdC bonds have achieved about TABLE 16.4 Atomic basin (AB), bond bundle (BB), and bond wedge (BW) condensed valance electron density values for atoms in the R, TS, and P states, i.e., before, during, and after the reaction. ρv [e] TS

P

Region

R

C1 (C4) AB

4.038

4.074

0.036

0.887

3.983

0.055

1.354

C2 (C3) AB

4.014

4.036

0.022

0.544

4.047

0.033

0.818

C5 (C6) AB

4.032

4.050

0.018

0.441

3.974

0.057

1.425

H1 (H2) AB

0.945

0.953

0.008

0.838

0.960

0.015

1.542

H3 (H5) AB

0.937

0.944

0.007

0.720

0.968

0.031

3.299

H4 (H6) AB

0.948

0.954

0.005

0.541

0.970

0.022

2.273

H7 (H8) AB

0.942

0.948

0.007

0.710

0.986

0.045

4.729

H10 (H9) AB

0.942

0.948

0.007

0.711

0.980

0.039

4.093

Total

33.596

33.814

0.217

0.647

33.738

0.142

0.421

C1dC2 BB

3.258

2.981

0.277

8.502

2.161

1.097

33.670

C1dC5 BB

0.000

0.312

0.312

–

1.971

1.971

C1dH3 BB

2.118

2.161

0.043

2.054

2.054

0.064

3.008

C1dH4 BB

2.170

2.140

0.030

1.397

1.954

0.216

9.951

C2dC3 BB

2.387

2.785

0.398

16.665

3.401

1.014

42.480

C2dH2 BB

2.144

2.137

0.007

0.328

2.132

0.012

0.558

C5dC6 BB

3.306

2.970

0.336

10.175

1.973

1.333

40.332

C5dH8 BB

2.131

2.142

0.011

0.500

2.011

0.120

5.614

0.147

6.921

0.311

0.926

ΔR

ΔR

%ΔR

C5dH9 BB

2.131

2.136

0.005

0.250

1.983

Total

33.596

33.772

0.176

0.524

33.907

ΔR values indicate the difference from the R state.

%ΔR

–

426

16. A bond bundle case study of Diels-Alder catalysis

30% of their respective total changes—for example, the C1dC2 bond, which loses 1.1 electrons through the reaction, has lost 0.28 of these at the TS. In contrast, the forming C ⋯ C bonds initially change less rapidly. Over the course of the reaction these bonds gain 1.97 electrons, but only 0.31 of these at the TS (16%). We conclude that the TS energy is dominated by bond rearrangement within the reacting molecules as a precursor to the intermolecular bond formation occurring on the downhill side of the reaction profile. For this reaction, bond rearrangement within the reacting molecules is rate controlling. The effects of an applied field to this rate controlling charge rearrangement can be discerned from Table 16.3. For the  z field, at the TS the butadiene C1dC2 and ethylene C5dC6 bond bundle valance electron counts decrease by 0.044 and 0.105 electrons, respectively, while the C2dC3 value increases by 0.032 valance electrons. For these bonds, such changes represent a shift in valence charge of between 10% and 30% of their respective ΔR TS values (Table 16.4). For example, at the TS the C5dC6 bond loses 0.105 valance electrons due to the z field, in addition to the 0.34 valance electrons already lost relative to the reactant state. The redistribution resulting from the + z field (which Meir et al. found not to raise or lower the reaction barrier [6]) also includes a lowered butadiene C1–C2 valance electron count, but in this case the C2–C3 and C5–C6 counts shift opposite the direction due to the reaction. So, the more catalyzing field for the forward reaction acts to promote the charge rearrangement of the rate controlling process, while the effects of the + z field are in this regard contradictory. Evoking Hammond’s Postulate [42], we conclude that the catalyzing field makes the TS occur at an earlier point along the reaction path. The evolution of the bond bundles through the reaction of Scheme 16.2 offers some basis to speculate as to the relative rates of intramolecular bond rearrangement versus intermolecular bond formation. Shown in Fig. 16.7 are the bond bundle surfaces of the reactant complex, as well as the TS and product state. Although intermolecular C ⋯ C bond paths are present in the R state, their bond bundles have not yet formed. (We have shown that bond paths need not map to maximum basins in P and hence need not correspond to bond bundles [2,4]) As noted, these bond bundles are slow to form, reaching only 16% of their final occupancy at the TS, beyond which they grow spontaneously to their final  2 electron occupancy. Evidently, there is an energy cost associated with the initial phase of bond formation. In this regard, bond bundle formation is like nucleation and growth where one bond grows at the expense of those with which it shares a boundary, and QTAIM’s bond paths can be reinterpreted as sites of bond—or potential bond—nucleation. This interpretation necessitates a critical bond bundle occupancy marking a dividing line between unstable and stable bond growth. Hence, the change in bond bundle energies accompanying the motion of their surfaces becomes the critical parameter mediating the relative stability of a molecule or solid’s structure and bonding. QTAIM originated from Bader’s prescient question, “Where are the atoms in molecules?” It has since progressed, as it began, by necessarily coupling the topological structural elements of ρ to preexisting chemical concepts. This coupling process can be premeditated, with researchers actively looking for the traces of some as yet “undiscovered” chemical concept in ρ (e.g., aromaticity), or it can be serendipitous. It occurs explicitly when objects are given names, like “bond paths,” and implicitly though the context in which they are used in research. It does not, however, happen in the opposite direction; discoveries are tied to existing chemical ideas rather than standing on their own.

4. Results and discussion

427

Bond bundle surfaces for the reactant complex (R0 ), transition state (TS), and product (P) optimized geometries. In the R column, the CdC single bond surfaces are shaded blue. In the TS column, the surfaces of the bonds formed in the reaction are emphasized and shaded blue/purple. In the P column, the C]C double bond surfaces are shaded red.

FIG. 16.7

It is as if QTAIM’s sole purpose is to provide quantum-mechanically rigorous weights to keep the hard-won models and theories of chemistry from floating away. This is certainly a noble cause, but also one that results at times in ρ’s features being “dumbed down” to match the conceptual simplicity of their chemical counterparts. The infinite geometric richness of atomic basin surfaces, for example, plays second fiddle to the scalar quantity of atomic basin energy, even though the latter is determined in part by the former. In our research, we were “tying down” bonds, and in the process necessarily discovered the bond bundle surface for which there is no chemical analog. The motions of these surfaces appear to be crucial to the making and breaking of bonds and cannot be lost by forcing the bond bundle through a “bond”-shaped hole. So, for the first time, chemistry will have to borrow concepts from QTAIM rather than the other way around. And hence, we really are moving beyond QTAIM.

428

16. A bond bundle case study of Diels-Alder catalysis

5. Conclusions The bridge between QTAIM and other branches of experimental and molecular and condensed phase sciences rests largely on its ability to produce atomic regional properties that can be readily compared with other results [1,3]. Well-defined regional energies are a quality of any volume bounded by zero flux surfaces. Taking this property to the limit, the gradient bundle decomposition method is the differential partitioning of ρ into infinitesimal zero-flux surface bounded regions used to compute, using ρ and other scalar fields, the gradient bundle condensed energy, charge density, and other condensed properties. The topology of the condensed charge density also defines the surfaces of unique charge density bonding regions called bond bundles. Here, we showed that the bond bundle decomposition method allows for the direct qualitative and quantitative inspection of the distribution and redistribution of charge density that accompany static and dynamic chemical bonding. Applied to the toy problem of electric field catalysis of a Diels-Alder reaction, we observed that property shifts between bond wedges and bond bundles are more dynamic in both sign and magnitude than those of atomic basins. Further, we demonstrated that these shifts allow for immediate chemical interpretation. Specifically, we observed that intramolecular bond rearrangement occurring early in the modeled Diels-Alder reaction is a necessary precursor to intermolecular bond formation, the majority of which occurs after the transition state configuration. The early bond rearrangement we identified to be the process controlling the energy of the reaction’s transition state. We found that the catalyzing field achieves its effect by increasing the magnitude of the charge rearrangement associated with the rate controlling process and thereby lowering the TS energy. These results are promising but limited. However, the data utilized here are being generated for more complex and scientifically interesting systems. Moving forward, we plan on incorporating methods like principal component analysis and machine learning-based regression in order to uncover charge density structure-property relationships either too subtle or occurring at too large a scale (i.e., emergent properties) to be detected through direct inspection, with in-progress investigations looking again into the charge density electric field response, but in a biological context. The biochemical community is making great strides in understanding the significant role of electric fields in enzymatic catalysis [25], which involves countless chemical scenarios not unlike that treated in this work. We hope to aid in the process of discovery with the dual ability of local direct inspection within and large-scale correlation across enzyme active sites.

Acknowledgments This work was supported by the National Science Foundation grant CHE-1903808 and by the Office of Naval Research grants N00014-05-C-0241 and N00014-10-1-0838.

References [1] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. [2] T.R. Wilson, M. Rajivmoorthy, J. Goss, S. Riddle, M.E. Eberhart, Observing the 3D chemical bond and its energy distribution in a projected space, ChemPhysChem 20 (2019) 3289.

References

429

[3] C.F. Matta, R.J. Boyd (Eds.), The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007. [4] T.R. Wilson, M.E. Eberhart, Quantum theory of atoms in molecules in condensed charge density space, Can. J. Chem. 97 (2019) 757. [5] T.R. Wilson, A.N. Alexandrova, M.E. Eberhart, Electron density geometry and the quantum theory of atoms in molecules, J. Phys. Chem. A 125 (2021) 10622. [6] R. Meir, H. Chen, W. Lai, S. Shaik, Oriented electric fields accelerate Diels–Alder reactions and control the endo/ exo selectivity, ChemPhysChem 11 (2010) 301. [7] S. Yu, P. Vermeeren, T.A. Hamlin, F.M. Bickelhaupt, How oriented external electric fields modulate reactivity, Chem. A Eur. J. 27 (2021) 5683. [8] N.G. L
eonard, R. Dhaoui, T. Chantarojsiri, J.Y. Yang, Electric fields in catalysis: from enzymes to molecular catalysts, ACS Catal. 11 (2021) 10923. [9] S. Shaik, D. Mandal, R. Ramanan, Oriented electric fields as future smart reagents in chemistry, Nat. Chem. 8 (2016) 1091. [10] S.D. Fried, S. Bagchi, S.G. Boxer, Extreme electric fields power catalysis in the active site of ketosteroid isomerase, Science 346 (2014) 1510. [11] J.S.M. Anderson, P.W. Ayers, J.I.R. Hernandez, How ambiguous is the local kinetic energy? J. Phys. Chem. A 114 (2010) 8884. [12] T.E. Jones, M.E. Eberhart, The bond bundle in open systems, Int. J. Quantum Chem. 110 (2010) 1500. [13] T.E. Jones, M.E. Eberhart, S. Imlay, C. Mackey, G.B. Olson, Better alloys with quantum design, Phys. Rev. Lett. 109 (2012), 125506. [14] T.E. Jones, M.E. Eberhart, S. Imlay, C. Mackey, Bond bundles and the origins of functionality, J. Phys. Chem. A 115 (2011) 12582. [15] W.H.E. Schwarz, K. Ruedenberg, L. Mensching, Chemical deformation densities. 1. Principles and formulation of quantitative determination, J. Am. Chem. Soc. 111 (1989) 6926. [16] A. Morgenstern, T. Wilson, J. Miorelli, T. Jones, M.E. Eberhart, In search of an intrinsic chemical bond, Comput. Theor. Chem. 1053 (2015) 31. [17] J.C. Slater, The virial and molecular structure, J. Chem. Phys. 1 (1933) 687. [18] K. Ruedenberg, M.W. Schmidt, Physical understanding through variational reasoning: electron sharing and covalent bonding, J. Phys. Chem. A 113 (2009) 1954. [19] J.I. Rodrı´guez, P.W. Ayers, A.W. Gotz, F.L. Castillo-Alvarado, Virial theorem in the Kohn-Sham DFT formalism: accurate calculation of the atomic QTAIM energies, J. Chem. Phys. 131 (2009), 021101. [20] H.M. Aitken, M.L. Coote, Can electrostatic catalysis of Diels–Alder reactions be harnessed with PH-switchable charged functional groups? Phys. Chem. Chem. Phys. 20 (2018) 10671. [21] M. Zhang, F. Zhang, H. Xu, Z. Su, The regulation of hydroboration of olefins by oriented external electric field, New J. Chem. 42 (2018) 18402. [22] Y.K. Lei, J. Zhang, Z. Zhang, Y.Q. Gao, Dynamic electric field complicates chemical reactions in solutions, J. Phys. Chem. Lett. 10 (2019) 2991. [23] C. Acosta-Silva, J. Bertran, V. Branchadell, A. Oliva, Kemp elimination reaction catalyzed by electric fields, ChemPhysChem 21 (2020) 295. [24] S.D. Fried, S.G. Boxer, Electric fields and enzyme catalysis, Annu. Rev. Biochem. 86 (2017) 387. [25] M.R. Hennefarth, A.N. Alexandrova, Advances in optimizing enzyme electrostatic preorganization, Curr. Opin. Struct. Biol. 72 (2022) 1. [26] A. Morgenstern, M. Jaszai, M.E. Eberhart, A.N. Alexandrova, Quantified electrostatic preorganization in enzymes using the geometry of the electron charge density, Chem. Sci. 8 (2017) 5010. [27] J. Lameira, R.P. Bora, Z.T. Chu, A. Warshel, Methyltransferases do not work by compression, cratic, or desolvation effects, but by electrostatic preorganization, Proteins: Struct. Funct. Bioinf. 83 (2015) 318. [28] S.C.L. Kamerlin, P.K. Sharma, Z.T. Chu, A. Warshel, Ketosteroid isomerase provides further support for the idea that enzymes work by electrostatic preorganization, Proc. Natl. Acad. Sci. 107 (2010) 4075. [29] A.C. Aragone`s, N.L. Haworth, N. Darwish, S. Ciampi, N.J. Bloomfield, G.G. Wallace, I. Diez-Perez, M.L. Coote, Electrostatic catalysis of a Diels–Alder reaction, Nature 531 (2016) 88. [30] M.T. Blyth, M.L. Coote, A PH-switchable electrostatic catalyst for the Diels–Alder reaction: progress toward synthetically viable electrostatic catalysis, J. Org. Chem. 84 (2019) 1517.

430

16. A bond bundle case study of Diels-Alder catalysis

[31] J.B. Siegel, et al., Computational design of an enzyme catalyst for a stereoselective bimolecular Diels-Alder reaction, Science 329 (2010) 309. [32] G. te Velde, F.M. Bickelhaupt, E.J. Baerends, C. Fonseca Guerra, S.J.A. van Gisbergen, J.G. Snijders, T. Ziegler, Chemistry with ADF, J. Comput. Chem. 22 (2001) 931. [33] C. Fonseca Guerra, J.G. Snijders, G. te Velde, E.J. Baerends, Towards an order-N DFT method, Theor. Chem. Accounts 99 (1998) 391. [34] E.J. Baerends, et al., ADF, Theoretical Chemistry, Vrije Universiteit, Amsterdam, The Netherlands, 2016. [35] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865. € rmann, COSMO: a new approach to dielectric screening in solvents with explicit expressions [36] A. Klamt, G. Sch€ uu for the screening energy and its gradient, J. Chem. Soc. Perkin Trans. 2 (1993) 799. [37] A. Klamt, Conductor-like screening model for real solvents: a new approach to the quantitative calculation of solvation phenomena, J. Phys. Chem. 99 (1995) 2224. [38] A. Klamt, V. Jonas, Treatment of the outlying charge in continuum solvation models, J. Chem. Phys. 105 (1996) 9972. [39] N.L. Allinger, X. Zhou, J. Bergsma, Molecular mechanics parameters, J. Mol. Struct. THEOCHEM 312 (1994) 69. [40] Tecplot Inc, Tecplot 360 2013R1, Tecplot Inc, 2013. https://www.tecplot.com. [41] T.R. Wilson, M.E. Eberhart, Bondalyzer, First Principles Materials Design and Software, Golden, CO, 2021. https://github.com/moltheorygroup/BondalyzerTecplotAddon. [42] G.S. Hammond, A correlation of reaction rates, J. Am. Chem. Soc. 77 (1955) 334.

C H A P T E R

17 Applications of the quantum theory of atoms in molecules and the interacting quantum atoms methods to the study of hydrogen bonds Jos
e M. Guevara-Velaa, Alberto Ferna´ndez-Alarco´nb, and Toma´s Rocha-Rinzab a

Departamento de Quı´mica Fı´sica Aplicada, Universidad Auto´noma de Madrid, Madrid, Spain b Instituto de Quı´mica, Universidad Nacional Auto´noma de M
exico, Mexico City, M
exico

1. Introduction Noncovalent interactions (NCI) are central in physical chemistry [1]. The evidence on macroscopic grounds which indicates the existence of such interactions is certainly quite compelling [2], for example, the mere existence of condensed phases. These strong evidences impose relationships from which the energetics of NCIs can be estimated from macroscopic thermodynamic features such as boiling temperatures and enthalpies [2], a fact which indicates the tremendous significance of these contacts. Indeed, these interactions govern the conformation, structure, dynamics of molecules, molecular clusters, liquids, and solid-state systems [3–12]. For example, NCIs are critical for the assembly of materials in organic, inorganic, and organometallic chemistry [13–19]. NCIs can also be of paramount importance for the modulation of reactivity, for example, in catalysis [20–22]. The relevance of NCIs has warranted continuous efforts to achieve a thorough understanding of these interactions via novel experimental methods and theoretical approaches. Regarding computational methodologies, there has been an impressive progress in the area of molecular dynamics [23–29] and electronic structure theory both in wave-function approximations and density-functional theory (DFT) [30–34] to the extent that A.J. Stone [2] wrote in his seminal monograph on this

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00010-4

431

Copyright # 2023 Elsevier Inc. All rights reserved.

432

17. Applications of the QTAIM and IQA methods to the study of H-bonds

matter: At the time of the first edition [1995], the balance was already shifting from experiment to computation as the prime source of information about intermolecular forces, and this shift has continued. Experiment will continue to have an essential role, but its task now is more to check the validity of calculations and to refine calculated potentials, rather than to provide the principal source of data for them. Nevertheless, there is still a need to get a further understanding on the origin of NCIs and some of their inherent properties such as nonadditivity on the basis of interatomic interactions. In this regard, the quantum theory of atoms in molecules (QTAIM) [35] and the interacting quantum atoms (IQA) [36, 37] energy partitions are methods of wave-function analyses that have been successfully exploited in a wide variety of chemical and physical phenomena, such as electronic correlation [38, 39], the nature of distinct types of chemical bonds [40–42], and chemical reactivity [43, 44]. The success of QTAIM and IQA in these endeavors is partially based on the fact that these methodologies allow the study of different sorts of chemical interactions on the same sound quantum mechanical basis [45–49]. Hence, the rigorous formalism of these methods of wave-function analysis developed for the study of covalent bonding can be utilized for the examination of NCIs without any loss of generality. Herein, we will focus on the application of IQA and QTAIM on the investigation of arguably the most important noncovalent interaction, the hydrogen bond (HB). For this purpose, this chapter is divided as follows. First, we give a brief introduction to the QTAIM and IQA approaches. Then, we go over the applications of these methods of wave-function analyses into the study of hydrogen bonding. We consider a few general results, nonadditive effects, the influence of π systems, excited states together with doubly and triply hydrogen-bonded systems. We do not pretend to give an exhaustive description of all the successful employments of QTAIM and IQA to the examination of hydrogen bonds, but rather we hope that this brief account would give an accurate view of the state of the art in this area and directives for further progress in the field.

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition We survey in this section the IQA and QTAIM methods of wave-function analyses. Our description of these approaches is rather brief, because there are many excellent books and reviews on the subject [35, 50–53]. We consider first the QTAIM, as the foundations of the IQA energy partition reside on divisions of the three-dimensional (3D) space, for example, that one defined by QTAIM.

2.1 Quantum theory of atoms in molecules QTAIM is an approach of wave-function analyses that provides a validation of the hypothesis of molecular structure based on laws and theorems of quantum mechanics [35]. More specific to the work presented here, one may obtain useful information about the nature of the interactions taking place within an electronic system via the QTAIM topological analyses

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition

433

of the electron density, ρ(r). This scalar field can be obtained either experimentally, or via an approximation of the wave function jΨ i of an N-electron system as ρðrÞ ¼ hΨ j^ ρðrÞjΨ i,

(17.1)

N X δðr  r i Þ:

(17.2)

in which the operator ρ^ðrÞ reads ^ ρ ðrÞ ¼

i¼1

Due to the properties of the Dirac delta function and the indistinguishably of electrons in jΨ i, one can easily obtain Z (17.3) ρðrÞ ¼ N jΨ ðx,x2 ,…,xn Þj2 dsdx2 dx3 …dxN , wherein si denotes the spin coordinate of electron i and xi embodies the spin and spatial coordinates of electron i, that is, xi ¼ (ri, si). The Born postulate and Eq. (17.3) indicate that ρ(r) equals the probability density to find any electron in r, and therefore ρ(r) is normalized to Z ρðrÞdr ¼ N: (17.4) The establishment of the electron density as a physical quantity with a definite value for every point in 3D space allows us to examine its topological properties. The topology of ρ(r) is determined by the attractive forces exerted by the nuclei on the electrons. This condition gives the scalar field ρ(r) its main topological feature, which is the presence of local maxima at nuclear positions [54]. The topological properties of ρ(r) can be conveniently examined in terms of its critical points (CP), that is, the points wherein rρ(r) equals 0. These points are usually indicated as rc. The CPs of a scalar field can be characterized by the Hessian matrix evaluated in rc, A(rc). The diagonalization of A(rc) is generally used to label CPs of ρ(r) with an ordered pair (r, s) in which r and s are denoted as rank and signature, respectively. The rank is the number of nonzero eigenvalues of A(rc), while the signature is the algebraic sum of the signs of such eigenvalues. Stable molecular structures in QTAIM (vide infra) are characterized by the occurrence of CPs of ρ(r) for which r ¼ 3. Every type of rc whose ranks equals 3 is identified with an element of molecular structure. The position of the nuclei typically coincides with (3, 3) CPs, while a (3, 1) CP indicates a chemical bond between two atoms. Rings and cages are evidenced in QTAIM by (3, +1) and (3, +3) CPs, respectively. Hence, (3, 3), (3, 1), (3, +1), and (3, +3) are referred as nuclear (NCP), bond (BCP), ring (RCP), and cage (CCP) critical points. The number of CPs in a stable molecular system satisfy the Poincar
e-Hopf rule [35]: NCP  BCP + RCP  CCP ¼ 1:

(17.5)

The flux of rρ(r), that is, trajectories σ: R ! R3 which satisfy σ 0 ðtÞ ¼ rρðσðtÞÞ,

(17.6)

provide (i) a definition for molecular structure and (ii) a partition of the 3D space which can be exploited later to establish the IQA energy partition. BCPs have an important role on this aspect. First, the unstable manifolds of BCPs, that is, all points in space for which the trajectories of rρ(r) start at a given BCP, are trajectories of ρ(r) which end typically in an NCP. These

434

17. Applications of the QTAIM and IQA methods to the study of H-bonds

trajectories are denoted as bond paths (BP) because they connect NCPs, which share a common BCP. With these considerations in mind, molecular structure in QTAIM is based on the concept of molecular graph, which is defined as the set of BPs together with that of CPs of ρ(r). Stable molecular graphs (i.e., molecular structures) are those that change neither the number or type of CPs of ρ(r) nor the connectivity of the NCPs via BPs, under infinitesimal displacements of the nuclei. This condition implies that a stable molecular graph cannot encompass CPs with ranks smaller than three and that the intersection of the stable and unstable manifolds of CPs of ρ(r) is transverse [55]. The stable manifolds of NCPs or atomic basins, denoted usually as ΩA, represent an exhaustive partition of the 3D space into nonoverlapping regions, [ ΩA ¼ R3 and ΩA \ ΩB ¼ Ø for ΩA 6¼ ΩB : (17.7) A

We note that any well-behaved scalar field f: R3 ! R may induce a partition based on the stable manifolds of the flux of rf(r) for the maxima of f(r). Ditto for the unstable manifolds of the minima of f(r) in the dynamical system defined by rf(r) [50]. Hence, other scalar fields apart from the electron density, for example, the electron localization function [56], have been used in chemistry to divide the 3D space of an electronic system. The boundaries among atomic basins or interatomic surfaces (IS) are the stable manifolds of BCPs. Every point r into an IS satisfies the zero-flux condition ^ ¼ 0, rρðrÞ  nðrÞ

(17.8)

^ is the normal vector to an IS at point r. in which nðrÞ Finally, the atomic basins ΩA are identified with atoms in chemistry and the QTAIM defines a procedure for the computation of the atomic properties of ΩA. For example, one can calculate the number of electrons within ΩA as Z ρðrÞdr, (17.9) NðΩA Þ ¼ ΩA

and hence the charge of the same atomic basin reads qðΩA Þ ¼ ZΩA  NðΩA Þ:

(17.10)

One may also consider the further moments of the atomic charge distribution. For instance, the first moment in the charge distribution of an atom ΩA is computed via the average of the vector r ΩA , over the charge density in the atomic basin, while taking as origin the nucleus of the atom, Z r ΩA ρðr ΩA Þdr ΩA : (17.11) MðΩA Þ ¼  ΩA

Thus, if X ΩA represents the position of the nucleus of atom ΩA, the dipole moment of a neutral molecule is given by Z X ZΩA X ΩA  rρðrÞdr μ ¼ ΩA

¼

X ΩA

qðΩA ÞX ΩA +

X MðΩA Þ: ΩA

(17.12)

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition

435

One physical observable of paramount importance in QTAIM is the kinetic energy. We consider the kinetic energy densities K(r1) and G(r1) given by Z N Ψ ? ðx1 ,x2 ,…,xn Þr21 Ψ ðx1 ,x2 ,…,xn Þds1 dx2 dx3 …dxN , (17.13) Kðr1 Þ ¼  2 Z N Gðr 1 Þ ¼ r1 Ψ ? ðx1 ,x2 ,…,xn Þ  r1 Ψ ðx1 ,x2 ,…,xn Þds1 dx2 dx3 …dxN , (17.14) 2 wherein ri and r2i indicate the gradient and the Laplacian with respect to the spatial coordinates of electron i. The definition of the kinetic energy operator N 1X r2 , K^ ¼  2 i¼1 i

implies that

We can also show that

(17.15)

D E Z K^ ¼ KðrÞdr:

(17.16)

D E Z K^ ¼ GðrÞdr:

(17.17)

For this purpose, we can consider the identity, r21 jΨ j2 ¼ Ψ ? r21 Ψ + Ψ r21 Ψ ? + 2r1 Ψ  r1 Ψ ? :

(17.18)

By integrating over all spin coordinates and every spatial coordinate except that of one electron, and multiplying by  N4 , we obtain 1  r21 ρðrÞ ¼ KðrÞ  GðrÞ, 4

(17.19)

in which we exploited the hermiticity of the Laplacian operator. Due to the Gauss’ divergence theorem, the integration of the LHS of the previous equation over the whole 3D space yields Z Z 1 1 r21 ρðrÞdr ¼  rρðrÞ  ds ¼ 0, (17.20)  4 4 because lim rρ(r) ¼ 0. Thus, Eq. (17.17) follows immediately after integration of expression r!∞

(17.19) in the 3D space and the consideration of Eq. (17.16). We note here that the integration of formula (17.19) over an atomic basin yields an identical result to expression (17.20), Z r2 ρðrÞdr ¼ 0, (17.21) ΩA

due to the zero-flux condition (formula 17.8). Hence, the integration of either K(r) or G(r) over ΩA provides the kinetic energy of this atomic basin, Z Z KðrÞdr ¼ GðrÞdr: (17.22) KðΩA Þ ¼ ΩA

ΩA

436

17. Applications of the QTAIM and IQA methods to the study of H-bonds

We observe thus, how the atoms defined in QTAIM are physically appealing regions to carry out the partition of 3D space for which the kinetic energy is well defined and computed by using either of the kinetic energy densities K(r) or G(r) in Eqs. (17.13) and (17.14), respectively. The formalism of QTAIM also proves the local form of the atomic virial theorem [35], 1 2 r ρðrÞ ¼ 2GðrÞ+VðrÞ, 4

(17.23)

wherein V (r) is the electronic potential energy density. Newly, the integration of expression (17.23) over an atomic basin ΩA leads to 0 ¼ 2KðΩA Þ+VðΩA Þ,

(17.24)

and, therefore, the energy of an atom in a molecule or a molecular cluster is given by 1 EðΩA Þ ¼ KðΩA Þ ¼ VðΩA Þ: 2

(17.25)

Finally, we consider the delocalization index between two atomic basins ΩA and ΩB as a measure of the number of pairs of electrons shared between them and hence of covalency. We start by considering the electron population in ΩA as an expectation value of the number of electrons found in this basin, NðΩA Þ ¼

N X npΩA ðnÞ,

(17.26)

n¼0

in which pΩA ðnÞ is the probability of finding n electrons in ΩA. Let us suppose that two atoms A and B with NA and NB electrons interact and that nA electrons are transferred from ΩA to ΩB. Let us further assume that once this electronic transfer takes place, no exchange of electrons between these basins occurs. We have under these circumstances, pΩA ðN A  nA Þ ¼ 1 ! NðΩA Þ ¼ N A  nA ,

(17.27)

pΩB ðN B + nA Þ ¼ 1 ! NðΩB Þ ¼ N B + nA ,

(17.28)

σ 2 ðNΩA Þ ¼

NA + NB X

ðn  NðΩA ÞÞ2 pΩA ðnÞ

n¼0

(17.29)

¼ 0, and σ 2 ðNΩB Þ ¼

NA + NB X n¼0

ðn  NðΩB ÞÞ2 pΩB ðnÞ

(17.30)

¼ 0,

wherein σ ðN ΩA Þ and σ ðN ΩB Þ are the variance of the electronic populations in ΩA and ΩB, respectively. More importantly for the matter at hand, we have X covðN ΩA ,NΩB Þ ¼ ðnΩA  NðΩA ÞÞðnΩB  NðΩB ÞÞpðnΩA ,nΩB Þ nΩA ,nΩB (17.31) ¼ 0, 2

2

where pðnΩA ,nΩB Þ is the probability that there are simultaneously nΩA and nΩB electrons in ΩA and ΩB, respectively. The condition covðNΩA ,NΩB Þ ¼ 0 is indicative of a purely ionic bond

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition

437

Distribution of electrons for completely symmetric electron densities of (a) H+2 and (b) H2 with independent electrons.

FIG. 17.1

between ΩA and ΩB. Conversely, the exchange of electrons between ΩA and ΩB is the distinctive feature of covalency between these two atoms. Indeed, the QTAIM vision of chemical bonding establishes that the covalent bond order between ΩA and ΩB is given by their delocalization index (DI), which reads δðΩA ,ΩB Þ ¼ 2covðNΩA ,N ΩB Þ:

(17.32)

Fradera and coworkers [57] showed that δ(ΩA, ΩB) corresponds to the bond order that one would expect from the Lewis model of bonding for simple diatomic molecules. To illustrate these ideas, we consider the electronic distributions of the one-electron system H+2 and the uncorrelated description of H2 (Fig. 17.1). According to expression (17.32), the delocalization index for H2+ is δðHA ,HB Þ

¼ 2covðN HA ,N HB Þ X ðnHA  NðHA ÞÞðnHB  NðHB ÞÞpðnHA ,nHB Þ ¼ 2 nHA ,nHB

(17.33)

1 ¼ , 2 in which NðHA Þ ¼ NðHB Þ ¼ 12. We interpret this result as the occurrence of half a bonding pair in the molecule H2+. The same procedure for the H2 molecule for which N(HA) ¼ N(HB) ¼ 1 yields δ(HA, HB) ¼ 1, i.e., there is a bonding pair in the dihydrogen molecule. We note that electron correlation tends to diminish the fluctuation among electron populations. This condition implies that the probability of the occurrence of any of the ionic configurations in Fig. 17.1b is less than p ¼ 1/4. Therefore, electron correlation shows a tendency to decrease the delocalization index and covalency between two given basins ΩA and ΩB [57].

2.2 Interacting quantum atoms energy partition The IQA energy partition comprises a division of the electronic energy with several attractive features such as orbital invariance and independence with respect to (i) the parameters which define the approximation of the vector state and (ii) unphysical states of the systems under consideration. Furthermore, the components of the IQA division of the electronic energy are physically meaningful quantities, which are straightforward to interpret [36, 37]. The elements necessary to carry out the IQA energy partition are

438

17. Applications of the QTAIM and IQA methods to the study of H-bonds

• the one particle reduced density matrix Z ρ1 ðr 1 ; r 10 Þ ¼ N Ψ ? ðx10 ,x2 ,…,xn ÞΨ ðx1 ,x2 ,…,xn Þds1 ds10 dx2 dx3 …dxN , and; • the pair density

(17.34)

0

ρ2 ðr A , r B ÞjΨ h ρ2 ðr A ,r B Þ ¼ hΨ j^

1 X @in which ρ^2 ðr A , r B Þ ¼ δðr i  r A Þδðr j  r B ÞA i6¼j

Z ¼ NðN  1Þ

(17.35)

jΨ ðxA ,xB ,x3 ,…,xn Þj2 dsA dsB dx3 dx4 …dxN ,

of the molecule or molecular cluster under investigation together with • a partition of the 3D space as that provided by QTAIM. Such partition can be specified with the functions 8 1 if rΩA , > > > < (17.36) ωΩA ðrÞ ¼ > > > : 0 if r62ΩA : Because of expression (17.7), X ωΩA ðrÞ ¼ 1, 8r: (17.37) ΩA

The scalar fields (17.34) and (17.35) allow the computation of the electronic energy [58], Z Z 1 ρ2 ðr 1 ,r 2 Þ 0 ^ E¼ dr 1 dr2 + V nn : (17.38) hðr 1 Þρ1 ðr 1 ; r 1 Þdr1 + 2 r12 r1 0 ¼r1 Here, Vnn is the nuclear repulsion and ^h is the monoelectronic part of the electronic Hamiltonian, X ZB 1 ^ , (17.39) hðr 1 Þ ¼  r21  rB1 2 B which acts only on the unprimed variable r1 and the prime is removed before integration. Eq. (17.37) allows us to rewrite expression (17.38) as XZ ωΩA ðr 1 Þ^hðr 1 Þρ1 ðr 1 ; r 10 Þdr1 E ¼ ΩA

r1 0 ¼r1

Z 1 X ρ ðr 1 ,r 2 Þ ωΩA ðr 1 ÞωΩB ðr 2 Þ 2 dr1 dr 2 + V nn 2 ΩA ,ΩB r12 X X X X A ΩA A ΩB A ΩA ¼ T ΩA + VΩ + VΩ + VΩ en en ee +

ΩA

ΩA

ΩA 6¼ΩB

1 X ΩA ΩB 1 X ZA ZB + V + , 2 Ω 6¼Ω ee 2 Ω 6¼Ω rAB A

B

A

B

ΩA

(17.40)

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition

in which T ΩA ¼ 

1 2

Z r1 0 ¼r 1

A ΩB ¼ ZB VΩ en

A ΩB ¼ VΩ ee

2  δAB 2

ωΩA ðr 1 Þr21 ρ1 ðr 1 ; r 10 Þdr 1 , Z ωΩA ðr 1 Þ

Z

ρðr 1 Þ dr1 , r1B

ωΩA ðr 1 ÞωΩB ðr 2 Þ

ρ2 ðr 1 ,r 2 Þ dr 1 dr2 : r12

439

(17.41)

(17.42) (17.43)

The symbol δAB represents the Kronecker delta. By gathering the terms that depend on one atomic basin on one hand and two atomic basins on the other, we can write X Ω 1 X ΩA ΩB A Eself + E , (17.44) E¼ 2 Ω 6¼Ω int ΩA A

B

wherein ΩA A A ΩA A ΩA + VΩ + VΩ , EΩ en ee self ¼ T A ΩB EΩ ¼ int

ZA ZB A ΩB B ΩA A ΩB + VΩ + VΩ + VΩ : en en ee rAB

(17.45) (17.46)

ΩA ΩB A The quantities EΩ are straightforward to interpret. The former is the net energy self and Eint of ΩA, which includes the kinetic energy along with the nucleus-electron and electronelectron contributions to the electronic energy within that basin. The latter denotes the interaction energy between basins ΩA and ΩB and comprises the repulsive electron-electron and nucleus-nucleus contacts along with the attractive nucleus-electron interactions in which one particle is within ΩA and the other in ΩB. We recall at this point that ρ2(r1, r2) can be split in Coulombic and exchange-correlation contributions,

ρ2 ðr 1 , r 2 Þ ¼ ρðr1 Þρðr2 Þ + ρxc 2 ðr 1 , r 2 Þ ¼ ρJ2 ðr 1 , r 2 Þ + ρxc 2 ðr 1 , r 2 Þ,

(17.47)

A ΩB and hence, we can divide V Ω as ee

ΩA ΩB ΩA ΩB ΩA ΩB ¼ Vee;cl + Vxc Vee Z ρJ ðr 1 ,r 2 Þ 1 ωΩA ðr 1 ÞωΩB ðr 2 Þ 2 ¼ dr 1 dr 2 r12 2 Z 1 ρxc ðr 1 ,r 2 Þ + dr 1 dr2 : ωΩA ðr 1 ÞωΩB ðr 2 Þ 2 2 r12

(17.48)

In turn, these considerations allow to split the IQA interaction energy between ΩA and ΩB as ΩA ΩB A ΩB EΩ ¼ VclΩA ΩB + Vxc int ZA ZB ΩA ΩB ΩA ΩB ΩB ΩA ΩA ΩB ¼ + Ven + Ven + Vee;cl + Vxc : rAB

(17.49)

440

17. Applications of the QTAIM and IQA methods to the study of H-bonds

A ΩB A ΩB The terms V Ω and V Ω represent the ionic and covalent contributions of the bonding xc cl A ΩB between ΩA and ΩB, respectively. Indeed, there is a close relationship between V Ω and xc δ(ΩA, ΩB) in which the vanishing of one of these quantities results in the vanishing of the other [50]. Eqs. (17.34)–(17.49) summarize the IQA energy partition. We point out now two important issues:

• The IQA method can be implemented with different divisions of the 3D space, that is, with other definitions of the function ωΩA ðrÞ, which satisfy relation (17.37), for example, those related with interpenetrating densities in fuzzy atoms [59, 60]. • Unlike QTAIM, the IQA energy partition can be utilized in nonstationary points of the PES under consideration. This circumstance arises because IQA does not rely on the fulfillment of the virial theorem as QTAIM does. The IQA approach was originally implemented for Hartree-Fock, complete active space self-consistent field and full configuration interaction wave functions. Recently, it has been coupled with other methods of correlated wave functions, mainly those based on MøllerPlesset perturbation theory [61–63] and coupled cluster approximations both for ground [64–66] and excited states [67]. These developments are based in the orbital representation of ρ1 ðr 1 ; r 10 Þ and ρ2(r1, r2), X ρ1 ðr 1 ; r 10 Þ ¼ Dpq ϕ?p ðr 10 Þϕq ðr 1 Þ, (17.50) pq

ρ2 ðr 1 ,r 2 Þ ¼

X pq

dpqrs ϕ?p ðr 1 Þϕq ðr 1 Þϕ?r ðr 2 Þϕs ðr 2 Þ,

(17.51)

wherein D and d are the one- and two-electron matrices, respectively, in the basis set {ϕp} subjacent to the Fock space of the system under consideration, typically comprised by the canonical Hartree-Fock orbitals. The determination of the matrix elements Dpq and dpqrs associated to a given wave-function approximation of a determined electronic state allows to carry out the IQA partition in a well-established way. Thus, every method used to compute approximate electronic wave function is susceptible of being coupled with the IQA energy partition. We bring to mind here that the computation of molecular properties of nonvariational methods of electronic structure theory is based on the determination of a Lagrangian function in terms of the parameters that define the approximate wave function and Lagrange multipliers [68]. One might compute one- and two-electron Lagrangian matrices from which it is possible to divide the electronic energy under the IQA formalism as done in Refs. [66, 67]. To conclude this section, we consider briefly the coupling of DFT with the IQA energy partition. Given the relevance of DFT in electronic structure theory, it has been of interest to incorporate DFT in different schemes of wave function analyses. The requirements of the availability of ρ1 ðr 1 ; r 10 Þ and ρ2(r1, r2) seem at first glance to preclude the IQA partition of Kohn-Sham (KS) electronic energies. A more careful examination reveals that all the components of these electronic energies apart from the KS exchange-correlation energy EKS xc can be divided as discussed earlier in a straightforward way [69]. Concerning the partition of EKS xc , scaling arguments like those employed in QTAIM using the virial ratio have been employed for the division of EKS xc into intra- and interatomic terms [69]. For these purposes, we consider first the exchange-correlation energy EKS xc in KS theory,

2. Review of the quantum theory of atoms in molecules and the interacting quantum atoms energy partition

Z

Z EKS xc

¼ ¼

ρKS x ðr 1 ,r 2 Þ dr1 dr 2 r12

ρðrÞEðrÞdr + a0 EKS xc;loc

+

441

(17.52)

a0 EKS x ,

in which EðρðrÞ,rρðrÞ,…Þ ¼ EðrÞ is a nonhybrid exchange-correlation functional, a0 is the fraction of Hartree-Fock exchange and ρKS x (r1, r2) is the exchange density formed from the spin integration of products of KS spin-orbitals, χ R(x), XZ KS χ R ðx1 Þχ ?R ðx2 Þχ S ðx2 Þχ ?S ðx1 Þds1 ds2 : ρx ðr 1 ,r 2 Þ ¼  (17.53) RS

By considering a partition of the 3D space like that defined by QTAIM, we can divide the quantity EKS x in intra- and interatomic terms, in a similar way to Eq. (17.44), ¼

EKS x

X Ω Ω 1 X ΩA ΩB A A Ex;KS + E 2 Ω 6¼Ω x;KS ΩA B

A

¼

X ΩA

¼

A ΩA EΩ x;KS

X Ω ,add A Ex;KS ,

1 X ΩA ΩB + E 2 Ω 6¼Ω x;KS B

! (17.54)

A

ΩA

wherein A ΩB EΩ x,KS

2  δAB ¼ 2

Z ωΩA ðr 1 ÞωΩB ðr 2 Þ

ρKS x ðr 1 ,r 2 Þ dr 1 dr 2 : r12

(17.55)

ΩA ,add and a similar result holds We note that EKS x can be decomposed in additive quantities Ex;KS KS for Exc ,  XZ ΩA ,add KS ωΩA ðrÞρðrÞEðrÞdr + a0 Ex;KS Exc ¼ ΩA (17.56) X A ,add EΩ ¼ xc;KS : ΩA

We define now the ratios, λΩA

¼

¼

A ,add EΩ xc;KS A ,add EΩ x;KS Z ωΩA ðrÞρðrÞEðrÞdr A ,add EΩ x;KS

(17.57) + a0 ,

442

17. Applications of the QTAIM and IQA methods to the study of H-bonds

and the scaled quantities, 1 ΩA ΩB A ΩB  EΩ xc,KS ¼ ðλΩA + λΩB ÞEx,KS : 2

(17.58)

Finally, we get X 1 X A ΩA A ΩB  EΩ +  EΩ xc,KS xc,KS 2 ΩA Ω 6¼Ω A

B

¼

A ,add X EΩ xc;KS

ΩA

A ,add EΩ x;KS

1 X + 4 Ω 6¼Ω A

A ΩA EΩ x,KS A ,add EΩ xc;KS

B

A ,add EΩ x;KS

+

B ,add EΩ xc;KS B ,add EΩ x;KS

! +

A ΩB EΩ x,KS

ΩA ,add 1 X Exc;KS 1 X ΩA ΩB A ΩA ¼ EΩ + E x;KS Ω ,add A 2 ΩA Ex;KS 2 Ω 6¼Ω x;KS B

A

ΩB ,add 1 X Exc;KS 1 X ΩB ΩA B ΩB + EΩ + E x;KS Ω ,add B 2 ΩB Ex;KS 2 Ω 6¼Ω x;KS A B X Ω ,add A ¼ Exc;KS

! (17.59) !

ΩA

¼ EKS xc , in accordance with expression (17.56). Eq. (17.59) expresses a partition of EKS xc in intra- and interatomic terms in a similar fashion than formula (17.44) and hence it enables to use the IQA energy partition with KS electronic energies [69].

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA Hydrogen bonds are interesting and complex interactions. Their energetics are generally between covalent bonds and van der Waals contacts. Thus, and as mentioned earlier, the correct study of H-bonds requires a theory that takes into account different types of chemical interactions on an equal footing. Therefore, QTAIM [70–73] and IQA [74–76] have been extensively used to study systems in which H-bonds play an important role. This situation occurs to the extent that the presence of a bond path is invoked as a feature to identify of H-bonds in the IUPAC definition of this interaction [77]. We start this survey of the application of IQA and QTAIM to the study of HBs by discussing a few generalities.

3.1 Some general results By virtue of the ability of IQA and QTAIM to characterize the chemical interactions within an electronic system, these two approaches of wave-function analyses have been exploited to study the inherent properties of HBs. For this purpose, several archetypal H-bonded systems have been examined throughout the years such as dimers, larger molecular clusters, and crystals. The IQA and QTAIM study of hydrogen-bonded dimers [78–90], for example, (HF)2 and (H2O)2 have revealed that for the three atoms mainly involved in the H-bond –D–H⋯ A– (A and D indicate the HB acceptor and donor, respectively), we have that

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

443

FIG. 17.2 Electronic charge transfer which takes place in the water dimer.

• For equilibrium energies in which one can apply the virial theorem, the hydrogen-bonded atom is considerably destabilized, while D and A are stabilized as a result of the formation of the H-bond interaction. • There is an electronic charge transfer from the hydrogen bond acceptor to the hydrogen bond donor. The electron populations of atoms A and D increase, while those of the hydrogen-bonded atom and the eventual H atoms covalently bonded to A have the opposite behavior as schematized in Fig. 17.2. One might visualize such electronic charge transfer as follows. When the hydrogen bond starts taking place, the lone pairs of D interact with the H-bonded atom and, hence, there is an electronic charge transfer from the H-bond acceptor to the H-bond donor, which ultimately resides on the D atom of the HB donor. Such charge transfer increases the electronegativity of the A atom in the HB acceptor and, thus, the atoms which are bonded covalently to A lose electrons to the extent that the electronic population of A increases. • Generally, the exchange interaction energy between the proton and acceptor donor cancels the corresponding deformation energies and hence the electrostatic component seems to be the one governing the interaction [90]. To further understand this statement, we note that the formation energy of the dimer D⋯A,

can be written as

D + A Ð D⋯A,

(17.60)

  ΔE ¼ ED⋯A  ED + EA    D  A D⋯A ¼ ED  E + EA self + Eself + Eint

(17.61)

¼

ED def

+

EA def

+

ED⋯A int ,

wherein EX is the electronic energy of species X. Hereof, ED⋯A which is partitioned using the IQA formalism, X Ω 1 X ΩA ΩB A Eself + E ðin which ΩA , ΩB  A⋯DÞ, (17.62) ED⋯A ¼ 2 Ω 6¼Ω int ΩA A

B

can be rewritten in the following way. The quantity EA self includes all the self and the interaction energies within A, X Ω 1 X ΩA ΩB A ¼ E + E ðin which ΩA , ΩB  A within the A⋯D complexÞ, EA self self 2 Ω 6¼Ω int Ω A

A

B

(17.63) D⋯A and ditto for ED represents the IQA pairwise interaction between the self. The quantity Eint atoms of A and B,

444

17. Applications of the QTAIM and IQA methods to the study of H-bonds

ED⋯A ¼ int

X ΩA ΩB

A ΩB EΩ int

ðin which ΩA A and ΩB DÞ:

(17.64)

Likewise Eq. (17.49), the interaction energy between D and A can also be split in a classical and an exchange-correlation contribution, ¼ ED⋯A + ED⋯A : ED⋯A int cl xc

(17.65)

Finally, the deformation energy for A, EA def , is defined as A A EA def ¼ Eself  E :

(17.66)

ED def.

A similar equation holds for The deformation energy of the monomers D or A in the molecular cluster D⋯A is the energetic cost of taking either D or A in its isolated equilibrium configuration to that within the cluster A⋯D. The insertion of Eq. (17.65) in expression (17.61) yields A D⋯A + ED⋯A : ΔE ¼ ED def + Edef + Ecl xc

(17.67)

D A⋯D Generally, EA < 0, and EA⋯D < 0. It occurs frequently that ED def > 0, Edef > 0, Ecl xc def A⋯D + EA j and hence the exchange-correlation contribution to the interaction energy def  jExc and the deformation energies nearly cancel each other and hence H-bonds seem to be guided by electrostatics. Nevertheless, the IQA energy partition reveals that covalent interactions are very important to H-bonding as revealed by several experimental parameters such as bathochromic shifts in infrared spectroscopy as well as NMR coupling constants and chemical shifts [90]. • There are certain directives for the identification and characterization of H-bonds based on QTAIM wave-function analyses. Koch and Popelier [85] put forward a series of requirements for an XH⋯ Y interaction to be considered as a hydrogen bond. These criteria include (i) the molecular graph of the system under examination, (ii) the values of ρ(r) and r2ρ(r) at the corresponding BCP, (iii) the interpenetration of the hydrogen and the acceptor atom, and (iv) several QTAIM properties of the bonded hydrogen such as diminution of electron population, dipolar polarization (Eq. 17.11), and a considerable destabilization [85]. These criteria have been used to recognize hydrogen bond interactions from mere van der Waals contacts [91–93]. Nevertheless, there are important exceptions to the fulfillment of these criteria for interactions, which are still identified as hydrogen bonds. For example, HBs are not always associated to a BCP. Fig. 17.3 shows the molecular graph of 1,2ethanediol, 1,3-propanediol, and 1,4-butanediol. Although there is experimental evidence that the three systems present an intramolecular H-bond, 1,2-ethanediol does not exhibit a BCP and BPs associated to this HB [94]. These interactions can be better examined with the Noncovalent Index, another tool in the realm of Quantum Chemical Topology [95, 96]. Likewise, other workers have pointed out that the criteria of Popelier and Koch to identify H-bonds are too lenient for the acknowledgment of an interaction as an HB, for example, those present in several configurations of (CH4)2 [97, 98]. Finally, we point out that there are some topological properties of the electron density such as the positive definite kinetic energy G(r) and the potential energy density V (r), which have been correlated with structural and energetic features of hydrogen bonds [86]. In particular, a relation between

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

445

FIG. 17.3

Molecular graphs of 1,2-ethanediol, 1,3-propanediol, and 1,4-butanediol. The first-mentioned system does not have a bond critical point or bond path associated with an intramolecular hydrogen bond. Reprinted with permission from J.R. Lane, J. Contreras-Garcı´a, J.-P. Piquemal, B.J. Miller, H.G. Kjaergaard, J. Chem. Theory Comput. 9 (2013) 3263–3266. Copyright 2013 American Chemical Society.

the formation energy of a hydrogen bond, EHB, and the potential energy density at the corresponding BCP, rBCP, has been put forward [86] 1 EHB ¼ Vðr BCP Þ: 2

(17.68)

Expression (17.68) has been utilized to estimate the formation energies of different hydrogen bonds [86]. Nonetheless, it cannot be applied for hydrogen bonds such as that in 1,2-ethanediol (Fig. 17.3) for which there is not an associated BCP. Even when this is not the case, Eq. (17.68) should not be used uncritically. For example, the value of EHB via formula (17.68) is always a negative quantity, and some contacts which resemble HBs (e.g., some C–H⋯ O interactions in Ref. [99]) are repulsive in nature. Another issue is the transferability of expression (17.68), for different H-bonds. For instance, the description of H⋯ F hydrogen bonds requires a different scaling factor [100]. The IQA and QTAIM methods of wave-function analyses have also been used to discern whether an interaction is H-bonded or not. For example, Ferna´ndez-Alarco´n et al. [98] concluded that the interaction within the clusters (H2S)2 and (H2Se)2 on one hand is substantially different from the hydrogen bond in (H2O)2 on the other. One piece of evidence is the fact that the classical contribution in (H2S)2 and (H2Se)2 is substantially larger than its exchangecorrelation counterpart as opposed to the situation in (H2O)2 (Fig. 17.4). This result is

446

17. Applications of the QTAIM and IQA methods to the study of H-bonds (H2O)2

(H2S)2

(H2Se)2

14.08

7.43

7.79

–1.27 –0.84

–6.66

–8.06 –8.63

–9.32 –9.47

–12.43

–19.09

Eclass

Exc

Eint

ΔEdef

FIG. 17.4 IQA interaction components according to Eq. (17.67) for (H2O)2, (H2S)2, and (H2Se)2. The data are reported in kcal mol1. Reproduced from A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. Casals-Sainz, E. Francisco, A. ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. 23 (2021) 10097–10107, with permission from the PCCP Costales, A Owner Societies.

consistent with the observation that the interaction between the monomers in (H2O)2 is considerably more anisotropic in (H2O)2 that it is in (H2S)2 and (H2Se)2 as revealed by the potential energy curves in Fig. 17.5.

3.2 Nonadditive effects of hydrogen bonding Another feature of hydrogen bonds which has been extensively studied by IQA and QTAIM is its characteristic nonadditivity. Hydrogen bonds might strengthen each other. Consider again the electronic charge transfer that takes place in (H2O)2. As discussed earlier, such electron transfer has two important consequences: (i) it increases the electron population of the oxygen atom within molecule D in Fig. 17.6 and (ii) it rises the positive charge of the H-atoms in the HB acceptor, molecule A, in the same figure. Such effects increase the susceptibility of (i) the O atom in monomer D in (H2O)2 to accept an extra HB and (ii) the H atoms in molecule A in the left part of Fig. 17.6 to participate in further H-bonds. In other words, the oxygen atom in molecule D is a better HB acceptor and the referred hydrogens are more liable to be hydrogen bonded than the corresponding atoms in an isolated water molecule. These charge transfers are usually invoked to explain the cooperativity of the H-bonds in (H2O)3, that is, the interactions among the water molecules in this cluster are stronger than the isolated HB in (H2O)2 [101]. Albrecht and Boyd [102] used QTAIM to investigate the underlying reasons to the cooperative effects of hydrogen bonds in small cyclic (H2O)n (n ¼ 25) clusters. They found that jq(Hbridging)j and jq(O)j increase with the number of water monomers in the cluster, whereas jq(Hnonbridging)j remains virtually unaltered (Fig. 17.7A). The polarization of these atoms is

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

447

FIG. 17.5

Potential energy curves of (H2O)2, (H2S)2, and (H2Se)2 as function of the angle θ shown on top. Reproduced from A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. ´ . MarCasals-Sainz, E. Francisco, A. Costales, A tı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. 23 (2021) 10097–10107, with permission from the PCCP Owner Societies.

related to an increase in the dipole moment of a water monomer computed with expression (17.12) as shown in Fig. 17.8. Because the bridging electrons lose electron density while the oxygens in the cluster rise their electron population as the clusters grow, the former type of atoms gets more destabilized, whereas the latter atomic basins exhibit the opposite behavior as more water monomers are added into the system (Fig. 17.7B). The stabilization of the O atoms overcomes the instability of the bridging hydrogen atoms in the molecular clusters. Later, Guevara-Vela et al. [103] studied cooperative effects of hydrogen bonding in the small water clusters (H2O)n (with n ¼ 26), comprised by a single homodromic cycle using the IQA energy partition. These workers found that the hydrogen bond cooperativity in these systems is accompanied by a continuous increase of the magnitude of the interaction energy 2 O⋯H2 O 2O and the deformation energy of the interacting monomers EH EH int def as water molecules FIG. 17.6 Left. Electronic transfer in (H2O)2 which strengthens the H-bond donor capacity of molecule A and the H-bond acceptor character of molecule D. Right. Such charge transfer is reflected in H-bond cooperativity in (H2O)3, that is, the hydrogen bonds in this cluster are stronger than that in (H2O)2.

448

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.7 Change in (A) atomic charges and (B) atomic energies for the atoms in the clusters (H2O)nn ¼ 15. Reprinted with permission from L. Albrecht, R.J. Boyd, J. Phys. Chem. A 116 (2012) 3946–3951. Copyright 2012 American Chemical Society.

are added to the molecular cluster (Fig. 17.9). Furthermore, the exchange-correlation compo2 O⋯H2 O is the most important contribution to the intermolecular interaction energy nent of EH int among small water clusters (H2O)n formed by a homodromic ring (Fig. 17.10). This observation is consistent with the results from NMR and IR spectroscopy, which evidence the importance of covalency in HBs (vide supra). Hydrogen bonds might also weaken each other. These effects are referred as hydrogen bond anticooperativity and they have been explained by considering the factors shown in Fig. 17.11. Because of the charge transfer shown in Figs. 17.2 and 17.6, the hydrogen bonds in Fig. 17.11A and B weaken each other. In the first case, the middle water molecule in Fig. 17.11A accepts electron charge from the monomer in the left. Such charge transfer impairs that associated with the hydrogen bond in the right part of the same figure and vice versa. We have a similar effect in Fig. 17.11B. The electron density transfers from the double H-bond acceptor hinder one another as well. Another argument that points out that the H-bonds of a double H-bond donor weaken each other is given in Fig. 17.11C. The two hydrogen bonds formed are reminiscent of a pair of electric dipoles, which are nearly parallel to each other, and thus, they destabilize the system [104]. A similar reasoning holds for the hydrogen bonds of double HB acceptors. These reasons lead to the widespread notion that double hydrogen FIG. 17.8

Dipole of a water molecule within small H2O clusters. Reproduced from J.M. GuevaraVela, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, J. Herna´ndez-Trujillo, O. Christiansen, E. Francisco, ´ . Martı´n Penda´s, T. Rocha-Rinza, Chem. Eur. J. A 19 (2013) 14304–14315, with permission from John Wiley and Sons, Inc.

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

449

FIG. 17.9 Formation, deformation, and IQA interaction energies per water monomer as a function of the numbers of molecules in the water clusters (H2O)n (wherein n ¼ 26) containing a single homodromic cycle in their structure. Reproduced from J.M. Guevara-Vela, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, J. Herna´ndez-Trujillo, O. Christiansen, E. ´ . Martı´n Penda´s, T. Rocha-Rinza, Francisco, A Chem. Eur. J. 19 (2013) 14304–14315, with permission from John Wiley and Sons, Inc.

bond donors or acceptors are related with H-bond anticooperativity. As discussed earlier, water clusters are systems, which have been extensively used to discuss hydrogen bond nonadditive effects. The smallest water cluster which presents double HB donors and acceptors is (H2O)6 and, therefore, different configurations of the water hexamer (Fig. 17.12) have been used to address simultaneously cooperative and anticooperative effects of hydrogen bonding. Ref. [105] considered the formation energy of different structures of the water hexamer via the IQA energy partition and a generalized version of Eq. (17.61). The formation energy of a molecular cluster ⋯F ⋯G⋯H⋯ in the process, ⋯ + G + H + I + ⋯Ð⋯ G ⋯ H ⋯ I ⋯

(17.69)

can be written as FIG. 17.10

Different contributions (electrostatic, Eele; exchange, VX; and classical, Vcl) to the interaction energy per H2O monomer in the systems considered in Fig. 17.9. Reproduced from J.M. Guevara-Vela, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, J. Herna´ndez-Trujillo, O. ´ . Martı´n Penda´s, Christiansen, E. Francisco, A T. Rocha-Rinza, Chem. Eur. J. 19 (2013) 14304–14315, with permission from John Wiley and Sons, Inc.

450

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.11

Double hydrogen bond: (A) donor and (B) acceptor. The electronic charge transfers occurring in the hydrogen-bonds of (A) and (B) impair one another. (C) Double H-bond donor in which the hydrogen bonds formed resemble a repulsive interaction of nearly parallel dipole interactions. Reproduced from J.M. Guevara-Vela, E. Romero´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Montalvo, V.A. Mora-Go´mez, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, E. Francisco, A Chem. Chem. Phys., with permission from the PCCP Owner Societies.

FIG. 17.12

Different structures of (H2O)6 addressed in Ref. [105] to examine cooperative and anticooperative effects of hydrogen bonding. Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, V.A. Mora-Go´mez, R. Cha´vez´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys., with permission from Calvillo, M. Garcı´a-Revilla, E. Francisco, A the PCCP Owner Societies.

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

ΔE ¼

X 1 X G⋯H EGdef + E : 2 H6¼G int G

451 (17.70)

Moreover, the RHS of Eq. (17.70) can be put as a sum of pairwise contributions, 0 0 1 0 1 1 ΔE

¼

B EG⋯H C G B EG⋯H C H C 1 X XB int int BEG⋯H + B X CE + B X CE C def A int def @ @ A @ J ⋯G J ⋯H 2 G H6¼G Eint Eint A

1 X X G⋯H 0 ¼ E : 2 G H6¼G int

J 6¼G

J 6¼H

(17.71)

One can show the equivalence of expressions (17.70) and (17.71) by considering that 0 1 2

1

X X B EG⋯H C B Xint CEG @ J ⋯G A def E G H6¼G int J 6¼G

0 +

1

B EG⋯H C H B Xint CE @ J ⋯H A def Eint J 6¼H

X X 9 8 EG⋯H EG⋯H > > int int > > = < X G6¼H 1 X H6¼G G H X X ¼ E + E def > 2> EJint⋯G def EJint⋯H > > H ; : G J 6¼G J 6¼H X ¼ EGdef : G

(17.72)

The exploitation of Eq. (17.71) reveals that double hydrogen-bond donor and acceptors are indeed associated with anticooperative effects of hydrogen bonds. The weakest hydrogen bonds in H2O clusters involve double HB donors, which donate hydrogen bonds to double HB acceptors (Fig. 17.13A). Guevara-Vela et al. [105] also found that the strongest type of hydrogen bonds found in the examined water clusters involves double HB acceptors acting as hydrogen bond donors to double HB donors (Fig. 17.13B). We have previously discussed in Fig. 17.11 that double HB acceptors are associated with anticooperative effects of hydrogen bonds based on the charge transfer occurring on these interactions. One can also use the same

FIG. 17.13 (A) Weakest and (B) strongest hydrogenbond types in water clusters. In the former case, a double HB donor donates a hydrogen bond to a double HB acceptor, while in the latter, a double HB acceptor donates a hydrogen bond to a double HB donor. Adapted from J.M. Guevara-Vela, E. Romero-Montalvo, V.A. Mora-Go´mez, R. Cha´vez-Calvillo, ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys., with permission from the PCCP M. Garcı´a-Revilla, E. Francisco, A Owner Societies.

452

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.14

Charge transfer involved in the occurrence of (A) double hydrogen bond donors and (B) double hydrogen bond acceptors. While the hydrogen bonds enclosed in rectangles weaken each other, those encompassed by ovals strengthen one another. Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, V.A. Mora-Go´mez, R. Cha´vez´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys., with permission from Calvillo, M. Garcı´a-Revilla, E. Francisco, A the PCCP Owner Societies.

sort of arguments to explain the relative strength of the hydrogen bond shown in Fig. 17.13B. As Fig. 17.14A displays, double HB donors receive electron density from two species. This circumstance increases the electron density within double HB donors and makes molecule D in Fig. 17.14A more liable to accept a hydrogen bond. In other words, while double HB donors are bad hydrogen bond donors, they are good hydrogen bond acceptors. Likewise, double HB acceptors donate electron density to two species. This condition increases the acidity of the hydrogens of monomer A in Fig. 17.14B and, therefore, these atoms are more susceptible to form hydrogen bonds. Conversely, whereas double HB acceptors are bad hydrogen bond acceptors, they are good hydrogen bond donors. Castor-Villegas et al. [106] put forward a hierarchy of hydrogen bond strength within water clusters in terms of the single or double character of the hydrogen bond donors and acceptors entailed in these systems (Table 17.1). The earlier arguments can also be used to rationalize: • Interactions between ions and protic solvents and how these ions alter the network of hydrogen bonds of these solvents. For example, the upper part of Fig. 17.15 shows the interaction of the F ion with one and with three water molecules. The electron charge transfer from the fluoride ion to the water molecule reduces the magnitude of the F ⋯ H2O interaction energy from 29.8 kcal mol1 with only one water molecule to 22.5 kcal mol1 in the F@(H2O)3 cluster. Once again, the electronic charge transfers which occur from the same species impair themselves. Similarly, the electronic charge transfer from the water molecules in the first solvation shell of Fig. 17.15D hinder the Li+ ⋯ OH2 interactions one another as can be seen from its comparison with Fig. 17.15C [107]. However, the flow of charge from the first solvation shell to Li+ increases the acidity of the protons of these water molecules and strengthen their interaction with the second solvation shell (Fig. 17.15D). These effects account for the observation of studies of water dynamics around different anions and cations, for example, those of Tielrooj et al. [108]. There are also other studies in which QTAIM is exploited to understand the effect of coordination of metals to hydrogen bonds. For example, Stasyuk et al. [109] considered the effect of alkali metal cations on the strength of individual H-bonds in DNA base pairs and they used QTAIM tools such as DIs to assess the relative strength of hydrogen bonds in a given complex.

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

TABLE 17.1 Type of HB

453

Scale of hydrogen bond formation energies within water clusters put forward in Ref. [106].

Description

(1)

(i) The H atom involved in the hydrogen bond belongs to a double HB donor and (ii) the oxygen that participates in the interaction acts as a double HB acceptor

(2)

A tetracoordinated water molecule either (i) donates an HB to a tricoordinated double HB acceptor or (ii) accepts an HB from a tricoordinated double HB donor

(3)

A tetracoordinated water molecule interacts with a monomer which is a single HB donor and a single HB acceptor

(4)

(i) The hydrogen of a double HB donor is bonded to the oxygen of a single HB acceptor or (ii) the oxygen of a double acceptor interacts with a hydrogen of a single donor

(5)

A hydrogen bond is formed between two double HB donors or two double HB acceptors

(6)

Both water molecules are tetracoordinated

(7)

A tetracoordinated H2O molecule either (i) donates an HB to a tricoordinated double HB donor or (ii) accepts an HB from a tricoordinated double HB acceptor

(8)

The oxygen of a single HB donor interacts with a hydrogen of a single HB acceptor

(9)

(i) A hydrogen of a double HB acceptor is in contact with the oxygen of a single donor or (ii) the O atom of a double donor interacts with a hydrogen of a single acceptor

(10)

The oxygen of a double HB donor interacts with a hydrogen of a double HB acceptor

The hierarchy is presented in increasing order of magnitude of H-bond formation energies. ´ . Martı´n Penda´s, T. Rocha-Rinza, A. Ferna´ndez-Alarco´n, J. Reproduced from V.M. Castor-Villegas, J.M. Guevara-Vela, W.E. Vallejo Narva´ez, A Comput. Chem. 41 (2020) 2266–2277, with permission from John Wiley and Sons, Inc.

• The bifunctional catalytic activity of water clusters in inorganic and organic chemistry. Hydrogen-bonded molecules such as HCN might form chains of the type δ+



H  C≡N⋯H  C≡N⋯H  C≡N⋯H  C≡Nδ :

(17.73)

FIG. 17.15 Molecular clusters: (A) F ⋯ H2O, (B) F@(H2O)3, (C) Li+ ⋯ H2O, and (D) Li+@(H2O)6. The electron 0 are reported in kcal mol1. Reproduced charge transfers are denoted with arrows and the different interactions EG⋯H int from A. Sauza-de la Vega, T. Rocha-Rinza, J.M. Guevara-Vela, ChemPhysChem 22 (2021) 1269–1285, with permission from John Wiley and Sons, Inc.

454

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.16

(A) General structure of a bifunctional catalyst. These systems possess a moiety, which can activate an electrophile and ditto for a nucleophile. (B) Water clusters acting as bifunctional catalysts in the formation of H2SO4 in acid rain. (C) Profile of electronic energy for the generation of sulfuric acid via the reactions of SO3 with H2O, (H2O)2, ´ . Martı´n Penda´s, and (H2O)3. Reproduced from E. Romero-Montalvo, J.M. Guevara-Vela, W.E. Vallejo Narva´ez, A. Costales, A M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Chem. Commun. 53 (2017) 3516–3519, with permission from The Royal Society of Chemistry.

Likewise the results of Albrecht and Boyd [102], the magnitudes of the negative and positive charge of the terminal N and H atoms in this chain increase with the number of monomers, until an asymptotic value is reached. The hydrogen bond chain is evocative of a bifunctional catalyst (Fig. 17.16A) in which one moiety of the catalyst might activate an electrophile (i.e., the terminal H) and another could activate a nucleophile (i.e., the terminal N). Water clusters can also present such reactivity (Fig. 17.16B) and they can activate an SO3 molecule and a water monomer in the formation of H2SO4 so that the activation energy for the formation of sulfuric acid in acid rain is considerably abated (Fig. 17.16C) [110]. There is a similar effect in the generation of H2CO3 from H2O and CO2 and in the hydrolysis of epoxides in neutral media as reported by Sauza de la Vega et al. [111] as shown in Fig. 17.17. The Gibbs free energy of activation, Gact, for the formation of glycols from the hydrolysis of epoxides in neutral aqueous media is considerably decreased as more water monomers are added into the system. We note that the largest decrease in Gact is observed when we add the third water molecule in the system. This circumstance permits the simultaneous activation of both nucleophile and electrophile of the reaction as well as the antiperiplanar attack of the nucleophilic oxygen in the process. Indeed, bifunctional catalysts allow the reaction to occur via the simultaneous activation of nucleophiles and electrophiles as well as the proper arrangement of these reactants throughout the transformation [20].

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

455

FIG. 17.17

(A) Gibbs free energy profiles for the formation of propane-1,2-diol from the hydrolysis of 2-methyl oxirane within its molecular cluster with n water molecules (n ¼ 15) at T ¼ 333.15 K. The reaction occurs on the more substituted carbon atom. Note the abatement of Gact with the increase of the size of the system and in particular with the change from n ¼ 2 to n ¼ 3. (B) Molecular clusters of oxirane with two and three water molecules. The cluster C2H4O⋯ (H2O)3 presents the activation of both nucleophile and electrophile in the reaction and its structure permits the antiperiplanar interaction between the nucleophilic oxygen and one carbon atom of oxirane. Reproduced from A. Sauza-de la Vega, H. Salazar-Lozas, W.E. Vallejo Narva´ez, M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Org. Biomol. Chem. 19 (2021) 6776–6780, with permission from the Royal Society of Chemistry.

3.3 H-bonds and π systems We have considered up to this point mostly HBs which involve only σ bonds. It is well known that HBs might entail π systems as acceptors and the QTAIM has also been exploited to characterize such HBs, with similar conclusions to the consideration of hydrogen bonds, which occur via σ bonds [70, 112]. We have also discussed the strengthening and weakening of HBs via σ bonds. Nevertheless, the strength of HBs can also be affected by their coupling with conjugated systems, for example, in resonance-assisted hydrogen bonds (RAHB). These interactions are important in several areas of organic chemistry, biochemistry, electron diffraction, nuclear magnetic resonance, and IR spectroscopy [113–116]. In an RAHB, the proton donor and proton acceptor are connected via a conjugated π system. The widespread explanation of the considerable strength of RAHBs is built on a chemically appealing argument of equalization of bond lengths throughout the π system via a tautomeric equilibrium. For instance, we have in malondialdehyde,

ð17:74Þ

Notwithstanding, numerous computational investigations indicate that there is not a correlation between the electron delocalization of the system and the energetics of these interactions. For example, although the influence of electron donating groups on an RAHB can be rationalized reasonably well by means of mesomeric structures, this is not the case for electronwithdrawing groups [117]. Neither chemical shifts nor coupling constants in NMR indicate a significant contribution from the delocalization of the π electrons to the energetics of this interaction [118]. We point out that among the different schemes for the partition of the interaction energies, QTAIM and IQA are particularly suitable to investigate the interplay between π and hydrogen bonds because both approaches address these two interactions on the same

456

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.18 Delocalization indices in the open and hydrogen-bonded forms of (A) malondialdehyde and (B) 3-amine propenal. The enclosed values in rectangles represent those DIs which decrease as a consequence of the formation of the resonance-assisted hydrogen bond and vice versa for the values encompassed within ovals. Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, A. ´ . Martı´n Penda´s, T. Rocha-Rinza, Costales, A Phys. Chem. Chem. Phys. 18 (2016) 26383–26390, with permission from the PCCP Owner Societies.

quantum mechanical basis. QTAIM studies of Grabowski et al. concluded that the strengthening of RAHBs is not related with the equalization of bonds throughout the π system [119–121]. Guevara-Vela et al. carried out an IQA/QTAIM study of malondialdehyde, 3-amine propenal as well as the dimers of formic acid and formamide as archetypes of RAHB systems. The results of this investigation revealed that although there is an equalization of the delocalization indices as a consequence of the formation of the RAHB (Fig. 17.18), the total number of delocalized electrons decreases as a result of the formation of the hydrogen bonds in these systems (Table 17.2). In addition, the formation of these RAHBs is accompanied by an increase of the magnitude of the intraatomic component of the IQA exchange-correlation energy, while the interatomic component exhibits the opposite behavior, that is, it becomes less stabilizing. Hence, the strengthening of the RAHB does not occur as a result of π resonance effects. The IQA classical component has a preponderant role in the RAHBs of malondialdehyde, 3-amine propenal, and the dimers of formic acid and formamide [122]. H-bond cooperativity and anticooperativity can also occur via networks of π bonds [123]. Romero-Montalvo et al. investigated H-bond nonadditivity between pairs of H-bonds connected by a π conjugated system (Fig. 17.19). Potential energy curves show that the TABLE 17.2 Changes in the sum of delocalization indices as well as the intra- and interatomic contributions to the IQA exchange-correlation energy as a result of the formation of RAHB in malondialdehyde, 3-amine propenal, and the dimers of formic acid and formamide. System

1 2

P

ΩA 6¼B ΔDIðΩA ,ΩB Þ

P

ΩA ΩA ΔV xc

1 2

P

ΩA ΩB ΩA 6¼B V xc

Malondialdehyde

 0.10

 31.77

16.81

3-Amine propenal

 0.07

 23.71

17.00

(HCOOH)2

 0.03

 10.37

6.62

(HCONH2)2

 0.01

 3.19

2.07

´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. 18 (2016) Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, A. Costales, A 26383–26390, with permission from the PCCP Owner Societies.

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

457

FIG. 17.19 Hydrogen-bonded systems presenting cooperativity (A) or anticooperativity (B) and (C) throughout a network of π-conjugated bonds. Reproduced from E. Romero-Montalvo, J.M. Guevara-Vela, ´ . Martı´n Penda´s, T. Rocha-Rinza, A. Costales, A Phys. Chem. Chem. Phys. 19 (2017) 97–107, with permission from the PCCP Owner Societies.

H-bonds in system 1co,RAHB strengthen each other, whereas those in 1an,RAHB and 2an,RAHB weaken one another. In other words, the H-bonds in 1co,RAHB present cooperative effects, while those in 1an,RAHB and 2an,RAHB exhibit anticooperativity. These effects are evidenced by the facts that (i) the energy of formation of the HBs in1co,RAHB in the presence of the other HB in the molecule (curves marked with vertical dashes in the top part of Fig. 17.20) are larger in magnitude than those in its absence (curves marked with asterisks in the top part of Fig. 17.20) and (ii) vice versa for 1an,RAHB and 2an,RAHB (curves marked both with vertical dashes ans asterisks in the bottom part of Fig. 17.20). Such cooperative and anticooperative effects are consistent with the mesomeric structures of 1co,RAHB, 1an,RAHB, and 2an,RAHB shown

FIG. 17.20

Potential energy curves for the formation of H-bonds in the systems shown in Fig. 17.19. The curves marked with vertical dashes represent the formation energy for a hydrogen bond in the presence of the other H-bond in the molecule. Ditto for the curves marked with asterisks and the absence of the other H-bond in the molecule. ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Reproduced from E. Romero-Montalvo, J.M. Guevara-Vela, A. Costales, A Phys. 19 (2017) 97–107, with permission from the PCCP Owner Societies.

458

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.21 Mesomeric structures which evidence the cooperativity of the H-bonds of 1co,RAHB (A) and the anticooperativity of the H-bonds in 1an,RAHB (B) and 2an,RAHB (C). Reproduced from E. Romero-Montalvo, ´ . Martı´n J.M. Guevara-Vela, A. Costales, A Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. 19 (2017) 97–107, with permission from the PCCP Owner Societies.

in Fig. 17.21. We note that for 1co,RAHB, the π system reinforces both H-bonds, while the H-bonds in 1an,RAHB and 2an,RAHB has to compete for the electrons in the π system. Table 17.3 shows that the IQA additive energies of the hydrogen-bonded atoms as well as those of the hydrocarbon chains play an important role in the nonadditive effects of 1co,RAHB, 1an,RAHB, and 2an,RAHB. A very strong anticooperative effect of hydrogen bonding can also occur via a network of π bonds. Consider the species

ð17:75Þ

which results from the deprotonation of 3-amine acrolein and a posterior tautomeric equilibrium because an enolate is less basic than an enamine. The arrow pushing in structure (17.75) TABLE 17.3 Changes in the IQA additive energies after the formation of the hydrogen bonds shown in Fig. 17.20. OH⋯O ΔEadd

⋯CH¼CHC⋯ ΔEadd

ΔE

1co,RAHB (Fig. 17.20, top left, curve marked with vertical dashes)

20.33

16.38

3.96

1co,RAHB (Fig. 17.20, top left, curve marked with asterisks)

15.48

13.29

2.19

1co,RAHB (Fig. 17.20, top right, curve marked with vertical dashes)

24.25

11.10

13.15

1co,RAHB (Fig. 17.20, top right, curve marked with asterisks)

20.64

9.25

11.38

1co,RAHB (Fig. 17.20, bottom left, curve marked with vertical dashes)

21.86

12.91

8.96

1co,RAHB (Fig. 17.20, bottom left, curve marked with asterisks)

28.43

7.62

20.80

2co,RAHB (Fig. 17.20, bottom right, curve marked with vertical dashes)

18.21

6.74

11.47

2co,RAHB (Fig. 17.20, bottom right, curve marked with asterisks)

20.93

9.27

11.66

HB

1

The results are reported in kcal mol . ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. 19 (2017) Reproduced from E. Romero-Montalvo, J.M. Guevara-Vela, A. Costales, A 97–107, with permission from the PCCP Owner Societies.

459

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

FIG. 17.22

Delocalization indices in the open and H-bonded form of the conjugated base of 3-amine propenal. The enclosed values in rectangles represent those DIs which decrease as a consequence of the formation of the resonanceassisted hydrogen bond and vice versa for the DI values enclosed in ovals, Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, A. del Rı´o Lima, ´ . Martı´n Penda´s, M. Herna´ndez-Rodrı´guez, T.R. A Rinza, Chem. Eur. J. 23 (2017) 16605–16611, with permission from John Wiley and Sons, Inc.

suggests that the electron-withdrawing nature of the HN¼CH–CH group would increase the electron population of the proton donor shown in the H-bond in the formula. Concomitantly, the proton acceptor in the same hydrogen bond would decrease its electron population, and hence, its basicity. Both effects contribute to diminish the formation of the H-bond in this system to the extent that the energy for formation of that HB is ΔEform ¼ 2.6 kcal mol1. This result is unexpected because this hydrogen bond involves charged species, a condition which is related to significant H-bond formation energies, which can be as large as 40 kcal mol1, for example, in [F⋯ H⋯ F]. This effect is known as resonance-inhibited hydrogen bond (RIHB) [124–126]. Contrarily to the RAHB, RIHB does not lead to an equalization of the bond lengths of the pseudocycle in the system, but rather to a larger difference between the DIs of single and double bonds in the system as a result of the formation of the H-bond (Fig. 17.22). The formations of RIHB and RAHB have similar effects concerning the delocalization indices as well as the intra- and interatomic contributions of the IQA exchange-correlation component to the electronic energy. Nevertheless, the effects of the RAHB are considerably larger than they are in RIHB (Table 17.4).

3.4 Analysis of H-bonds in excited states by means of QTAIM and IQA Hydrogen bonds are also relevant in electronic states different from the ground state. For instance, RAHBs can take place in excited states. Hereof, salicylideneaniline (SA) presents a phenomenon called excited state intramolecular proton transfer (ESIPT) in which a hydrogen atom is transferred from one part of the molecule in the S1 state but the process is forbidden in S0 (Fig. 17.23). This proton transfer occurs via an RAHB in the system. The QTAIM analysis of atomic energies reveals that the phenol ring in SA increases considerably its energy in the TABLE 17.4 Comparison of delocalization indices as well as the intra- and interatomic contributions to the IQA exchange-correlation energy associated to the formation of resonance-assisted and resonance-inhibited hydrogen bonds in 3-amine propenal, and its conjugated base, respectively. P

ΩA 6¼B ΔDIðΩA ,ΩB Þ

P

ΩA ΩA ΔV xc

P

ΩA ΩB ΩA 6¼B V xc

H-bond

1 2

RAHB

 0.08

 23.9

17.5

0.01

 0.7

4.0

RIHB

1 2

´ . Martı´n Penda´s, M. Herna´ndez-Rodrı´guez, T.R. Rinza, Chem. Eur. Reproduced from J.M. Guevara-Vela, E. Romero-Montalvo, A. del Rı´o Lima, A J. 23 (2017) 16605–16611, with permission from John Wiley and Sons, Inc.

460

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.23

(A) Resonance-assisted hydrogen bond that leads to the excited state intramolecular proton transfer in salicylideneaniline. (B) Potential energy curves for the proton transfer in salicylideneaniline in S0 and S1. Reproduced from L. Guti
errez-Arzaluz, F. Cort
es-Guzma´n, T. Rocha-Rinza, J. Peo´n, Phys. Chem. Chem. Phys. 17 (2015) 31608–31612, with permission from the PCCP Owner Societies.

excited state and that situation is substantially ameliorated in the ground state. The difference of ΔE for the proton transfer in the excited and in the ground state in Fig. 17.23B, ΔΔE

¼ ΔEðS1 Þ  ΔEðS0 Þ X ¼ ΔEΩA ðS1 Þ  ΔEΩA ðS0 Þ ΩA

X ¼ ΔΔEΩA ,

(17.76)

ΩA

can be decomposed in contributions from atomic basins or functional groups. The more negative the value of ΔΔEG for a given moiety G, the larger its contribution to the difference in reactivity observed between the ground and excited states. There is a considerable negative value of ΔΔEG for C6H4– (Fig. 17.24). To explain this result, Ref. [127] considered several aromaticity indices, for example, FIG. 17.24 Partition of salicylideneaniline considered in Ref. [127] and corresponding values of ΔΔEG according to Eq. (17.76). Reproduced from L. Guti
errez-Arzaluz, F. Cort
es-Guzma´n, T. Rocha-Rinza, J. Peo´n, Phys. Chem. Chem. Phys. 17 (2015) 31608–31612, with permission from the PCCP Owner Societies.

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

461

FIG. 17.25 Aromaticity index θ0 (Eq. 17.75) for the phenol ring throughout the excited state intramolecular proton transfer considered in Fig. 17.23. The larger the value of θ0 , the less aromatic the system. Reproduced from L. Guti
errez-Arzaluz, F. Cort
es-Guzma´n, T. Rocha-Rinza, J. Peo´n, Phys. Chem. Chem. Phys. 17 (2015) 31608–31612, with permission from the PCCP Owner Societies.

0

θ ¼

X

DI

ΩA

 DI

 0 2

!1=2 ,

(17.77)

ΩA G

in which DI0 is the sum of DIs of a carbon atom in C6H6 in its ground state and DIΩA is the corresponding quantity of a C atom in ring G. Fig. 17.25 shows that there is a considerable impairment of the aromaticity of the phenol ring in SA due to photoexcitation. This loss of aromaticity permits the RAHB in Fig. 17.23A to take place and, therefore, the ESIPT occurs barrierlessly as opposed to the situation in ground state. Moreover, QTAIM results show that the electronic distribution around the migrating H species is intermediate between that of a bare proton and a hydrogen atom. There are other hydrogen bonds in excited states which have been investigated using QTAIM and IQA. The computation of one- and two-electron matrices to obtain ρ1 ðr 1 ,r 10 Þ and ρ2(r1, r2) according to formulae (17.50) and (17.51) with EOM-CCSD theory allows to exploit these methods of wave-function analysis in excited states [67]. In particular, the IQA partition energy allows to determine the atoms and the chemical interactions of an electronic system, which contain the energy of an absorbed photon. For instance, the IQA method 00 established that the H-bond donor in the 1 1A excited state of (H2O)2 (in the equilibrium Cs structure of the ground state) contains most of the photoabsorbed energy in the system (Fig. 17.26B). Ditto for the H-bond acceptor and the 2 1A0 state [98]. The IQA energy partition is also instructive in the elucidation of the hypsochromic and bathochromic shifts of the 2 1A0 00 and 1 1A states of (H2O)2 with respect to the S0 !S1 excitation energy of the monomer. Fig. 17.27 shows that the red-shifted state has a larger interaction between the monomers than the ground state and vice versa for the blue-shifted state. Fig. 17.27 also indicates that the classical IQA component has an important contribution in the hypsochromic shift of the 2 1A0 excited state. Indeed, this circumstance can be understood using a simple dipole-dipole model of

462

17. Applications of the QTAIM and IQA methods to the study of H-bonds

00

FIG. 17.26 (A) Potential energy curves of the 1 1A and 2 1A0 excited states of (H2O)2 in the Cs equilibrium structure of the S0 state. (B) Deformation energies with respect to an isolated monomer in the ground state of the two H2O 00 monomers in the 1 1A and 2 1A0 electronic states of (H2O)2. The interaction energies between the monomers are also ´ . Martı´n shown. Reproduced from A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. Casals-Sainz, A. Costales, E. Francisco, A Penda´s, T.R. Rinza, Chem. Eur. J. 26 (2020) 17035–17045, with permission from John Wiley and Sons, Inc.

00

(A) Contributions of the IQA interaction energies in the 1 1A , 1 1A0 (S0), and 2 1A0 electronic states of 00 (H2O)2 at the equilibrium Cs configuration of the ground state. (B) Differences in the contributions for 2 1A0 and 1 1A ´ ´ state with respect to the ground state. Reproduced from A. Fernandez-Alarcon, J.M. Guevara-Vela, J.L. Casals-Sainz, A. ´ . Martı´n Penda´s, T.R. Rinza, Chem. Eur. J. 26 (2020) 17035–17045, with permission from John Wiley Costales, E. Francisco, A and Sons, Inc.

FIG. 17.27

interaction. As stated earlier, the monomer which contains most of the excitation energy within (H2O)2 in the 2 1A0 state is the HB acceptor. After this molecule gets photoexcited, its electric dipole moment (computed with expression 17.12) changes its direction and it points away from the hydrogens (top of Fig. 17.28). This configuration of dipole moments remains throughout the whole potential curve of the 2 1A0 state. The situation is, nevertheless, 00 different for the 1 1A excited state of (H2O)2. Once the dipole of the H-bond donor gets

3. The chemical nature of hydrogen bonds as revealed by QTAIM and IQA

463

FIG. 17.28

Magnitude and orientation of dipole moment calculated with expression (17.12) within 00 the monomers in the electronic states 2 1A0 , 1 1A , and 1 1A0 of (H2O)2 in the Cs equilibrium configuration of S0. Reproduced from A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. Casals-Sainz, A. Costales, E. Francisco, ´ . Martı´n Penda´s, T.R. Rinza, Chem. Eur. J. 26 (2020) A 17035–17045, with permission from John Wiley and Sons, Inc.

inverted, it rotates and increases considerably its magnitude as the monomers in (H2O)2 get close to each other. These circumstances increase the attractive interaction of the water molecules in this electronic state (half part of Fig. 17.28). In short, the QTAIM and IQA methods show how the interaction energies between the different molecules lead to the red- and blue00 shifted states 1 1A and 2 1A0 of (H2O)2 and that these shifts can be qualitatively understood on the basis of a simple dipole-dipole model.

3.5 Doubly and triply H-bonded systems The IQA energy partition has also been used to test models aimed to qualitatively compare the association of hydrogen-bonded systems, for example, the Jørgensen secondary interaction hypothesis (JSIH) [128]. This model has been very successful to explain the relative association of different clusters such as homodimers of amides and imides as well as complexes of the type AAA-DDD, AAD-DDA, and ADA-DAD (A stands for acceptor and D for donor of hydrogen bonds). The JSIH establishes that the relative strength of similarly related H-bonded complexes is based on the balance between the attractive and repulsive contacts in the border of the interacting molecules. For example, the JSIH indicates that amide homodimers will be more strongly bound than the corresponding imide homodimers because there is an extra repulsion with the carbonyl groups, which are not involved in the hydrogen bond in the imide dimer (Fig. 17.29A). Likewise, the JSIH correctly predicts that the order of association of AAA-DDD, AAD-DDA, and ADA-DAD complexes is AAA-DDD > AAD-DDA > ADA-DAD (Fig. 17.29B). Notwithstanding, the JSIH cannot be applied when the numbers of attractive or repulsive interactions in two pairs of investigated complexes to be compared is the same or even sometimes, the JSIH gives a wrong ordering of the strength of H-bonded clusters. For example, JSIH predicts that the stability of the heterodimer amide-imide will be between that of the amide and imide homodimers. Nonetheless, the heterodimer amide-imide is more stable than both amide and imide homodimers (Fig. 17.30). The IQA analyses of the imide homodimers and the imide-amide heterodimers reveal that indeed, the oxygen of the spectator carbonyl, OS has repulsive interactions with the interacting molecules. Nevertheless, the carbon in the same carbonyl group has an attractive interaction with the interacting molecule, which

464

17. Applications of the QTAIM and IQA methods to the study of H-bonds

FIG. 17.29

Jørgensen secondary interaction hypothesis applied for the interactions within (A) amide and imide homodimers and (B) H-bonded complexes of the type AAA-DDD, AAD-DDA, and ADA-DAD (A and D stand for hydrogen bond acceptors and donors, respectively). Solid and dashed arrows represent attractive and repulsive interactions, respectively. Reproduced from W.E. Vallejo Narva´ez, E.I. Jim
enez, M. Cantu´-Reyes, A.K. Yatsimirsky, M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Chem. Commun. 55 (2019) 1556–1559, with permission from the Royal Society of Chemistry.

FIG. 17.30

IQA interaction energies in selected pair of atoms in (A) the homodimer of 3,3-dimethyl glutarimide and (B) the heterodimer of 3,3-dimethyl glutarimide and 2-piperidone. Reproduced from W.E. Vallejo Narva´ez, E.I. Jim
enez, E. Romero-Montalvo, A. Sauza-de la Vega, B. Quiroz-Garcı´a, M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Chem. Sci., 2018, 9, 4402, with permission from the Royal Society of Chemistry.

overwhelms the repulsion of OS. These results indicate that the atoms which are further away from the region of intermolecular interaction can have an important effect on the stability of the resulting complex. These results provide evidence that the relative stability of homo- and heterodimers of amides and imides as well as triply-bounded hydrogen bonds can be better understood by considering the acidity and basicity of the functional groups involved in the interaction, in particular for the strongest acids and bases in the system [129, 130]. In this regard, the assessment of the energetics of individual H-bonds in doubly and triply hydrogen-bonded systems is an important and still open problem in supramolecular chemistry. QTAIM in conjunction with NBO [131] have been used to compute the formation energies of individual hydrogen bonds in RNA and DNA base pairs [132, 133]. Indeed, this problem is far from trivial and many subtleties may arise. For example, QTAIM on one hand

References

465

along with NBO and EDA analyses on the other predict contrary effects and dependencies of the electron donating and withdrawing character and positioning of different substituents on the formation energies of AAA-DDD systems [134].

4. Summary We considered herein the application of two important methods of quantum chemical topology, namely, the quantum theory of atoms in molecules and the interacting quantum atoms energy partition, to the study of hydrogen bonding. We reviewed briefly a series of topics within the conceptual framework of these methods of wave-function analyses, which are useful for the study of noncovalent interactions. These subjects include topological analyses of the electron density and other scalar fields, computation of atomic properties, delocalization indices as well as pairwise decompositions of the formation energy of a molecular cluster among other themes. We presented a series of general results along with a discussion of σ- and π-nonadditive effects of hydrogen bonding, the relevance of these interactions in excited states as well as the examination of doubly and triply H-bonded complexes with the earlier mentioned tools. Although we do not present an all-encompassing monograph (it would be easy to enumerate a list of relevant topics, which were not included herein), we hope that the reader could appreciate a precise panorama of the state of the art in this area and thereby reckon new directions for progress in the field.

Acknowledgment J.M.G.-V. gratefully acknowledges financial support from DGAPA/UNAM (grant 11730).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

J. Israelachvili, Intermolecular and Surface Forces, Academic Press, Burlington, MA, 2011. A.J. Stone, The Theory of Intermolecular Forces, Oxford University Press, Oxford, 2013. G.R. Desiraju, Chem. Commun. (1997) 1475–1482. D. Braga, F. Grepioni, Acc. Chem. Res. 33 (2000) 601–608. E. Cabaleiro-Lago, J. Hermida-Ramo´n, A. Pen˜a-Gallego, E. Martı´nez-Nu´n˜ez, A. Ferna´ndez-Ramos, J. Mol. Struct. THEOCHEM 498 (2000) 21–28. S.J. Stuart, A.B. Tutein, J.A. Harrison, J. Chem. Phys. 112 (2000) 6472–6486. M.A. Spackman, J.J. McKinnon, CrystEngComm 4 (2002) 378–392. J.G. Hill, J.A. Platts, H.-J. Werner, Phys. Chem. Chem. Phys. 8 (2006) 4072–4078. G.J.O. Beran, J. Chem. Phys. 130 (2009) 164115. Y.-P. Sun, F. Hennies, A. Pietzsch, B. Kennedy, T. Schmitt, V.N. Strocov, J. Andersson, M. Berglund, J.-E. Rubensson, K. Aidas, F. Gel’mukhanov, M. Odelius, A. F€ ohlisch, Phys. Rev. B 84 (2011) 132202. K.R.J. Lovelock, R. Soc. Open Sci. 4 (2017) 171223. K. Alhameedi, A. Karton, D. Jayatilaka, S.P. Thomas, IUCrJ 5 (2018) 635–646. W.H.D. Jeu, Philos. Trans. R. Soc. Lond. Ser. A 309 (1983) 217–229. A.T. Coomber, D. Beljonne, R.H. Friend, J.L. Br
edas, A. Charlton, N. Robertson, A.E. Underbill, M. Kurmoo, P. Day, Nature 380 (1996) 144–146. X. Zhang, I. Burgar, M.D. Do, E. Lourbakos, Biomacromolecules 6 (2005) 1661–1671. Y. Geng, H.-B. Li, S.-X. Wu, Z.-M. Su, J. Mater. Chem. 22 (2012) 20840. C. Sutton, C. Risko, J.-L. Br
edas, Chem. Mater. 28 (2015) 3–16.

466 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64]

17. Applications of the QTAIM and IQA methods to the study of H-bonds

A.V. Vologzhanina, Crystals 9 (2019) 478. C. Wang, Energy Environ. Mater. 3 (2020) 441–452. T. Okino, Y. Hoashi, Y. Takemoto, J. Am. Chem. Soc. 125 (2003) 12672–12673. A. Nosal-Wierci
nska, Electroanalysis 26 (2014) 1013–1023. A. Shiotari, S.E.M. Putra, Y. Shiozawa, Y. Hamamoto, K. Inagaki, Y. Morikawa, Y. Sugimoto, J. Yoshinobu, I. Hamada, Small 17 (2021) 2008010. T. Panczyk, T.D. Ros, G. Pastorin, A. Jagusiak, J. Narkiewicz-Michalek, J. Phys. Chem. C 118 (2014) 1353–1363. A. Nash, H.L. Birch, N.H de Leeuw, J. Biomol. Struct. Dyn. 35 (2016) 564–573. M. Sivanandam, K. Saravanan, P. Kumaradhas, J. Biomol. Struct. Dyn. 36 (2017) 3246–3264. A. Boags, P.-C. Hsu, F. Samsudin, P.J. Bond, S. Khalid, J. Phys. Chem. Lett. 8 (2017) 2513–2518. R. Ghanghas, A. Jindal, S. Vasudevan, J. Phys. Chem. B 122 (2018) 9757–9762. D.N. Kenie, M.C. Sekhar, M. Raveendra, K. Sivakumar, J. Mol. Liq. 297 (2020) 111796. D.K. Panda, B. Bhargava, J. Mol. Liq. 335 (2021) 116139. T. Li, P. Ayers, S. Liu, M. Swadley, C. Aubrey-Medendorp, Chem. Eur. J. 15 (2008) 361–371. M.S. Gordon, Q.A. Smith, P. Xu, L.V. Slipchenko, Annu. Rev. Phys. Chem. 64 (2013) 553–578. M. Zhang, T. Li, CrystEngComm 16 (2014) 7162–7171. J. Řeza´c, Y. Huang, P. Hobza, G.J.O. Beran, J. Chem. Theory Comput. 11 (2015) 3065–3079. S. Das, J. Preiß, J. Plentz, U. Br€ uckner, M. von der L€ uhe, O. Eckardt, A. Dathe, F.H. Schacher, E. T€auscher, U. Ritter, A. Csa´ki, G. Andr€a, B. Dietzek, M. Presselt, Adv. Energy Mater. 8 (2018) 1801737. R. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. ´ . Martı´n Penda´s, E. Francisco, J. Chem. Theory Comput. 1 (2005) 1096–1109. M.A. Blanco, A ´ . Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Comput. Chem. 28 (2006) 161–184. A J.L. McDonagh, A.F. Silva, M.A. Vincent, P.L.A. Popelier, J. Phys. Chem. Lett. 8 (2017) 1937–1942. ´ . Martı´n Penda´s, ChemPhysChem J.L. Casalz-Sainz, J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A 18 (2017) 3553–3561. P.L.A. Popelier, P.I. Maxwell, J.C.R. Thacker, I. Alkorta, Theor. Chem. Acc. 138 (2018) 12. V. Tognetti, L. Joubert, J. Chem. Phys. 138 (2013) 024102. ´ . Martı´n Penda´s, E. Francisco, M.A. Blanco, C. Gatti, Chem. Eur. J. 13 (2007) 9362–9371. A J.E. Barquera-Lozada, J. Phys. Chem. A 120 (2016) 8450–8460. J. Munarriz, E. Velez, M.A. Casado, V. Polo, Phys. Chem. Chem. Phys. 20 (2018) 1105–1113. L.J. Farrugia, C. Evans, D. Lentz, M. Roemer, J. Am. Chem. Soc. 131 (2008) 1251–1268. J.F.V. der Maelen, J.A. Cabeza, Inorg. Chem. 51 (2012) 7384–7391. I. Cukrowski, J.H de Lange, M. Mitoraj, J. Phys. Chem. A 118 (2014) 623–637. ´ . Martı´n Penda´s, J.L. Casals-Sainz, E. Francisco, Chem. Eur. J. 25 (2018) 309–314. A S. Hayashi, T. Nishide, E. Tanaka, W. Nakanishi, Molecules 26 (2021) 2936. ´ . Martı´n Penda´s, Molecules 25 (2020) 4028. J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A F. Cort
es-Guzma´n, R.F.W. Bader, Coord. Chem. Rev. 249 (2005) 633–662. P.L.A. Popelier, The Chemical Bond, Wiley-VCH Verlag GmbH & Co. KGaA, 2014, pp. 271–308. C.F. Matta, The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design, WileyVCH John Wiley Distributor, Weinheim, Chichester, 2007. P. Coppens, Science 158 (1967) 1577–1579. R. Thom, Structural Stability and Morphogenesis, Chapman and Hall/CRC, Boulder, 2018. A. Savin, R. Nesper, S. Wengert, T.F. F€assler, Angew. Chem. Int. Ed. 36 (1997) 1808–1832. X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1998) 304–314. R. McWeeny, Methods of Molecular Quantum Mechanics, Academic Press, London, 1992. F.L. Hirshfeld, Theor. Chim. Acta 44 (1977) 129–138. P. Bultinck, C.V. Alsenoy, P.W. Ayers, R. Carbo´-Dorca, J. Chem. Phys. 126 (2007) 144111. J.L. McDonagh, M.A. Vincent, P.L. Popelier, Chem. Phys. Lett. 662 (2016) 228–234. A.F. Silva, P.L.A. Popelier, J. Mol. Model. 24 (2018) 201. ´ . Martı´n Penda´s, J. Comput. Chem. J.L. Casals-Sainz, J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A 41 (2020) 1234–1241. ´ . Martı´n Penda´s, T. Rocha-Rinza, Comput. Theor. Chem. R. Cha´vez-Calvillo, M. Garcı´a-Revilla, E. Francisco, A 1053 (2015) 90–95.

References

467

´ . Martı´n Penda´s, T. Rocha-Rinza, Theor. Chem. Acc. 135 [65] I. Ruiz, E. Matito, F.J. Holguı´n-Gallego, E. Francisco, A (2016) 209. ´ . Martı´n Penda´s, T. Rocha-Rinza, J. [66] F.J. Holguı´n-Gallego, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, E. Francisco, A Comput. Chem. 37 (2016) 1753–1765. ´ . Martı´n-Penda´s, T. [67] A. Ferna´ndez-Alarco´n, J.L. Casals-Sainz, J.M. Guevara-Vela, A. Costales, E. Francisco, A Rocha-Rinza, Phys. Chem. Chem. Phys. 21 (2019) 13428–13439. [68] T. Helgaker, P. Jørgensen, J. Olsen, Molecular Electronic-Structure Theory, Wiley, Chichester, New York, 2000. [69] E. Francisco, J.L. Casals-Sainz, T. Rocha-Rinza, A. Martı´n Penda´s, Theor. Chem. Acc. 135 (2016) 170. [70] S.J. Grabowski, J.M. Ugalde, J. Phys. Chem. A 114 (2010) 7223–7229. [71] S.J. Grabowski, J. Phys. Chem. A 115 (2011) 12789–12799. [72] Y. Mo, J. Phys. Chem. A 116 (2012) 5240–5246. [73] G. Sa´nchez-Sanz, C. Trujillo, I. Alkorta, J. Elguero, ChemPhysChem 21 (2020) 2557–2563. [74] I. Alkorta, I. Mata, E. Molins, E. Espinosa, Chem. Eur. J. 22 (2016) 9226–9234. [75] S. Ebrahimi, H.A. Dabbagh, K. Eskandari, Phys. Chem. Chem. Phys. 18 (2016) 18278–18288. ´ . Martı´n Penda´s, ChemPhysChem 19 (2018) 973–987. [76] D. Sua´rez, N. Dı´az, E. Francisco, A [77] E. Arunan, G.R. Desiraju, R.A. Klein, J. Sadlej, S. Scheiner, I. Alkorta, D.C. Clary, R.H. Crabtree, J.J. Dannenberg, P. Hobza, H.G. Kjaergaard, A.C. Legon, B. Mennucci, D.J. Nesbitt, Pure Appl. Chem. 83 (2011) 1637–1641. [78] M.T. Carroll, R.F. Bader, Mol. Phys. 65 (1988) 695–722. [79] R.J. Boyd, S.C. Choi, Chem. Phys. Lett. 120 (1985) 80–85. [80] R.J. Boyd, S.C. Choi, Chem. Phys. Lett. 129 (1986) 62–65. [81] M.T. Carroll, C. Chang, R.F. Bader, Mol. Phys. 63 (1988) 387–405. [82] J. Cheeseman, M. Carroll, R. Bader, Chem. Phys. Lett. 143 (1988) 450–458. [83] D.K. Taylor, I. Bytheway, D.H.R. Barton, C.A. Bayse, M.B. Hall, J. Org. Chem. 60 (1995) 435–444. [84] P. Popelier, R. Bader, Chem. Phys. Lett. 189 (1992) 542–548. [85] U. Koch, P.L.A. Popelier, J. Phys. Chem. 99 (1995) 9747–9754. [86] E. Espinosa, E. Molins, C. Lecomte, Chem. Phys. Lett. 285 (1998) 170–173. [87] J.A. Platts, Phys. Chem. Chem. Phys. 2 (2000) 973–980. [88] J.A. Platts, Phys. Chem. Chem. Phys. 2 (2000) 3115–3120. [89] J. Poater, X. Fradera, M. Sola`, M. Duran, S. Simon, Chem. Phys. Lett. 369 (2003) 248–255. [90] A. Martı´n Penda´s, M.A. Blanco, E. Francisco, J. Chem. Phys. 125 (2006) 184112. [91] M. Goswami, E. Arunan, Phys. Chem. Chem. Phys. 11 (2009) 8974–8983. [92] M. Goswami, E. Arunan, Phys. Chem. Chem. Phys. 13 (2011) 14153–14162. [93] E. Arunan, D. Mani, Faraday Discuss. 177 (2015) 51–64. [94] J.R. Lane, J. Contreras-Garcı´a, J.-P. Piquemal, B.J. Miller, H.G. Kjaergaard, J. Chem. Theory Comput. 9 (2013) 3263–3266. [95] E.R. Johnson, S. Keinan, P. Mori-Sa´nchez, J. Contreras-Garcı´a, A.J. Cohen, W. Yang, J. Am. Chem. Soc. 132 (2010) 6498–6506. [96] J. Contreras-Garcı´a, E.R. Johnson, S. Keinan, R. Chaudret, J.-P. Piquemal, D.N. Beratan, W. Yang, J. Chem. Theory Comput. 7 (2011) 625–632. [97] V. Duarte-Alaniz, T. Rocha-Rinza, G. Cuevas, J. Comput. Chem. 36 (2014) 361–375. ´ . Martı´n Penda´s, T. [98] A. Ferna´ndez-Alarco´n, J.M. Guevara-Vela, J.L. Casals-Sainz, E. Francisco, A. Costales, A Rocha-Rinza, Phys. Chem. Chem. Phys. 23 (2021) 10097–10107. [99] M. Jabło
nski, G. Monaco, J. Chem. Inf. Model. 53 (2013) 1661–1675. [100] I. Mata, I. Alkorta, E. Espinosa, E. Molins, Chem. Phys. Lett. 507 (2011) 185–189. [101] K. Liu, J.D. Cruzan, R.J. Saykally, Science 271 (1996) 929–933. [102] L. Albrecht, R.J. Boyd, J. Phys. Chem. A 116 (2012) 3946–3951. [103] J.M. Guevara-Vela, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, J. Herna´ndez-Trujillo, O. Christiansen, E. Francisco, ´ . Martı´n Penda´s, T. Rocha-Rinza, Chem. Eur. J. 19 (2013) 14304–14315. A [104] T. Steiner, Angew. Chem. Int. Ed. 41 (2002) 48–76. [105] J.M. Guevara-Vela, E. Romero-Montalvo, V.A. Mora-Go´mez, R. Cha´vez-Calvillo, M. Garcı´a-Revilla, E. ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. Phys. Francisco, A

468

17. Applications of the QTAIM and IQA methods to the study of H-bonds

´ . Martı´n Penda´s, T. Rocha-Rinza, A. [106] V.M. Castor-Villegas, J.M. Guevara-Vela, W.E. Vallejo Narva´ez, A Ferna´ndez-Alarco´n, J. Comput. Chem. 41 (2020) 2266–2277. [107] A. Sauza-de la Vega, T. Rocha-Rinza, J.M. Guevara-Vela, ChemPhysChem 22 (2021) 1269–1285. [108] K.J. Tielrooij, N. Garcia-Araez, M. Bonn, H.J. Bakker, Science 328 (2010) 1006–1009. [109] O.A. Stasyuk, M. Sola`, M. Swart, C.F. Guerra, T.M. Krygowski, H. Szatylowicz, ChemPhysChem 21 (2020) 2112–2126. ´ . Martı´n Penda´s, M. Herna´ndez[110] E. Romero-Montalvo, J.M. Guevara-Vela, W.E. Vallejo Narva´ez, A. Costales, A Rodrı´guez, T. Rocha-Rinza, Chem. Commun. 53 (2017) 3516–3519. [111] A. Sauza-de la Vega, H. Salazar-Lozas, W.E. Vallejo Narva´ez, M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Org. Biomol. Chem. 19 (2021) 6776–6780. [112] B. Oliveira, R. Arau´jo, A. Carvalho, E. Lima, W. Silva, M. Ramos, A. Tavares, J. Mol. Struct. THEOCHEM 775 (2006) 39–45. [113] R. Markovi
c, A. Shirazi, Z. Dzˇambaski, M. Baranac, D. Mini
c, J. Phys. Org. Chem. 17 (2004) 118–123. [114] S. Goswami, A. Manna, S. Paul, A.K. Das, K. Aich, P.K. Nandi, Chem. Commun. 49 (2013) 2912–2915. [115] K.T. Mahmudov, A.J.L. Pombeiro, Chem. Eur. J. 22 (2016) 16356–16398. [116] M. Saccone, M. Pfletscher, S. Kather, C. W€ olper, C. Daniliuc, M. Mezger, M. Giese, J. Mater. Chem. 7 (2019) 8643–8648. ´ . Martı´n Penda´s, Mol[117] J.M. Guevara-Vela, M. Gallegos, M.A. Valentı´n-Rodrı´guez, A. Costales, T. Rocha-Rinza, A ecules 26 (2021) 4196. [118] C. Trujillo, G. Sa´nchez-Sanz, I. Alkorta, J. Elguero, O. Mo´, M. Ya´n˜ez, J. Mol. Struct. 1048 (2013) 138–151. [119] S.J. Grabowski, J. Leszczynski, Chem. Phys. 355 (2009) 169–176. [120] F. Fuster, S.J. Grabowski, J. Phys. Chem. A 115 (2011) 10078–10086. [121] Z. Huang, Y. Dai, L. Yu, H. Wang, J. Mol. Model. 17 (2011) 2609–2621. ´ . Martı´n Penda´s, T. Rocha-Rinza, Phys. Chem. Chem. [122] J.M. Guevara-Vela, E. Romero-Montalvo, A. Costales, A Phys. 18 (2016) 26383–26390. [123] V. Bertolasi, L. Pretto, G. Gilli, P. Gilli, Acta Crystallogr. Sect. B Struct. Sci. 62 (2006) 850–863. [124] E. Masumian, A. Nowroozi, Theor. Chem. Acc. (2015) 134. ´ . Martı´n Penda´s, M. Herna´ndez-Rodrı´guez, T.R. [125] J.M. Guevara-Vela, E. Romero-Montalvo, A. del Rı´o Lima, A Rinza, Chem. Eur. J. 23 (2017) 16605–16611. [126] E. Masumian, A. Nowroozi, P. Nikparsa, F. Zargari, Chem. Phys. 549 (2021) 111255. [127] L. Guti
errez-Arzaluz, F. Cort
es-Guzma´n, T. Rocha-Rinza, J. Peo´n, Phys. Chem. Chem. Phys. 17 (2015) 31608–31612. [128] W.L. Jorgensen, D.L. Severance, J. Am. Chem. Soc. 113 (1991) 209–216. [129] W.E. Vallejo Narva´ez, E.I. Jim
enez, M. Cantu´-Reyes, A.K. Yatsimirsky, M. Herna´ndez-Rodrı´guez, T. RochaRinza, Chem. Commun. 55 (2019) 1556–1559. [130] W.E. Vallejo Narva´ez, E.I. Jim
enez, E. Romero-Montalvo, A. Sauza-de la Vega, B. Quiroz-Garcı´a, M. Herna´ndez-Rodrı´guez, T. Rocha-Rinza, Chem. Sci. 9 (2018) 4402–4413. [131] J.P. Foster, F. Weinhold, J. Am. Chem. Soc. 102 (1980) 7211–7218. [132] H. Szatylowicz, A. Jezierska, N. Sadlej-Sosnowska, Struct. Chem. 27 (2015) 367–376. [133] A. Halder, D. Data, P.P. Seelam, D. Bhattacharyya, A. Mitra, ACS Omega 4 (2019) 7354–7368.
[134] A.H.D. Lopez, G.F. Caramori, D.F. Coimbra, R.L.T. Parreira, E.H. da Silva, ChemPhysChem 14 (2013) 3994–4001.

C H A P T E R

18 Recent advances on halogen bonds within the quantum theory of atoms-in-molecules Vincent Tognetti and Laurent Joubert Normandy University, COBRA UMR, Universit
e de Rouen, INSA Rouen, CNRS, Mont St Aignan Cedex, France

1. Introduction Chemistry keeps on evolving and creating new areas of research, including those dealing with its deepest roots. Indeed, while bonds are at the very heart of its gist, open questions and challenges remain for the correct description of chemical bonding. Noticeably, new types are regularly reported, in particular in the field of so-called “weak” bonds. More precisely, as recalled by Schneider in a recent survey, “with courageous simplification, one might assert that the chemistry of the last century was largely the chemistry of covalent bonding, whereas that of the present century is more likely to be the chemistry of noncovalent binding” [1]. Even if some of them, like the ubiquitous hydrogen bonds, have been studied for decades and decades, new types have emerged or re-emerged, such as tetrel [2,3] and halogen bonds [4]. The history of these last ones is actually quite ancient since, in 1814, Colin and Gaultier de Claubry [5] already succeeded in forming the I2…NH3 complex. But the field felt asleep before being revigorated in the two last decades, experiencing a surprising booming both from the experimental and theoretical points of view. This renaissance was triggered by the many various applications spanned by halogen bonds, which encompass, among others, materials science [6,7], drug design and medicinal chemistry [8,9], supramolecular chemistry [10–12], crystal engineering [13], and catalysis [14,15]. As for quantum chemistry [16–20], almost all available energy decomposition analyses (EDAs) [21,22] were invoked to unravel the nature of these counterintuitive bonds, revealing how much subtle they might be.

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00001-3

469

Copyright # 2023 Elsevier Inc. All rights reserved.

470

18. Recent advances on halogen bonds

In this chapter, we deliberately chose to focus on Bader’s Quantum Theory of Atoms-inMolecules (QTAIM) [23,24], a subfield of quantum chemical topology (QCT) [25], which affords a rigorous and extensive framework to unveil the physicochemical effects hidden in atomic interactions. Certainly, other approaches are as appealing, and we are not seeking here to draw comparisons between them. Even more, we do not intend either to present a comprehensive review of the application of QTAIM to halogen bonds. Instead, we have chosen to restrict our account on specific QTAIM tools, such as the Interacting Quantum Atoms (IQA) approach [26–29], some of them being less standard that those currently used in more routine QTAIM works. Following the same philosophy, specific issues, which are from our point of view underrated in the literature, will be emphasized, at the expense of more consensual elements. In all cases, we will try to present the main concepts in a self-consistent way, so that researchers unfamiliar with QTAIM may follow the whole discussion without the need to refer to the more specialized literature. To this aim, this chapter will be divided into the following sections: – the first one will briefly review QTAIM basics; – the next one will present which kinds of issues QTAIM may address, and we will discuss in details the general IQA decomposition and how it may answer the questions raised by halogen bonds; – then some specific results using such methodology will be presented, before general conclusions. As already mentioned, we do not intend to be exhaustive (the field actually generates a plethoric literature that we will not try here to consistently review), but rather to point out some useful tools and to open a vivid discussion on their relevance, their assets and drawbacks to investigate the nature of chemical bonds.

2. QTAIM basics In a nutshell, QTAIM uses the electron density gradient vector field to identify relevant points in real space (from a mathematical point, this is ℝ3) and to partition it into so-called QTAIM atoms. Such an approach thus considerably differs from methods based on the polyelectronic wavefunction, which operate on the relevant Hilbert space. There are two main possible analysis scales in QTAIM. The first one is local: it inspects the !
! so-called critical points (CPs), which are points where the electron density gradient rρ r vanishes. CPs can be classified in terms of rank (r) and signature (s) of the electron density Hessian matrix (which collects all second derivatives of the electron density with respect to space coordinates): r is the number of nonzero eigenvalues, while s is the algebraic sum of the eigenvalues signs. Electron density maxima are thus (3,3) CPs and electron density minima are (3,+3) CPs (they are also called attractors). (3,1) CPs correspond to a minimal electron density value in one direction and to a maximum in the perpendicular plane, whereas the electron density is maximal in one direction and minimal in the orthogonal plane at (3,+1) CPs.

3. Specific QTAIM tools for halogen bonds

471

At a stationary point on the potential energy surface (PES), (3,1) CPs are coined “bond critical points” (BCPs). To understand this terminology, it is useful to introduce the concept of gradient path (GP). A GP is a curve in real space whose tangent line at any point is collinear ! ! to rρ r . For a given BCP, there are two special GPs that originate at this BCP and that terminate at one attractor. When these two attractors are nuclei, the union of these two GPs defines the bond path (BP) between them. In the orthodox QTAIM interpretation (that has created useful controversies [30–35]), two atoms are bonded if and only if there exists a BP between them, and thus if there exists a BCP between them. We now describe how atoms are properly defined by QTAIM. The bundle of all GPs ending at a given attractor forms a volume in the three-dimensional space, called a QTAIM basin. A QTAIM atom is then simply the union of a basin with its attractor. Such basins are separated !

by surfaces, called “interatomic surfaces” (IASs). At each point on an IAS, the flux of rρ through the IAS vanishes. As a consequence, basins are not overlapping and constitute an exhaustive partition of real space. Looking at basin properties is part of the regional analysis scale. Obviously, other real space partitions exist (for instance, based on the electron localization function (ELF) [36]). One of the QTAIM advantages is that it requires (at least for its simplest applications) only the knowledge of the electron density as the basic ingredient, which can be determined both theoretically and experimentally (for instance by X-ray diffraction experiments).

3. Specific QTAIM tools for halogen bonds 3.1 General aims With these tools in hand, we can wonder what the main questions a researcher would like to tackle using a QTAIM analysis are. Below is a restricted selection of relevant ones. They are related either to prediction or explanation, thus covering both qualitative and quantitative purposes: 1. 2. 3. 4.

What is the strength of this bond? Why does this bond form? Could this strength be predicted? What is the nature of this bond?

At first sight, some questions might seem redundant. For instance, one may think that the nature of the bond is linked to the cause for its formation. As we will discuss later, the factors that govern the approach of two fragments when they are still far from each other are not necessarily the same as those that control the interaction at the equilibrium geometry. Similarly, one may find a descriptor that is correlated to the bond strength but that does not properly account for the bond nature. Indeed, correlation has not be confounded with causality [37]. Each of these issues will be studied in specific sections: question 1 will be scrutinized in Section 3.2, question 2 in Section 4.1, question 3 in Section 4.2, and the last one in

472

18. Recent advances on halogen bonds

Section 4.3. First, we will show that defining a bond strength for a complex made of two molecular fragments is not an unambiguous task.

3.2 The many faces of interaction energies Indeed, several nonequivalent energies can be calculated to estimate interaction between the halogen bond donor R  X and acceptor (in general a Lewis base L). The first one is the interaction energy: Eint ¼ EðR  X…LÞ  ERX…L ðR  XÞ  ERX…L ðLÞ

(18.1)

where ERX…L(R  X) (respectively ERX…L(L)) denotes the energy of fragment R  X (respectively L) at the geometry it adopts in the complex. Energy is here to be meant as the sum of the electronic energy and the nuclei repulsion energy, as defined within the Born-Oppenheimer approximation. Eq. (18.1) can actually be evaluated in two main ways: the first one uses for ERX…L(R  X) and ERX…L(L) the basis sets B of R  X and L, respectively, which can be written as: BðRXÞ

B ð LÞ

Eint ¼ EðR  X…LÞ  ERX…L ðR  XÞ  ERX…L ðLÞ

(18.2)

In the case of localized finite basis sets such as Gaussians, this approach is prone to basis set superposition errors (BSSEs), which can be partly cured by resorting to the Boys-Bernardi [38] (BB) counterpoise method: BðRX…LÞ

EBB int ¼ EðR  X…LÞ  ERX…L

BðRX…LÞ

ðR  XÞ  ERX…L

ðL Þ

(18.3)

As a consequence, one has always EBB int > Eint. From our experience, for halogen bonds, BSSEs remain quite low (order of magnitude 0.1 kcal mol1 or below) when triple-ζ basis sets with diffuse functions are used. Such energies should be distinguished from the complexation energies where the energies of the fragments are computed at their own equilibrium geometries: BðRXÞ

Ecompl ¼ EðR  X…LÞ  ERX

BðLÞ

ðR  X Þ  E L

ðLÞ

(18.4)

It can be noticed that such a definition does not allow applying BB corrections in a safe way. From these equations, one can further calculate the deformation energy that is always a positive quantity measuring the energy that has to be brought to each fragment in order to put them in their geometries inside the complex: BðRXÞ

BðRXÞ

Edeform ¼ Ecompl  Eint ¼ ERX…L ðR  XÞ  ERX

BðLÞ

BðLÞ

ðR  XÞ + ERX…L ðLÞ  EL

ðLÞ

(18.5)

0K

Stricto sensu, none of these energies represent internal energies at 0 K (U ) since zero-point energy (ZPE) vibrational contributions are missing. Including ZPEs (mostly achieved using the harmonic oscillator approximation on the optimized geometries), one can estimate complexation internal energies at 0 K according to: BðRXÞ

K ¼ EðR  X…LÞ  ERX U0compl  ZPEðLÞ

BðLÞ

ð R  X Þ  EL

ðLÞ + ZPEðR  X…LÞ  ZPEðR  XÞ (18.6)

3. Specific QTAIM tools for halogen bonds

473

Using the appropriate thermodynamic partition functions, one can subsequently evaluate UTcompl at any temperature T. Furthermore, assuming an ideal gas behavior (PV ¼ RT for each molecule or fragment), the complexation enthalpy at a given temperature simply equals: HTcompl ¼ UTcompl  RT

(18.7)

Adding entropic contributions on top of enthalpy straightforwardly affords complexation Gibbs energies GTcompl: GTcompl ¼ HTcompl + T ½SðR  XÞ + SðLÞ  SðR  X…LÞ

(18.8)

which is termed standard complexation Gibbs energies when evaluated at 25°C and with a pressure equal to 1 bar. As the formation of a halogen bond results from the condensation of two molecules, the term intro brackets in Eq. (18.8) is in general positive, so that GTcompl > HTcompl. It might have important impact on the stability of the complex, as discussed by Murray and Politzer [39]. For instance, only slightly exothermic complexation (HTcompl ⋦ 0) might become endergonic (GTcompl ≿ 0) when entropic effects are taken into account. It should be noticed that all these energies can be calculated in gas phase or in condensed phase (solvent or solid). In such cases, specific terms (for instance solvation energies) can be also analyzed, but we will not discuss them in this chapter. One of the main drawbacks of such quantities is that they cannot be univocally calculated in case of intramolecular halogen bonds [40–42] (fragments are then not uniquely defined) and that they do not give insight into the role of each atom (or functional group) with respect to the bond. To this last aim, energy decompositions are required. Several ones are currently available in various codes, each of them featuring its own strong and weak points. In this chapter, we have decided to focus on the IQA scheme, mainly developed by the Penda´s’ team at the University of Oviedo in Spain. Our purpose is not here to present a detailed survey of this method. Instead, we refer the interested reader to the definitive review [26] recently published by Penda´s and co-workers.

3.3 The IQA decomposition Contrarily to other energy decompositions, IQA offers an atomic resolution of molecular energies. An alternative atomic decomposition exists within QTAIM: the venerable (and older) virial energy decomposition. However, it is less general since it can only be applied at stationary points (in practice energy minima and transition states) on the PES, while IQA can be performed at any point on the PES. Only IQA thus allows following “point by point” a reaction path (or a dynamical trajectory). It should however be emphasized that IQA (as well as the virial approach) concentrates on electronic energies E, so that it will give no insight at all on the ZPE, thermal and entropic contributions. Up to now, we are not aware of available atomic decompositions of these components, albeit they can be essential, as already mentioned, in a correct interpretation of Gibbs complexation energies. In practice, IQA can be performed either on post-Hartee-Fock wavefunctions or after density functional theory (DFT) calculations. In the following, we will briefly discuss the master equations within both frameworks, and we will highlight the possible discrepancies between

474

18. Recent advances on halogen bonds

them. We start with the exact polyelectronic wavefunction of N electrons for which the exact ground state molecular energy reads in atomic units: 1 X X ZA ZB (18.9) EGS mol ¼ T e ½ρ1  + Ene ½ρ + Eee ½ρ2  + 2 A A6¼B RAB where ρ, ρ1 and ρ2 are the electron density, the first-order and second-order reduced density matrices (RDMs), respectively. The last term in Eq. (18.9) represents the repulsion energy between each pair of nuclei that are here considered as pure classical point ! charges  (here gener! ! !   ically referred to by ZA and ZB, located at RA and RB with RAB ¼ RA  RB ). If one introduces the infinite set of natural orbitals (NOs), {φNO i }, that diagonalize ρ1 with , the electronic kinetic energy T can be exactly expressed by: (partial) occupations nNO i e ð ∞     1 X  ! NO ! 2 3 NO n r φ r  dr (18.10) ¼ T e ½ρ1  ¼ T e φNO  i i 2 i¼1 i ℝ3

The energy Ene resulting from the total interaction between the electrons and the nuclei simply reads: 1 0 ð   X ZB ! B  Cd3 r (18.11) Ene ½ρ ¼ ρ r @ ! ! A B  r  RB  3 ℝ

Finally, the total repulsion between all electrons is exactly:   ! ð ðP ! r 1, r 2   d3 r1 d3 r2 Eee ½ρ2  ¼ !  !  r 1  r 2 3 3

(18.12)

ℝ ℝ



!

!



owdin’s normalization convenwhere P r 1 , r 2 is the diagonal element of the 2-RDM with L€ tion for the second-order RDM (integrating to N(N1)/2). In case an exhaustive and nonoverlapping of real space into basins ΩA is used, Ð partition P Ð each integral can be decomposed following ℝ3 ¼ A ΩA. We will now consider here that each basin contains one and only one nucleus (nonnuclear attractors are here excluded for the sake of simplicity), so that ΩA is the volume associated with topological atom A, which is the union of nucleus of charge ZA and its basin ΩA. Eq. (18.9) then becomes: ! ∞   2   Xð 1 X X Xð ZB ! NO  NO !  3 EGS ¼ n r φ r d r +  ρ r ! !  d3 r   mol i i   2 Ω Ω A A B A A i¼1  r  RB    ! ! X X ð ð P r 1, r 2 3 3 1 X X ZA ZB   d r1 d r2 + + (18.13) !  ! 2 A A6¼B RAB r  r  B A ΩA ΩB

1

2

475

3. Specific QTAIM tools for halogen bonds

All

PP A B

double sums can be divided into A ¼ B and A 6¼ B terms: ! ∞   2   Xð 1 X Xð ZA ! GS NO  NO !  3 ni rφi r  d r+  ρ r ! !  d3 r Emol ¼   2 A ΩA A ΩA i¼1  r  RA  |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A Te

+

X Xð A A6¼B ΩA



EA ne

 ! ð ð P !   r , r X 1 2 ZB !   d 3 r 1 d3 r 2  d3 r + ρ r  !  ! ! !   r 1  r 2 A  r  RB  ΩA ΩA |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 

EA ee



! ð ð 2P ! r 1, r 2 1 XX 1 X XZA ZB   d3 r1 d3 r2 + + !  ! 2 A A6¼B 2 A A6¼B RAB  r 1  r 2 |fflffl{zfflffl} ΩA ΩB |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} EAB nn

(18.14)

EAB ee

A where TA e is the electron kinetic energy of atom A, Ene the attraction energy of electrons of A by A their own nucleus, Eee is the bielectronic repulsion inside atom A, while EAB ee represents the total repulsion energy between electrons in atom A with those inside other atoms B. Besides, Ð ρð!r ÞZB 3 P P  the Q ¼  A A6¼B ΩA  !r R!B  d r term can be conveniently written in a more symmetric way

in order to make atoms A and B play equivalent roles according to: 0

1

    B C Bð ρ ! C ð ρ ! r r ZA Z X X B B C 1 3 3 B C     Q¼ ! d r + !  d rC ! !   2 A A6¼BB ΩB  r  RA  B ΩA  r  R B  C @ A |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} EAB en

(18.15)

EAB ne

with EAB en representing the attraction energy of electrons in atom A by nucleus in atom B, and that of electrons in atom B by nucleus in atom A. EAB ne Eq. (18.14) can thus be summarized by: X
 1 X X
AB  A A AB AB TA Een + EAB (18.16) EGS mol ¼ e + Ene + Eee + ne + Eee + Enn |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 A A6¼B|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A EA self

EAB inter

Hence, the final form of the IQA decomposition is (the ½ factor avoiding any double counting): X 1 X X AB EA E (18.17) EGS mol ¼ self + 2 A A6¼B inter A

476

18. Recent advances on halogen bonds

This fundamental equation can also be written in a purely additive form: X 1 X AB A A EA E EGS mol ¼ IQA with EIQA ¼ Eself + 2 A6¼B inter A Note that the energy decomposition in the virial approach is also additive (EGS mol ¼

(18.18) P A

EA virial),

A but we have shown in a recent paper [43] that EA virial and EIQA may strongly differ (and, of A course, they are also different from Eself). This additive form can be useful for instance to decompose global descriptors from conceptual DFT into atomic contributions [44]. As a related digression, we recently showed [45] that interaction energies in halogen bonds were highly μ2 , where μ is the electronic potential and η correlated to Parr’s electrophilicity index [46] ω ¼ 2η the molecular hardness. Using finite difference linearization (FDL), (
 μ ¼ EmolðN+1Þ  EmolðN1Þ =2 (18.19) η ¼ EmolðN+1Þ + EmolðN1Þ  2Emol

where Emol(N+1) (respectively Emol(N1)) is the molecular energy when one electron is added to (respectively removed from) the molecule at its geometry with N electrons (vertical processes at constant external potential). All three molecular energies (Emol, Emol(N+1), Emol(N1)) can be partitioned according Eq. (18.18), so that:   8 X > μA with μA ¼ EA  EA =2 μ¼ > IQA ð N+1 Þ IQA ð N1 Þ < A (18.20) X A A > > ηA with ηA ¼ EA :η ¼ IQAðN+1Þ + EIQAðN1Þ  2EIQA A

This additive IQA decomposition thus paves the way toward the atomic description of electrophilicity and nucleophilicity, which are key-concepts in halogen bonds (see Section 4.1). However, it fully hides the interactions between atoms, a fact that makes the additive interpretation less intuitive from a standard chemical point of view, and, in some way, less useful. It should be stressed by the decomposition in Eq. (18.17) is in principle exact and only involves mono-atomicanddiatomictermssinceonlyone-bodyandtwo-bodyoperatorsenterthemolecular   !

!

Hamiltonian. In practice, nevertheless, the big challenge is the actual computation of P r 1 , r 2 . Efficient implementations have been reported at various Møller-Plesset levels by the Popelier group [47–50] and at the coupled cluster theory by the Rocha-Rinza and Penda´s groups [51–54]. We recently proposed [55] a less computationally expensive alternative within the framework of Reduced Density Matrix Functional Theory (RDMFT) based on the M€ uller functional [56]. term can be accurately estimated by (assuming real-valued natural orbitals): For instance, the EA ee     ! ð ð ρ ! r1 ρ r2 1 A   d3 r 1 d3 r 2 Eee ¼ !  ! 2  r 1  r 2 Ω 1 0A ΩA         ! ! ! ! ∞ X ∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð ð φNO r 1 φNO r 2 φNO r 1 φNO r 2 X i i j j C B NO   nNO  @αA + βA d3 r1 d3 r2 A i nj !  !  r 1  r 2 i¼1 j¼1 Ω Ω A

A

(18.21)

3. Specific QTAIM tools for halogen bonds

477

where αA and βA values are constant only depending on atom type. More specifically, all hydrogen atoms will use the same αH and βH parameters, all carbon atoms will share the common αC and βC values, and so on. Interestingly, such an approach affords a post-HF quality at a rather moderate computing cost. Let us now discuss such energy decomposition within Kohn-Sham (KS) density functional theory [57]. In this framework,  KS 

1 X X ZA ZB (18.22) + Ene ½ρ + J ½ρ + Exc ½ρ + EGS mol ¼ T s φi 2 A A6¼B RAB where Ts denotes the electronic kinetic energy of the fictitious noninteracting electronic KS system (which by definition yields the same electron density as the real interacting electronic system) according to the finite sum (at variance with Eq. 18.10 that involves do not exactly vanish in molecular systems in principle an infinite summation since nNO i [58,59]): Ts



φKS i



N=2 ð   1X   ! KS ! 2 3 ¼ rφi r  d r 2 i¼1 ℝ

(18.23)

3

φKS i

where occupied KS orbitals (here doubly-occupied in the case of closed-shell chemical species) allow building the KS wavefunction ψ KS through a Slater determinant, which in general differs from the exact (correlated) wavefunction ψ exact: h i  ψ KS ∝ det φKS (18.24) i 1iN=2 In Eq. (18.22), J stands for the so-called Hartree energy, which is the “classical” electrostatic part of the bielectronic repulsion:     ! ð ðρ ! r1 ρ r2 1   d3 r 1 d3 r 2 (18.25) J ½ρ ¼ !  ! 2  r 1  r 2 3 3 ℝ ℝ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} ! ! Jð r 1 , r 2 Þ Finally, the Exc[ρ] exchange-correlation (xc) functional gathers the so-called correlation kinetic energy (difference between the exact electronic kinetic energy and the one of the KS system) and the nonclassical contribution to the bielectronic contribution. Traditionally, exchange-correlation is split into an exchange and a correlation terms. While there is in principle not a unique way for such a partition, the most widespread is that based on the so-called exact exchange (XX) definition (which also naturally emerges from the G€ orling-Levy perturbational approach): 8

D E

b > > < EXX ½ρ ¼ ψ KS V ee ψ KS  J ½ρ (18.26)

D E D E > > be + V b ee

ψ exact  ψ KS jT be + V b ee jψ KS : Ec ½ρ ¼ ψ exact

T b e and V b ee are, respectively, the electronic kinetic operator and the bielectronic repulwhere T sion operator entering the molecular Hamiltonian. In principle, EXX can be expressed as an

478

18. Recent advances on halogen bonds

electron density functional. However, using Eq. (18.24), the exact exchange density functional can be alternatively written as a functional of the occupied KS orbitals (here assumed realvalued):         ! ! ! ! ð ðX N=2 X N=2 φKS r 1 φKS r 2 φKS r 1 φKS r 2  KS 

i i j j   d3 r 1 d3 r 2 EXX ½ρ ¼ EXX φi ¼ !  ! r  r   i¼1 j¼1 1 2 ℝ3 ℝ3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ! ! eXX ð r 1 , r 2 Þ

(18.27)

This is the same functional as that in Hartree-Fock (HF) theory. However, the HF exchange energy and the KS EXX[{φKS i }] energy are in general different, because HF and KS orbitals are not identical. We can now partition the energy using the same approach as in the case of post-HF wavefunctions. For instance, the Ene[ρ] is the same in both frameworks, and can thus be writ P P
AB 1 ten as EA Een + EAB ne + 2 ne . Processing the other terms similarly, we obtain: A A6¼B

EGS mol ¼

Xð

N=2   X  1 X 1 X X
AB  ! KS ! 2 3 EA Een + EAB r φi r  d r + ne + ne 2 i¼1 2 A A6¼B A ΩA A |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} TA s ð ð   X ð ð ! !  3 3 1 XX ! ! + J r 1 , r 2 d r1 d r2 + 2J r 1 , r 2 d3 r1 d3 r2 2 A A6¼B A ΩA ΩA ΩA ΩB |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

+

Xð ð A

EA J

EAB

J ð ð    1 XX ! ! ! ! 3 3 eXX r 1 , r 2 d r1 d r2 + 2eXX r 1 , r 2 d3 r1 d3 r2 + Ec ½ρ 2 A A6¼B ΩA ΩA ΩA ΩB |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}



1 X X nn E + 2 A A6¼B AB

EA XX

EAB XX

(18.28)

It is common practice to gather “classical” electrostatic terms together according to: ( A A Ecl ¼ EA ne + EJ (18.29) AB AB AB nn EAB cl ¼ Een + Ene + EJ + EAB The last unspecified term is the correlation functional. In almost all published DFT IQA results, mathematically local functionals (in the sense they involve only one space position) are used, since up  to now  the nonlocal approaches such as Van Voorhis [60] (that incorporate ! !0

kernels of the h r , r

type) or double-hybrids functionals [61,62] (that include a second-

order perturbation term in the spirit of MP2 approaches) were not employed. We will thus

3. Specific QTAIM tools for halogen bonds

479

exclusively consider meta-GGA (m-GGA) formulas that gather the first three rungs of Perdew’s Jacob’s ladder of DFT: ð    !     ! ! ! ¼ e ρ r , rρ r , τ r d3 r (18.30) EmGGA mGGAc c ℝ3

  ! where τ r denotes the KS kinetic energy density in its Lagrangian (i.e., definite positive)   ! form and emGGAc r the correlation energy density. The integral in Eq. (18.30) can be easily partitioned into atomic contributions. Eq. (18.28) then becomes: with mGGAc EGS ¼ mol

X
A

 1 X X
AB  A A A TA E + EAB e + Ecl + EXX + EmGGAc + XX 2 A A6¼B cl

(18.31)

which can be put, using obvious notations, in a form reminiscent of Eq. (18.17): with mGGAc EGS ¼ mol

X 1 X X AB,cl,XX with mGGAc EA + E self 2 A A6¼B inter A

(18.32)

Note that Eq. (18.32) can also put in an additive way in the spirit of Eq. (18.18). It is also interesting to study the long-range behavior of the interatomic term using multipolar expansion of the interaction distance. To first order, the classical electrostatic term is     ! ð ðρ ! ð ð     r1 ρ r2 1 NA NB ! !   d3 r 1 d3 r 2  ρ r 1 ρ r 2 d 3 r 1 d3 r 2 ¼ (18.33) !  ! RAB R AB r  r  ΩA ΩB

1

2

ΩA ΩB

where NA is the electron population in basin ΩA. Similarly,   ð ρ ! ð   r ZB ZB ZB N A ! 3   ρ r d3 r ¼ ! d r  !  RAB ΩA RAB ΩA  r  RB 

(18.34)

so that: EAB cl  

ZB N A ZA NB NA NB ZA ZB  + + RAB RAB RAB RAB

(18.35)

QTAIM atomic charges are defined by: qA ¼ ZA  N A

(18.36)

Eq. (18.35) is then no more than: EAB cl 

qA qB RAB

(18.37)

480

18. Recent advances on halogen bonds

In other words, the first term in the multipolar expansion of the classical interaction term [63] is the pure point charge interaction. The exchange contribution can be treated in the same vein [64], involving orbital overlaps Sij over each atomic basin: EAB XX

1  RAB

N N ð ð X 2 2 X

ΩA 0 ΩB

        ! ! ! ! φKS r 1 φKS r 2 φKS r 1 φKS r 2 d 3 r 1 d3 r 2 i i j j

i¼1 j¼1

10

1

CB C B CB N N ð C B N N ð C C B B         2 2 2 2 X X X X 1 B 3 CB 3 C KS ! KS ! KS ! KS ! ¼ φi r φj r d r C B φ i r φj r d r C B CB i¼1 j¼1 C RAB B i¼1 j¼1 CB C B ΩA ΩB A @ @ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

(18.38)

SBij

SA ij

The delocalization index (DI), which is a kind of QTAIM bond index [65–67], reads (according to Bader and Fradera [68] and Cioslowki and Mixon [69], with obvious simplified notations): 0 10 1 X X A@ δAB ¼ 2@ (18.39) SA SBij A ij i, j

i, j

Accordingly, the first term in the multipolar expansion of the exchange interatomic energy reads (see also Refs [70,71]): EAB XX  

δAB 2RAB

(18.40)

This result hints that EAB XX is directly linked to covalency. Nevertheless, higher-order multipoles are sometimes needed to reach convergence, even if, from our experience, Eqs. (18.37), (18.40) usually provide more than good estimates for the interaction energies involved in halogen bonds. In practice, standard current DFT calculations do not actually use 100% exact exchange, but a mixing between nonlocal XX and GGA or mGGA exchange functionals in order to optimize error cancellations between an approximate correlation functional and an approximate exchange functional. Then, in order to recover the full self-consistent field (SCF) energy, some with mGGAc terms are required. Several corrections can be found in Refs modifications to the EA self [72,73], which will be not further detailed. Up to now, the correlation energy contribution has not been investigated in details. Understanding correlation effects is actually of outmost importance in chemistry. It is thus valuable to examine what IQA can tell us about its (inter)atomic components. In KS DFT, the correlation energy functional collects two contributions: the so-called correlation kinetic energy and the correlation bielectronic repulsion energy. As already stated, the first one originates from the fact that the kinetic energy Ts computed from the KS orbitals is that of the fictitious noninteracting system, which can be shown, by construction, to be always lower (at the molecular level) than the true electronic kinetic energy Te. The molecular correlation kinetic Tc is

3. Specific QTAIM tools for halogen bonds

481

simply equal to Te  Ts. Using Eqs. (18.10), (18.23), it can be partitioned into atomic contributions according to [74]: ! ð N=2  ! ∞  2   X X 1  ! KS ! 2 3 A NO  NO !  3 T c,Bader ¼ n rφi r  d r (18.41) r φi r  d r 2 ΩA i¼1 i i¼1 While exact, this formula is not very useful in practice since it requires a (often timeconsuming) post-HF calculation to get NOs. In a pioneering work, built upon relationships due to Perdew and Levy, Rodr
ıguez et al. [75] proposed an alternative definition of atomic correlation kinetic energy based on: ð      !   ! ! ! ! ¼  exc r + ρ r r  rvxc r d3 r (18.42) TA c,Rodr
ıguez ΩA

where exc and vxc denote the exchange-correlation energy density and the exchangecorrelation potential (i.e., the G^ ateaux derivative of the exchange-correlation functional with respect to the electron density), respectively. This approach is that implemented in the popP ular ADF software. It should be stressed that if the exact exc and vxc are used, the A T A c,Bader and P A A T c,Rodr
ıguez molecular (i.e., total) values are equal (and positive). However, there is no theA oretical reason that, for any atom, TA c, Bader and Tc, Rodr
ıguez are identical (and not necessarily positive) [74]. While Rodr
ıguez’ approach is powerful and deeply rooted in physics, it however requires the calculation of the exchange-correlation potential, a task that can be cumbersome for complex functionals. In a recent paper [74], we thus proposed to use, instead, approximate expressions based on electron density functionals such as those based on Shannon entropies: ð        ! ! ! ¼ A + Bρ r (18.43) + Cρ r ln ρ r d3 r TA c,Shannon ΩA

In summary, there are now several valuable methods that are available to estimate atomic correlation kinetic energies. We now turn our attention to bielectronic correlation. In case a metaGGA correlation functional is used, as shown in Eqs. (18.30)–(18.32), no correlation term appears in the diatomic interaction contributions. However, correlation between atoms can be crucial in many cases, epitomized by complexes maintained by (London) dispersion [76]. Some studies have actually given evidence that dispersion could be a leading term in halogen bond strength. A simple way to introduce this missing contribution is to use Grimme’s pairwise dispersion correction: EdispGD

1 XX CAB ¼ f AB 66 2 A B6¼A RAB

! ¼

1 X X AB E 2 A A6¼B dispGD

(18.44)

where fAB is a damping function and CAB 6 the dispersion coefficient that is directly related to atomic polarizabilities through the Casimir-Polder expression. It should be added that higher order terms (such as those proportional to 1/R8AB) could be similarly incorporated, but we will not explicitly use them in this chapter. Let us also note that in cases the associated

482

18. Recent advances on halogen bonds

exchange-correlation functional already incorporates some dispersion, the EAB dispGD values may differ from a more exact treatment. An alternative way [77] is then to use HF and MP2 calculations in order to evaluate the following difference: AB AB EAB dispMP2 ¼ EinterMP2  EinterHF

(18.45)

All of these formulas show that the IQA scheme allows in principle a full atomic decomposition of the molecular energy, even at the DFT level. But it also proposes the partition of Emol into physical components such as kinetic, electrostatic, covalency and dispersion energies. As a final remark, it should be underlined that these IQA decompositions are based on a nonrelativistic treatment. We refer the interested reader to the insightful papers by Anderson, Rodr
ıguez and Ayers [78–80] for relativistic extensions of QTAIM. Note also that only ground state energies were considered here, but IQA could be in principle applied to excited states [81], even if such applications still remain scarce.

3.4 IQA decomposition and halogen bond strength Let us now come back to the interaction and the complexation energies (the differences being caused by the choice of the geometries considered for the fragments R  X and L) between the halogen bond acceptor and donor. It can be decomposed according to: 9 8 X X AB Einter jRX…L + > > > > > > ARX BL > > > > > > > > > >     X X > > = < A A B B Eself j  E j E j  E j + + self self self … … RX L RX RX L L Eint,compl ¼ BL ARX > > > > > > > > > >     XX > >1 X X 0 0 0 0 1 > > AA AA BB BB > > > E j  E j E j  E j + … … ; inter RX L inter RX inter RX L inter L > :2 2 ARX A0 RX

BL B0 L

(18.46) The first term in the right-hand side represents the interaction energy between each atom in R  X with each atom in L. This is a measure of the pure interaction between the two fragments. The two following terms account for the variation in the self-energies of all atoms upon the complex formation. Finally, the two last terms are the variations of the interactions energies inside each fragment (difference between their values in the complex and in each isolated fragment). All atoms might thus have a nonnegligible contribution to the interaction energies. However, two atoms have a particular position. Indeed, in virtue of the Poincar
e-Hopf relationship, a (3,1) CP exists between R  X and L at any separation distance [82]. In general, it links halogen atom X and the atom bearing the lone pair in the Lewis acid (for instance nitrogen in the FCl…NH3 complex) that we will denote X0 . Hence, at the equilibrium geometry, there is a bond between X and X0 . This privileged interaction will be called the primary interaction. All interactions between an atom in R  X with another one in L will be called secondary.

4. Some QTAIM results about halogen bonds

483

With obvious notations, Eq. (18.46) can thus be rewritten: X sec ondary X X X primary primary Eint,compl ¼ Einter + ΔEself + Einter + ΔEA ΔERX ΔELinter (18.47) self + inter + |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} A6¼X,X0 primary

Etot

where the two first contributions in the right-hand side can be collected in a term (Eprimary ) that tot gathers the mains effects of atoms X and X0 : in a sense, this is a kind of X  X0 bond energy [83]. Thanks to the IQA decomposition, one can go even further by evaluating the role of covalency (exchange-correlation) and ionicity (classical electrostatics) to this bond. This last contribution is linked to charge transfers between atoms, a contribution that is assumed to be important in halogen bonds [84]. One way for calculating the energy of such process is to use constrained DFT, as applied by Řeza´c and de la Lande [85], who estimated it close to 10% of the interaction energy. If one wants to remain in the IQA context, it is useful to first introduce the pro-fragment density according to:       ! ! ! r + ρRX…L r (18.48) ρpro r ¼ ρRX…L RX L (respectively ρRX…L ) denotes the electron density of the isolated where ρRX…L RX L fragment R  X (respectively L) at its geometry within the R  X…L complex. We recall that the classical electrostatic interaction energies only involve the electron density. We can thus roughly evaluate the contribution of charge transfer (CT) to the X  X0 bond energy by [77]: h i 0 XX0 XX0 ρpro (18.49) EXX CT ¼ Ecl ½ρRX…L   Ecl Once more, this example proves that IQA can provide various relevant energy components to unravel the main physicochemical processes at stake in such bonding patterns. The next section will now give some examples for the use of the presented concepts to study the nature of halogen bonds and its mechanism of formation.

4. Some QTAIM results about halogen bonds 4.1 What drives the bond formation? The official IUPAC definition of the halogen bond reads [86]: “A halogen bond occurs when there is evidence of a net attractive interaction between an electrophilic region associated with a halogen atom in a molecular entity and a nucleophilic region in another, or the same, molecular entity.” Several real-space functions have been proposed to visualize these electrophilic areas within the halogen atom. The most popular one is certainly that based on the molecular electrostatic potential (MEP):  0 ð ρ !   X r ZA !  3 0    (18.50) MEP r ¼ ! ! !0  d r !  r  r  A  r  RA 

484

18. Recent advances on halogen bonds

In R  X isolated species, a region with positive MEP values generally develops in the outer part of the halogen atom along the R  X line for chlorine, bromine, iodine and astatine. This is very rarely the case with fluorine [87,88], explaining (perhaps too crudely) why the formation of halogen bonds with fluorine is generally precluded. This area of positive MEP values has been called “σ-hole” by Politzer and co-workers [89,90], and it has become (thanks to or despite its simplicity) the most popular paradigm to account for halogen bond formation. Alternatively, conceptual DFT [91,92] is also a framework of choice to decipher electrophilicity and nucleophilicity. In particular, the electrophilic (respectively nucleophilic) regions     !

can be characterized by Fukui functions f + r

!

respectively f  r

. The information

encoded in these two local descriptors can be conveniently summarized by the dual descriptor f(2) [93], which is often complementary [94] to the MEP: 0 !1           ∂2 ρ r ! ! ð2Þ ! + !  ! @ A r ¼ r r r f  f  ρ  ρ (18.51) ¼ f LUMO HOMO r ∂N 2 ! vð r Þ where the last expression, involving the electron density of the lowest unoccupied KS molecular orbital (LUMO) and of the highest occupied one (HOMO), neglects orbital relaxation when the number of electrons is varied (at constant external potential). Zones that are mainly electrophilic (respectively nucleophilic) correspond to positive (respectively negative) values of the dual descriptor. The σ-hole clearly appears [95] as a region with positive f (2) values, confirming its electrophilic character. Besides, we showed that such domains can be integrated [96], affording useful descriptors to predict halogen bond strength. The last function used tolocate the σ-hole historically belongs to QTAIM: this is the elec !

tron density Laplacian r2 ρ r . Let us recall that electron density concentration is character  ! ized by negative values of r2 ρ r , while there is electron depletion where the laplacian takes positive values. Electrophilic (respectively nucleophilic) regions, and hence σ-holes, are thus expected to be these latter (respectively first) ones. This gives rises to the so-called “lumphole” model [97]. A connection between the MEP and Laplacian viewpoints can be roughly established using Bader and Gatti’s source function [98]. Indeed, in virtue of one of the Green’s theorems, the electron density at any point can be decomposed into atomic contributions according to:  0 ð r2 ρ !   r X 1 !   3 0 (18.52) ρ r ¼ 0 d r ! !  4π A r  r   Ω A

so that [99]:  00  ! 2 ð ð   X r ρ r X ZA 1 !   0  3 0 3 00   + MEP r ¼ ! 0 ! ! ! !00  d r d r !  4π  r  r  r  r  A  r  RA  A ΩA 3 ℝ

hinting that positive MEP values are related to positive Laplacian values.

(18.53)

4. Some QTAIM results about halogen bonds

485

These three tools evaluated from the intrinsic properties of the fragments can be used to predict halogen bond strength.

4.2 Prediction models In practice, the maximal value of MEP on a chosen isodensity surface (usually 0.001 a.u., leading to MaxMEP0.001) is determined and was shown to be rather well linearly correlated to interaction energies. However, we demonstrated [45] that when different families are gathered, the coefficient of determination is not so high (typically about 0.85), this descriptor being outperformed by others, such as those stemming from conceptual DFT (for instance, Parr’s electrophilicity index or the domains of constant sign of the dual descriptor). Importantly, these three approaches (MEP, dual descriptor, and Laplacian) affords descriptors evaluated on the isolated halogen fragment that may qualitatively account for the halogen bond formation and that are in general quantitatively correlated to interaction and complexation energies through monolinear regressions. Another way of predicting these energies is very well known: it uses the properties at the X  X0 bond critical point. Many linear relationships between the electron density value (and, to a less extent, other local properties) at this BCP have been reported in the literature. From a theoretical perspective, they require the geometry optimization of the complex, which thus gives its total molecular energy. From this point of view, such linear models might appear not very useful since calculation is made on the formed complex. However, they also allow for a direct estimation of interaction energies from experimental data using the electron density obtained from X-ray diffraction experiments.

4.3 Looking at interactions along the bond formation path As obvious from Section 3, IQA provides a huge amount of information. Indeed, for a molecule constituted by M atoms, there are M(M  1)/2M2 atom pairs, leading to the same number of EAB inter terms (which can even further be decomposed into classical and exchange-correlation contributions). Determining the most relevant contributions along the bond formation path could become intractable by hand or might be biased by intuitive selection. A tool for an automatic selection of the most valuable features would be preferable. Recently, Popelier and coworkers proposed a general approach to achieve this goal. This procedure, called the Relative Energy Gradient (REG) analysis, actually allows ranking energy components, and it was successfully applied to various kinds of chemical problems, such as nucleophilic substitutions, catalytic effects, gauche effects and hydrogen bonds [100–103]. REG follows the variations of each IQA quantity Ei (which can be self or interaction energies, electrostatic or exchange-correlation contributions, and even additive IQA energies) along a control coordinate ξ, which is often chosen as Fukui’s intrinsic reaction coordinate (IRC) when one looks at the minimum energy reaction path. This path is in practice discretized by taking representative snapshots along it. Monolinear regressions (here performed in its least

486

18. Recent advances on halogen bonds

squares flavor) for each descriptor i against the total molecular energy EGS mol give relationships of the following form: EGS Ei ðξÞ  mREG i mol ðξÞ + ci R2i

(18.54)

coefficient of determination. Energy components featuring positive associated with the values act in the same direction as the molecular energy. For instance, in the case of mREG i REG values will allow identifying an activation barrier, EGS mol(ξ) will increase, and positive mi the energies that contribute to create this barrier. Obviously only components with enough large R2i values should be considered. More precisely, regressions are performed on so-called segments that define successive regions in the reaction path. In a recent paper, Alkorta et al. [104] performed a REG analysis on the halogen bond formation at the MP4(SDQ)/6-31 + G(2d,2p) post-HF level of theory for complexes where X0 is a nitrogen atom (the Lewis base being NH3, HCN or N2). To this aim, they carried out relaxed energy scans on the interfragment distance. Two segments were then defined: the first one (SEG1) corresponds to a compression process between the complex’s equilibrium distance to shorter distances where the molecular energy monotonically increases. The second one (SEG2) describes the formation of the complex from an infinite separation distance to the equilibrium geometry: along this segment, the molecular energy is thus strictly decreasing. The detailed REG analysis proved that (we are quoting the authors’ conclusions) [104]: “(i) the exchange energy between X and N drives the complex formation, and that (ii) the complexes’ compression is best described by the destabilization of N, which is essentially steric energy. If the REG analysis is confined to correlation energy only then it turns out that (i) the interatomic correlation energy between X and N drives the complex formation, and that (ii) complex compression is best described by the destabilization of the through-space correlation energy between X and N”. These statements also stress that IQA offers a deep and unambiguous quantification of chemical concepts that could appear “fuzzy” such as steric hindrance. Indeed, Popelier and co-workers unveiled the strong correlation between steric effects and the IQA intraatomic (a.k.a. self ) atomic contributions [105,106]. The REG analysis was also instrumental in explaining the directionality of halogen bonds. It has been recognized that they are highly directional with a bond angle very close to 180 degrees in most cases. To unravel, the origin of this linearity, Orangi et al. [107] studied comd bond angle as control coordinate ξ, plexes of the F3C  X…NH3 type. They used the CXN which they made vary from 180 to 90 degrees in steps of 10 degrees (keeping the X…NH distance fixed, but relaxing the other structural parameters). They found that while “the interatomic electrostatic interactions play important roles in the bond directionality of halogen bonds, however, they are hidden in the different interatomic interactions that change in opposite ways, and cancel each other out as the halogen bond angle varies” [107]. Quite unexpectingly, the primary interaction between the halogen and the nitrogen atom only plays a minor role in the bond directionality. Conversely, the “atomic self-energies are the most important participants” to account for the linear bond angle. Such conclusions are fully corroborated by our previous work [108] and also by Penda´s and coworkers [109], “demonstrating that although σ-holes might be qualitatively helpful, much care has to be taken in ascribing the stability of these systems to electrostatics. It is

References

487

clearly shown that electron delocalization is essential to understand the stability of the complexes” [109], where delocalization effects are, as already discussed, the manifestation of exchange(-correlation) effects. From our point of view, such findings clearly (and definitely) prove the unbiased power and physical relevance of the IQA decomposition.

5. Conclusions In this chapter, more devoted to conceptual issues (selected with some subjectivity) than to numerical results, we advocated the use of Bader’s quantum theory of atoms-in-molecules to study halogen-bonded complexes. In particular, the interacting quantum atoms decomposition was described within wavefunction and density functional theories. Some particular issues were tackled. The first one was related to the many faces of bond strengths: we showed that several definitions and quantities can be evaluated and that they are not equivalent. The second question was to identify which physical properties might be responsible for the bond formation initiation, and we discussed three different tools for characterizing the intrinsic halogen atom electrophilicity in the isolated fragments (molecular electrostatic potential, dual descriptor, and electron density Laplacian). These functions in general afford descriptors that are nicely correlated to bond strengths. Then we follow the reaction path for the bond formation. It then appeared that the pure electrostatic viewpoint does not hold any longer. The relative energy gradient method revealed the importance of covalency, the role of electronic correlation and of intraatomic (self ) components. Such results definitively prove that halogen bonds are certainly not as simple as elementary models (such those based on the σ-hole concept) may suggest when one looks at properties around the equilibrium geometry. More generally, we gave evidence that such tools afford a general framework to quantify electrostatic and covalent effects and, more generally, to deeply discuss the nature of these interactions without the need to refer to any arbitrary reference. Noteworthy, this approach can be applied in principle to any type of bonds, paving the way toward a more widespread use of IQA in the chemistry community.

Acknowledgments The authors would like to gratefully acknowledge the LABEX SynOrg (ANR-11-2ABX-0029) and the Normandy Region for funding and support (project AGAC), as well as the Centre R
egional Informatique et d’Applications Num
eriques de Normandie (CRIANN) for computing resources.

References [1] [2] [3] [4] [5] [6] [7]

H.-J. Schneider, Angew. Chem. Int. Ed. 48 (2009) 3924. S. Scheiner, Phys. Chem. Chem. Phys. 23 (2021) 5702. A. Bauza´, T.D. Mooibroek, A. Frontera, Angew. Chem. Int. Ed. 52 (2013) 12317. G. Cavallo, P. Metrangolo, R. Milani, R. Pilati, A. Priimagi, G. Resnati, G. Terraneo, Chem. Rev. 116 (2016) 2478. J. Colin, G. De Claubry, Anal. Chim. 90 (1814) 87. F. Meyer, P. Dubois, CrystEngComm 15 (2013) 3058. M. Saccone, L. Catalano, J. Phys. Chem. B 123 (2019) 9281.

488 [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

18. Recent advances on halogen bonds

R. Wilcken, M.O. Zimmermann, A. Lange, A.C. Joerger, F.M. Boeckler, J. Med. Chem. 56 (2013) 1363. Y. Lu, T. Shi, Y. Wang, H. Yang, X. Yan, X. Luo, H. Jiang, W. Zhu, J. Med. Chem. 52 (2009) 2854. G. Berger, J. Soubhye, F. Meyer, Polym. Chem. 6 (2015) 3559. K. Rissanen, CrystEngComm 10 (2008) 1107. P. Metrangolo, F. Meyer, T. Pilati, G. Resnati, G. Terraneo, Angew. Chem. Int. Ed. 47 (2008) 6114. A. Mukherjee, S. Tothadi, G.R. Desiraju, Acc. Chem. Res. 47 (2014) 2514. A. Bruckmann, M.A. Pena, C. Bolm, Synlett (2008) 900. D. Bulfield, S.M. Huber, Chem. A Eur. J. 22 (2016) 14434. M.H. Kola´r, P. Hobza, Chem. Rev. 116 (2016) 5155. S. Kozuch, J.M.L. Martin, J. Chem. Theory Comput. 9 (2013) 1918. K.E. Riley, P. Hobza, Phys. Chem. Chem. Phys. 15 (2013) 17742. L.P. Wolters, F.M. Bickelhaupt, ChemistryOpen 1 (2012) 96. P. Varadwaj, A. Varadwaj, H. Marques, Inorganics 7 (2019) 40. M.J.S. Phipps, T. Fox, C.S. Tautermann, C.-K. Skylaris, Chem. Soc. Rev. 44 (2015) 3177. J. Andr
es, P.W. Ayers, R.A. Boto, R. Carbo´-Dorca, H. Chermette, et al., J. Comput. Chem. 40 (2019) 2248. R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, New York, 1990. P.L.A. Popelier, Atoms in Molecules: An Introduction, Pearson Education, Harlow, 2000. P.L.A. Popelier, Quantum Chemical Topology: On Bonds and Potentials, vol. 115, Springer, 2005. J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A.M. Penda´s, Molecules 25 (2020) 4028. M.A. Blanco, A.M. Penda´s, E. Francisco, J. Chem. Theory Comput. 1 (2005) 1096. E. Francisco, A.M. Penda´s, M.A. Blanco, J. Chem. Theory Comput. 2 (2005) 90. A.M. Penda´s, E. Francisco, M.A. Blanco, J. Comput. Chem. 26 (2005) 344. M. Jabłonski, J. Phys. Chem. A 116 (2012) 3753. M. Jabło
nski, ChemistryOpen 8 (2019) 497. S. Shahbazian, Chem. A Eur. J. 24 (2018) 5401. A.M. Penda´s, E. Francisco, M. Blanco, C. Gatti, Chem. A Eur. J. 13 (2007) 9362. V. Tognetti, L. Joubert, J. Chem. Phys. 138 (2013), 024102. P.L.A. Popelier, P.I. Maxwell, J.C.R. Thacker, I. Alkorta, Theor. Chem. Acc. 138 (2019) 12. B. Silvi, A. Savin, Nature 371 (1994) 683. S.R. Johnson, J. Chem. Inf. Model. 48 (2008) 25. S.F. Boys, F. Bemardi, Mol. Phys. 19 (1970) 553. P. Politzer, J.S. Murray, CrystEngComm 15 (2013) 3145. D.L. Widner, Q.R. Knauf, M.T. Merucci, T.R. Fritz, J.S. Sauer, E.D. Speetzen, E. Bosch, N.P. Bowling, J. Org. Chem. 79 (2014) 6269. M. Yahia-Ouahmed, V. Tognetti, L. Joubert, Theor. Chem. Acc. 135 (2016) 45. M. Yahia-Ouahmed, V. Tognetti, L. Joubert, Comput. Theor. Chem. 1053 (2015) 254. V. Tognetti, L. Joubert, ChemPhysChem 18 (2017) 2675. R. Inostroza-Rivera, M. Yahia-Ouahmed, V. Tognetti, L. Joubert, B. Herrera, A. Toro- Labb
e, Phys. Chem. Chem. Phys. 17 (2015) 1779. G. Hoffmann, V. Tognetti, L. Joubert, J. Phys. Chem. A 124 (2020) 2090. R.G. Parr, L. von Szentpa´ly, S. Liu, J. Am. Chem. Soc. 121 (1999) 1922. J.L. McDonagh, M.A. Vincent, P.L.A. Popelier, Chem. Phys. Lett. 662 (2016) 228. M.A. Vincent, A.F. Silva, J.L. McDonagh, P.L.A. Popelier, Int. J. Quantum Chem. 118 (2017), e25519. M.A. Vincent, A.F. Silva, P.L.A. Popelier, Z. Anorg. Allg. Chem. 646 (2020) 1244. A.F. Silva, L.J. Duarte, P.L.A. Popelier, Struct. Chem. 31 (2020) 507. J.L. Casals-Sainz, J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A.M. Penda´s, J. Comput. Chem. 41 (2020) 1234. R. Cha´vez-Calvillo, M. Garc
ıa-Revilla, E. Francisco, A.M. Penda´s, T. Rocha-Rinza, Comput. Theor. Chem. 1053 (2015) 90. F.J. Holgu
ın-Gallego, R. Cha´vez-Calvillo, M. Garc
ıa-Revilla, E. Francisco, A.M. Penda´s, T. Rocha-Rinza, J. Comput. Chem. 37 (2016) 1753. J.L. Casalz-Sainz, J.M. Guevara-Vela, E. Francisco, T. Rocha-Rinza, A.M. Penda´s, ChemPhysChem 18 (2017) 3553.

References

[55] [56] [57] [58] [59] [60] [61] [62] [63] [64]

[65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100]

489

V. Tognetti, A.F. Silva, M.A. Vincent, L. Joubert, P.L.A. Popelier, J. Phys. Chem. A 122 (2018) 7748. A. M€ uller, Phys. Lett. A 105 (1984) 446. V. Tognetti, L. Joubert, Phys. Chem. Chem. Phys. 16 (2014) 14539. K.J.H. Giesbertz, R. van Leeuwen, J. Chem. Phys. 139 (2013), 104109. V. Tognetti, P.-F. Loos, J. Chem. Phys. 144 (2016), 054108. O.A. Vydrov, T. Van Voorhis, Phys. Rev. Lett. 103 (2009), 063004. S. Grimme, J. Chem. Phys. 124 (2006), 034108. E. Br
emond, I. Ciofini, J.C. Sancho-Garc
ıa, C. Adamo, Acc. Chem. Res. 49 (2016) 1503. S. Cardamone, T.J. Hughes, P.L.A. Popelier, Phys. Chem. Chem. Phys. 16 (2014) 10367. M. Rafat, P.L.A. Popelier, Topological atom-atom partitioning of molecular exchange energy and its multipolar convergence, in: The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug, Wiley, Weinheim, Germany, 2007. M. Garc
ıa-Revilla, P.L.A. Popelier, E. Francisco, A.M. Penda´s, J. Chem. Theory Comput. 7 (2011) 1704. C. Outeiral, M.A. Vincent, A.M. Penda´s, P.L.A. Popelier, Chem. Sci. 9 (2018) 5517. A.M. Penda´s, E. Francisco, Phys. Chem. Chem. Phys. 20 (2018) 16231. X. Fradera, M.A. Austen, R.F.W. Bader, J. Phys. Chem. A 103 (1999) 304. J. Cioslowski, S.T. Mixon, J. Am. Chem. Soc. 113 (1991) 4142. J. Jara-Cort
es, J. Herna´ndez-Trujillo, J. Comput. Chem. 39 (2017) 1103. Z. Badri, C. Foroutan-Nejad, Phys. Chem. Chem. Phys. 18 (2016) 11693. P. Maxwell, A.M. Penda´s, P.L.A. Popelier, Phys. Chem. Chem. Phys. 18 (2016) 20986. E. Francisco, J.L. Casals-Sainz, T. Rocha-Rinza, A.M. Penda´s, Theor. Chem. Acc. 135 (2016) 170. L. Patrikeev, L. Joubert, V. Tognetti, Mol. Phys. 114 (2016) 1285. J.I. Rodr
ıguez, P.W. Ayers, J. Chem. Phys. 131 (2009), 021101. K.E. Riley, J.S. Murray, J. Fanfrl
ık, J. Řeza´c, R.J. Sola´, M.C. Concha, F.M. Ramos, P. Politzer, J. Mol. Model. 19 (2013) 4651. V. Tognetti, L. Joubert, Theor. Chem. Acc. 135 (2016) 124. J.S.M. Anderson, J.I. Rodr
ıguez, P.W. Ayers, D.E. Trujillo-Gonza´lez, A.W. G€ otz, J. Autschbach, F.L. CastilloAlvarado, K. Yamashita, Chem. A Eur. J. 25 (2019) 2538. J.S.M. Anderson, J.I. Rodr
ıguez, P.W. Ayers, A.W. G€ otz, J. Comput. Chem. 38 (2017) 81. J.I. Rodr
ıguez, E.A. Uribe, M.I. Baltazar-M
endez, J. Autschbach, F. Castillo-alvarado, I. Gutierrez-Gonza´lez, Chem. Phys. Lett. 660 (2016) 287. D. Ferro-Costas, A.M. Penda´s, L. Gonza´lez, R.A. Mosquera, Phys. Chem. Chem. Phys. 16 (2014) 9249. V. Tognetti, M. Yahia-Ouahmed, L. Joubert, J. Phys. Chem. A 118 (2014) 9791. V. Tognetti, L. Joubert, Following halogen bonds formation with Bader’s atoms-in- molecules theory, in: R. Chauvin, et al. (Eds.), Challenges and Advances in Computational Chemistry and Physics, vol. 22, 2016, p. 435. J. Thirman, E. Engelage, S.M. Huber, M. Head-Gordon, Phys. Chem. Chem. Phys. 20 (2018) 905. J. Řeza´c, A. de la Lande, Phys. Chem. Chem. Phys. 19 (2017) 791. G.R. Desiraju, P.S. Ho, L. Kloo, A.C. Legon, R. Marquardt, P. Metrangolo, P. Politzer, G. Resnati, K. Rissanen, Pure Appl. Chem. 85 (2013) 1711. P. Metrangolo, J.S. Murray, T. Pilati, P. Politzer, G. Resnati, G. Terraneo, CrstEngComm 13 (2011) 6593. K. Eskandari, M. Lesani, Chem. A Eur. J. 21 (2015) 4739. T. Clark, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 3 (2013) 13. T. Clark, M. Hennemann, J.S. Murray, P. Politzer, J. Mol. Model. 13 (2007) 291. P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793. H. Chermette, J. Comput. Chem. 20 (1999) 129. C. Morell, A. Grand, A. Toro-Labb
e, J. Phys. Chem. A 109 (2005) 205. F. Gu
egan, P. Mignon, V. Tognetti, L. Joubert, C. Morell, Phys. Chem. Chem. Phys. 16 (2014) 15558. B. Pinter, N. Nagels, W.A. Herrebout, F. De Proft, Chem. A Eur. J. 19 (2013) 519. V. Tognetti, C. Morell, L. Joubert, J. Comput. Chem. 36 (2015) 64. K. Eskandari, H. Zariny, Chem. Phys. Lett. 492 (2010) 9. R.F.W. Bader, C. Gatti, Chem. Phys. Lett. 287 (1998) 233. V. Tognetti, L. Joubert, Theor. Chem. Acc. 134 (2015) 90. J.C.R. Thacker, P.L.A. Popelier, Theor. Chem. Acc. 136 (2017) 86.

490 [101] [102] [103] [104] [105] [106] [107] [108] [109]

18. Recent advances on halogen bonds

J.C.R. Thacker, P.L.A. Popelier, J. Phys. Chem. A 122 (2018) 1439. J.C.R. Thacker, M.A. Vincent, P.L.A. Popelier, Chem. A Eur. J. 24 (2018) 11200. O.J. Backhouse, J.C.R. Thacker, P.L.A. Popelier, ChemPhysChem 20 (2019) 555. I. Alkorta, A.F. Silva, P.L.A. Popelier, Molecules 25 (2020) 2674. B.C.B. Symons, D.J. Williamson, C.M. Brooks, A.L. Wilson, P.L.A. Popelier, ChemistryOpen 8 (2019) 560. P.I. Maxwell, P.L.A. Popelier, J. Comput. Chem. 38 (2017) 2459. N. Orangi, K. Eskandari, J.C.R. Thacker, P.L.A. Popelier, ChemPhysChem 20 (2019) 1922. O.A. Syzgantseva, V. Tognetti, L. Joubert, J. Phys. Chem. A 117 (2013) 8969. F. Jim
enez-Gra´valos, M. Gallegos, A.M. Penda´s, A.S. Novikov, J. Comput. Chem. 42 (2021) 676.

C H A P T E R

19 The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state Bruno Landeros-Riveraa and Julia Contreras-Garcı´ab a

Departamento de Fı´sica y Quı´mica Teo´rica, Facultad de Quı´mica, UNAM, M
exico City, M
exico b Sorbonne Universit
e, CNRS, Laboratoire de Chimie Th
eorique, Paris, France

1. Introduction The electron density, ρ(r), has undoubtedly become one of the cornerstones of modern quantum chemistry. It belongs to the realm of quantum mechanical observables, whose eigenvalues are interpreted as probability densities and, actually, can be measured by X-ray diffraction experiments. For a N-electron system, in non-relativistic time-independent quantum mechanics, ρ(r) is the probability of finding an electron in an infinitesimal region of space around a certain position given by the vector r. At first glance, the fact that ρ(r) provides only statistical information about the electrons position may seem daunting, as it did to some of the founders of quantum mechanics. Nevertheless, it turned out that ρ(r) contains more information than was initially expected. On the one hand, Hohenberg and Kohn [1] demonstrated in their famous theorems that for a given system (under the Born-Oppenheimer approximation) the ground-state electron density unequivocally determines the potential and, therefore, possess information about all of its properties. This work derived in the celebrated Density Functional Theory (DFT), which nowadays is one (if not the most) important quantum mechanical method for studying molecular and periodical systems [2]. One of DFT’s most attractive feature is that, at least in principle, it just requires the trace of the first-order density matrix, in contrast to the post-Hartree-Fock methods for which the second-order density matrix is essential. Thus, this made it possible to apply DFT to relatively large systems, at a reasonable cost/benefit balance.

Advances in Quantum Chemical Topology Beyond QTAIM https://doi.org/10.1016/B978-0-323-90891-7.00006-2

491

Copyright # 2023 Elsevier Inc. All rights reserved.

492

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

On the other hand, Bader and coworkers [3] conceived a way to (partially) reconcile the classical view of matter—where atoms are connected to each other by sharing electrons in fixed regions of space—with the stochastic worldview of quantum mechanics. This work led to the development of the quantum theory of atoms in molecules (QTAIM) [4], which is based on the topology of ρ(r), i.e., the behavior of its gradient, laplacian and critical points. In QTAIM, the space is partitioned in “basins” that are identified with atoms, and where some properties such as charge or energy densities can be integrated. This partition allows to quantify the contribution of individual atoms (or groups of atoms) in a molecule to the global properties of the system, thus recovering the notion of transferability or group additivity [5]. Besides, chemical structure is defined by the set of critical points, which are classified by their rank (number of non-zero Hessian eigenvalues) and signature (algebraic sum of the Hessian eigenvalues signs). Another important concept is that of the molecular graph, which is constructed from the bond critical points (those with rank and signature equal to 3 and
1, respectively), along with the gradient lines that pass through these critical points and connect two atoms (bond paths). In most of the cases, there is a one-to-one correspondence between the QTAIM molecular graphs and the Lewis structures [6]. Furthermore, a physical base is given to the models of atomic shells and lone pairs by means of the laplacian of ρ(r), which indicates the presence of local charge concentration or depletion [7]. Hence, QTAIM supplies a solid physical ground for several basic chemical concepts that were apparently at odds with the principles of quantum theory. Although attention has been called to understand the limits of QTAIM because of its alleged reductionist view [8], in contrast to other scientific areas where reductionism has been troublesome [9], Bader’s research laid the groundwork for the emergence of quantum chemical topology (QCT) [10]. In QCT, other scalar and vectorial fields besides the electron density are studied, and it has led to the exploration of new chemical phenomena, generating and improving in this way fundamental chemical concepts. Despite the success of QTAIM for the description of chemical bonding, one of its pitfalls is the unsatisfactory account for some types of non-covalent interactions. Specific interactions, such as hydrogen bonds, can be accurately characterized by bond critical points because of their local nature. This means that, as in the case of a covalent bond, they can be attributed to two particular atoms and, hence, they show preference for certain orientations (directionality). However, there are many systems that do not have the possibility of creating new covalent bonds or forming specific interactions, but are nevertheless capable to agglomerate by the presence of other sort of closed-shell interactions. For instance, benzene is liquid at ambient conditions because of the formation of stacking interactions. Even if QTAIM predicts the existence of some intermolecular bond critical points in stacked benzene dimers [11], it has been shown that stacking is governed by London dispersion effects [12]. Contrary to hydrogen bonds, dispersive forces are delocalized, that is, they are not restricted to atomic pairs but arise from the simultaneous interaction of several atoms or functional groups. Besides, they are not delimited to a “bonding region” but are extended in large intermolecular areas, and its strength increases with the molecular size. For example, the boiling point of linear alkanes is proportional to the number of methyl groups. Thus, modeling dispersive interactions by localized bond critical points seems to be unsuitable. The one-to-one correspondence between the molecular graphs and Lewis structures is not achievable for systems where delocalized interactions have a predominant role.

2. Theoretical background

493

In view of the above, for an exhaustive analysis of non-covalent interactions it is desirable to have a theoretical framework that, on the one hand, recovers the advantages of QTAIM for the description of localized interactions and, on the other hand, is able to identify the presence of delocalized interactions which are elusive for QTAIM. With this in mind, it will be shown in the following sections that there exists a visualization tool with such qualities called the Non-Covalent Interaction Index (NCI) [13]. First, the principles of the NCI theoretical background will be introduced, followed its applications to three areas of interest for chemistry: biology, reactivity and solid state. Additionally, it will be shown that NCI is flexible enough to be applied to different type of theoretical and experimental electron densities, without loss of rigor.

2. Theoretical background The non-covalent interaction index (NCI) [13] is based on the analysis of the reduced density gradient, s(r), which is defined as sðrÞ ¼

jrρðrÞj 1=3

2ð2π 2 Þ

ρðrÞ4=3

This is a dimensionless function that measures the deviation of ρ(r) with respect to the uniform electron gas (UEG). In the UEG, the electrons are “spread” out homogenously in space, along with a positive charge that is also uniformly distributed so as to neutralize the system. Thus, for this hypothetical system the electron density is finite and constant everywhere. Since jrρ(r)j of UEG vanishes all over space, this model has served as a basis for the local density approximation in DFT, for which an exact expression of the exchange energy is known. Nevertheless, in real molecules ρ(r) behaves far from being uniform. The nuclei (which are normally treated as punctual positive charges) act as attractors of ρ(r)—i.e., the electron density is maximum in the nuclear positions as a consequence of the coulombic attraction between the protons and the electrons. Accordingly, ρ(r) decays exponentially to zero in regions far from the nuclei. This feature of ρ(r) is essential for chemistry as, for instance, it is one of the fundamentals of crystallography that has allowed this powerful technique to locate atomic positions in periodic systems since more than 100 years ago. Also, this fact is crucial to QCT because it has permitted the application of the formal language of dynamical systems to the characterization of a great variety of interactions in chemical systems. As mentioned above, s(r) takes into account this inhomogeneous distribution of ρ(r) in real space caused by the external field of a given nuclear arrangement. Therefore, it is expected that this function contains important information about phenomena such as bonding, London dispersion or steric clashes, which are thought to emerge as manifestations of subatomic particle interactions. As can be expected, s(r) has had a special role in DFT [14], in particular in the development of generalized gradient approximation density functionals. Nevertheless, it was not until 10 years ago that its potential application for the analysis of atomic and molecular interactions was exploited. To understand this capacity of s(r), some of its basic characteristics in different regions of the molecule must be examined. It is important to mention that the qualitative

494

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

behavior of s(r) in molecular densities constructed by the overlap of independent atomic densities (promolecular densities) is identical to that of electron densities obtained from non-spherical models (e.g., from quantum mechanical computations or multipole models adjusted to experimental structure factors). Owing to this fact, NCI constructed from promolecular densities can be easily extended to analyze huge systems (proteins, for example) that otherwise would be very difficult to study because of the demanding quantum mechanical computational costs. In the electron density tails, far from the nuclei, s(r) shows very large positive values because ρ(r) decays faster than its gradient. At the other end, near the nuclei, high values of ρ(r) predominate and, thus, s(r) approaches zero. These zones are not of interest for examining non-covalent interactions. The important areas for the NCI analysis are found in low s(r) and low ρ(r) values. They can be easily casted as graphs of s(r) vs. ρ(r). For instance, the s(ρ(r)) graph of the methane molecule (constructed with promolecular electron densities) is depicted in Fig. 19.1A. The typical pattern of a covalent bond is observed, the peak in Fig. 19.1A. The overall curve of this graph assumes the form s(ρ(r)) ¼ αρ(r)
1/3, where α is a constant. Notice that near ρ(r) ¼ 0.17 a.u., s(r) reaches zero. This point corresponds to the CdH bond critical point as defined in QTAIM, where the electron density (and consequently s(r) too) vanishes. In the case of the methane dimer (Fig. 19.1B), an extra “peak” (marked with a red square) is observed at low s(r) and ρ(r) values, which is absent in the case of the isolated molecule. These peaks found in the s(ρ(r)) graphs can be considered as “signatures” of interactions. As well, a 3D representation of the regions where non-covalent interactions are present can be obtained from s(r) isosurfaces at low ρ, normally in the range s ¼ 0.3
0.7. Furthermore, the strength and classification of the type of non-covalent interaction can be obtained by the analysis of the function sign (λ2)*ρ(r), where sign (λ2) is the sign of the second

FIG. 19.1 The s(ρ(r)) plots of (A) methane and (B) methane dimer. The peak corresponding to the non-covalent interactions is marked with a red rectangle in the case of the dimer. Also, NCI isosurfaces s(r) ¼ 0.3 of (C) methane and (D) methane dimer.

2. Theoretical background

495

eigenvalue of the electron density Hessian matrix. The second eigenvalue is chosen because it reflects the charge accumulation or depletion in the plane perpendicular to an interaction region. In cases where λ2 > 0 the electron density is depleted, which is correlated with steric repulsion. On the other hand, when λ2 < 0 the electron density is accumulated, as will be expected for strong non-covalent interactions such a hydrogen bond, but also for covalent bonds (although at larger s(r) values). Finally, regions where ρ(r) 0 are typical of weak interactions (van der Waals forces), irrespective of sign (λ2). Taking this into account, the 2D plots (NCI diagrams) or 3D NCI isosurfaces are constructed to identify the regions and type of non-covalent interactions in real space. For instance, the NCI isosurface of the methane molecule is depicted in Fig. 19.1C. Small blue thick disks in between carbon and hydrogen indicate the presence of covalent bonding. In the NCI isosurface of the methane dimer (Fig. 19.1D), additionally to CdH covalent bonds, a green flat surface is observed, which corresponds to the weak van der Waals forces between the two molecules. An important feature of these green surfaces is that it is not located in between two particular atoms, but reflects de delocalized nature of dispersive interactions. As mentioned in the Introduction, this is one of the main advantages of the NCI with respect to other methods. While it recovers the information obtained, for example, from the QTAIM critical points (where rρ(r) ¼ 0, as in the CdH bond critical point mentioned before), NCI is also able to describe interactions where more than two atoms are involved, and which are not well represented by pairwise contacts. The later characteristic is related to the existence of critical points of s(r) beyond those of QTAIM. These “extra” critical points fulfill r2 ρðrÞ 4 ðrρðrÞÞ2 ¼0 ¼ ρðrÞ 3 ρ2 ð r Þ This non-QTAIM NCI critical points are found in regions where necessarily the laplacian is positive and are thus associated with weak closed-shell delocalized interactions. The phenol dimer is another illustrative example of the application of the NCI analysis (Fig. 19.2). For this case, a cutoff was applied so that all values with ρ(r) > 0.07 a.u. are not considered for the construction of the NCI isosurfaces. The reason is that with this threshold value (0.05 is recommended for SCF-derived electron densities), only non-covalent interactions will be examined since covalent ones appear at larger values. The three cases mentioned above are observed in this system: a blue small disk, where λ2 < 0 and is associated with the strong OdH⋯O hydrogen bond formed between the two molecules; a red surface in the

FIG. 19.2 NCI isosurface s ¼ 0.3 of the phenol dimer, constructed from a promolecular electron density.

496

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

middle of the aromatic rings, where λ2 > 0 and is correlated with the steric clash of the ring structures; and flat surface between the aromatic rings, where ρ(r)0 and represent the stacking interactions. Lastly, a quantitative analysis can be obtained from the integration of functions of ρ(r) in the NCI interacting region. In the case of promolecular electron densities, this region is defined by those points of space with small s(r) values where the electron density has an important contribution from various molecules. This is possible because under the promolecular approximation the monomer’s densities are unambiguously defined. Thus, we can define and NCI region (ΩNCI) without resorting to isosurfaces. More specifically, for two interacting systems A and B, the points ri belong to the NCI region (ΩNCI) whenever they satisfy:
ri  ΩNCI sðri Þ < sref ðri Þ ρðri Þ < γ ref ρA ðri Þ ρðri Þ < γ ref ρB ðri Þ where sref and γ ref are threshold values that make sure that s(ri) and that it is due to an intermolecular interaction from both A and B. Exploration of this thresholds has shown that γ ref ¼ 0.95 and sref ¼ 1.0 are good choices for the evaluation of all relevant interactions. For the case of non-spherical electron densities, partition of the electron density of a complex into monomeric densities is not trivial. Instead, the reduced density gradient is employed for the NCI region definition. Here sref is not set as a constant value but is a function computed from a reference system (usually, a monomer). This makes sense since, as was demonstrated in Fig. 19.1, the s(ρ(r)) function of the complex shows the same behavior as that of the monomer, plus the appearance of some extra peaks that indicate the presence of non-covalent interactions. In this way, only the points ri that belong to those peaks will be taken into account. If the interacting molecules are different (i.e., A 6¼ B), sref takes the lowest value between the s(ρ(r)) curves of both monomers. Thus, the NCI region is defined as: n ri  ΩNCI sðri Þ < sref ðri Þ sðri Þ < scutoff ðri Þ ρðri Þ < ρcutoff The cutoffs are applied to get rid of the interference of covalent bonds, as in the case of the phenol dimer (Fig. 19.2). The main disadvantage of this NCI region definition is that it requires additional calculations for the monomers. Therefore, the use of promolecular densities for the NCI integrals is preferable, in particular when studying supramolecular or biological systems. The integration of the electron density in promolecular-defined NCI regions has been shown to correlate with interaction energies computed from classical or quantum mechanical methods [15]. Moreover, ρ(r) can be integrated in different sign (λ2)*ρ(r) intervals to study the contribution of each type of non-covalent interactions. A typical example [16] would be to integrate in the following intervals (given in a.u.): from
0.1 to
0.02 for strong attractive interactions, from
0.02 to 0.02 for weak interactions, and from 0.02 to 0.1 for repulsive interactions. The following sections will show some applications of the NCI methodology in different branches of the study of chemistry.

3. Ligand-protein interactions This section is devoted to exemplify the application of NCI to ligand-protein interactions. In a work that involved classical molecular dynamics simulations (MD) and NCI calculations,

3. Ligand-protein interactions

497

FIG. 19.3

Models of (A) D614G, (B) H49Y, (C) T573I and (D) WT. The shifted amino acids are overlapped in the zoomed images.

Sixto-Lo´pez et al. [17] analyzed the structural behavior of the three most abundant mutations of SARS-CoV-2 (H49Y, D614G and T573I) that were circulating in the Mexican population during the first months of the COVID-19 pandemic. These are point-mutations that cause residue changes in the transmembrane spike (S) glycoprotein, that is vital for the virus to enter a host cell. The S protein is trimeric, and each of its monomers is formed by two subunits, S1 and S2. S1 is responsible for the virus binding to the human angiotensin-converting enzyme-2 (ACE2), while S2 contains the membrane fusion machinery. The wild-type protein (WT) and mutant models employed in the MD simulation are depicted in Fig. 19.3. For each mutant, the first and final letters indicate the original and shifted amino acids, respectively, and the number indicates the position within the protein sequence. For instance, in T573I, the threonine residue found at position 573 in the WT, is exchanged for an isoleucine. Since the former amino acid is polar while the latter is non-polar, structural modifications are expected for this type of mutation. The three examined mutations do not occur at the receptor-binding domain, which is the region that recognizes the ACE2 human receptors. Nevertheless, it is known that ligand or receptor recognition can be affected from outside binding site mutation due to structural modifications [18]. For the simulation, 100 ns MDs were run with the WT and the three mutants. Clustering methods were then used to group the structures from the final 80 ns of the MD according to geometrical similarities. A representative structure from the most populated clusters of WT and the three mutants was selected for molecular docking studies with three drugs that showed in vitro anti-SARS-CoV-2 activity, i.e., cepharanthine [19], hydroxychloroquine

498

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

[20] and nelfinavir [21]. The first two drugs interact only with amino acids that belong to S1, while the latter can be attached to both S1 and S2 subunits. Thus, a total of 12 protein-ligand complexes were studied. The final results of the molecular docking were used as input for 50 ns MD simulation of the protein-ligand complexes to understand the effect of mutations in the ligand’s binding affinity. The same clustering method was applied to group the structures of the final 20 ns of the MDs. In three cases, the ligands diffused out of the protein so the complex was not formed. In the remaining 9 protein-ligand complexes, NCI analysis was applied to a representative structure of the most populated clusters of the protein-ligand systems in order to obtain a deeper insight of the intermolecular interactions responsible for the complex’s formation. The NCI isosurfaces of the studied systems, constructed from promolecular electron densities, are shown in Fig. 19.4. Additionally, integrals of ρ(r) over different sign (λ2)*ρ(r)

FIG. 19.4 NCI isosurfaces (drawn at s ¼ 0.5) of (A) Cepharanthine-D614G, (B) Cepharanthine-H49Y, (C) Cepharanthine-T573I, (D) Cepharanthine-WT, (E) Hydroxychloroquine-H49Y, (F) Hydroxychloroquine-T573I, (G) Nelfinavir-H49Y, (H) Nelfinavir-T573I and (I) Nelfinavir-WT complexes. Blue and green regions represent strong attractive and weak dispersive interactions, respectively. The localized interactions are marked in a red square. The regions of π-interactions are also indicated. The amino-acid residues forming localized and delocalized interactions with the ligand are depicted in yellow and magenta, respectively.

3. Ligand-protein interactions

499

intervals were performed to study the contribution of strong, weak and repulsive interactions involved in the protein-ligand complexation. The NCI isosurfaces and the ρ(r) integrals were computed with the NCIPLOT4 software [15], using the following input: 2 protein.xyz drug.xyz LIGAND 2 5.0 RANGE 3 –0.1 –0.02 –0.02 0.02 0.02 0.1. The first line in the input indicates the number of systems involved, which is 2 in this case (the protein and the drug). The second and third lines indicate the name of the files containing the coordinates of both systems, using the conventional XYZ file format. The keyword LIGAND is inserted so that only intermolecular interactions between the ligand and the protein are considered (intramolecular interactions are dropped out). The first number next to LIGAND indicates the file that contains the ligand coordinates (drug.xyz in this case), while the second number establishes a cutoff radio around the ligand for the amino-acid residues that will be considered in the NCI surface generation and the integral computations. Finally, RANGE indicates the number of intervals of sign (λ2)*ρ(r) that will be used in the NCI integrals (3 in this case), and the following lines indicate the intervals values. The first, second and third intervals correspond to strong, weak and repulsive interactions, respectively. As can be seen in Fig. 19.4, there is a plethora of non-covalent interactions in the 9 proteinligand complexes that goes from weak non-specific delocalized van der Waal interactions, to very strongly localized hydrogen bonds. To simplify the visualization, the residues that interact by delocalized or localized interactions are depicted in magenta or yellow, respectively. Localized interactions are marked with a red square, and π-interactions are indicated as well. For instance, the Nelfinavir-H49Y complex is governed almost exclusively by dispersive interactions. The Hydroxychloroquine-H49Y system is dominated by stacking interactions (π-π stacking and CdH⋯π too). The controversial hydrogen-hydrogen bond (H⋯H) is present in two cepharantine, and in one hydroxychloroquine complexes. It appears as a small green disk, which means that it is a weak, albeit localized interaction. A similar result is observed for the non-canonical CdH⋯O hydrogen bond that is also found in three complexes. The most abundant strong interaction is the NdH⋯O hydrogen bond, found in four complexes. A complete list of all residue-ligand interactions is given in Table 19.1. The value of the NCI integrals, decomposed in strong and weak interactions (repulsive ones were negligible) are compared with the mean Free energies of binding computed with the MMGBSA approach for all the structures obtained from the last 20 ns of the MD (Fig. 19.5). For this method, the Free energy of binding is decomposed in electrostatic, van der Waals and solvent (polar and non-polar) interaction terms. It is outstanding that the NCI integrals and the Free energies of binding show the same trend, given that the latter is an average of different structures. Furthermore, according to the MMGBSA calculation the van der Waals interactions are predominant, which is supported by the NCI integral decomposition. These outcomes can be rationalized in terms of the non-covalent interactions observed in the

500

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

TABLE 19.1 The localized and delocalized interactions found for each protein-ligand complex within the NCI isosurfaces. The atom-atom intermolecular distances of the localized interactions are shown as a complementary guide for its strength classification. System

Localized interaction/residue (strength/distance)

Delocalized interaction/residue

Cepharanthine D614G

˚ ); H⋯H/A352 H⋯H/S349 (weak/2.94 A ˚) (weak/2.28 A

Van der Waals/A344; Van der Waals/N354; Van der Waals/R346; Van der Waals/N450; Van der Waals/F347; Van der Waals/L452; Van der Waals/A348; Van der Waals/I468

H49Y

˚ ); H⋯H/L452 N–H⋯O/Y449 (strong/2.05 A ˚ ); H⋯H/T470 (weak/2.26 A ˚) (weak/2.13 A

C–H⋯π/Y449; Van der Waals/L492; Van der Waals/N450; Van der Waals/Q493; Van der Waals/L452; Van der Waals/S494; CdH⋯π/ F490

T573I

˚) C–H⋯O/F342 (weak/2.91 A

Van der Waals/L335; Van der Waals/L368; Van der Waals/F338; Van der Waals/S371; Van der Waals/G339; Van der Waals/S373; Van der Waals/N343; CdH⋯π/F374; Van der Waals/ D364; CdH⋯π/W436; Van der Waals/V367

WT

˚ ); N–H⋯O/E484 (very strong 1.71 A ˚) CdH⋯O/E484 (weak/3.05 A

Van der Waals/E484; CdH⋯π/F490; Van der Waals/G485; Van der Waals/Q493; Van der Waals/Y489

Hydroxychloroquine π⋯π (parallel displaced)/F338; Van der Waals/ G339; CdH⋯π/A372; CdH⋯π/F342; CdH⋯π/ F374; Van der Waals/L368

H49Y

T573I

˚) H⋯H/A522 (weak/2.11 A

C–H⋯π/I332; Van der Waals/A522; Van der Waals/N360; Van der Waals/T523; Van der Waals/C361; Van der Waals/V524; Van der Waals/C391; Van der Waals/C525; Van der Waals/P521

Nelfinavir H49Y

Van der Waals/F817; Van der Waals/D936; Van der Waals/L821; Van der Waals/S939; Van der Waals/N824; Van der Waals/S940; Van der Waals/K825; Van der Waals/T941, Van der Waals/Q935

T573I

˚ ); N–H⋯O/K310 (very strong/2.05 A ˚ ); OdH⋯O/I312 (very strong/1.96 A ˚ ); CdH⋯O/ NdH⋯O/I312 (strong/2.14 A ˚ ); OdH⋯O/D663 Y313 (weak/2.33 A ˚ ); NdH⋯O/Q954 (very (strong/2.22 A ˚) strong/1.88 A

Van der Waals/E309; Van der Waals/D950; Van der Waals/G311; Van der Waals/V951; Van der Waals/Y313; Van der Waals/N953; Van der Waals/I664; Van der Waals/Q957; CdH⋯π/ P665

WT

˚) N–H⋯O/I666 (very strong/1.87 A

Van der Waals/I312; Van der Waals/G667; Van der Waals/Q314; Van der Waals/A668; Van der Waals/L611; Van der Waals/V1040; Van der Waals/Q613; Van der Waals/D1041; Van der Waals/A647

3. Ligand-protein interactions

501

FIG. 19.5

Free energy of binding calculated with the MMGBSA approach (top) and NCI integrals (bottom) for the 9 protein-ligand complexes. The NCI integrals are decomposed in strong attractive (blue) and weak dispersive interactions (orange). Repulsive interactions were negligible in all cases.

NCI isosurfaces. The Nelfinavir-H49Y complex, where van der Waals forces are dominant, shows the lowest value of Free energy of binding and NCI integral. On the other hand, the Nelfinavir-H49Y system, which not only has abundant dispersive interactions but also has the biggest number of strong hydrogen bonds, possesses the largest Free energy of binding and NCI integral. In summary, this example demonstrates the capacity of NCI to assess the presence and extent of the non-covalent interactions that mediate the formation of protein-ligand complexes, by means of promolecular electron densities. Additionally, the NCI integrals provide a quantitative analysis that can be contrasted with interaction energy calculations, and explain them in terms of the strong, weak and repulsive interactions localized by the NCI isosurfaces. Thus, NCI is a potential tool for evaluating intermolecular interactions in biological systems that goes far beyond the simple geometrical examination commonly used after classical molecular dynamic simulations.

502

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

4. Interplay between inter and intramolecular interactions in reaction mechanisms Even though reaction mechanisms are generally associated with bond rupture and formation, non-covalent interactions can play an important role in some cases in, for example, stabilizing intermediates or transition states [22]. This section exemplifies the use of NCI to analyze the interplay between inter and intramolecular interactions in a ring-opening metathesis polymerization (ROMP) [23], in which 2 intermediates and 1 transition state are present in each reaction cycle. ROMP is a synthetic methodology useful for developing high-performance polymeric materials from functionalized cyclic structures, that have interesting applications such as carbon capture and storage (CO2 separation processes) [24]. Za´rate-Saldan˜a et al. [25] studied the monomer synthesis and ROMP of two norbornene-derivatives (1a and 1b), mediated by ruthenium-alkylidene catalysts (Fig. 19.6). The difference between 1a and 1b was the substituent R at the “bridge position” of the molecule, which is small in the former (R ¼ –H) and voluminous in the later (R ¼ –Si(CH3)3). The reason for analyzing this substituent effect is that bulky groups such as 1b tend to enhance the permeability properties of the synthetized polymeric membranes [26]. This phenomenon is a consequence of the augmentation of the free volume (the space in a membrane that is not occupied by the polymeric chains, where the gas molecules can be diffused), which is caused by the steric repulsion between the voluminous functional groups. Notwithstanding, the polymerization of 1b was not achievable with any of the used catalysts, nor with temperature (40–100°C) or time (2–72 h). In order to explain this fact, the investigation was complemented with DFT simulations of the reaction pathways, along with a topological analysis of all the involved species withing the QTAIM framework.

FIG. 19.6 Monomer synthesis and further ROMP polymerization of the two norbornene derivatives (top), mediated by ruthenium-alkylidene catalysts (bottom).

4. Interplay between inter and intramolecular interactions in reaction mechanisms

503

FIG. 19.7 Reaction pathways for the metathesis of 1a and 1b by catalyst II.

The reaction pathway for the ROMP of 1a and 1b, mediated by catalyst II (called the Grubbs second-generation catalyst), is depicted in Fig. 19.7. First, a van der Waal complex (2) is formed between the norbornene-derivative monomer and the catalyst, followed by two intermediates: the π-complex (3) and the metallacyclobutane (5). The two intermediates are related by a transition state (TS4). Finally, 5 is transformed to the final a carbene complex 6. Once this reaction cycle is completed, the process is repeated n times to form a polymer of n repeating units. More details will be given in the following paragraphs. A similar reaction pathway has been reported for many ROMP processes [27]. According to the DFT calculations, the reasons why the reaction does not proceed for 1b are the absence of a π-complex 3b, a higher energy barrier for the corresponding transition state (i.e., the energy barrier of TS 4b was three times higher than TS 4a), and the formation of the metallacyclobutane structure 5b is endergonic, in contrast to 5a which is exergonic. While these results were rationalized in terms of the bond critical points and delocalization indexes of the atoms involved in the bond rupture/formation process, the role of the noncovalent interactions was underestimated. This was especially due to the fact that the forces involved were of the van der Waals type, not properly described by the QTAIM critical points because of their delocalized nature. As will be seen below, NCI is able to account for every type of interactions present in the ROMP pathway. With this in mind, two isosurfaces will be examined: one at s ¼ 0.5 for noncovalent interactions, and another at s ¼ 0.3 for covalent bonding patterns. For the former, a cutoff was applied to the electron density (at ρ¼0.05) so that only weak interactions are observed. Both isosurfaces were computed from the electron densities of the studied systems that were obtained from geometry optimizations carried out at a PBE-D3(BJ)/def2-TZVP

504

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

NCI isosurfaces of 2a at s ¼ 0.5 (left) and s ¼ 0.3 (right). Some relevant non-covalent interactions are marked for s ¼ 0.5. The three carbons involved in the ROMP bond rupture/formation are labeled and depicted as orange spheres for s ¼ 0.3.

FIG. 19.8

level of theory. Thus, this section corresponds to an example of the application of the NCI index to a ρ(r) calculated with quantum mechanical methods. The NCI isosurfaces of the van der Waals complex 2a are shown in Fig. 19.8. From the s ¼ 0.5 isosurface, the three types of strong, weak and repulsive interactions discussed before can be observed. There is a noncanonical CdH⋯Cl hydrogen bond (blue disk circled in red), and a CdH⋯π interaction formed between the two aromatic rings of the catalyst. Besides, the presence of dispersive forces all over the complex is clear. These interactions were not taken into consideration in the original QTAIM analysis. As expected, there is a red cylinder in the middle of all rings and a larger red surface in the middle of the norbornene-derivative both indicative of steric clashes. Besides, another ring-shape steric clash surface is observed between the chlorine and the ruthenium atoms which is reminiscent of the RudCl strong interaction that can be seen in Fig. 19.8 right. In this figure, the atoms involved in bond rupture/formation are highlighted by large spheres: Ruthenium in pink, and three carbons in orange. C3 is the carbon of phenyl group of the catalyst, and C1 and C2 are the carbons that form the double bond of 1a that will be broken at the end of the reaction. Notice that all the bonding regions described by these isosurfaces are found where expected, including the C1]C2 double bond. All covalent bonds formed between non-hydrogen atoms look like blue cylinders, while the bonds involving hydrogen atoms are similar to those reported in Fig. 19.1. Moreover, in the C3dRu bond the blue cylinder comprises only half of the bond and widens near the ruthenium atom. This feature is characteristic of an organometallic bond. The same behavior is observed for the other carbonruthenium bond of the catalyst. Additionally, the blue disk found between the Ru and Cl atoms is indicative of non-covalent bonds (dative, ionic, etc.). Also, the blue spherical surfaces around the chlorine atoms reflect their ionic character. It is important to note that only weak interactions are observed between 1a and the catalyst. The next step in the ROMP pathway is the π-complex formation. In this structure, the Ru atom is coordinated with the π-electrons of the C1]C2 double bond. From the NCI isosurface at s ¼ 0.5 of the π-complex 3a (Fig. 19.9), some changes in the intermolecular interactions can be appreciated. The orientation of the aromatic rings has changed, so the CdH⋯π interaction has transformed into a π⋯π one. The most notorious change is observed now in the

4. Interplay between inter and intramolecular interactions in reaction mechanisms

505

FIG. 19.9 NCI isosurfaces of 3a at s ¼ 0.5 (left) and s ¼ 0.3 (right). Some relevant non-covalent interactions are marked both isosurfaces. The three carbons involved in the ROMP bond rupture/formation are labeled and depicted as orange spheres for s ¼ 0.3.

“intermolecular” region between 1a and catalyst II. The interactions have gone from completely dispersive in 2a to a combination of strong attractive and repulsive in 3a. This effect is clearer at the s ¼ 0.3 isosurface. A flat rectangular surface appears in-between 1a and catalyst II. The blue regions at the extreme of this surface correspond to the coordination bonds between C1 and C2, respectively, with the metallic center. The mid red zone indicates a steric repulsion caused by the proximity of these two molecules. The NCI isosurfaces of the transition state TS 4a are shown in Fig. 19.10. From the isosurface with s ¼ 0.5, it can be noticed that the π⋯π interaction strengthens (the green surface has a larger area) with respect to 3a. Also, the number of localized CdH⋯Cl hydrogen bonds increases (circled in red). Moreover, the dispersive interactions between the chlorine atoms and 1a become stronger. From the s ¼ 0.3 isosurfaces, a rectangular surface similar to that of 3a is observed. However, the mid red zone and the blue region found between C2 and C3 are enlarged. This outcome indicates that the transition state is more similar to 3a than to the other intermediate 5a.

FIG. 19.10 NCI isosurfaces of TS 4a at s ¼ 0.5 (left) and s ¼ 0.3 (right). The CdH⋯Cl hydrogen bonds are circled in red in s ¼ 0.5. The three carbons involved in the ROMP bond rupture/formation are labeled and depicted as orange spheres for s ¼ 0.3.

506

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

FIG. 19.11 NCI isosurfaces of 5a at s ¼ 0.5 (left) and s ¼ 0.3 (right). The new CdH⋯Cl hydrogen bond formed between the 1a structure and chlorine is circled in red in s ¼ 0.5. The three carbons involved in four-member ring are labeled and depicted as orange spheres for s ¼ 0.3.

In the following structure, 5a, both 1a and catalyst II are fused in a single intermediate called metallacyclobutane. As the name indicates, the union of these two molecules is given by the appearance of a four-member ring formed between C1, C2, C3 and Ru. In the s ¼ 0.5 isosurface (Fig. 19.11), it is observed that now one hydrogen atom of the 1a structure is able to form a new CdH⋯Cl hydrogen bond. The four-member ring can be easily appreciated in the s ¼ 0.3 isosurface. The initial bond between C1 and C2 still remains, but a new covalent bond between C2 and C3 emerges. Additionally, a new C1dRu bond appears, which shows the same behavior as the C3dRu bond described before. Thus, half of the bonds of the fourmember ring have a covalent origin, while the other two possess an organometallic nature, as the name “metallacyclobutane” suggests. Moreover, the red curved surface observed at the middle of the four-member ring indicates that the steric clash increases in this region, counteracting the stability gained by the formation of the C2-C3 and C1dRu bonds. Thus, the NCI method explains in this way why 5a is a metastable system that eventually evolves into a more stable structure. Finally, in the last step of the ROMP mechanism the metallacyclobutane “opens” to form the carbene complex 6a. This step is called cycloreversion that, in this case, proceeds without a transition state because of the instability of 5a caused by the steric repulsion described above. From the s ¼ 0.5 isosurface (Fig. 19.12), it is noticed that a complex mixture of strong, weak and repulsive interactions is found in the region in-between C1, C2 C3 and Ru. Besides,

FIG. 19.12 NCI isosurfaces of 6a at s ¼ 0.5 (left) and s ¼ 0.3 (right). The three carbons involved in four-member ring are labeled and depicted as orange spheres for s ¼ 0.3.

5. NCI applied to experimental electron densities

507

FIG. 19.13 NCI isosurfaces of 5b at s ¼ 0.5 (left) and s ¼ 0.3 (right). The most relevant steric clashes that are presumed to destabilize this structure are indicated.

new dispersive interactions are formed between one aromatic ring of the catalyst II and the oxygenated groups of the norbornene-derivative, which were originally too far away from each other to show an appreciable interaction. From the s ¼ 0.3 isosurface, it is clear that the original C1]C2 bond and C3dRu bonds (Fig. 19.8) are not present anymore. On the other hand, the C1dRu and C2]C3 bonds are kept. In addition, a flat surface, similar to that of the π-complex 3a, is found. This indicates that C2 and C3 are coordinated to the metallic center, thus forming a different type of π-complex. The bridge structure of 1a does not allow for the structure to open completely and break this interaction. Nevertheless, this is not strong enough to deactivate the catalyst and stop the polymerization process. As mentioned before, the fact that the ROMP was not feasible for 1b was ultimately attributed to the steric repulsion caused by the –Si(CH3)3 group. Although this was inferred by the DFT energetic analysis, no proof was given within the QTAIM framework. Nonetheless, this phenomenon is effectively described by the NCI method. The s ¼ 0.5 and s ¼ 0.3 isosurfaces of the metallacyclobutane structure 5b are shown in Fig. 19.13. Besides the steric clashes in the aromatic rings, in the bicyclic structure of the norbornene group, and the four-member ring— all of them described before for 5a—an extra repulsion is observed around the Si atom (depicted as a purple sphere) in 5b. This results highlights the inherent steric clash caused by the addition of a voluminous functional group such as Si(CH3)3. It is presumed that the presence of all these repulsive interactions is the cause of the instability of 5b with respect to 5a, i.e., a higher energy barrier for the transition state TS 4b and the endergonic formation of 5b vs. the exergonic formation of 5a. In summary, this section shows how the NCI method is able to describe all variety of inter and intramolecular interactions, such as van der Waals, hydrogen bonds, steric repulsion, ionic, organometallic, coordination and covalent bonds, that take place simultaneously in a reaction mechanism such as that of ROMP.

5. NCI applied to experimental electron densities One main advantage of analyzing scalar fields of quantum mechanical observables such as ρ(r) and its derivatives, in comparison to other methods like those based on orbitals, is that

508

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

they (at least in principle) can be measured in a laboratory. From the scientific point of view, it will always be preferable to study properties or phenomena that can be tackled by theory and experiments. In that way, the models employed to understand the behavior of the system of interest will be less arbitrary than those that are merely abstract and that cannot be corroborated empirically. As mentioned before, ρ(r) can be determined, to a great accuracy, by high-resolution X-ray diffraction experiments. Since s(r) depends on ρ(r) and its gradient, then the NCI method can be applied to electron densities derived from X-ray diffraction experiments, which is the topic of this section. The following case illustrates the NCI approach capability for explaining the differences in the electronic structure and the crystal packing of two related compounds. As we will see, the combination of crystallography and QCT methods are suitable for studying solid state materials, which is another active and important field of chemistry. Molcanov et al. [28] studied the degree of aromaticity and electron delocalization of four tetrachloroquinone derivatives, through a combination of X-ray charge density analysis and quantum mechanical calculations. In particular, the QTAIM critical points, atomic charges and the delocalization index were used to study the electronic structure and to measure the aromatic character of these systems. The experimental ρ(r) was determined by the Hansen-Coppens multipole model [29], in which the electron density of the unit cell is constructed from the sum of aspherical atomic densities. Additionally, the lattice energies were computed with the PIXEL method [30], which allows to decompose them in electrostatic, induction, dispersive and repulsion terms. The relevance of evaluating electron delocalization in the quinone crystals lies in their redox properties, which are being used to develop organic semiconductors [31]. Beyond the changes observed in the internal electronic distribution of the quinone derivatives, the variations in π-electron delocalization also produces notable modifications of their crystal packing. Two extreme examples (shown in Fig. 19.14) were chosen to demonstrate the effect of structural and electronic changes on the non-covalent interactions that emerge in both crystals. These are tetrachloroquinone (CQ), also known as chloranil, and tetrachlorohydroquinone (CHQ). From the Lewis structures of these two molecules (Fig. 19.14A), it is expected that only CHQ shows an aromatic character, while bonds in CQ are more localized. This trend was corroborated numerically in the original paper [28] by the analysis of the values of electron density at the bond critical points and the delocalization index. The NCI isosurfaces can reveal additional visual information of the differences between CQ and CHQ. The isosurfaces were constructed from the experimental electron density using MoPro [32], the same software employed for the multipole refinement. Note from the s ¼ 0.77 (Fig. 19.14B) that there is a clear distinct pattern in the carbon skeletons of CQ and CHQ. The “hole” found in the carbon atoms of the carbonyl group of CQ is more prominent. This resemble the “holes” expected for the laplacian of ρ(r) for this type of functional groups that are susceptible of a nucleophilic attack. In contrast, the NCI surface in the carbon skeleton of CHQ seems nearly isotropic. Upon further decrease down to s ¼ 0.20
NCI concentrates on minima [33] (Fig. 19.14C), the NCI surfaces enable to visualize the differences between single and double bonds on the carbon skeleton of CQ. They are almost localized in the middle of the CdC bond (around the BCP) in the single bonds, whereas in the C]C bonds, they have an elongated shape. Instead, an intermediate behavior is observed in the carbon skeleton of CHQ, i.e., the bonding regions are approximately half filled by the NCI surface (although both resonant forms are not exactly equivalent).

5. NCI applied to experimental electron densities

509

FIG. 19.14

(A) Lewis structures of CQ (left) and CQH (right), and its NCI isosurfaces at (B) s ¼ 0.77 and (C) s ¼ 0.20.

Regarding crystal packing (Fig. 19.15), the CQ crystal lacks the π⋯π stacking interactions that are normally found for similar substituted benzene systems like tetrachlorobenzene. Instead, the CQ molecules are connected in the crystal by O⋯C contacts between carbonyl groups belonging to neighboring molecules (Fig. 19.16). This agrees with the nucleophilic behavior of the carbonyl group mentioned above. In contrast, CHQ shows an offset face-to-face stacking and, additionally, each stacking layer is connected by OH⋯O hydrogen bonds, similarly to hydroquinone (Fig. 19.17). A cutoff was applied to the isosurfaces so that covalent interactions are not considered. In CQ, the O⋯C interactions are the strongest of all (Fig. 19.16). These contacts are observed as a small blue disk, surrounded by a wider green surface, indicating that the interaction is not completely localized but has a dispersive component too. Also, some halogen-halogen (Cl⋯Cl) bonds can be noticed. Apart from these, the rest of the interaction regions correspond to van der Waals forces. Instead, more specific interactions are observed in CHQ (Fig. 19.17). The strongest is the OH⋯O hydrogen bond, which is found as a blue round disk, characteristic of the strong localized interactions. Some strong Cl⋯Cl halogen bonds are also observed. Furthermore, there is a flat large green surface between the aromatic rings that confirms the existence of π⋯π interactions in CHQ. Intramolecular interactions are also present. OH⋯Cl intramolecular interactions are found where the

510

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

FIG. 19.15

Crystal packing of CQ (left) and CHQ (right).

FIG. 19.16

NCI isosurface (s ¼ 0.7) of CQ.

FIG. 19.17

NCI isosurface (s ¼ 0.7) of CHQ.

furthermost area shows a strong interaction character (blue color), while the zone closest to the ring reveals the steric clash (red) caused by the O-C-C-Cl pseudo-ring strain. Also, a green region is found between the blue and red zones, demonstrating that strong, weak and repulsive interactions can coexist in balance in a shared area of a molecular crystal as a consequence

References

511 FIG. 19.18 Pixel total lattice energies and its electrostatic, induction, dispersion and repulsion terms for CQ (blue) and CHQ (red). All values in kcal/mol.

of many atoms interactions. Moreover, the OH⋯Cl contact was not located by the QTAIM critical points, highlighting the capacity of NCI to reveal the presence of delocalized interactions. Finally, the PIXEL energies (Fig. 19.18) can be rationalized in terms of the NCI isosurfaces. In general, all the energetic components of CHQ are higher than those of CQ. The coulombic energy of CHQ is greater because of the presence of the strong OdH⋯O bonds, which are known to be mainly electrostatic. The induction and dispersion energies of CHQ are also larger because of the abundant of the green surfaces (van der Waals forces), in particular due to the presence of π⋯π interactions. The later interactions allow for closer packing, which causes the increase in the Pauli repulsion interactions. Thus, the presence of strong localized interactions (blue regions), along with the existence of larger areas of dispersive interactions (green zones) found by the NCI method, explain why the CHQ lattice energy is higher than that of CQ. This example demonstrates the potential of NCI to understand the differences in the electronic structure, packing and energetic behavior of related molecular crystals.

6. Concluding remarks As it could be observed, the NCI method is able to recover the proper description of QTAIM of the localized interactions, while it identifies better the regions where delocalized interactions are present. Also, it was shown that NCI can be applied to analyze in the same way an electron density determined from promolecular models, quantum mechanical calculations or X-ray diffraction experiments. This flexibility makes NCI a versatile tool for analyzing the role of non-covalent interactions in diverse areas of interest for chemistry such as biology, reactivity and solid state.

References [1] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (3B) (1964) B864. [2] P. Verma, D.G. Truhlar, Status and challenges of density functional theory, Trends Chem. 2 (4) (2020) 302–318.

512

19. The Non-Covalent Interactions index: From biology to chemical reactivity and solid-state

[3] R.F. Bader, S.G. Anderson, A.J. Duke, Quantum topology of molecular charge distributions. 1, J. Am. Chem. Soc. 101 (6) (1979) 1389–1395. [4] R.F. Bader, Atoms in Molecules : A Quantum Theory, Clarendon Press, 1990. [5] R.F. Bader, D. Bayles, Properties of atoms in molecules: group additivity, J. Phys. Chem. A 104 (23) (2000) 5579–5589. [6] P.L. Popelier, The QTAIM perspective of chemical bonding, in: G. Frenking, S. Shaik (Eds.), The Chemical Bond: Fundamental Aspects of Chemical Bonding, John Wiley & Sons, 2014, pp. 271–308. [7] R.F.W. Bader, R.J. Gillespie, P.J. MacDougall, A physical basis for the VSEPR model of molecular geometry, J. Am. Chem. Soc. 110 (22) (1988) 7329–7336, https://doi.org/10.1021/ja00230a009. [8] M.E. Eberhart, T.E. Jones, The two faces of chemistry: can they be reconciled? Found. Chem. 15 (3) (2013) 277–285. [9] M.H. Van Regenmortel, Biological complexity emerges from the ashes of genetic reductionism, J. Mol. Recognit. 17 (2004) 145–148. [10] P.L. Popelier, Quantum chemical topology, in: D.M. Mingos (Ed.), The Chemical Bond II, Springer, 2016, pp. 71–117. [11] O.A. Zhikol, O.V. Shishkin, K.A. Lyssenko, J. Leszczynski, Electron density distribution in stacked benzene dimers: a new approach towards the estimation of stacking interaction energies, J. Chem. Phys. 122 (14) (2005), 144104. [12] S. Ehrlich, J. Moellmann, S. Grimme, Dispersion-corrected density functional theory for aromatic interactions in complex systems, Acc. Chem. Res. 46 (4) (2013) 916–926. [13] E.R. Johnson, S. Keinan, P. Mori-Sa´nchez, J. Contreras-Garcı´a, A.J. Cohen, W. Yang, Revealing noncovalent interactions, J. Am. Chem. Soc. 132 (18) (2010) 6498–6506. [14] A. Zupan, J.P. Perdew, K. Burke, M. Causa`, Density-gradient analysis for density functional theory: application to atoms, Int. J. Quantum Chem. 61 (5) (1997) 835–845. [15] R.A. Boto, F. Peccati, R. Laplaza, C. Quan, A. Carbone, J.P. Piquemal, Contreras-Garcı´a, J., NCIPLOT4: fast, robust, and quantitative analysis of noncovalent interactions, J. Chem. Theory Comput. 16 (7) (2020) 4150–4158. [16] F. Peccati, NCIPLOT4 guide for biomolecules: an analysis tool for noncovalent interactions, J. Chem. Inf. Model. 60 (1) (2020) 6–10. [17] Y. Sixto-Lo´pez, J. Correa-Basurto, M. Bello, B. Landeros-Rivera, J.A. Garzo´n-Tiznado, S. Montan˜o, Structural insights into SARS-CoV-2 spike protein and its natural mutants found in Mexican population, Sci. Rep. 11 (1) (2021) 1–16. [18] L. Tolentino-Lopez, A. Segura-Cabrera, P. Reyes-Loyola, M. Zimic, M. Quiliano, V. Briz, J. Correa-Basurto, Outside-binding site mutations modify the active site’s shapes in neuraminidase from influenza A H1N1, Biopolymers 99 (1) (2013) 10–21. [19] S. Jeon, M. Ko, J. Lee, I. Choi, S.Y. Byun, S. Park, S. Kim, Identification of antiviral drug candidates against SARS-CoV-2 from FDA-approved drugs, Antimicrob. Agents Chemother. 64 (7) (2020). [20] S. Satarker, T. Ahuja, M. Banerjee, S. Dogra, T. Agarwal, M. Nampoothiri, Hydroxychloroquine in COVID-19: potential mechanism of action against SARS-CoV-2, Curr. Pharmacol. Rep. (2020) 1–9. [21] F. Musarrat, V. Chouljenko, A. Dahal, R. Nabi, T. Chouljenko, S.D. Jois, K.G. Kousoulas, The anti-HIV drug nelfinavir mesylate (Viracept) is a potent inhibitor of cell fusion caused by the SARSCoV-2 spike (S) glycoprotein warranting further evaluation as an antiviral against COVID-19 infections, J. Med. Virol. 92 (10) (2020) 2087–2095. [22] Y. Kobuke, T. Sugimoto, J. Furukawa, T. Fueno, Role of attractive interactions in endo-exo stereoselectivities of Diels-Alder reactions, J. Am. Chem. Soc. 94 (10) (1972) 3633–3635. [23] C.W. Bielawski, R.H. Grubbs, Living ring-opening metathesis polymerization, Prog. Polym. Sci. 32 (1) (2007) 1–29. [24] D. Kim, I. Hossain, Y. Kim, O. Choi, T.H. Kim, PEG/PPG-PDMS-adamantane-based crosslinked terpolymer using the ROMP technique to prepare a highly permeable and CO2-selective polymer membrane, Polymers 12 (8) (2020) 1674. [25] D. Za´rate-Saldan˜a, B. Landeros-Rivera, J.A. Cruz-Morales, S. Guti
errez, Metathesis of norbornene-derivatives bearing trimethylsilyl groups using Ru-alkylidene catalysts: an experimental and computational study, J. Organomet. Chem. 913 (2020), 121206.

References

513

[26] D.A. Alentiev, E.S. Egorova, M.V. Bermeshev, L.E. Starannikova, M.A. Topchiy, A.F. Asachenko, E.S. Finkelshtein, Janus tricyclononene polymers bearing tri (n-alkoxy) silyl side groups for membrane gas separation, J. Mater. Chem. A 6 (40) (2018) 19393–19408. [27] A. Song, J.C. Lee, K.A. Parker, N.S. Sampson, Scope of the ring-opening metathesis polymerization (ROMP) reaction of 1-substituted cyclobutenes, J. Am. Chem. Soc. 132 (30) (2010) 10513–10520. [28] K. Molcanov, C. Jelsch, B. Landeros, J. Herna´ndez-Trujillo, E. Wenger, V. Stilinovi
c, E.C. Escudero-Adan, Partially covalent two-electron/multicentric bonding between semiquinone radicals, Cryst. Growth Des. 19 (1) (2019) 391–402. [29] N.K. Hansen, P. Coppens, Testing aspherical atom refinements on small-molecule data sets, Acta Crystallogr. Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 34 (6) (1978) 909–921. [30] A. Gavezzotti, Calculation of lattice energies of organic crystals: the PIXEL integration method in comparison with more traditional methods, Z. Krist.–Cryst. Mater. 220 (5–6) (2005) 499–510. [31] P. Alemany, E. Canadell, Y. Geng, J. Hauser, P. Macchi, K. Kr€amer, S.X. Liu, Exploring the electronic structure of an organic semiconductor based on a compactly fused electron donor-acceptor molecule, ChemPhysChem 16 (7) (2015) 1361–1365. [32] B. Guillot, L. Viry, R. Guillot, C. Lecomte, C. Jelsch, Refinement of proteins at subatomic resolution with MOPRO, J. Appl. Cryst. 34 (2) (2001) 214–223. [33] J. Andr
es, S. Berski, J. Contreras-Garcı´a, P. Gonza´lez-Navarrete, Following the molecular mechanism for the NH3+ LiH! LiNH2+ H2 chemical reaction: a study based on the joint use of the quantum theory of atoms in molecules (QTAIM) and noncovalent interaction (NCI) index, J. Phys. Chem. A 118 (9) (2014) 1663–1672.

Index Note: Page numbers followed by f indicate figures, t indicate tables, and s indicate schemes.

A AACID. See Asymmetric magnetically induced current density (AACID) Ab-initio electron correlation methods, 285 ab initio lattice-QCD computational studies, 79 ACID. See Anisotropy of the induced current density (ACID) Adiabatic approximation, 88 Affinity, 74 AGD. See Atomic graph descriptor (AGD) AIM. See Atoms in molecules (AIM) AIM-based partitioning scheme, 96 Alchemical transmutation, 86–88 Alkene insertion step, 59–60 AMD. See Antisymmetrized molecular dynamics (AMD) AMS. See Amsterdam Modeling Suite (AMS) Amsterdam Density Functional (ADF) software package, 361–362 Amsterdam Modeling Suite (AMS. www.scm.com), 38–40, 361–362 Angiotensin-converting enzyme-2 (ACE2), 496–497 Anisotropy of the induced current density (ACID), 335–336, 342–344, 343f, 351–352 ANO-RCC. See Atomic Natural Orbital (ANO-RCC) Antimatter molecules, 81–82 Anti-symmetric-stretch, 158 Antisymmetrized molecular dynamics (AMD), 79–81 Applications of ED SF electron delocalization and aromaticity (see Electron delocalization and aromaticity) hydrogen bonds, 293–296 organometallic complexes, metal-metal and metal-ligand interactions (see Metal-metal and metal-ligand interactions in organometallic complexes) Aromaticity index, 306–315, 461f Asymmetric magnetically induced current density (AACID), 344, 345f, 351–352 Asymmetric synthetic reactions predictions, 160 Atom-centric interpretations, 416 Atomic basins, 89–90, 359–360, 407–408 Atomic basin visualization (QTAIM-DI-VISAB), 67

Atomic chart, 409 Atomic gradient coordinate system, 408–409 Atomic graph (AG), 391–393, 392f, 393–394t, 398–401, 400t, 405 Atomic graph descriptor (AGD), 401 Atomic hypothesis, 73 Atomic members, 82–83 Atomic multipole moments, 171–173 dipole moment, carbon monoxide, 172 photochromism, 172–173 quadrupole moment tensor, 172 Atomic Natural Orbital (ANO-RCC), 126 Atomic properties, 65–66 atomic charge, 65–66 atomic polarizability, 527 atomic quadrupole moments, 65–66, 66f Atomic self-energies, 486 Atomic spin graphs, 394–396 Atoms in molecules (AIM), 73–74 electronic matter, 78 electronic Schr€ odinger equation, 74 MC-QTAIM context, 91–95 molecular non-BO wavefunctions, 88 molecular systems, 75 partitioning scheme, 96 properties and interaction modes, 74 QTAIM context, 88–91

B Bader atomic volumes, 7–8 Basal wave function, 336 Basin particle number distributions, 93–94 Basin property fluctuations, 93–94 Basis set superposition errors (BSSEs), 472 BCPs. See Bond critical points (BCPs) BH-OPV. See Bulk-heterojunction OPV (BH-OPV) Bielectronic repulsion, 475 BO molecular dynamics, 60–61 Bond-anharmonicity, 142–147 Bond bundles (BB), 360, 363, 407–408 condensed charge density, 409–410 qualitative and quantitative properties, 410

541

542

Index

Bond-centric interpretations, 416 Bond critical points (BCPs), 3–4, 40–41, 42f, 113–114, 530–531 cage critical points (CCPs), 3–4, 113–114, 118s, 376, 408 equivalent, 216 non-nuclear critical maxima (NNM), 375–385 nuclear critical points (NCPs), 40–43, 41–42f, 113–114, 142–144, 376 ring critical points (RCPs), 3–4, 41–43, 41–42f, 113–114, 118s, 141, 141s, 145f, 146–147, 154s, 376 Bonded nuclei, 113 Bond energy parameter, 79–81 Bond-flexing, 142–146 Bonding Evolution Theory, 69 Bonding forces, 68–69 Bonding region, 492 Bond-path bond-path curvature (BPC), 530–531 bond-path-flexibility, 142 bond-path framework set B applications, 118–126 bond-path lengths (BPL), 116, 132–134, 132t, 134t, 140 bond-path precession K applications, 142 bond-path-rigidity, 142 Bond-path curvature (BPC), 530–531 Bond-path-flexibility, 142 Bond-path framework set B applications, 118–128 Bond-path lengths (BPL), 134t Bond-path precession K applications excited states bond-path-flexibility, 142 bond-path-rigidity, 142 normal modes, 142–152 switches, 152 Bond-path-rigidity, 142 Bond wedge, 410 Born-Oppenheimer (BO) approximation, 23–24 Boys-Bernardi (BB) counterpoise method, 472 Bulk-heterojunction OPV (BH-OPV), 40–41

C Cahn-Ingold-Prelog (CIP), 111, 160 Canonical algorithm, 32–33 Carbon capture, 502 Carbon-Hydrogen bond activation mechanisms, 59–60, 61f Cardinal number, 84 Casimir-Polder expression, 481–482 CASPT2 energy, 187–188 CASSCF. See Complete Active Space Self Consistent Field (CASSCF) Catastrophe, 56 Catastrophe points, 56 Catastrophe process, 59, 394–396

Catastrophe theory (CT) configuration space, 56–57, 57f functions topology, 56 hypothetical planar triatomic molecule, 56 points, 56 three-dimensional control space, 56 topological analysis, 53 ω parameter, 57 CCs. See Charge concentrations (CCs) CD. See Charge depletion (CD) CDFT. See Conceptual density functional theory (CDFT) Charge concentrations (CCs), 62 Charge density accumulation, 111–112 Charge depletion (CD), 62, 390–391 C-H BCP bond-path K profiles, 152 CHD. See Cyclohexadiene (CHD) Chemical bond, 532 Chemical-bonding schemes, 65 Chemical bonding, vector-based perspective historical context, 111–113 non-scalar, 112–113 Chemical change, 53 Chemical glue, 26 Chemical interpretation, 88–89 Chemical reaction (CR), 69, 517 Chemical topology. See Quantum chemical topology (QCT) Chemical transferability, 281–283 Chemiluminescence, 519 Chirality-helicity equivalence, 112, 157 Chloranil, 508 CIP. See Cahn-Ingold-Prelog (CIP) Clark and Davidson approach, 205 Classical “force field” description, 32 Clockwise (CW) torsions, 153–155 Closed-shell BCP, 114 Close-packed spherical arranged geometrical model, 79–81 Clustering methods, 497–498 Clusterization, 96 “Coarse-grained” quantum model, 79–81 “Columbic” systems, 82–83 Columbic three-body species, 83–84 Commercial and open-source quantum chemistry tools, 34–35 Competitive and non-competitive reactions, 129–130 Complementary information of semibullvalene, 67 Complete Active Space Self Consistent Field (CASSCF), 315–319 Complexation energies, 472 Complexation enthalpy, 473 Complexation Gibbs energies, 473 Complexation internal energies, 472–473

Index

Conceptual density functional theory (CDFT), 363–364 Configuration interaction, 26 Configuration state functions (CSF), 186 Conical intersection (CI), 533–534 Continuous set of damped gauge transformation (CSDGT), 336 Control-space variables, 65 Conventional scalar-based chemistry, 112 Cope rearrangement, 67 Correlation bielectronic repulsion energy, 480–481 Correlation kinetic energy, 477–478, 480–481 Correlation methods, 26 Coulomb interactions, 79–81, 83–84, 86 Coulomb operator, 29 Counterclockwise (CCW) torsions, 153–155 Coupled cluster, 26 Coupled cluster wave function, 188–189 Covalent bonding, 469 Critical points (CPs), 470 Crystal packing, 509–511 CSDGT. See Continuous set of damped gauge transformation (CSDGT) CSF. See Configuration state functions (CSF) CT. See Catastrophe theory (CT) CTOCD-DZ. See Current density-diamagnetic current set to zero (CTOCD-DZ) Current density. See Magnetically induced current density Current density-diamagnetic current set to zero (CTOCD-DZ), 336–338 Current density tensor, 341–342 Cycloaddition, 417 Cyclohexadiene (CHD), 140 1,3-cyclohexadiene (CGD), 533–534

D Daltonian atom, 73–74 Dalton’s atomic theory, 73–74 Deactivation processes, 517 Decoupling, 86–88 Deformation energy, 472 Delocalization indices (DIs), 66–67, 93–94, 280, 480 Density functional theory (DFT), 215–216, 473–474, 491 Coulomb and external functionals, 28 Hohenberg-Kohn (HK theorem) theorems, 27 Kohn-Sham equations, 29–30 Kohn-Sham (KS) formulation (see Kohn-Sham formulation) Kohn-Sham (KS) functional (see Kohn-Sham functional) N integro-differential nonlinear coupled equations, 29 variational calculus, 28–29

543

Density-functional theory (DFT), 431–432 Descriptor functions BCP, 4–5 covalent bonds, 6 diatomic molecules, 4–5, 5f kinetic energy density, 6 Destabilizing, 416 DFT. See Density functional theory (DFT) DHCL. See Dihydrocostunolide (DHCL) Diarylethene (DTE) switch molecule, 152 Diatomics, 378–381 Diels-Alder catalysis, 408 Diels-Alder cycloaddition, 68–69 Diels-Alder electric field catalysis, 421–427 Diels-Alder reactions, 416, 428 Diels-Alder-type reactions, 64 Dihydrocostunolide (DHCL), 140 1,3-Dipolar cycloaddition, 64 Dipole-bound electron model, 381–382 Dirac-Coulomb equation, 248–249, 255 Dirac delta function, 432–433 Dirac equation, 249 Directional electric-field effects, 152 Divergence Theorem, 12–13 Domain natural orbitals (DNOs), 214 Double/single/ionic character, 113 Dumbed down bonds, 427

E EBC. See Extended Bader’s conjecture (EBC) ECPs. See Effective core potentials (ECPs) ED SF studies in chemical bonding advantage, 276–277 bond path, 277 C atom, 275–276 CC bond critical point (bcp), 275f HB bcp, crystals, 279 Interacting Quantum Atom (IQA) approach, 277 local source, 277 negative sources, 278 positive sources, 278 positive vs. negative local sources, 278–279 QTAIM, 275–276 SF profiles, 279–280 trivial tautological reconstruction, 275–280 EEFs. See External electric fields (EEFs) Effective core potentials (ECPs), 119 E-field, 132–136, 133f, 137f Ehrenfest force density, 241 Ehrenfest forces, 68, 126–128 Born-Oppenheimer approximation, 226 conservative force, 239 electronic, 226

544 Ehrenfest forces (Continued) electronic wave-function, 226 properties Coulombic, 231–232, 231t molecular graph, 233, 234f and QTAIM partitionings, 232 stress tensor, 233 symmetry-inequivalent atom types, 232, 232t second-derivative tensor, 225 and stress tensor, 227 Ehrenfest Hessian, 238–239 and Ehrenfest potential, 241 eigenvalues, 233, 235–236t Ehrenfest Lagrange points (BCP) data, 233, 235–236t Ehrenfest partitioning electron-density, 229–231 symmetric molecules, 230 traditional QTAIM, 229–230 zero-flux condition, 230 Ehrenfest potential definition, 233–234 electron hydrogenic atom, 237 hydrogenic atom, 238 literature, 234–237 nonconservative nature, 238–239 one-electron atoms, 238 Electric field charge density, 416–417 Electromagnetic plane waves, 519–520 Electromagnetic radiation, 516, 524–525 Electron, 21 Electron delocalization, 53–54, 280, 306–315, 335–336, 343, 351–352 Electron delocalization and aromaticity “aromatic” Lewis structures, 311 BAB, 309–311, 309s benzene, 310–311 bond ellipticity, 313 C atoms, 306–308 detect electron delocalization effects, 309 electron correlation effects, 308–309 electron delocalization features comparison, 310 lone pair (LP), 311 naphthalene, 309–311 NHCs, 311 “nonaromatic” Lewis structure, 311 π-electron delocalization, 312f, 313 SFLAI, 313–315 Electron density (ED), 53–54, 74 chemical bonding, 275–280 chemical transferability, 281–283 differences benzene, 166–167 excimer formation, 167–171

Index

formaldehyde, 167 electron delocalization, 280 evolution alkene insertion, 59–60 BO molecular dynamics, 60–61 catastrophe mechanisms, 59 chemical process, 61–62 coordination chemistry, 59–60 cycloaddition reaction, 59 hydroformylation reaction, 59–60 iminol-amide tautomerism, 59 internal reaction coordinate, 60–61 isomerization reactions, 59, 60f keto-enol prototropisms, 59 metallicity, 61 noncovalent interactions (NCI), 61–62 photo-induced hydrogen, 60–61 SN2 reaction, 59 Li atom, 284f locality and nonlocality in chemistry, 271 molecules and crystals, 274–275 reference points, 281 software and computational details, 283–284 topology analysis, 2 to benzene molecule, 2, 2f eigenvalues, 3 features, 3, 3t nuclear attractor, 3–4 quantum chemical topological analysis, 3 solid electron, 1–2 Electron-electron repulsion, 88–89 Electronic accumulation, 115–116 Electronic charge density distribution, 111–112 Electronic energy, 27 Electronic structure problem eigenvalue-eigenvector representation, 24–25 electronic energy, 25 electronic Hamiltonian operator, 24 external potential, 24 Hamiltonian nuclei coordinates, 24 many-electron wave-function determination, 25–26 molecular energy, 24 molecule’s electronic structure, 25 single determinantal description, 26–27 Electronic structure software community, 34 Electronic wavefunctions, 74 Electron localization function (ELF), 389–390 Bond Evolution Theory (BET), 63–64 Catastrophe Theory (CT), 63–64 cycloaddition process, 64 Diels-Alder-type reactions, 64

Index

ELF/NCI cross interpretative approach, 64 measures, 63–64 molecular graph, 63–64 Pauli repulsion, 63–64 synergistic combination, 63–64 topological analysis, 64 topological changes, 64 topological stability domains, 64 unconventional back-donation, 64 Electron number delocalization, 91 Electron number distribution, 91 Electron number distribution functions (EDFs) calculation, 209 multidomain sectors, 207 RSRSs, 208 Electron-pair reorganization, 66 Electron reorganization, 65 Electron spin density (ESD), 271–272 Ab-initio electron correlation methods, 285 applications, water triplet 3B1 state, 315–320, 316f, 319f atomic ESD SF contributions to s(r) magnetic components, 287–289 relaxation components, 287–289 local source function, 286t PND, 285 positive and negative ESD local source contribution LSS, 285–286, 286t reference points, ESD SF studies, 286 software and computational details, 289 Electrophilicity, 476 Electrostatic catalysis progresses, 416 ELF. See Electron localization function (ELF) ELF/NCI topological approach, 64 Ellipticity profile, 136, 137f Endergonic formation, 507 Energetic evolution, 54 Energy functional, 27–28 EOM-CCSD, 188–189 Ethylene basin boundaries, 422f basin values, 420 bond rearrangement, 426 charge densities, 418–419t charge rearrangement, 426 interatomic charge transfer, 421 kinetic energy density, 420–421 NEF, 424–425 numbering and orientation, 421 orientation of, 418 valence charge densities, 423–424t Ethylene electric field response, 418–421 Exact exchange (XX), 477–478

545

Exact relativistic method, 248 Exchange-correlation density, 66 Excimer formation atoms/molecules, 167 benzene and naphthalene, 167–168 bonding properties, 170–171 electron distribution, 168–169, 171 intermolecular separations, 167–168, 168t internuclear separations, 170, 170t mechanism, 168–169 stabilization, 169–170 Excited electronic states (EEs), 165. See also Quantum chemical topology (QCT) atomic multipole moments, 171–173 electron density differences, 166–171 excited-state aromaticity, 173–176 organic molecules, 166 Excited-state aromaticity electron delocalization, 175 hypoxanthine, 176 PAHs reported, 175, 175f polycyclic arenes, 173–175 salicylideneaniline, 175–176 Excited state intramolecular proton transfer (ESIPT), 459–460 Exergonic formation, 507 Exotic atoms, 82–83 Exotic molecular system, 84–85 Extended Bader’s conjecture (EBC), 85 External conversion (EC), 517 External electric fields (EEFs), 370–371, 370f, 418

F FCI. See Full Configuration Interaction (FCI) Finite difference method (FDM), 364–365 First Hohenberg-Kohn theorem, 382–385, 385s Fluorescence, 515 Fluorine-hydrogen bond path (F13-H36), 534–535 FMO. See Frontier molecular orbital theory (FMO) FMR. See Fragments of molecular response (FMR) approach FNOs. See Fragment natural orbitals (FNOs) Foundational quantum mechanical description, 21 Fractional electric charges, 79 Fragment natural orbitals (FNOs), 194, 212–214 atomic, 218 nickel atom, 219 Fragments of molecular response (FMR) approach, 364–365 Frontier molecular orbital theory (FMO), 366–367 Fukui functions, 484 Full Configuration Interaction (FCI), 27

546

Index

G Gamma density, 84, 86–88 Gauge-including magnetically induced current (GIMIC), 336–337 Gauss’ divergence theorem, 435–436 Gaussian curvature, 409 Gaussian functions, 26 Gaussian quadrature tetrahedral, 409 Gauss’ theorem, 13 GB. See Gradient bundle (GB) General-purpose reactivity indicator (GPRI), 367 Geometry critical points, 55–56 diffusion functions, 56 gradient trajectories, 55 Hessian matrix, 54–55 hydrogen bonding interaction, 56 molecular geometries, 55–56 molecular graph, 55–56 molecular system, 54–55 QTAIM bonding scheme, 55–56 The Theory of Atoms in Molecules, 54–55 unitary matrix, 54–55 Geometry optimization, 32–33 GIMIC. See Gauge-including magnetically induced current (GIMIC) GitHub repositories, 34 Gold cluster complexes, 41–43 Gold nanoclusters, 41–43, 42f GPRI. See General-purpose reactivity indicator (GPRI) GPs. See Gradient paths (GPs) Gradient approximation density function, 493–494 Gradient bundle (GB), 407 atomic basins, 359–360 change size and shape, 363–367 charge density analysis, 410–411 condensed deformation property, 413 condensed properties, 410–412 condensed scalar properties, 412f decomposition, 408–409 definition, 360 electric fields, 370–371 gradient paths (GPs), 359–360, 361f irreducible bundle (IB), 360 kinetic energy density, 413 kinetic energy distributions, 362–363 methods, 361–362 normalized electron count, 363, 364f oxygen atom in methanol molecule, 360f predict reactivity, reactant state charge density, 367–370, 367f QDBs, 360 QTAIM, 359

quantitative condensed property analysis, 413–416 visual qualitative inspection, 411–413 Gradient paths (GPs), 359–360, 371 Gradient vector field, 113–114 Grid-based QTAIM method, 38–40, 39f Grimme’s empirical 3-center dispersion correction, 131–132 Ground-state electronic wavefunction, 86–88 Grubbs second-generation catalyst, 503

H Hadronic molecules, 79 Hadron spectroscopy, 79 Halogenabenzene/NH3 complexes, 119–123, 121–122f Halogen bonds, 469 additive form, 476 applications, 469 bond formation path, 485–487 bond strength, 482–483 electrophilicity, 476 electrostatic terms, 478–479 exchange-correlation potential, 481 exothermic complexation, 473 formation, 483–485 gas phase, 473 Grimme’s pairwise dispersion correction, 481–482 interaction energies, 472–473, 480 intramolecular, 473 IQA, 470, 473–482 meta-GGA (m-GGA), 478–479 Møller-Plesset levels, 476–477 nucleophilicity, 476 physical components, 482 physicochemical processes, 483 post-HF calculation, 481 post-HF wavefunctions, 478 prediction models, 485 pro-fragment density, 483 QCT, 470 QTAIM, 470–471 atomic charges, 479 tools, 471–483 REG analysis, 486 Hamilton Action Principle, 247 Hamiltonian operator, 21–22 Hansen-Coppens multipole model, 508 Hard-wall atoms properties, 11–12 volume associated atom, 11–12 Harmonic-like morphology, 144–145 Harmonic oscillator, 85–86 Hartree energy, 477 Hartree-Fock approximation, 537

Index

Hartree-Fock determinant, 26–27 Hartree-Fock equations, 26 Hartree-Fock (HF) theory, 25–26, 478 Hartree-Fock wave-function, 25 Heavy radioisotope, 86 Hellmann-Feynman forces, 68–69 Hellmann-Feynman theorem, 31 HER. See Hydrogen evolution reaction (HER) Hermitian operators, 13–14 Hessian matrix, 132–134 High-performance grid-based QTAIM algorithms/ software Amsterdam Modeling Suite (AMS), 38–40 atomic integration, 38 computational efficiency, 37–38 grid-based methods, 38–40 grid point associating, 38–40 reducing QTAIM computing time, 37–38, 47 steepest-ascent path construction procedure, 38–40 zero-flux surfaces, 38–40 High performance liquid chromatography (HPLC), 289–290 Hill-Wheeler method, 79–81 Hohenberg-Kohn (HK theorem) theorems Bright Wilson justification, first Hohenberg-Kohn theorem, 382–385 HPLC. See High Performance Liquid Chromatography (HPLC) HPLC molecular recognition processes, 320–321 Hydroformylation reaction, 59–60 Hydrogen bonds (HB), 293–296, 442 anticooperativity, 448–449 bifunctional catalytic activity, 453, 454f characterization, 444 charge transfer, 452f contributions, 449f cooperative effects, 447–448 cooperativity/anticooperativity, 456–458, 457f cyclic clusters, 446–447 deformation energies, 462f delocalization indices, 456t dipole, 448f double, 450f doubly and triply, 463–465 electronic states, 459–460 electronic transfer, 447f energy curves, 447f, 457f EOM-CCSD theory, 461–463 equilibrium energies, 443 exchange-correlation component, 447–448 formation energies, 453t Gibbs energy, 455f interaction energy, 443

547

intra-/interatomic contributions, 459t ions/solvents, 452 IQA additive energies, 458t IQA interaction energies, 462f, 464f magnitude and orientation, 463f mesomeric structures, 455–456, 458f molecular clusters, 453f nonadditive effects, 446–454 pairwise contributions, 451 properties, 442–445 resonance-inhibited, 459–460f single homodromic cycle, 449f structures of, 450f tautomeric equilibrium, 458–459 types, 451–452 wave-function, 445–446 Hydrogen-bonds 3-D bond-path framework, 126–127 Hydrogen evolution reaction (HER), 43 Hydrogen fluoride clusters, 381 Hydrogen fuel, 43 Hydrogen transfer tautomerization process, 130

I Icosohedron method, 361 IGAIM. See Individual gauges for atoms in molecules (IGAIM) IGLO. See Individual gauge for localized orbitals (IGLO) method Individual gauge for localized orbitals (IGLO) method, 336 Individual gauges for atoms in molecules (IGAIM), 336 Induced dipolar moment, 523 Induced polarization, 522 Infinitesimal generator, 15–16 Integrated schemes, 69 Interacting boson model, 79–81 Interacting quantum atoms (IQA), 10–11, 67–68, 88–89, 432, 437–442 advantages, 177–178 applications, 178–185 approach, 277, 535 bond formation, 180–182 potential energy curves, 180–181, 181f quantitative information, 181–182 conical intersections, 182–183 cyclobutane, 184–185 decomposition, 180 DFT, 440–442 3D space, 440 electron-electron contacts, 439–440 electronic absorption, 179–180, 179t electronic energy, 438–439

548 Interacting quantum atoms (IQA) (Continued) features, 437–438 hydrogen contributions, 179–180 implementations of, 166 LiF molecule, 182, 182f molecular cluster, 438 monoelectronic operator, 176–177 nonrelativistic electronic energy, 176–177 nucleus-nucleus contacts, 439–440 partitioning, 178 self-energy, 177–178 wave functions, 440 Interatomic interaction energies, 67–68 Interatomic interactions, 486 Interatomic surfaces (IS), 360–361, 434, 471 Interatomic valance charge transfer, 421–424 Inter-basin covariance, 90 Inter-basin property fluctuation, 95 Intermediate level of organization, 78 Internal clusterization, 86–88 Internal deformations, 79–81 Inter-quark interaction, 79 Intra- and inter-basin contributions, 88–89, 91–93 Intrinsic reaction coordinate (IRC), 184f, 485–486 IQA. See Interacting quantum atoms (IQA) IR-active modes, 142 IRC. See Intrinsic reaction coordinate (IRC) IR-inactive modes, 152 Irreducible bundle (IB), 360 IR-responsivity, 152 Isolated fragment, 483

J Jahn-Teller effect, 68–69, 405 Joint distributions, 95 Jørgensen secondary interaction hypothesis (JISH), 463–464, 464f

K Kato-cusp condition, 382 Kinetic balance, 249 Kinetic energy, 369 distributions, 362–363 operators, 409 Kohn-Sham formulation density, 477 DFT calculations, 30 equations, 28–29 formalism, 29 molecular spin-orbitals, 29 operator, 28–29 Kohn-Sham functional, 28–29 Kronecker delta, 438–439

Index

L LAO. See London atomic orbitals (LAO) Laplacian of electron density, 390, 390f, 394, 397f atomic Graph (AG), 62 charge concentrations (CCs), 62–63 charge depletion (CDs), 62 photoinduced electron transfer, 63 polarization, 63 scalar function, 62 transition mechanisms, 62 VSCC evolution, 62–63, 62f VSEPR model, 62 Laplacian of the conditional pair density (LCPD), 390–391 LCPD. See Laplacian of the conditional pair density (LCPD) Lead-molecule-lead array, 41–43 Li atom, 282–283 Ligand-protein interactions, 496–501 Light-driven molecular motors, 534–535 Light radioisotope, 86 Linear polarization, 521 Liquid-drop model of nucleus, 79–81 Locality and nonlocality in chemistry, 271 Localization index, 66–67 Localized negative muon’s one-particle density, 86–88 Local spin analysis, 217–218 Local topological features, 65 Local zero-flux equation, 84, 86–88 London atomic orbitals (LAO), 336–337 Lowest unoccupied KS molecular orbital (LUMO), 484 Luminescent chemical reactions, 515 Lumphole model, 484

M Machine learning-based regression, 428 Magnetically induced current density AACID, 344 ACID, 342–343 aromaticity vs. magnetic indices, 337–338 basal wave function, 336 charge density, 336 CSGT, 336 CTOCD-DZ method, 337–338 current density and stagnation graph, 339f current density tensor, 341–342 density and velocity operators, 336 diatropic, 337–338, 338f electronic velocity, 336 gauge origin, 336 GIAO method, 336 GIMIC, 336 IGAIM, 336

Index

invariants, current density tensors, 350 magnetically induced Lorentz force density, 349–350 organic aromatic molecules, 338 Ryleigh-Schr€ odinger perturbation theory, 336 scalar fields, 351 topology, 339–341, 340f, 351–352 TVCD, 347–349, 348f vorticity of J(r) (5J(r)), 345–346, 346f Magnetically induced Lorentz force density, 349–350 Magnetic component, 287–289 Many-electron wave-function determination, 25–26 Maximum stress tensor projections, 157t MC-QTAIM. See Multi-component QTAIM (MC-QTAIM) MDW. See Multideterminant wave functions (MDW) Mean basin electron population, 89–90 m-electron cluster, 89–90 MEP. See Minimum energy path (MEP); Molecular electrostatic potential (MEP) MESP topography. See Molecular electrostatic potential (MESP) topography Metallacyclobutane, 506 Metal-ligand interaction, 397, 398t Metal-metal and metal-ligand interactions in organometallic complexes Co-Co delocalization indices, 303 H-acceptor atom, 297–299 Me-Me bonds, 304–305 metal-metal bcp, 301–303 metal-metal internuclear axis, 303 Mn-C bonds, 297–299 Mn-Mn bond formation, 299–301 Mn-Mn internuclear, 299f Mn-Mn midpoint, 301 transition metal π-hydrocarbyl complexes, 305–306 Metal-metal bonds, 275f, 278–279, 297–299, 301–305 Minimum energy path (MEP), 184–185 Mirror image, 81–82 Molecular dynamics simulations (MD), 496–497 Molecular electron density, non-nuclear maxima atomic virial theorem, 376 diatomics, 378–381 molecules and complexes, 381–382 nonnuclear attractor, 376–378 QTAIM, 375–376 topological properties, 376 zero-flux surfaces share, NNA, 378f Molecular electrostatic potential (MEP), 272–273, 483–484 applications chemical substitution, σ-hole regions in sin 4,40 bipyridine derivatives, 321–325, 321f, 324f HPLC technique, 320–321

549

atomic decompositions of V(r), 289–290 atomic SF contributions, 290–291, 290f ED isosurfaces, 289–290 QTAIM atomic contributions, 289–290 σ- and π-holes regions, 321f, 323f software and computational details, 291 Molecular electrostatic potential (MESP) topography, 65 Molecular equation BO approximation (see Born-Oppenheimer (BO) approximation) explicit time-dependence, 22 external electromagnetic fields, 22 partial differential equation, 22 simplification, 22 system’s kinetic energy operator, 22 Molecular expectation values, 88–89 Molecular graph, 3–4, 4f Molecular Hamiltonian operator, 22 Molecular nuclear problem BO approximation, 30 electronic structure problem, 30 geometry optimization, 32–33 Hellmann-Feynman theorem, 31 nuclear equation, 30 potential energy surfaces, 30–31 Schr€ odinger-like equation, 30 Molecular polarizability, 527 Molecular positronium, 83–84 Møller-Plesset perturbation theory, 26 Molybdenum disulfide (MoS2), 43 Monolayer (2D) materials, 43 Morpholino enamines, 59 MoS2-based materials bond information, 43–47, 45t crystal graphs, 46f DFT-QTAIM calculations, 47 doped and vacancy systems, 43, 44f GGA-PBE, 43 Mo-S bonds, 43–47 MoS2 bulk and monolayer systems, 43, 44f partial charge analysis, 47, 48f QTAIM bond information, 43–47 ring critical point (RCP), 43–47 S atoms, 43–47 topological properties, 45f TZP, 43 Vac/MoS2 system, 47 Zn/MoS2 system, 47 MoS2 monolayer primitive, 43 MRCI-SD method, 186–187 Multi-component QTAIM (MC-QTAIM) ab initio wavefunctions, 75 AIM (see Atoms in molecules (AIM))

550

Index

Multi-component QTAIM (MC-QTAIM) (Continued) developments, 83 exotic molecules, 96–97 local zero-flux equation, 86–88 nonelectronic systems, 77 particle fluctuation, 93–94 property partitioning, 91–93 QTAIM analysis results, 75 theoretical developments, 75 Multi-component wavefunctions, 85–86, 93–94 Multiconfigurational (MC) methods, 520–521 Multideterminantal description, 27 Multideterminant wave functions (MDW), 196–197, 203–204 Multipolar expression, 523 Muon spin resonance spectroscopies (μSR), 83

N Naphthalene, 309 Natural adaptive orbitals, 194, 210–212 Natural density partitioning algorithm AdNDP method, 214 chemical bond, 214 eigenvectors, 215 spin variables, 215 Natural orbitals (NOs), 474 Natural transition orbitals (NTOs), 117–118 NCI. See Non-covalent interaction index (NCI) NCP. See Nuclear critical point (NCP) Next-generation QTAIM (NG-QTAIM) applications (see Bond-path framework set B applications) bond-by-bond basis, 159–160 bond-path framework, 113–117 chirality, 111 chirality-helicity equivalence, 159 developments, 112 directional 3-D interpretation, 112, 159 “smoothing” methods, 159 theoretical developments, 111 vector-based quantum chemical theory, 111 NG-QTAIM. See Next-generation QTAIM (NG-QTAIM) NG-QTAIM bond-path framework set B bond critical points (BCP), 113–114 bond-paths, 113–114 bond torsion, 114 charge density, 115–116 chemical bond, 116 electronic charge density, 113 gradient vector field distribution, 116s Hessian matrix, 113–114 kinetic and potential energy densities, 114 molecular torsion, 114–115

non-negligible ellipticity, 115 stress tensor, 114 topological stabilities, 115–116 wave-function, 113 NG-QTAIM bond-path precession K applications (see Bond-path precession K applications) bond critical points (BCP), 140–141 bond-path IR-non-responsivity, 141–142 bond-path length (BPL), 140–141 bond-path torsion, 140–141 eigenvector, 142 geometric bond length (GBL), 140–141 maximum degree of alignment, 141–142 ring critical point (RCP), 141 NG-QTAIM 3-D bond-path framework set B chemical bond, 116 eigenvalues, 116 eigenvector, 116 eigenvector-following paths, 116–117 electronic charge density accumulation, 116 natural transition orbitals (NTOs), 117 p and q path-packet expressions, 117–118 NG-QTAIM Uσ-space stress tensor trajectory Tσ(s) applications (see Uσ-space stress tensor trajectory Tσ(s) applications) BCP shift vector, 152–153 charge density accumulation, 153 computational protocol, 153–154 constant bond-path torsion, 153 CW and CCW, 153–154 molecular graphs, 154s QTAIM partitioning, 152–153 S and R classifications, 153 S and R stereoisomers, 153–154 torsional BCP, 153–154 NHC. See N-heterocyclic carbenes (NHC) N-H/C-H bond rearrangement, 68 N-heterocyclic carbenes (NHC), 311 NICS. See Nucleus-independent chemical shifts (NICS) Nifedipine (NIF), 529–530 N integro-differential nonlinear, 29 NLF. See Nucleon localization function (NLF) NMR. See Nuclear magnetic resonance (NMR) NNA. See Non-nuclear attractor (NNA) NNM. See Non-nuclear maxima (NNM) No electric field (NEF), 418 Non-BO calculations, 88 Nonclassical 7-norbornyl cation, 67 Noncompetitive/competitive torquoselectivity, 61 Non-covalent interaction index (NCI), 61–62, 431–432, 493 application, 495–496 disadvantage, 496

Index

experimental electron densities, 507–511 free energies, 499–501, 501f isosurfaces, 494–495, 498–499, 504f localized and delocalized interactions, 500t monomeric densities, 496 non-QTAIM, 495 PIXEL energies, 511 promolecular-defined, 496 protein-ligand complexes, 501 quantitative analysis, 496 RANGE, 499 signatures, 494 surface generation, 499 types, 494–495 Noncovalent intermolecular bonds, 40–41 Nonlinear optical properties, 515 Nonlinear polarization, 522 Non-nuclear attractor (NNA), 147–152, 233, 281, 376–382, 385s Non-nuclear maxima (NNM), 375–385 Nonoverlapping partition, 474–475 Nonrelativistic quantum chemistry, 248 Non-vanishing fluctuations, 89–90 Non-zero bond-path torsion, 115 Normal hydrogen molecule, 86–88 Novel theoretical framework, 74 NTOs. See Natural transition orbitals (NTOs) Nuclear configurations, 53 Nuclear critical point (NCP), 42f, 113–114 Nuclear magnetic resonance (NMR), 335–336 Nuclear molecules, 79–81 Nuclear Quadrupole Double Resonance (NQDR), 529–530 Nuclei, 23 Nuclei-centered spheres, 409–410 Nucleon densities, 79–81 Nucleon localization function (NLF), 79–81 Nucleophilic attack, 508 Nucleophilicity, 476 Nucleophilic oxygen, 453 Nucleus-independent chemical shifts (NICS), 335–336 Numerical algorithms, 38

O “OFF” functioning, 156 One-electron density, 53–54 One-electron “Fock” operator, 25–26 one-electron system, 437 One-particle electron density, 86–88 One-particle property density, 93–94 “ON” functioning, 156 “Openness” of atomic basins, 91 Open quantum subsystem (OQS), 193–194

551

application, 204 formalism, 207 local spins, 206–207 orthogonal spin-orbitals, 197–200 sector density operators, 195–196 subsystem density operator, 194–195 Open Source vs. commercial software development, 34 OPV. See Organic photovoltaics (OPV) OQS. See Open quantum subsystem (OQS) Orbital-like packet shapes, 116 Organic photovoltaics (OPV) bulk-heterojunction OPV (BH-OPV), 40–41 charge transfer properties, 40–41 donor-acceptor pair, 40–41 electronic properties, 40–41 Laplacian value, 40–41 P3HT and PCBM active layer, 40–41 stable structures, 40–41, 41f technology alternative, 40–41 Organic reactions, 66 Origin-independent energy-based (OIEB) approximation, 538

P Pair density, 93–94 Parallelization, 38–40 Partial differential equation, 22 Partial source function reconstructed density (PSFRD), 291–293, 319–320 Partial source function reconstructed potential (PSFRP), 291–293 Particle number fluctuations, 95 Particles distinguishability, 91–93 Partitioning methodology, 96 Path-packets, 136, 138–139, 138f PCBM. See [6,6]-Phenyl-C 61-butyric acid methyl ester (PCBM) Pendas-Hernandez-Trujillo method, 229 Penta-2,4-dieniminium cation (PSB3), 126 Perdew-Burke-Ernzerhof (PBE), 417 PES. See Potential Energy Surface (PES) Phenomenological models, 73–74 [6,6]-Phenyl-C 61-butyric acid methyl ester (PCBM), 40–41 Phosphorescence, 515 Photochemical reactions (PR), 515 energetic profiles, 518f excitation of the ground state, 516–517 excited states, 517–519, 532 issues, 521 Jablonski diagram, 516f nonradiative process, 518 radiative processes, 518

552 Photochemical reactions (PR) (Continued) semiclassical approach, 519 wavefunction, 520 Photochemistry, 515 QTAIM applications, 529–530 topological approaches, 530–538 Photo-induced hydrogen, 60–61 Photoinduced structural change, 63 Photoisomerization, 534–535 Photo-isomerization dynamics, 156 Photo promoted chemical reactions, 515 P3HT. See Poly(3-hexylthiophene) (P3HT) P3HT/PCBM structure, 42f P3HT/PCBM topological features, 40–41 Poincare-Hopf relationship, 4, 482 Poisson equation, 240–241 Polarization, 521–523 Polarized neutron diffraction (PND), 285 Poly(3-hexylthiophene) (P3HT), 40–41 Polycyclic aromatic hydrocarbons (PAHs), 166–167 and benzene, 173–174 Polyelectronic wavefunction, 473–474 Positron annihilation spectroscopies, 83 Positronic densities, 85–86 Positronium atom, Ps, 83–84 Potential energy surface (PES), 31–32, 471 Primary interaction, 482 Principal approximations, 34–35 Principle of Stationary Action, 247 Propellane molecule, 124 Property density, 88–89, 91–93 Property partitioning, 91–93 Protonium atom, 83–84 Pseudoatoms, 376, 378–382, 385 Pseudopotentials treatment, 254 Pseudo-problem, 74 PSFRD. See Partial source function reconstructed density (PSFRD) PSFRP. See Partial source function reconstructed potential (PSFRP) Pt-based catalyst, 43 Pure antimatter, 81–82 Purely electronic system arbitrary electron, 75–76 atomic basins, 75–76 Bader’s conjecture (BC), 76 BC reveals, 77 conditional truth, 77–78 Coulomb’s potential, 77 critical point (CP), 76 Daltonian atoms, 77–78 electronic population, 76 electronic wavefunction, 75–76

Index

electron-nucleus potential, 77 equilibrium geometries, 76 high-energy excited states, 77 main classes, 78 molecular electron densities, 76 nucleus and monotonic, 76 partitioning scheme, 77 pseudo-atoms, 77–78 QTAIM computational studies, 77 QTAIM’s paradigm, 84 quantum mechanics, 76 X-ray diffraction, 76

Q Q-atomic basin system transforms, 85–86 QCD. See Quantum chromodynamics (QCD) QCT. See Quantum chemical topology (QCT) QDBs. See Quantum dividable basins (QDBs) QTAIM. See Quantum theory of atoms in molecules (QTAIM) QTAIM bond-path framework sets, 117 QTAIM-DI-VISAB. See Atomic basin visualization (QTAIM-DI-VISAB) QTAIM electron density topology, 38–40 QTAIM integrated properties, 38, 39t QTAIM-photochemistry atomic basin, 527 atomic polarizability, 527 dipolar approximation, 526–527 dipole approximation, 525–526 dipole moment, 528–529 electronic polarizability, 528 energy contribution, 526 molecular polarizability, 527 perturbation densities, 526 polarizabilities, 525–529 transition density, 525–526 transition probabilities, 525–529 wave function, 526 QTAIM’s bond paths, 426 QTAIM scalar measures, 140 QTAIM standard software, 38 Quantitative 3-D rendering, 118–119 Quantum chemical topology (QCT), 9–10, 193, 269, 389–390, 492, 523, 538 applications, 166 conventional, 205 for EEs, 165–166 scalar and vector fields, 10, 10–11t use, 165–166 Quantum chemistry DFT (see Density functional theory (DFT)) electronic structure problem, 24–27

Index

molecular equation, 22–24 nuclear problem, 30–33 principal approximations, 34–35 quantum application, 34–35 software packages, 34 Quantum chromodynamics (QCD), 79 Quantum dividable basins (QDBs), 360 Quantum mechanics, 13, 21 Quantum particles, 91–93 Quantum stress tensor. See Stress tensor Quantum theory of atoms in molecules (QTAIM), 269, 282–283, 315–317, 336, 351–352, 359, 371, 375–376, 381, 432–437, 492 atomic charge distribution, 434–435 atomic virial theorem, 436 atomic volumes, benzene, 7–8, 8f atom’s quantum properties determination, 37 BCP, 433–434 canonical atomic properties, 8 zero flux, 9 concepts, 10–11 conceptual chemistry, 37 critical points (see Critical points (CPs)) delocalization index, 436–437 descriptor functions, 4–6 electron density topology, 1–4 external nonuser-friendly software, 37 fundamentals, 1 Hard-wall atoms, 11–13 integrated/atomic properties, 6–8 IS, 434 key role, 37–38 kinetic energy, 435 mathematical foundations, 75 molecular virial theorem, 8 “multi-component” quantum systems, 75 NCP, 434 population data, 216 real-world applications catalyst materials, 43–47 nanostructures, wires, 41–43 organic photovoltaics (OPV), 40–41 themolecular systems and chemical reactivity, 9

R RCP. See Ring critical points (RCP) RDM. See Reduced density matrices (RDM) 2-RDM, general non-diagonal form, 90 Reaction axis, 416–417 Reaction evolution, integrated properties chemical reactions, 67–68 force analysis, 68–69

553

IQA analysis, 67–68 localization and delocalization indices, 66–67 local vs. integrated properties, 65–66 Reaction mechanisms arbitrary delocalization, 53–54 catastrophe processes, 53–54 catastrophe theory, 56–57 chemical change, 57–58 conformational change, 53–54 definition, 54 electron density evolution, 59–61 electron localization function (ELF), 63–64 energetic evolution, 54 energy barrier, 503 geometry and structure, 54–56 geometry optimizations, 503–504 integrated properties evolution, 65–69 Laplacian of electron density evolution, 62–63 MESP evolution, 64–65 monomer synthesis, 502f one-electron density, 53–54 reaction pathways, 503, 503f ROMP, 504–505 stabilizing intermediates/transition states, 502 structural evolution, 57–58 transition state, 505 two-electron density, 53–54 Reaction/relaxation spin density, 287 Real-space clustering, 78 Real-space resonance structure (RSRS), 208 Reduced density matrices (RDM), 194 Reduced Density Matrix Functional Theory (RDMFT), 476–477 REG. See Relative Energy Gradient (REG) Regional virial theorem, 84–85 Relative energy gradient (REG), 68 analysis, 485–486 method, 68 Relativistic effects on atoms in molecules QTAIM topology, 261–262 SR-ZORA method, 260 zero-flux condition, 260 corrections, 250 direct and indirect, 250 nonrelativistic quantum proper subsystem, 251–254 and proper quantum subsystem action principle, 251 straight-forward approach, 251 QTAIM properties, 250 Relativistic QTAIM, 245–263 Resonance-assisted hydrogen bonds (RAHB), 455 Resonating group method (RGM), 79–81

554 Response of molecular fragments (RMF) approach, 364–366 RGM. See Resonating group method (RGM) Ring critical points (RCP), 42f, 113–114 Ring-opening metathesis polymerization (ROMP), 502 RMF. See Response of molecular fragments (RMF) approach RSRS. See Real-space resonance structure (RSRS) Ryleigh-Schr€ odinger perturbation theory, 336

S Salicylideneaniline, 460f S and R stereoisomers, 156 “SARC-ZORA-TZVP” basis set, 119 Scalar fields, 65 Scalar-relativistic ZORA (SR-ZORA), 256 ZORA and, 256 SCF. See Self-consistent-field (SCF) Schr€ odinger equation, 16–17, 21, 22f, 29, 88 Schr€ odinger’s wave function, 6 Secondary interaction, 482 Second-quantized based methods, 79–81 Second-row homonuclear diatomics, 68 Sector density operators MDW, 196–197 SDW, 195–196 Sector spin analysis, 219 Self-consistency, 29 Self-consistent-field (SCF), 29, 480 SF. See Source function (SF) approach SFLAI. See Source Function Local Aromaticity Index (SFLAI) Shared-shell BCP, 114 Shell model, 79–81 Single determinantal description, 26–27 Single-determinant wave functions (SDWs), 194 first-and second-order, 201 spin-orbitals, 202 Single-point calculation, 33 Single-point SCF calculations, 131–132 Slater determinant, 25–26 Solvated electrons (s.e.), 381–382 Source function (SF) approach electron density (ED), 270–272 (see also electron density (ED)) ESD, 271–273 (see also Electron spin density (ESD)) MEP, 272–273 (see also Molecular electrostatic potential (MEP)) Poisson equation, 270 PSFRD/PSFRP, 291–293, 292f QCT methods, 269 QTAIM, 269, 271, 273 scalar fields, 269, 273 values, 273–274

Index

Source Function Local Aromaticity Index (SFLAI), 313–315 Species, 74 Spectroscopic phenomenon, 515 Spherical polar coordinate system, 408 Spin-orbit coupling, 258–259 Spin polarization of atomic valence shell atomic graph, 391–393, 392f, 393–394t atomic polarization vs. atomic graph, 401–403 atomic spin graphs, 394–396 catastrophe process, 394–396 classification, 396f copper (II) quadrupole moment, 402f descriptors, 389–390, 402t energy polarization, 402f excited states, atomic graphs, 398–401, 400t, 401f Laplacian of electron density, 390, 390f, 395f, 397f Laplacian of the attractive potential energy, 404f metal valence shell, metal-ligand interaction, 397, 398t molecular orbital approach, 389–390 photo-induced CIC flattening, 399f QCT, 389–390 VSEPR, 391 Spin-resolved Ni sectors, 219 SR-ZORA approximation, 119–123 S-S bond formation, 41–43 Stagnation lines (SL), 340 Standard optimization methods, 33 Stark-Einstein law, 516 Static polarization, 536–537 Steepest ascent path, 38–40 Steiner-Kato-cusp condition, 382 Steric repulsion, 506–507 Stress tensor, 126–128, 130f, 532–538 Cauchy’s first law of motion, 227 hydrogen atom, 228, 228f kinetic energy, 228 quantum system, 227–228 Structural evolution bifurcation and conflict mechanisms, 58–59f catastrophe mechanisms, 57 catastrophe points, 57 conflict geometry, 58 definition, 57 internal reaction coordinate (IRC), 58 ring structure, 58 Structural preferences, 124–125 S vs. R stereoisomers, 113 Symmetric planar ethene molecule, 132–134 Symmetric-stretch, 158 Symmetric wave functions, 437f Systematic variation, 25 System’s electron density, 38

555

Index

T Tautomeric equilibrium, 458–459 Tetrachlorohydroquinone, 508 Tetrachloroquinone, 508 “Theory of everything,”, 73–74 Thom’s Catastrophe theory, 69 Topological structural transformation, 85–86 Torsional coordinate (TC), 533 Total charge-density distribution, 111–112 Trace of the vorticity of the current density (TVCD), 347–348, 348f, 351–352 Traditional magnetic field effects, 157 Transition probabilities, 524–525 Transition state (TS), 66 Transition state inward (TSIC), 129–130 Transition state outward (TSOC), 129–130 Triangulated GBs construction, 361, 362f Triple-zeta with polarization (TZP), 417 Truly exotic species, 81–82 TS. See Transition state (TS) TSIC. See Transition state inward (TSIC) TSIC and TSOC ring-opening reactions, 158 TSOC. See Transition state outward (TSOC) Tunneling pathways, 158–159 TVCD. See Trace of the vorticity of the current density (TVCD) Two-body interaction potentials, 79–81 Two-electron bond, 86–88 Two-electron operators, 88–89 Two-particle operators, 91–93 Tying down bonds, 427 Type-a heteronuclear diatomic systems, 380–381

U UHF. See Unrestricted Hartree-Fock (UHF) Uniform electron gas (UEG), 493 Universal “indices of clusterization,”, 96 Unrestricted Hartree-Fock (UHF), 315–319 Uσ-space stress tensor trajectory Tσ(s) applications bonding environments and structural preferences eigenvector projection space, 155 normal modes of benzene, 155 isoenergetic phenomena chirality-helicity equivalence, 156–157

flip rearrangements, 158–159 intramolecular mode coupling, 158 QTAIM vs. stress tensor, 158 torquoselectivity, 158 molecular devices C-O ring-opening photo-reactions, 155 fatigue and photochromism, 155 light-driven rotary molecular motor, 156 quinone-based switches, 156

V Valence shell (VS), 391 Valence shell charge concentration (VSCC), 62–63 Valence-shell electron-pair repulsion (VSEPR), 62, 391 van der Waal interactions, 499 Variance of a property, 90 vdW interaction. See van der Waal interactions Vector coordinate, 22 Vectorization, 38–40 Vibrational model, 79–81 Vibrational relaxation (VR), 517 Virial theorem, 376, 385 Vorticity of J(r) (5J(r)), 345–346, 346f VSEPR. See Valence-shell electron-pair repulsion (VSEPR) VSEPR theory, 362, 371

W

Water triplet 3B1 state, 315–320, 316f Wave-behavior approach, 519–520 Wave function, 21 Weak bonds, 469

X X-ray wavelength, 82–83

Z Zero-flux surfaces (ZFSs), 359–360, 364, 365f, 366, 371–372, 428 Zeropoint energy (ZPE), 472–473 Zeroth-Order Regular Approximation (ZORA), 255–258 ZFSs. See Zero-flux surfaces (ZFSs) ZORA. See Zeroth-Order Regular Approximation (ZORA)