Accurate Structure Determination of Free Molecules [1st ed.] 9783030604912, 9783030604929

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Lecture Notes in Chemistry 105

Jean Demaison Natalja Vogt

Accurate Structure Determination of Free Molecules

Lecture Notes in Chemistry Volume 105

Series Editors Barry Carpenter, Cardiff, UK Paola Ceroni, Bologna, Italy Katharina Landfester, Mainz, Germany Jerzy Leszczynski, Jackson, USA Tien-Yau Luh, Taipei, Taiwan Eva Perlt, Bonn, Germany Nicolas C. Polfer, Gainesville, USA Reiner Salzer, Dresden, Germany

The series Lecture Notes in Chemistry (LNC), reports new developments in chemistry and molecular science - quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge for teaching and training purposes. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research. They will serve the following purposes: • provide an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas, • provide a source of advanced teaching material for specialized seminars, courses and schools, and • be readily accessible in print and online. The series covers all established fields of chemistry such as analytical chemistry, organic chemistry, inorganic chemistry, physical chemistry including electrochemistry, theoretical and computational chemistry, industrial chemistry, and catalysis. It is also a particularly suitable forum for volumes addressing the interfaces of chemistry with other disciplines, such as biology, medicine, physics, engineering, materials science including polymer and nanoscience, or earth and environmental science. Both authored and edited volumes will be considered for publication. Edited volumes should however consist of a very limited number of contributions only. Proceedings will not be considered for LNC. The year 2010 marks the relaunch of LNC.

More information about this series at http://www.springer.com/series/632

Jean Demaison Natalja Vogt •

Accurate Structure Determination of Free Molecules

123

Jean Demaison Section of Chemical Information Systems University of Ulm Ulm, Germany

Natalja Vogt Section of Chemical Information Systems Institute of Theoretical Chemistry University of Ulm Ulm, Germany Chemistry Department Lomonosov Moscow State University Moscow, Russia

ISSN 0342-4901 ISSN 2192-6603 (electronic) Lecture Notes in Chemistry ISBN 978-3-030-60491-2 ISBN 978-3-030-60492-9 (eBook) https://doi.org/10.1007/978-3-030-60492-9 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to Barbara Mez-Starck

Foreword

Loosely speaking the structure of molecules is determined by the forces between the atoms. In practice, the potential energy surface of a polyatomic molecule is calculated by ordering the atomic nuclei in the field created by the electrons. This basic concept is known as the Born–Oppenheimer approximation applied to molecular physics. As a result, the molecular structure is related to the minimum energy configuration (including zero-point motion) in this landscape which describes the chemical bonds. Therefore, molecular structure and chemical bond are related to each other in an intimate way where these two aspects are the two sides of the same coin. Thus, knowing a precise molecular structure warrants a detailed understanding of the chemical bonds. From such a chemical view point, there is also a strong relation between structure and reactivity. As such there is a fundamental need to determine accurate molecular structures as this is the basis for our understanding of chemistry as a whole which is the basis of the world around us. Binding energies in molecules range over many orders of magnitude between strong covalent bonds and the weak interactions in complexes which are bound only by van der Waals forces. These variations lead to dramatic changes in bond distances in particular and the molecular structure in general. But also in more ordinary molecules, a large variety of different isomers, i.e., different connectivities of atoms within a molecule, are realized in nature. One prominent example of isomers detected in the interstellar medium, the space in-between stars, concerns the common molecule acetic acid (CH3COOH) with its isomers methyl formate (HCOOCH3) and glycolaldehyde (HCOCH2OH). The properties of these molecules are vastly different. The chemical richness becomes even larger when considering species of the same chemical connectivity but with different orientations of chemical subgroups as for example the anti and syn conformers of methyl formate. These examples illustrate the relation between chemical variation and structural richness. It also demonstrates the need for accurate structures of a large variety of species. Moreover, it shows the need to account for all effects (forces) to arrive at reliable molecular structures. This chemical richness explains why the structure determination of free molecules described in this book is still a very active field of research even though a tremendous body of data is already available in databases. vii

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Another reason lies in the difficulty to derive accurate structures from ab initio calculations. One might be tempted to state that today’s computing power should be sufficient to calculate the structure of any relevant molecule. However, experimentally derived parameters such as the rotational constants are determined with much higher precision than any possible calculation even for much smaller molecules than mentioned above. Likewise, also perturbative corrections need to be accounted for in state-of-the-art descriptions of molecules. It is the comprehensive task of this book to bridge the gap between the accuracy of experimental work and the versatility as well as speed of theoretical work. Both descriptions are very detailed and elaborative. The thorough discussion of all the necessary steps to compare results from both sides will make this book valuable to interested readers coming from either side of this approach. In this endeavour to establish a bridge between these two shores, a comprehensive and coherent description of receiving the accurate structure data of free molecules is achieved. Stephan Schlemmer I. Physikalisches Institut Universität zu Köln Cologne, Germany

Preface

In the frame of the Born–Oppenheimer approximation, the equilibrium structure of a molecule corresponds to the minimum of the potential hypersurface. Thus, it has a clear physical definition, and it is the only structure permitting the comparison between molecules. This is quite important because the molecular geometry is required to explain molecular behavior (reactivity, electric and magnetic properties, etc.). As described in the introduction, it was possible since the beginning of the twentieth century to determine the equilibrium structure of a molecule. However, it was a difficult and time-consuming task. Furthermore, it was limited to very small molecules (typically up to three independent parameters; examples are OCS and SO2), and the result was not always accurate (a nice illustration is HCO+ discussed in Chap. 6). For these reasons, empirical methods trying to approximate the equilibrium structure have been developed. At first sight, they are appealing and they are still extensively employed. However, although they often give satisfactory results for very small molecules, they rapidly become unreliable for medium-sized molecules. Starting from the end of the seventies, a lot of progress has been made. First, from the experimental point of view, the introduction of Fourier transform technique and computer-controlled experiments permitted to record spectra much faster and with a better sensitivity, in particular making possible the measurement of some important isotopologues (13C) in natural abundance. But the most important advancement was the considerable increase of power of the computers, and the development of ab initio methods permitting since the nineties to optimize a molecular geometry with an accuracy sometimes better than the experimental one (typically, a few thousandth of an Å). Another essential application of ab initio methods is the calculation of anharmonic force fields, which allow us to correct experimental internuclear distances determined by gas-phase electron diffraction and/or high-resolution spectroscopy and which is the basis of the semiexperimental procedure. This last technique is one of the most powerful methods to determine an accurate equilibrium structure. Coupled with the mixed regression, it has recently enabled the determination of accurate equilibrium

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structures of rather large molecules (up to 66 independent structural parameters) from spectroscopic data. The principal aim of this book is to bridge the gap between the different methods, and the chapters cover the different aspects of structure determination of an isolated molecule: quantum chemistry, spectroscopy, and gas-phase electron diffraction. Particular emphasis is put on the recent developments and on the ways to estimate the accuracy. This book is aimed at advanced undergraduate or graduate students and at research workers in all these fields. The authors are grateful to Jacques Liévin for reading Chap. 2, to Alberto Lesarri for reading Chaps. 4 and 6, to Agnès Perrin for reading Chap. 5, to Norman Craig for reading Chap. 8, and to Jürgen Vogt for reading Chap. 10. Ulm, Germany Ulm, Germany/Moscow, Russia

Jean Demaison Natalja Vogt

Acknowledgements

We are very grateful to the staff of the Springer Nature editorial office: Editors Dr. Zachary Evenson and Dr. Adelheid Duhm for their thoughtful guidance and senior editorial assistant Mrs. Elke Sauer for her permanent support of this book project, as well as to Ms. Banu Dhayalan and Priyadharshini Subramani from the book production service for their efforts in the preparation of this book. Ulm, Germany Ulm, Germany/Moscow, Russia

Jean Demaison Natalja Vogt

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 2.3 Born–Oppenheimer Approximation and Electronic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Hartee–Fock (HF) Method . . . . . . . . . . . . . . . . . . . . . 2.5 Post Hartree–Fock Methods . . . . . . . . . . . . . . . . . . . . 2.6 Choice of the Method . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Choice of the Basis Sets . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Description of the Basis Sets . . . . . . . . . . . . . 2.7.2 Core Correlation . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Diffuse Functions . . . . . . . . . . . . . . . . . . . . . 2.7.4 Convergence of the Basis Set . . . . . . . . . . . . 2.8 Relativistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Correction to the Born–Oppenheimer Approximation . 2.10 Born–Oppenheimer Equilibrium Structure, reBO . . . . . . 2.11 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Lower Level Ab Initio Methods . . . . . . . . . . . . . . . . . 2.13 Density Functional Theory (DFT) . . . . . . . . . . . . . . . 2.13.1 Local Density Approximation (LDA) . . . . . . . 2.13.2 Generalized Gradient Approximation (GGA) . 2.13.3 Meta-GGA . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.4 Hybrid DFT . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.5 Double-Hybrid Density Functionals (DHDFT) 2.14 Calculation of the Force Field . . . . . . . . . . . . . . . . . . 2.15 Semi-empirical Methods . . . . . . . . . . . . . . . . . . . . . .

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2.16 Molecular Mechanics (MM) . . . . . . . . . . . . . . . . . . . . 2.16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.2 Calculation of the Energy . . . . . . . . . . . . . . . . 2.16.3 Accuracy and Limitations . . . . . . . . . . . . . . . . 2.17 Combined Quantum/Classical (QM/MM) Methods . . . . 2.18 Quantum Theory of Atoms in Molecules (QTAIM or AIM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1 Ratio of the Magnitudes of the Gravitational and Electrostatic Forces Between a Proton and an Electron . . . . . . . . . . . . . . . . . . . . . . . 2.19.2 Nuclear Size . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.3 The Product of Two Gaussians is Another Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.4 Relativistic Correction Due to the Dependence of the Electron Mass on Velocity . . . . . . . . . . 2.19.5 Lennard-Jones Potential . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Vibrational Energy . . . . . . . . . . . . . . . . . . 3.2.1 Harmonic Oscillator . . . . . . . . . . . . . . 3.2.2 Anharmonic Oscillator . . . . . . . . . . . . 3.3 The Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . 3.4 Vibrating Rotor . . . . . . . . . . . . . . . . . . . . . . . 3.5 Centrifugal Distortion . . . . . . . . . . . . . . . . . . . 3.6 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Electronic Correction . . . . . . . . . . . . . . . . . . . 3.8 Definition of the Different Structures . . . . . . . . 3.8.1 Experimental Equilibrium Structure, re 3.8.2 Effective Structure, r0 . . . . . . . . . . . . . 3.8.3 Substitution Structure, rs . . . . . . . . . . . 3.8.4 Zero-Point Structure rz (or ra0 ) . . . . . . . 3.9 Higher-Order Effects . . . . . . . . . . . . . . . . . . . . 3.9.1 Dunham Expansion . . . . . . . . . . . . . . 3.9.2 Breakdown of the Born–Oppenheimer Approximation . . . . . . . . . . . . . . . . . . 3.9.3 Effect of the Size of the Nuclei . . . . . . 3.10 Direct Potential Fit (DPF) . . . . . . . . . . . . . . . . 3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rotation of the Polyatomic Molecule . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Classical Kinetic Energy of the Rigid Rotor . . . . . . . . . . . . . 4.3 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Rotational Hamiltonian of the Rigid Rotor . . . . . . . . . . . . . . 4.5 Symmetric Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Rigid Rotor Approximation . . . . . . . . . . . . . . . . . . . 4.5.2 Centrifugal Distortion . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Determination of the Axial Rotational Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Linear Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Rotational Spectra of Linear and Symmetric Tops in Excited Vibrational States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Asymmetric Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Rigid Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Centrifugal Distortion . . . . . . . . . . . . . . . . . . . . . . . 4.9 Rovibrational Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Electronic Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Determination of the Rotational Constants . . . . . . . . . . . . . . 4.12 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12.1 Frequency-Domain Microwave Spectroscopy . . . . . . 4.12.2 Time-Domain Microwave Spectroscopy . . . . . . . . . . 4.12.3 Millimeterwave and Submillimeterwave Spectroscopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Vibrations of Polyatomic Molecules . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Classical Kinetic Energy of the Rigid Rotor . . . 5.3 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Molecular Symmetry . . . . . . . . . . . . . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . 5.4.2 Subgroup G1 . . . . . . . . . . . . . . . . . . . 5.4.3 Operations of Second Kind . . . . . . . . . 5.4.4 Molecular Point Groups . . . . . . . . . . . 5.4.5 Classification of the Vibrational Modes 5.5 Coriolis Interaction . . . . . . . . . . . . . . . . . . . . . 5.6 Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Internal Coordinates . . . . . . . . . . . . . . . . . . . . 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Equilibrium Structures from Spectroscopy . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Vibrational Dependence of the Rotational Constants . . . . 6.3 Determination of the Rotational Constants . . . . . . . . . . . 6.3.1 Ground-State Constants . . . . . . . . . . . . . . . . . . 6.3.2 Rotational Constants in a Vibrationally Excited State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Empirical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Effective Structures (r0) . . . . . . . . . . . . . . . . . . 6.4.3 Substitution-Like Structures . . . . . . . . . . . . . . . 6.5 Mass-Dependent Structures . . . . . . . . . . . . . . . . . . . . . . 6.6 Experimental Equilibrium Structure . . . . . . . . . . . . . . . . 6.7 Semiexperimental Equilibrium (se) Structure . . . . . . . . . 6.8 Extrapolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Structure of Weakly Bound Complexes . . . . . . . . . . . . . 6.9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.2 Classification of Non-covalent Bonds . . . . . . . . 6.9.3 Rotational Constants and Structure . . . . . . . . . . 6.9.4 Distance Between the Monomers, Pseudo-Diatomic Approximation . . . . . . . . . . . . 6.9.5 Use of the Quadrupole Coupling Constants . . . . 6.10 Accuracy of Equilibrium Structures . . . . . . . . . . . . . . . . 6.11 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Calculation of the Anharmonic Force Field and of the Rovibrational Correction . . . . . . . . . . 6.11.2 Fit of the Internal Coordinates to the Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13.1 Kraitchman’s Equations for an Asymmetric Top Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13.2 Chutjian’s Equations for Substitution of Two Equivalent Atomsa . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Molecular Structures from Gas-Phase Electron Diffraction . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Short Description of a Conventional Gas-Phase Electron Diffraction Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Main Theoretical Expressions . . . . . . . . . . . . . . . . . . . . . . 7.5 Significance of Structural Parameters (ra, rg, and re) . . . . . .

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Methodology of Equilibrium Structure Determination . . . . . Determination of Molecular Structure in Terms of Potential Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Harmonic Approximation . . . . . . . . . . . . . . . . . . . 7.7.2 Anharmonic Approximation . . . . . . . . . . . . . . . . . 7.7.3 Cumulant–Moment Method . . . . . . . . . . . . . . . . . . 7.8 The Use of Curvilinear Internal Coordinates . . . . . . . . . . . . 7.9 Large-Amplitude Motions . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Combined Analysis of Data from Different Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11 Accuracy of Structure Determinations . . . . . . . . . . . . . . . . 7.12 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.13 The Way to Estimate Semiexperimental Equilibrium Structures (rese ) for a Large Number of Molecules . . . . . . . . 7.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Models of Chemical Bonding and “Empirical” Methods . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bond Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Covalent Radii . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Polarity of the Bond . . . . . . . . . . . . . . . . . . . . . 8.2.3 Bond Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Van der Waals Radii . . . . . . . . . . . . . . . . . . . . 8.3 Valence-Shell Electron-Pair Repulsion (VSEPR) Model . 8.4 Ligand Close-Packing (LCP) Model . . . . . . . . . . . . . . . . 8.5 Empirical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Relationship Between Stretching Force Constant and Bond Length . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Dihedral Angles . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Other Correlations . . . . . . . . . . . . . . . . . . . . . . 8.6 Appendix: Electronegativity (v) . . . . . . . . . . . . . . . . . . . 8.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Pauling Scale . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Allred-Rochow Scale . . . . . . . . . . . . . . . . . . . . 8.6.4 Other Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.5 Group Electronegativity . . . . . . . . . . . . . . . . . . 8.6.6 Electronegativity Equalization . . . . . . . . . . . . . . 8.6.7 Partial Atomic Charge . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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179 179 180 182 184 188

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Least-Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Nonlinear Least-Squares Method . . . . . . . . . . . . . . . . 9.2.1 Principle of the Method . . . . . . . . . . . . . . . . 9.2.2 Practical Solution . . . . . . . . . . . . . . . . . . . . . 9.3 Weighted Least Squares Method . . . . . . . . . . . . . . . . 9.3.1 Choice of the Weights . . . . . . . . . . . . . . . . . 9.3.2 Practical Solution . . . . . . . . . . . . . . . . . . . . . 9.4 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Condition Number . . . . . . . . . . . . . . . . . . . . 9.4.2 Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 “Jackknifed” Residual . . . . . . . . . . . . . . . . . . 9.5 Iteratively Reweighted Least-Square Method . . . . . . . 9.6 Effect of Fixed Parameters . . . . . . . . . . . . . . . . . . . . . 9.7 Mixed Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Correlated Least Squares . . . . . . . . . . . . . . . . . . . . . . 9.9 Merged Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10.1 Autocorrelation of the Errors . . . . . . . . . . . . . 9.10.2 Diagnostics of Autocorrelation . . . . . . . . . . . 9.10.3 Effect of the Autocorrelation on the Value of the Parameters . . . . . . . . . . . . . . . . . . . . . 9.10.4 Effect of the Autocorrelation on the Standard Deviation of the Parameters . . . . . . . . . . . . . 9.11 Accuracy of the Structural Parameters . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Databases with Information on Molecular Structure 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The MOGADOC Database . . . . . . . . . . . . . . . . 10.3 SpringerMaterials . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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265 265 266 269 271

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Chapter 1

Introduction

Abstract The introduction is first a summary of the history of the field of molecular structure determination. Then, the goal of the book is explained and an abstract of the different chapters is given.

The structure of a molecule (molecular geometry, electron density distribution, flexibility) contains the features responsible for its physical and chemical properties. In other words, molecular geometry is required to explain molecular behavior. This is particularly important in biochemistry. There is indeed considerable evidence that odors, enzymatic activity, etc., are all subject to stereochemical control. Molecule (from new latin molecula = small mass) is the smallest unit, consisting of groups of atoms, into which a substance can be divided without a change in its chemical structure (Oxford dictionary). Although the concept of molecular structure was developed during the nineteenth century, Franklin (1774) devised an experiment permitting to determine the size of the oleic acid molecule using what would be later called a Langmuir monolayer, i.e., a one-molecule thick layer of oil spread onto a surface of water. In 1808, John Dalton’s experiments showed that matter consisted of elementary particles, or atoms. These atoms combine in fixed relative proportions to form molecules (Proust 1806). The first attempt to explain the structure of the molecules is the electrostatic theory of Davy (1812) around 1806 and later Berzelius (1814). Group of atoms are bound together by electrostatic attraction. In 1834, Dumas (1837) has shown that this theory is not applicable to organic molecules. But it was Avogadro (1811), who made the distinction between molecule and atom. In 1858, Couper (1858) and, then, Kekulé (1865) were the first to state that carbon is tetravalent and to form a concrete idea of molecular structure. However, Butlerov (1861) was the first to use the term “chemical structure” and he restated, clarified, and enlarged upon the ideas of Couper and Kekulé. von Bayer (1885) published his theory of ring strain whose main conclusion is that bond angles may deviate from the tetrahedral value of 109° 28 . In 1874, the concept of tetrahedral carbon was simultaneously introduced by Le Bel (1874) and Van’t Hoff (1874). In 1873, van der Waals (1873) proposed his famous equation describing the behavior of real © Springer Nature Switzerland AG 2020 J. Demaison and N. Vogt, Accurate Structure Determination of Free Molecules, Lecture Notes in Chemistry 105, https://doi.org/10.1007/978-3-030-60492-9_1

1

2

1 Introduction

gas and first published in his dissertation. It was used to determine the radius of molecules (van der Waals radius for atoms). However, the existence of molecules was contested by reputed scientists such as Ernst Mach and Marcellin Berthelot until the experimental work of Perrin (1913) on the Brownian motion. The golden age of molecular structure is the first half of the twentieth century with the discovery of X-rays by Röntgen in (1895) and the diffraction of X-rays by crystals by Friedrich et al. (1912). This permitted, for the first time, to determine interatomic distances in crystals. The structure of diamond was obtained in 1913 by Bragg father and son (1913). Debye (1929) was the first to determine the experimental structure of a molecule (CCl4 ) in gas phase by X-ray diffraction. At about the same time, around 1930, the first gas-phase electron diffraction experiment was performed by Mark and Wierl (1930). Although the accuracy was not high, many molecules were studied in a short time, and the results were used to develop the concept of chemical bond by Pauling (1931) (see also Pauling 1960) and others. Since that time, a lot of progress has been made, both experimentally and theoretically, and nowadays, gasphase electron diffraction is one of the best methods to determine the structure of a wide range of molecules (both small and relatively large, both polar and non-polar, both in ground and excited electronic states, etc.). Starting from 1945, neutron diffraction experiments were also carried out. Thermal neutrons (i.e., with a wavelength of about 0.1 nm) are used. The big inconvenient of this method is that it requires a nuclear reactor and is therefore rarely used. On the other hand, its advantage is that the neutrons are diffracted by the nuclei contrary to the other methods. Therefore, light atoms as hydrogen may contribute strongly to the diffracted intensity (Shull 1995). In the diffraction methods, the elastic scattering, i.e., occurring without exchange of energy between the radiation and the molecules, is used (see Sect. 7.4 for electron diffraction). When a molecule absorbs a photon of frequency ν, the molecule goes from energy level E 0 to level E 1 and Planck’s law gives E 1 −E 0 = hν, i.e., absorption or emission of photons is possible for some well-defined wavelengths. It is usual to classify spectra according to the type of molecular energy that is modified by the process: • Electronic spectra: The electric field of the radiation enters in interaction with the dipole moment of the moving electrons and the fixed nuclei. The transitions are found in the visible and ultraviolet ranges. • Vibrational spectra: The electric field of the radiation enters in interaction with the oscillating dipole moment formed from the center of gravity of the nuclei and the center of gravity of the electrons. In other words, the vibration must modify the electric dipole moment. The transitions are observed in the infrared range.

1 Introduction

3

• Rotational spectra: The electric field of the radiation enters in interaction with the rotating permanent dipole moment of the molecule. The transitions are in the microwave (or centimeter wave) and millimeter wave ranges. To a good approximation, it is possible to assume that the energy of the molecule is the sum of independent electronic (E e ), vibrational (E V ), and rotational (E R ) energies. In other words, the Hamiltonian of the molecule may be written H = He + HV + HR We will see that the separation of the electronic Hamiltonian, He , is an extremely good approximation which, furthermore, considerably simplifies the problem. On the other hand, the separation of the vibrational and rotational motions is only approximate, and it will be necessary to take into account their interaction. After the Second World War appeared nuclear magnetic resonance (NMR) (Abragam 1961) and microwave spectroscopy (Townes and Schawlow 1955). NMR spectroscopy is widely used to identify molecules. Although it provides information about the structure, it generally does not deliver geometrical parameters. Nevertheless, there are two exceptions worth to be noted: (i) the NMR spectrum of a molecule dissolved in a nematic solvent gives relations between molecular parameters. However, solvent effects limit the accuracy of this method (Diehl 1992), and (ii) in saturated X–C–C–Y units, the spin–spin coupling constant over three bonds 3 J XY depends primarily on the dihedral angle τ( X–C–C–Y) which can thus be obtained using the empirical Karplus equation (1959). Microwave spectroscopy is generally considered to be the most precise technique for obtaining molecular geometries in gas phase. This spectroscopy (as well as high-resolution infrared spectroscopy) now determines rotational constants with a precision close to 1 in 108 . If there were a simple relation between experimental rotational constants and equilibrium geometry, it would be possible to determine the structure of molecules with a tremendous accuracy. However, we will see that many factors limit this accuracy. In the early days, the resolution of infrared spectra did not permit to observe the rotational fine structure but, starting from the sixties, the advent of Fourier transform and laser spectroscopies allowed the determination of accurate rotational constants. This is particularly useful for molecules with no dipole moment or with a very small dipole moment. More recently, synchrotron radiation was used to measure rotational spectra in the submillimeter wave range with a good sensitivity. In quantum chemistry, the first successful attempt to calculate the structure of a molecule was by Burrau (1927) on H2 + . It was followed, the same year, by the calculation of the bond length in H2 by Heitler and London (1927). The apparition of the computer and its considerable increase in power are also extremely important because it permitted the development of ab initio methods, which can now provide accurate estimates of equilibrium structures. In many cases, high-level ab initio calculations deliver structures faster than the experimental methods and, often, with a better accuracy. Ab initio methods permit also to calculate the anharmonic force field, which

4

1 Introduction

is then used to obtain semiexperimental equilibrium structures both by gas-phase electron diffraction and by spectroscopy or by combined use of these methods. There are many reviews devoted to structure determination. However, most of the time, they limit themselves either to a single experimental technique or to ab initio calculations. There are very few papers where a critical comparison of the experimental and ab initio techniques is made and where the interplay of these methods is emphasized. Furthermore, although the techniques are described in great detail, there is no thorough discussion of the accuracy that can be really achieved. The first question to answer is which accuracy is desirable. There are three reasons militating in favor of a high accuracy: • Theoreticians need accurate structures to check their calculations. Although the accuracy of ab initio calculations varies wildly, it is possible to define a range from 0.3 pm1 to better than 0.1 pm. For instance, Ruden et al. (2004) computed the structure of CO and, using higher-order corrections to the usual coupled cluster method, they obtained 112.84 pm in fair agreement with the experimental value, r e = 112.8230(1) pm (Authier 1993). It demonstrates that the progress of ab initio methods remains dependent of experimental results. • Inspection of the range of a few bond lengths shows that it is rather small. For instance, it is only 6 pm for the CH and NH bonds. Thus, to compare the structures of different molecules, an accuracy significantly better than 1 pm is required. • The energy of a molecule is sensitive to its structure. Molecular mechanics programs used to calculate the properties of large molecules are parameterized against a small number of small molecules whose structure is assumed to be accurate. For instance, the distortion of a C–C single bond by 2 pm “costs” 0.6 kJ mol−1 ; the distortion of a ∠(CCC) bond angle by 2°, about 0.4 kJ mol−1 ; and the torsional distortion of a CCCC chain by 5°, about 0.2 kJ mol−1 (Hargittai and Levy 1999). In order to be able to determine the relative energy of a molecule with an acceptable accuracy (a few kJ mol–1 ), it is necessary to scale the molecular mechanics programs (Burkert and Allinger 1982) with molecules whose molecular geometry is very accurately known. The next step is to see at which condition such an accuracy can be achieved. For this goal, we will first analyze the approximations, which are made during the derivation of the molecular Hamiltonian. It will be one of the goals of Chap. 2. It deals with the concepts of potential energy surfaces and equilibrium molecular structures. It also describes quantum chemical computations of structures and force fields. Chapter 3 is devoted to diatomic molecules. As they are much simpler, a more sophisticated theory may be used and the bond length is determinable with a much higher accuracy. Furthermore, it is a good introduction to the more complicated case of polyatomic molecules. Chapter 4 is a summary of rotational spectroscopy and the determination of rotational constants from experimental spectra, and Chap. 5 is a short introduction to

1 Along

the pm unit, the Å unit is also used in this book for practical reason (1 Å = 100 pm).

1 Introduction

5

vibrational spectroscopy. It is limited to the introduction of the notions permitting to understand how the vibrations affect the structure determination. Chapter 6 explains the different ways to determine the structure of a molecule by spectroscopy. Chapter 7 is dedicated to gas-phase electron diffraction. This method in combination with spectroscopy and/or ab initio calculations is a powerful technique for determination of molecular structure. Chapter 8 describes some “empirical” methods useful to obtain information on the structure of a molecule. In particular, the valence-shell electron-pair repulsion (VSEPR) model and ligand close-packing (LCP) model are discussed. Chapter 9 deals with the least-squares method. The dangers of ill-conditioning, outliers, and leverage points are discussed in detail. The iteratively least-squares method and the mixed regression are proposed as solutions. The accuracy of the parameters is also examined. Chapter 10 presents the established databases containing information about molecular structures of free molecules. Determining an accurate molecular structure is time consuming and expensive. Furthermore, it requires a sophisticated equipment. For these reasons, it is useful to have at his disposal easily accessible databases.

References Abragam A (1961) The principles of nuclear magnetism. Clarendon Press Authier N, Bagland N, Lefloch A (1993) The 1992 evaluation of mass-independent Dunham parameters for the ground state of CO. J Mol Spectrosc 160:590–592 Avogadro A (1811) Essai d’une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons. Journal de Physique, de Chimie, D’histoire Naturelle et des Arts. 73:58–76 Berzelius JJ (1814) An attempt to establish a pure scientific system of mineralogy, by the application of the electro-chemical theory and chemical proportions. John Black, London, p 1814 Bragg WH, Bragg WL (1913) The structure of diamond. Proc Roy Soc London Ser A 89:277–291 Burkert U, Allinger NL (1982) Molecular mechanics. American Chemical Society, Washington, D.C. Burrau Ø (1927) Berechnung des Energiewertes des Wasserstoffmolekül-ions (H2 + ) im normalzustand. Kgl Danske Vid Selsk Math-Fys Medd 7(14) Butlerov AM (1861) Einiges über die chemische Struktur der Körper. Zeitschrift Für Chemie 4:549– 560 Couper AS (1858) Sur une nouvelle théorie chimique. Annales de Chimie et de Physique, Série 3(53):469–489 Dalton J (1808) A new system of chemical philosophy. Bickerstaff, Manchester Davy H (1812) Elements of chemical philosophy. Bradford and Inskeep, Philadelphia Debye P, Bewilogua L, Ehrhardt P (1929) Zerstreuung von Röntgenstrahlen an einzelnen Molekeln. Phys Z 30:84–87 Diehl P (1992) Nuclear magnetic resonance spectroscopy and precise molecular geometries. In: Domenicano A, Hargittai I (eds) Accurate molecular structures: their determination and importance. Oxford University Press, pp 299–321 Dumas J-B (1837) Leçons sur la philosophie chimique professées au collège de France en 1836, 2nd edn. Bechet Jeune, Paris, 1837. Gauthier-Villars, Paris

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Franklin B, Brownrigg W, Farish MR (1774) Of the filling of waves by means of oil. Phil Trans R Soc London 64:445–460 Hargittai I, Levy JB (1999) Accessible geometrical changes. Struct Chem 10:387–389 Heitler W, London F (1927) Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z Physik 44:455–472 Karplus M (1959) Contact electron-spin coupling of nuclear magnetic moment. J Chem Phys 30:11–15 Kekulé FA (1865) Sur la constitution des substances aromatiques. Bull Soc Chim 3:98–110 Le Bel AJ (1874) Sur les relations qui existent entre les formules atomiques des corps organiques et le pouvoir rotatoire de leurs dissolutions. Bull Soc Chim Paris 22:337–347 Mark H, Wierl R (1930) Über Elektronenbeugung am einzelnen Molekül. Naturwissenschaften 18:205–205 Pauling L (1931) The nature of the chemical bond. Application of results obtained fom the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J Am Chem Soc 53:1367–1400 Pauling L (1960) The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry. Cornell University Press, Ithaca, NY Perrin J (1913) Les Atomes. Félix Alcan, Paris Proust L-J (1806) Mémoire sur le sucre de raisin. Ann chim. 57:13 Röntgen W (1895) Ueber eine neue Art von Strahlen. Vorläufige Mitteilung, in: Aus den Sitzungsberichten der Würzburger Physik.-medic. Gesellschaft Würzburg, pp 137–147 Ruden TA, Helgaker T, Jørgensen P, Olsen J (2004) Coupled-cluster connected quadruples and quintuples corrections to the harmonic vibrational frequencies and equilibrium bond distances of HF, N2, F2, and CO. J Chem Phys 121:5874–5884 Shull C (1995) Early development of neutron scattering. Rev Mod Phys 67:753–757 Townes CH, Schawlow AL (1955) Microwave spectroscopy. McGraw-Hill, New York Van der Waals JD (1873) Over de Continuiteit van den Gas- en Vloeistoftoestand (on the continuity of the gas and liquid state). PhD thesis, Leiden von Bayer A (1885) Über Polyacetylenverbindungen. Ber Dtsch Chem Ges 18:2269–2281 von Laue M (1912) Münchener Ber., 303. In: Friedrich W, Knipping P, von Laue M (eds) InterferenzErscheinungen bei Röntgenstrahlen. Sitzungsberichte der Mathematisch-Physikalischen Classe der Königlich-Bayerischen Akademie der Wissenschaften zu München, pp 303–322 Van’t Hoff JH (1874) Sur les formules de structure dans l’espace. Archives Néerlandaises des Sciences Exactes et Naturelles 9:445–454

Chapter 2

Computational Methods

Abstract This chapter describes the different methods used in computational chemistry to determine the structure of a molecule. It starts with the presentation of the molecular Hamiltonian. Then, the different ab initio methods are reviewed in detail. The approximations are discussed, and a particular emphasis is put on their effect on the accuracy. The molecular mechanics methods and the combined quantum/classical methods are also briefly reviewed.

2.1 Introduction This chapter is not intended to be a thorough review of computational chemistry. It is an introduction to the methods that are used to compute a molecular structure. The number of books and reviews on this subject is considerable. A few of them are cited in the text. Some modern sources are: Allen and Császár (2011) and Helgaker et al. (2000). A more general presentation of the computational chemistry is in Schleyer (1998) and Cramer (2004). Although the first quantum mechanical determination of a molecular geometry was made as early as 1927 (see introduction), ab initio methods only became competitive with experimental techniques in the nineties thanks to new developments in computational methods and as a result of spectacular advances in computer technology. At present, their accuracy rivals and even surpasses that of experimental measurements. Furthermore, they are now an indispensable tool for experimentalists as they allow to considerably increase the accuracy of their results and to study much more complicated molecules. It will be the main subject of this book. However, ab initio methods are still time consuming and, moreover, limited to small molecules (perhaps one hundred atoms for the simplest ab initio methods). It would be extremely desirable to have a method that is at the same time fast and applicable to very large molecules such as drugs and proteins. For this reason, the molecular mechanics methods are also described. They use classical (Newtonian) mechanics to predict the energy of a molecule as a function of its conformation allowing among others the prediction of the structure. These © Springer Nature Switzerland AG 2020 J. Demaison and N. Vogt, Accurate Structure Determination of Free Molecules, Lecture Notes in Chemistry 105, https://doi.org/10.1007/978-3-030-60492-9_2

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methods are easy to understand and to apply and are still useful for very large molecules. Their natural extension are the combined quantum/classical (QM/MM) methods where the interesting part of the system is treated quantum mechanically while the effects of the surrounding are taken into account by molecular mechanics.

2.2 Molecular Hamiltonian (Bunker and Jensen 1998) A molecule is a collection of nuclei and electrons held together by certain forces and obeying the laws of quantum mechanics. The dominant forces are the Coulomb electrostatic forces. Gravitational forces are also present but a simple calculation shows that they are completely negligible compared to the electrostatic forces, see Appendix 2.19.1. For open shell molecules or when heavy nuclei are present, the electron spin magnetic moment may interact with other magnetic moments (generated by the orbital motion of the electrons or of the nuclei, or magnetic moments of the other electrons). These interactions are much smaller and will be neglected in order to simplify the presentation. When they are not negligible, they can be taken into account without difficulty.1 What is left is the Coulomb Hamiltonian. To write it, it is assumed that the nuclei are punctual masses and the relativistic effects are negligible. The first approximation is a very good one, see Appendix 2.19.2, and will be further discussed in Sect. 3.9.3. The second approximation is also a good one. However, when heavy nuclei are present, relativistic effects become important and should be taken into account when a structure is calculated ab initio. This approximation will be further discussed in Sect. 2.8. With these approximations, the Hamiltonian of a molecule with N nuclei and n electrons may be written H = TN + (Te + VNe + VNN + Vee ) = TN + He

(2.1)

with  N  N   α  α TN = −2 = − 2Mα 2Mα α=1 α=1  n  n  i 2  i = − Te = − m i=1 2 2 i=1

VNe

1 The

 N n  N n   Zα 1   Z α e2 =− = − 4π ε0 α=1 i=1 riα r α=1 i=1 iα

(2.2a)

(2.2b)

(2.2c)

interactions of the magnetic and electric moments of the nuclei with the other electric and magnetic moments in the molecule may lead to an hyperfine structure of the energy but they do not affect the following discussion.

2.2 Molecular Hamiltonian (Bunker and Jensen 1998)

VNN

⎤ ⎡ N N  N N  Z Z 1   Z α Z β e2 α β ⎦ = =⎣ 4π ε0 α=1 β>α Rαβ R αβ α=1 β>α ⎤ ⎡ n n n  n  1 1   e2 ⎦ Vee = =⎣ 4π ε0 i=1 j>i ri j r i j i=1 j>i

9

(2.2d)

(2.2e)

 is the Laplace operator, M α and Z α are the mass and atomic number of the nucleus α, m is the mass of the electron, r iα is the distance between electron i and nucleus α, and similar definitions hold for r ij and Rαβ . The terms in brackets on the right are in atomic units (au). It is common to use them in order to eliminate the fundamental physical constants from the electronic Hamiltonian. They are chosen such that:  = m = e2 4π ε0 = 1. The atomic unit of length is the Bohr: a0 =

4π ε0 2 = 5.29177210903(80) × 10−11 m me2

The atomic unit of energy is the Hartree, E H , which is the Coulomb repulsion between two electrons separated by 1 Bohr: 2 −18 J. E H = ma 2 = 4.3597447222071(85) × 10 0 The atomic unit of mass is the mass of the electron: me = 9.1093837015(28)× 10−31 kg. The CODATA (Committee on Data for Science and Technology) internationally recommended values of the fundamental constants may be found at: https://physics. nist.gov/cuu/Constants.

2.3 Born–Oppenheimer Approximation and Electronic Hamiltonian (Bunker and Jensen 2000) This Hamiltonian is too complicated to be solved exactly. As first proposed by Born and Oppenheimer (BO), the nuclear kinetic energy TN is first neglected. The justification of the BO approximation is that the heavy nuclei move much more slowly than the light electrons (it also assumes that the momentum of the electrons and the nuclei is of the same order of magnitude). In the remaining electronic Hamiltonian He , the nuclear positions R enter as parameters He ψ(e) = EBO ψ(e)

(2.3)

Varying the position of R in small steps, one obtains Ebo as a function of R. This is the potential energy (hyper)surface (PES): Ebo (R). Its global minimum corresponds to the equilibrium structure of the molecule. As the nuclear masses are absent in He ,

10

2 Computational Methods

E bo is isotopically invariant. In other words, all the isotopologues of a same molecule have the same PES and the same equilibrium structure. It can be shown that the BO approximation can be trusted when the PESs corresponding to the different electronic states are well separated: E 0BO  E 1BO  E 2BO  · · ·

(2.4)

It is generally a good approximation in the vicinity of the equilibrium position, except for a few molecules. For instance, the first two excited electronic states of NO3 are close to the ground state and all three levels interact via vibronic coupling (Deev et al. 2005). The PES and the equilibrium structure can be calculated by ab initio methods. This will be discussed in Sects. 2.6 and 2.10 together with the relativistic effects, Sect. 2.8, and the correction to the BO approximation, Sect. 2.9.

2.4 Hartee–Fock (HF) Method The electronic Hamiltonian is still too complicated to be solved exactly (except in a few special cases). Neglecting Vnn (which is a constant term), it may be written in atomic units He =

n  i=1

hi +

n  n  1 r i=1 j>i i j

(2.5)

The first part is a sum of monoelectronic terms which are easy to solve. The simplest approximation is to assume that each electron moves in the field created by the other electrons. The second term will be approximated by a monoelectronic operator u(r i ) which will take into account the mean repulsion effect of all the other electrons on the electron i He =

n  i=1

[h(ri ) + u(ri )] +



F(ri )

n  n n   1 − u(ri ) r i=1 j>i i j i=1



V

=

n 

F(ri ) + V = H0 + V

(2.6)

i=1

If V is small (it is expected to be much smaller than V ee ), it is a good approximation to replace He by H0 which is monoelectronic and, by application of the variational method, is easily solvable. This is called the Hartree–Fock (HF) method. The total HF energy is obtained by adding VNN to the eigenvalues of H0 .

2.4 Hartee–Fock (HF) Method

11

A monoelectronic wavefunction (called orbital) is attributed to each electron. Molecular orbital (MO) theory assumes that the atomic orbitals of the atoms in the molecule combine to produce molecular orbitals that are delocalized over the entire molecule. Generally, a basis set expansion technique is used. The many-electron wavefunction is written as a product of orthonormal one-electron functions called molecular orbitals (MOs) Ψ = ϕ1 ϕ2 · · · ϕn

(2.7)

However, the electrons being Fermions, this function, called Hartree product, should be antisymmetric upon exchange of two electrons. The simplest antisymmetric function is a determinant which may be written as follows for a closed-shell system, introducing the spin functions of the electron: α for spin +1/2 or β for spin −1/2    ϕ1 (r1 )α(1) ϕ1 (r1 )β(1) ϕ2 (r1 )α(1) ϕ2 (r1 )β(1) · · · ϕn / 2 (r1 )α(1) ϕn / 2 (r1 )β(1)     ϕ1 (r2 )α(2) ϕ1 (r2 )β(2) ϕ2 (r2 )α(2) ϕ2 (r2 )β(2) · · · ϕn / 2 (r2 )α(2) ϕn / 2 (r2 )β(2)    .. ..  1   (r ) = √  . .  n!   . .   .. ..     ϕ1 (rn )α(n) ϕ1 (rn )β(n) ϕ2 (rn )α(n) ϕ2 (rn )β(n) · · · ϕn / 2 (rn )α(n) ϕn / 2 (rn )β(n)

(2.8)

Each of these MOs is expressed as a linear combination of basis functions ϕi =



cik χk

(2.9)

k

The unknown cik coefficients are determined by the variational method, i.e., they are chosen as to minimize the energy (Roothaan 1951)  ∗ Ψ HΨ dτ E = min  ∗ ≥ E exact Ψ Ψ dτ

(2.10)

It is tempting to use Slater-type atomic orbitals (STOs) as basis functions because they provide a reasonable representation of atomic orbitals: STO = N x a y b z c r n−1 e−ςr χabc

(2.11)

The integers a, b, c control the angular momentum, n plays the role of principal quantum number, ς controls the width of the orbital, a large ς gives a tight function and a small ς a diffuse function. However, they are now rarely used because it is not convenient to calculate the multicentric integrals

12

2 Computational Methods

¨

χi∗ (r1 )χ j (r1 )

1 ∗ χ (r2 )χl (r2 )dτ1 dτ2 r12 k

(2.12)

For practical reasons, the χ k functions are Gaussian-type atomic orbitals (GTOs). GTO χα,abc (x, y, z) = N x a y b z c e−αr

2

(2.13)

a, b, c again control the angular momentum, and α controls the width of the orbital The main advantage of the GTOs is that the product of two GTOs is still a GTO. Therefore, a two-center integral of the type (2.12) can be reduced to a one-center integral; see Appendix 2.19.3. However, the GTOs have a wrong behavior at the nucleus where they should have a cusp because the potential energy of the electron and the nucleus becomes infinite as the distance becomes zero: in other words, the GTOs are too flat at r ~ 0 and fall off to fast at large r; see Fig. 2.1. One possible solution to this problem is to use a linear combination of primitive Gaussians to obtain contracted Gaussians CGTO χα,abc (x, y, z) = N

n 

ci x a y b z c e−αr

2

(2.14)

i=1

where the contraction coefficients, ci , are fixed constants within a given basis set. The number of primitive Gaussian functions is called degree of contraction. When one increases the number of primitive GTOs in (2.14), the result looks more and more like a STO, except at the nucleus where it can never attain the correct shape. This is the reason of the slow convergence of the energy when the size of the Fig. 2.1 Comparison of the 1s Slater wavefunction (unbroken line) to its approximation with one Gaussian (dotted line)

2.4 Hartee–Fock (HF) Method

13

basis set increases, it is particularly a problem when electron correlation is taken into account; see Sect. 2.7.4. For a molecule with n electrons, the n/2 orbitals of smallest energy (which is negative) are called occupied orbitals and the orbitals of higher energy are called virtual orbitals. Typical errors are 1% in bond distances (which are underestimated), the energy is too high and the dissociation energy is not correctly calculated. For instance, in the case of the HF theory, the two electrons of dihydrogen, H2 , spend half the time on the same atom and half the time on both atoms, even when the distance between the two atoms is infinite, which is obviously not possible. The main weakness of the HF method is that it does not take into account the instantaneous interactions between the electrons, called electronic correlation. The correlation of electrons of same spin is partially accounted for because they cannot belong to the same spin orbital (the determinant, (2.8) would be zero). On the other hand, electrons of opposite spin are allowed to approach each other closely. The difference between the HF energy and the exact non-relativistic energy is the correlation energy E(exact) = E(HF) + E(correlation)

(2.15)

There are different ways to estimate this correlation energy. One of the simplest ones is to use the second-order perturbation theory (the first order is the E(HF) energy), often called Møller–Plesset perturbation theory (Møller and Plesset 1934), abbreviated as MP2. As we will see in Sect. 2.6 and Table 2.1, this method significantly improves the situation but is not always accurate enough. Higher-order perturbation theory is possible (MP3, MP4) but not fully satisfactory. It is also possible to use configuration interaction methods that will be described below, Sect. 2.5, but they require large amounts of computer resources, even for small molecules. Although they are powerful, they are difficult to use. Another recent method giving excellent results is the coupled cluster theory. Finally, a completely different method, nonab initio, is the Density Functional Theory (DFT, see Sect. 2.13) which may give satisfactory results, particularly for large molecules.

2.5 Post Hartree–Fock Methods (Bartlett and Stanton 1994; Helgaker et al. 2000, 2004) For these methods the HF solution corresponds to a single Slater determinant. Assume a molecule with n electrons, the HF solution is 1 ψ0 = √ |χ1 χ2 · · · χn | n!

(2.16)

where the determinant is written in abbreviated form, | |, and the χ i are spin orbitals, i.e., the product of a molecular orbital and a spin function. The spin orbitals χ 1 , χ 2 ,

14

2 Computational Methods

… χ n are a subset of the larger set used in the variational procedure. The unused spin orbitals, called virtual orbitals, are noted χ a with a = n + 1, n + 2, … In configuration interaction (CI) methods, other determinants are constructed by replacing one or more occupied orbitals χ i , χ j , … within the HF determinant with a virtual orbital χ a , χ b , … A determinant where χ i is replaced by χ a will be called ψia , a determinant where χ i , χ j are replaced by χ a , χ b will be called ψiab j If ψ 0 is the HF wavefunction, a better approximation of ψ is ψ = c0 ψ0 +



cia ψia +

ia



ciabj ψiab j +···

(2.17)

i jab

where the first sum refers to single excitations applied to ψ 0 , the second one to double excitations, etc. The coefficients ci are determined by the variational method. If all the virtual orbitals and all degrees of excitation are included, the wavefunction is called full configuration interaction (CI) wavefunction. In this wavefunction, there are three categories of correlation corrections 1. excitations whose individual contributions are small, but their total contribution is large because of their great number. This is called dynamic correlation. It enables electrons to stay apart, and it is usually the largest part of the correlation energy. 2. excitations required to provide a correct zeroth-order description as dictated by spin-and orbital-symmetry considerations. It occurs for open-shell molecules and it is called static correlation. 3. excitations whose coefficients ci are large. It is called nondynamic correlation. The CI calculations are computer intensive; they are thus limited to small systems. Furthermore, they require a lot of experience. Practically, the series, (2.17), is truncated, usually after the double excitation term (CISD). One of the simplest methods to estimate this correlation energy is the many-body perturbation theory with V being the perturbation term. The second-order theory gives the Møller–Plesset 2 (MP2) method (Møller and Plesset 1934) which recovers about 90% of the correlation energy. To improve the accuracy, it is better to use the coupled cluster method which takes into account the instantaneous interactions between the electrons (Purvis and Bartlett 1982; Lee and Scuseria 1995). The best possible wavefunction may be written ψ = eT ψ0

(2.18)

where T = T1 + T2 + T3 + … is the cluster operator. T1 performs all singly excited substitutions T1 ψ0 =



tia ψia

ia

T2 performs all doubly excited substitutions

(2.19)

2.5 Post Hartree–Fock Methods …

15

T2 ψ0 =



ab tiab j ψi j

(2.20)

i jab

The coefficients t are called amplitudes. The most important contribution comes from the double excitations. It gives the coupled cluster double method (CCD). The next step is to include T1 eT = 1 + (T1 + T2 ) +

1 (T1 + T2 )2 + · · · 2!

(2.21)

which gives the CCSD method and reduces the error by a factor three to four. A better description is obtained by also taking into account, triple excitations, quadruple excitations, etc. The CCSDT method stops at triple excitations and further reduces the error by a factor three to four. If n is the number of basis functions and m the order of the clusters (m = 2 for CCSD, m = 3 for CCSDT, m = 4 for CCSDTQ, the computation time is proportional to n2m+2 . To reduce this cost, the connected triple excitations may be taken into account by perturbation theory (Raghavachari et al. 1989). It gives the CCSD(T) method whose cost is proportional to n7 (instead of n8 for CCSDT). The small error due to the perturbation calculation is nearly compensated by the error due to the neglect of quadruple excitations. Thus, the CCSD(T) method is faster and more accurate than the CCSDT method. The CC methods estimate accurately the dynamic correlation but, when the nondynamic correlation is large (coefficients > 0.2), the accuracy is reduced and it may be necessary to use CI methods. For systems presenting a strong multiconfigurational character, and for dealing with excited states, it is recommended to use multireferences approaches like the CASSCF (Complete Active Space SCF), CASPT2 (secondorder perturbation theory based on the multiconfiguration self-consistent field theory) or MRCI (Multi-reference configuration interaction) methods, for more details, see for instance Helgaker et al. (2000). Note that these methods are not easy to use and require some expertise. In conclusion, when the nondynamic correlation is small, we have at our disposal a hierarchy of approximations of increasing accuracy: HF < MP2 < CCSD < CCSD(T) < CCSDTQ < . . . Helgaker et al. (1997) and Bak et al. (2001) compared the performances of the different methods using small closed-shell molecules containing first-row atoms and whose experimental equilibrium structure is accurately known. They concluded that the models HF, MP2, and CCSD(T) give improved accuracy at increased computational cost and that the accuracy of the CCSD(T) model with a basis set of quadruplezeta quality is high and comparable to that observed in most experimental studies. The results are given in Table 2.1. In conclusion, CCSD(T) calculations with corepolarized quadruple-zeta basis sets (CVQZ or wCVQZ, see 2.7) provide an accuracy

16

2 Computational Methods

Table 2.1 Accuracy of the ab initio methods for bond lengths r (in pm) and bond angles ∠ (in degree), basis set CVQZ, all electrons correlated) Parameter

Method

Mean error

Standard deviation

Mean absolute error

Maximum absolute error

r

HF

−2.60

2.03

2.60

8.51

MP2

−0.18

0.59

0.46

1.70

∠a

CCSD

−0.67

0.66

0.67

2.45

CCSD(T)

−0.04

0.16

0.09

0.59

HF MP2 CCSD CCSD(T)

1.07

1.27

1.41

2.84

−0.24

0.27

0.30

0.52

0.01

0.37

0.27

0.69

−0.21

0.13

0.21

0.41

Source Bak et al. (2001) a The number of data is limited to 7 angles

of about 0.1–0.2 pm in the calculated distances between first-row atoms. However, it is still possible to achieve a higher accuracy as it is obvious that, at the quadruple zeta-level, the basis set is not yet converged. It is also possible to improve the accuracy being going beyond CCSD(T). The choice of the method and of the correct basis set will be discussed in the next sections. Another interesting observation from Table 2.1 is that the MP2 model is an inexpensive and useful alternative to the CCSD model.

2.6 Choice of the Method (Puzzarini and Barone 2009) To be sure to obtain a reliable structure, the use of CCSD(T) method is required as shown in Table 2.1. Actually, the relatively small errors in the CCSD(T) bond lengths result from a cancelation of errors in the perturbative treatment of the connected triples and the neglect of higher-order connected excitations, as shown in Table 2.2. This table also shows that the rate of convergence depends on the kind of bond: It is for instance much slower for F2 . Halkier et al. (1997) studied the performance of the CCSDT method, and they found that it was generally less accurate than the cheaper CCSD(T) one. However, the inclusion of higher excitations may be non-negligible for some molecules. The contributions of the connected quadruple and quintuple excitations have been studied by several authors. In particular, Ruden et al. (2004) made a thorough studies for a few simple diatomic molecules whose experimental structure is very accurately known; see Chap. 3. Their results are reported in Table 2.2. They show that the quadruple excitations increase the bond distance of 0.4 pm for F2 . The quintuple corrections are one order of magnitude smaller. The good performance of the CCSDTQ method was also verified on several small polyatomic molecules (Heckert et al. 2005). The calculation was based on the assumption of the additivity

2.6 Choice of the Method …

17

Table 2.2 Corrections to the CCSD bond distances (in pm) HF

N2

F2

CO

CCSD(T)–CCSDa

0.29

0.73

2.26

0.72

CCSDT–CCSD(T)b

0.00

−0.07

−0.04

0.02

CCSDTQ–CCSDTb

0.02

0.14

0.38

0.04

CCSDTQ–CCSD(T)b

0.02

0.07

0.34

0.06

CCSDTQ5–CCSDTQc

0.00

0.03

0.03

0.00

Source Ruden et al. (2004) a AV6Z, frozen core b VTZ, frozen core c VDZ, frozen core

of electron correlation which is well established for accurate energy calculations. It may be written in the following way E Total = E CCSD(T) (A) + E CCSDT (B) + E CCSDTQ (C) + E core (D)

(2.22)

with E CCSDT (B) = E CCSDT (B) −E CCSD(T) (B)

(2.23a)

E CCSDTQ (C) = E CCSDTQ (C)−E CCSDT (C)

(2.23b)

E core (D) = E ae [CCSD(T)/D] −E fc [CCSD(T)/D]

(2.23c)

The CCSD(T) energy is calculated with a basis set, A, as large as possible, E CCSDT (B) with a smaller basis set B, E CCSDTQ with a still smaller basis set C, all calculations being performed in the frozen core (fc) approximation (all electrons in the core orbitals kept frozen). Finally, the core correlation E core is calculated with a basis set D, which is, if possible, of quadruple-zeta quality; see Sect. 2.7.2. This last correction is the difference between the energy computed with all electrons correlated (ae) and the energy obtained in the frozen core approximation (fc). With the notable exception of CCSDT, all higher-order connected contributions are positive and converge monotonically. For this reason, CCSDT should not be used without the CCSDTQ correction. As this correction due to quadruple excitations is small, it can be calculated more easily by a perturbative treatment, this is the CCSDT(Q) method (Bomble et al. 2005; Kállay and Gauss 2005). Furthermore, as the contributions from connected quadruple excitations to bond distances are known to converge faster with basis-set size than those from connected triples, a small basis set may be used (Ruden et al. 2004). For some molecules, the correction CCSDTQ5– CCSDTQ is still not completely negligible. For instance, it is 0.03 pm for N2 and F2 (Ruden et al. 2004); see Table 2.2.

18 Table 2.3 Constrained optimization of the r(N–O) bond length (in pm) in trans-HONO using various levels of theory

2 Computational Methods Methoda

r(N–O)

CCSD

140.1

CCSD(T)

142.1

CCSDT

142.4

CCSDTQ

142.9

Reprinted With Permission from Demaison et al. (2006). Copyright 2006. American Chemical Society a The pVDZ basis of Ahlrichs for the middle two atoms and the VDZ basis of Ahlrichs for the two terminal atoms have been used. The frozen-core approximation is employed throughout

The fast convergence is true when the non-dynamical correlation is small. When it is large, the situation is completely different as can be seen on the example of HONO (and other NOx molecules) where the O-N bond is 0.8 pm too short at the CCSD(T) level; see Table 2.3. Another typical example is the Cl–N bond length in ClNO for which r[CCSD(T)_ae/CV5Z] = 196.1 pm, whereas the value of the experimental exp equilibrium structure is re = 197.263(7) pm, the latter value in nice agreement with the semiexperimental structure, reSE = 197.308(32) pm (Demaison et al. 2006). To estimate the importance of the nondynamical correlation, the coupled cluster T 1 diagnostic, which is implemented in several programs, may be used (Lee and Taylor 1989). Ideally, it should be smaller than 0.020.

2.7 Choice of the Basis Sets 2.7.1 Description of the Basis Sets There are a great number of basis functions available. • Minimal: one basis function for each atomic orbital (AO). • Double-zeta: two basis functions for each AO. It allows more flexibility in the radial size. One set is tighter (closer to the nucleus, i.e., a large ζ), the other set is looser. • Triple-zeta: three basis functions for each AO • Quadruple-zeta: four basis functions for each AO. • Etc. • A split-valence basis uses only one basis function for each core AO, and a larger basis for the valence electrons. Usually, polarization functions are added to the basis set. It is important for an accurate description of the bonding because the presence of other atoms distorts the environment of the electrons and removes its spherical symmetry. To polarize a basis function with angular momentum l, one adds at least a basis function of angular

2.7 Choice of the Basis Sets

19

momentum l + 1. p orbitals polarize if they are mixed with d orbitals. There should be 5 d functions: dx 2 −y 2 , dz 2 , d xy , d xz , and d yz called pure angular momentum functions. However, 6 d functions (called Cartesian) are regularly used: dx 2 , d y 2 , dz 2 , d xy , d xz , and d yz . The same situation is encountered with f orbitals: There are only seven pure angular momentum functions but ten Cartesian functions. Diffuse functions may also be added. They have a very small exponent and, thus, decay slowly with the distance from the nucleus. They are mainly of s and p type. They are necessary for a correct description of anions, weak bonds, and for electronegative atoms as fluorine. For correlated calculations, it is recommended to use a hierarchy of basis sets which provide a systematic approach to the complete basis set. One of the most popular are the correlation-consistent polarized valence basis sets of Dunning (1989): cc-pVnZ (often abbreviated as VnZ) where n = D, T, Q, 5, … is the cardinal number and represents the highest spherical harmonic. The cc-pVnZ basis sets are designed for correlation of valence electrons only. To correlate all electrons, the correlation– consistent core–valence cc-pCVnZ, basis sets (Woon and Dunning 1995) or the correlation-consistent weighted core-valence basis sets, cc-pwCVnZ (often abbreviated as wCVnZ) (Peterson and Dunning 2002) which significantly improve the convergence with n have to be used (Gaussians with large exponents are added to the VnZ basis sets). On the other hand, for an accurate description of the outer valence region, diffuse functions (Gaussians with small exponents) are added to the VnZ basis sets to give the aug(mented)-cc-pVnZ basis sets (often abbreviated as AVnZ) (Kendall et al. 1992). In the case of complexes, doubly and triply augmented sets may also be used. Thus the most general basis sets are aug-cc-pwCVnZ (AwCVnZ). For instance, for the carbon atom, cc-pVDZ consists of 3s2p1d, cc-pVTZ is 4s3p2d1f, and cc-pVQZ would be 5s4p3d2f1g, for aug-cc-pVTZ it is 5s4p3d2f and for cc-pwCVTZ it is 6s5p3d1f. An overview over the correlation-consistent basis sets for hydrogen and first-row elements is given in Table 2.4. For the second-row elements Al–Ar, the original d polarization functions in VnZ are not tight enough, the revised cc-pV(n + d)Z basis sets should be used (Dunning et al. 2001). Table 2.4 Composition of the correlation-consistent valence VnZ basis sets, the augmented basis sets AVnZ, the core-valence CVnZ basis sets and augmented core-valence ACVnZ basis sets (the total number of basis functions is given in parentheses) Cardinal number Atoms

Basis

D

T

Q

5

H–He

VnZ

2s1p (5)

3s2p1d (14)

4s3p2d1f (30)

5s4p3d2f1g (55)

AVnZ

3s2p (9)

4s3p2d (23)

5s4p3d2f (46)

6s5p4d3f2g (80)

Li–F

VnZ

3s2p1d(14)

4s3p2d1f (30)

5s4p3d2f1g (55)

6s5p4d3f2g1h (91)

AVnZ

4s3p2d (23)

5s4p3d2f 46)

6s5p4d3f2g (80)

7s6p5d4f3g2h (127)

CVnZ

4s3p1d (18)

6s5p3d1f (43)

8s7p5d3f1g (84)

10s9p7d5f3g1h (145)

ACVnZ

5s4p2d (27)

7s6p4d2f (59)

9s8p6d4f2g (109)

11s10p8d6f4g2h (181)

20

2 Computational Methods

It is important to have a look at the size of the basis sets because it determines the computation time. For the VnZ basis sets, the number of contracted functions N V (n) increases as the third power of the cardinal number n, see Table 2.4 which also shows that the core–valence sets are considerably larger than the valence sets and that the number of diffuse functions increase quadratically with the cardinal number. The atomic natural orbital (ANO) basis sets provide another contraction approach (Almlöf and Taylor 1987, 1990). The contraction coefficients are obtained by optimizing atomic energies. For atoms with many electrons, the standard basis sets become too large. Furthermore, relativistic effects, see Sect. 2.8, are no more negligible for the inner electrons. For these reasons, the Effective Core Potentials (ECP) have been developed (Dolg 2000). Their goal is to reduce the basis set size and the number of electrons as well as to include some relativistic effects (mass-velocity and Darwin, but the spin– orbit effect are neglected, see Sect. 2.8). Their success is due to the fact that the chemical bonding is mainly determined by the valence electrons, as for the frozen core approximation. The core electrons are replaced by an approximate effective pseudopotential. Most basis sets may be found in the basis–set library of program packages. They are also available on the EMSL basis set exchange web page: https://www.basissete xchange.org (Pritchard et al. 2019).

2.7.2 Core Correlation The (aug)-cc-pVnZ basis sets are not appropriate when all electrons are correlated because they do not provide sufficient flexibility in the core region (Martin 1995; Császár and Allen 1996). It is quite common in correlated calculations to use the frozen core approximation, in which the orbitals of the inner-shell electrons are constrained to remain doubly occupied in all configurations. Indeed, it significantly reduces the computational effort and, fortunately, the error due to freezing the core is nearly constant for molecules containing the same type of atoms; see Table 2.5. Furthermore, it is small for bond angles, see Table 2.6. Then, the core–core and core–valence correlations are calculated separately as a correction, see (2.22) and (2.23c). From Table 2.7, it appears that an accurate computation of the core correlation requires at least a CCSD(T)/wCVQZ level of theory, the wCVTZ basis set giving a too small correction (as the wCVQZ basis set is quite large, it may be advantageous to use a completely decontracted VTZ basis set supplemented by an appropriate (1p3d2f) primitive set. This basis set called Martin–Taylor basis set (denoted as MT) is significantly smaller without any loss in accuracy (Martin 1995)). On the other hand, the MP2/wCVQZ level gives a correction slightly too large for atoms of the row Li–F and is not accurate enough for heavier atoms. For instance, for the C–Cl bond in ClCN, the MP2 method gives −0.43 pm, whereas the CCSD(T) method gives −0.36 pm (Demaison et al. 2004). However, it is still possible to achieve a high accuracy, starting from CCSD(T)/wCVTZ and

2.7 Choice of the Basis Sets

21

Table 2.5 Core correlation r[CCSD(T)_fc/CVQZ] − r[CCSD(T)_ae/CVQZ] for a few bond lengths (in pm)a CH

CC

C=O

HC≡CH

0.12

HC≡CH

0.25

H2 CO

0.22

HC≡CF

0.12

HC≡CF

0.24

HCOOH cis

0.21

H2 C=CH2

0.14

H2 C=CH2

0.28

HCO+

0.21

CH3 F

0.14

CH2 =CHF

0.29

HOCO+

0.21

H2 CO

0.13

CHCl=CHF cis

0.29

CO

0.22

HCN

0.13

C2 H6

0.32

CO2

0.20

C2 H6

0.15

CH2 =CH–CN

0.33

CH3 CHO

0.21

CH4

0.15

CH3 CN

0.35

HNCO

0.21

a Computed

with MolPro 2009 (Werner et al. 2009, 2012)

Table 2.6 Core correlation for the bond angles at the CCSD(T) level of theory (ae − fc in degrees)a

Molecule

Angle

Basis set

H2 O

HOH

MTb

Value 0.12

NH3

HNH

MTb

0.17

H2 CO

HCO

MTb

0.02

CH3 F

HCF

CVQZ

0.03

N2 H2

HNN

CVQZ

0.12

HNO

HNO

CVQZ

0.07

H2 O2

HOO

CVQZ

0.09

HOOH

CVQZ

CSSC

wCVQZ

(CH3 )2 S2

0.13 −0.06

a Computed

with MolPro 2009 (Werner et al. 2009, 2012) unless otherwise stated b Martin (1995)

calculating the small effect of basis set extension at the MP2 level; see Table 2.7. For the second-row atoms, it is also possible to take advantage of the compensation of errors, the MP2 method overestimating the core correction and the wCVTZ basis set underestimating it; see Table 2.7.

2.7.3 Diffuse Functions Finally, the contribution of the diffuse functions has to be considered. Their effect may be large if the basis set is small (n ≤ 3), but it decreases rapidly when n increases and for n ≥ 5, it is negligible in most cases, even for the C–F bond length in CH3 F which is particularly sensitive to the effect of diffuse functions: for n = 4 (or Q), the C–F bond length is still lengthened by 0.29 pm but for n = 5 the lengthening

22

2 Computational Methods

Table 2.7 Calculation of the core correlation (fc−ae in pm) at various levels of theory for a few bondsa CCSD(T)

CCSD(T)

CCSD(T)

MP2

MP2

b

Molecule

Bond

wCVTZ

wCVQZ

wCV5Z

wCVQZ

wCVTZ

HCCH

CC

0.221

0.262

0.274

0.290

0.255

0.256

CH

0.107

0.126

0.132

0.130

0.113

0.124

BO

0.268

0.318

0.332

0.339

0.289

0.318

OH

0.074

0.085

0.088

0.090

0.079

0.085

BF

0.220

0.270

0.284

0.283

0.233

0.270

BF2 OH

BF3

BF

0.223

0.274

0.288

0.286

0.236

0.273

CS

CS

0.364

0.45

0.473

0.49

0.409

0.445

CCl2

CCl

0.457

0.49

0.56

0.494

0.524

CH3 Cl

CCl

0.370

0.38

0.46

0.416

0.414

(CH3 S)2

SS

0.462

0.540

0.662

0.558

0.566

(CH3 S)2

CS

0.405

0.464

0.529

0.466

0.468

a Computed

with MolPro 2009 (Werner et al. 2009, 2012) + MP2/wCVQZ − MP2/cc-wCVTZ

b CCSD(T)/wCVTZ

is reduced to 0.08 pm; see also Table 2.8. The van der Waals and hydrogen bonds are however an important exception. The main exception is when an electronegative atom as fluorine is present but, in such a case, the correction can be calculated at a lower level. For instance, in the particular case of HF, r[CCSD(T)/AVQZ] − r[CCSD(T)/VQZ] = 0.15 pm which is identical to r[MP2/AVQZ] − r[MP2/VQZ] = 0.15 pm. The diffuse functions are also very important to compute the properties of a weakly bound cluster molecule. A typical example is H2 O…HF (Demaison and Liévin 2008). Weakly-bound systems require the use of very large basis sets. This can be remedied by adding some functions between the subsystems where a higher electron density is expected. These functions are denoted as mid-bond functions. A typical example is the calculation of the potentials of He–He and He–Ar (Tao 1993).

2.7.4 Convergence of the Basis Set The basis set error due to the use of a finite basis set is called basis set incompleteness. As noted in Sect. 2.4, the convergence is slow because of the cusp condition; see Table 2.9 for a few examples. It is in particular quite slow for F2 . There are two main ways to improve the situation as well as an approximate way.

2.7 Choice of the Basis Sets

23

Table 2.8 Effect of diffuse functions on the geometrical parameters: AVnZ − VnZ (lengths in pm and angles in degrees) Molecule

Bond

VTZ

VQZ

V5Z

V6Z −0.04

HF

0.38

0.15

0.06

N2

0.02

0.02

0.01

0.00

F2

0.23

0.01

−0.01

−0.01

CO HC≡CH NH3 CH3 F

H2 O H2 O2

0.03

0.04

0.02

0.01

C≡C

0.05

0.04

0.01

0.01

C–H

0.03

0.02

0.01

0.01

N–H

0.06

0.04

0.03

0.02

HNH

0.72

0.34

0.02

C–F

0.78

0.29

0.08

C–H

0.03

0.02

0.02

HCF

−0.46

−0.17

−0.04

O–H

0.22

0.10

0.04

0.02

HOH

0.60

0.25

0.06

0.02

O–H

0.25

0.08

0.04

O–O

0.31

0.10

0.03

All calculations were performed at the CCSD(T) level of theory Reprinted from Journal of Molecular Structure, 1023. Demaison J, Császár AG. Equilibrium CO bond lengths, 7–14. Copyright 2012, with permission from Elsevier

Table 2.9 Convergence of the basis set for a few bond lengths (in pm) at the CCSD(T) level of theory HF

N2

F2

CO

AVTZ

92.10

110.40

141.81

113.60

QZ-TZa

−0.33

−0.35

−0.51

−0.42

5Z-QZa

−0.04

−0.10

−0.21

−0.09

6Z-5Za

0.00

−0.03

−0.08

−0.03

AV6Z

91.73

109.92

141.01

113.06

AV∞Zb

91.72

109.88

140.93

113.02

Source Ruden et al. (2004) a r[AVnZ] − r[AV(n − 1)Z] b Extrapolated value, see Sect. 2.7.4

2.7.4.1

Extrapolation to Complete Basis Set (CBS) (Puzzarini 2009)

It is established that the HF energy converges much faster than the correlation correction. The basis set limit may be estimated by extrapolating them separately. For the HF energy, the following empirical formula is often used with the correlation-consistent basis sets

24

2 Computational Methods ∞ n E HF = E HF + ae−bn

(2.24)

where n is the cardinal number of the basis set. This formula requires three calculations with different n. For the electron-correlation correction, the following equation may be used ∞ n = E corr − E corr

c n3

(2.25)

This equation only requires two calculations. The rate of convergence depends on the kind of bond. For instance, for the CH bond, convergence is almost achieved at n = 3. On the other hand, for the O· · · H bond (hydrogen bond), it is necessary to go at least up to n = 6 (Demaison and Liévin 2008). An interesting alternative method of extrapolation using the semiexperimental equilibrium rotational constants is presented in Sect. 6.8. The ground state rotational constants X 0 computed at the CCSD(T)/wCVTZ and CCSD(T)/wCVQZ levels of theory are extrapolated as a function of the structural parameters, r e (T) and r e (Q) computed at the same level of theory. The intersection of the line with the experimental X 0 of the parent species gives the extrapolated r e .

2.7.4.2

Explicitly Correlated Methods (Klopper et al. 2006; Werner et al. 2010)

They make explicit use of the interelectronic distance r ij . The correct Coulomb-cusp condition is obtained by multiplying the orbital product expansion by a correlation factor . A correlation containing only linear r 12 terms gives the R12 method, where the used wavefunction is ψ R12 = (1 + c12 r12 )ψ

(2.26)

Recently, it was demonstrated the superiority of writing the correlation factor in terms of Slater functions exp (γ r 12 ). It gives the F12 methods. With these methods, triple-zeta basis sets are enough to reach quintuple-zeta quality. See Tables 2.10 and 2.11 for a comparison of the CCSD(T) and CCSD(T)-F12 methods. In addition to the basis set incompleteness and the electron correlation, there are two further approximations limiting the accuracy.

2.8 Relativistic Effects (Pyykkö 1988)

25

Table 2.10 Comparison of the CCSD(T) and CCSD(T)-F12 methods Number of bonds

H–X

X–Y

X=Y

XTY

75

49

43

14

H, B–F, Al–Cl

B–F, Al–Cl

C, N, O, Si, S, Cl

B,C,N,P

Atoms

X, Y

CCSD(T)

VTZ

1.4

7.1

5.8

5.7

VQZ

0.4

2.2

1.7

1.3

V5Z

0.2

0.6

0.5

0.3

A V5Za

0.1

0.7

0.5

0.4

VTZ-F12

0.4

0.9

0.6

0.5

CCSD(T)-F12

The root-mean-square deviation (pm) is given relative to the CCSD(T)/AV6Z values Source Spackman et al. (2016) a A means aug functions on all atoms except hydrogen

Table 2.11 Convergence of the CCSD(T)-F12 method Molecule, bond

re

CCSD(T)_ae AwCVQZ

CVTZ-F12

CVQZ-F12

N2

109.76

109.82

109.70

109.68

HC≡N, C–H

106.51

106.57

106.52

106.52

HC≡N, C≡N

115.33

115.39

115.27

115.25

99.54

99.56

99.54

99.53

HNC, N–H

CCSD(T)-F12(ae)

HNC, N=C

116.85

116.93

116.82

116.81

N≡C–C≡N, C–C

138.33

138.49

138.43

138.42

N≡C–C≡N, C≡N

115.84

115.86

115.75

115.73

Bond lengths in pm Source Breidung and Thiel (2019)

2.8 Relativistic Effects (Pyykkö 1988) For the inner-electrons of heavy atoms, the relativistic effects become non-negligible. The best method to take them into account is to use a fully relativistic Dirac Hamiltonian. However, it is very demanding in computational resources and various methods have been developed to estimate relativistic effects. To first order, there are three relativistic contributions: • The relativistic dependence of the electron mass on velocity which leads to a decrease of the kinetic energy; see Appendix 2.19.4. • The Darwin term which is due to the fact that the instantaneous position of the electron cannot be defined more precisely than within a spherical volume of radius  m 0 c. It smears the effective potential felt by the electrons. It only exists for s orbitals.

26

2 Computational Methods

Table 2.12 Relativistic effects (distances in pm, angles, ∠, in degrees) Molecule

r a

Methodb

Molecule

r a

Methodb

N2

−0.02

DKH

SiH4

−0.07

DKH

CS

0.05

DKH

Br2 CO, CO

−0.05

DKH

CS2

0.02

DKH

Br2 CO, CBr

−0.23

DKH

F2

0.03

DKH

Br2 CO, ∠(BrCBr)

−0.1

DKH

Cl2

0.05

DKH

H2 O, OH

0.016

Breit

Br2

−0.32

DKH

H2 O, ∠(HOH)

−0.074

Breit

ClF

0.06

DKH

CH4

−0.013

DHF

0.13

DKH

SiH4

−0.066

DHF

−0.04

DKH

GeH4

−0.70

DHF

CF4

0.00

DKH

SnH4

−2.06

DHF

SiF4

−0.05

DKH

PbH4

−7.33

DHF

BrF BrCl

Source Demaison (2007) a r = r[relativistic) − r[non-relativistic] b DKH = Douglas–Kroll–Hess; Breit = Breit–Pauli; DHF = Dirac–Hartree–Fock

• The spin-orbit term due to the interaction of the spin magnetic moment of the electron with the effective magnetic field arising orbital motion of the electrons. The first two contributions introduce an energy correction and describe the socalled scalar relativistic effects, in opposition to the last one which induces energy splittings. When heavy atoms are present, the most widely used method is the pseudopotential approximation (Martin and Sundermann 2001; Peterson 2003) because it avoids the basis functions necessary for the description of the electronic core and for the inner nodal structure of the valence orbitals. It introduces scalar relativistic effects adjusted from relativistic atomic calculations. Since the effects of relativity are small for the great majority of usual atoms (they scale up to Z 4 ), perturbation theory may be successfully used. A number of approximate methods have been developed. One of the most widely used is the Douglas–Kroll–Hess (DKH) method (Douglas and Kroll 1974; Hess 1986; de Jong et al. 2001) which recovers most of the scalar relativistic effects. A few typical values are given in Table 2.12. For chlorine and lighter atoms, this correction is smaller than 0.07 pm (value for SiH4 ) but it becomes important for heavy atoms.

2.9 Correction to the Born–Oppenheimer Approximation (Handy and Lee 1996; Gauss et al. 2006) A first-order Born-Oppenheimer correction which is diagonal in the electronic state is easily calculated

2.9 Correction to the Born–Oppenheimer Approximation … Table 2.13 Diagonal born–oppenheimer contribution (distances in pm, angles in degrees)

27 DBOC

H2 O, r(OH)

0.003

H2 O, ∠(HOH)

0.015

H2

0.021

HF

0.002

N2

0.001

F2

0.000

Source Experimental, semi-experimental and ab initio equilibrium structures, Demaison J. Molecular Physics, 105: 3109–3138, Dec 10, 2007, reprinted by permission of the publisher Taylor&Francis Ltd, http://www.tandfonline.com

      E DBOC = ψ0(e) TN ψ0(e)

(2.27)

It is called adiabatic correction (or diagonal BO correction) and is a good approximation if E (0)  E (i) . It is usually small and mass dependent (proportional to 1/M). Table 2.13 gives a few values, which show that this correction is normally negligible. This correction slightly differentiates the geometries of the deuterated isotopologues of ethyne (C2 H2 ), by increasing the C≡C, C–H, and C–D bond lengths by 0.002, 0.015, and 0.008 pm, respectively (Liévin et al. 2011).

2.10 Born–Oppenheimer Equilibrium Structure, reBO First, it is important to understand how to determine the geometrical parameters of the molecule. The energy is a continuous function of the p internal coordinates defining the geometry of the molecule. It describes a hypersurface called potential energy surface (PES). The lowest-energy minimum or global minimum corresponds to the Born– Oppenheimer equilibrium structure, reBO . It has a positive curvature for distortions in any direction which is characterized by a positive-definite second-derivative matrix (Hessian). When the approximate position of the global minimum is not known, it is useful to first conduct a grid mapping of the parameter space. There are many different methods to find the exact position of the minimum. At first sight, the simplest one is to fit the energy with a polynomial function. For instance, for a triatomic molecule, one may use V (R1 , R2 , R3 ) =



Ci jk (R1 )i (R2 ) j (R3 )k

(2.28)

i jk

It is then easy to find the minimum as well as the force constants (see Sect. 2.14). The difficulty is that this method requires some skill.

28

2 Computational Methods

In most cases, a Taylor series expansion of the energy is made about a reference point R0 1 E = E(R0 ) + (R − R0 )T G + (R − R0 )T H(R − R0 ) 2

(2.29)

where T means the transpose, G is the gradient and H the Hessian. By differentiating to find the minimum, it gives R = R0 − H−1 G

(2.30)

The final solution is obtained by iteration. This is a Newton–Raphson method. The main difficulty is the calculation of the inverse of the Hessian. Variants, called quasi-Newton methods have been devised to simplify the calculation of H−1 .

2.11 Strategy To determine the structure of a molecule, if it is not too large and if a high accuracy is required, (2.22–2.23c) should be used together with the CBS extrapolation procedure, (2.24–2.25). The CCSD(T) method may be advantageously replaced by an explicitly correlated method such as CCSD(T)_F12 (Werner et al. 2010), see Sect. 2.7.4.2. When the molecule is too large, or when a very high accuracy is not required, the CCSD(T)_ae/wCVQZ level of theory gives satisfactory results, its mean error being only 0.1 pm (Coriani et al 2005); see also Table 2.14. This good result is mainly due to a compensation of errors because the introduction of connected quadruples (CCSDTQ) increases the bond length by 0.1–0.2 pm, whereas the basis set extension from quadruple zeta to sextuple zeta shortens it by about 0.1 pm. Some CCSD(T)_ae/CVQZ bond lengths are compared to the CCSD(T)_ae/CV5Z values and to the best equilibrium structure in Table 2.14. The CCSD(T)_ae/CVQZ bond length is larger than the equilibrium one by 0.04 pm with a maximum absolute deviation (MAD) of 0.04 pm (corresponding to a standard deviation of 0.06 pm). On the other hand, The CCSD(T)_ae/CV5Z bond length is smaller than the equilibrium one by 0.04 pm with a maximum absolute deviation (MAD) of 0.02 pm (corresponding to a standard deviation of 0.03 pm). There is one outlier: HF. Fluorine is extremely electronegative and, in such a case, diffuse functions are required or a basis set up to CV6Z has to be used. For instance, in the case of HF, r[CCSD(T)_ae/ACVQZ] = 91.73 pm is close to the r e value. For large molecules, it may happen that the CVQZ (or wCVQZ) basis set is too large. In such a case, a compound method, similar to (2.22) may be used. For instance, the Born–Oppenheimer (BO) equilibrium structure is optimized at the CCSD(T)_ae/wCVTZ level of theory and the small effect of further basis set enlargement (wCVTZ → wCVQZ) is then estimated at the MP2 level. The resulting reBO estimate is:

2.11 Strategy

29

Table 2.14 Comparison of the CCSD(T)_ae/CVQZ bond length between heavy atoms with the CCSD(T)_ae/CV5Z one and with the most accurate equilibrium structure (all values in pm) re

Ref.

CVQZ

CV5Z

r e − CVQZ

r e − CV5Z

CO

112.84

a

112.89

112.80

−0.05

0.04

CO2

116.00

b

116.04

115.98

−0.04

0.02

HCCH

120.28

c

120.34

120.28

−0.06

0.00

N2

109.75

d

109.81

109.71

−0.06

0.04

F2

141.08

d

141.13

140.92

−0.05

0.16

HF

091.71

d

091.58

091.54

0.13

0.17

HCN

115.32

e

115.38

115.30

−0.06

0.02

CH2 O

120.45

e

120.43

120.41

0.02

0.05

CH2 =CH2

133.08

e

133.12

133.06

−0.04

0.02

HNC

116.87

e

116.93

116.86

−0.06

0.01

HNO

120.81

e

120.85

120.77

−0.04

0.03

Median

−0.04

0.04

MAD

0.04

0.02

sf

0.06

0.03

Molecule

a Experimental:

Authier et al. (1993) Teffo and Ogilvie (1993) c Experimental: Tamassia et al. (2016) d CCSD(T)/V∞Z + core + T + Q: Ruden et al. (2004) e CCSD(T)/V∞Z + core + T + Q: Puzzarini et al. (2008) f Standard deviation calculated from MAD, see (9.36) b Experimental:

  reBO = re CCSD(T)− ae/wCVTZ + re (MP2− ae/wCVQZ) − re (MP2− ae/wCVTZ)

(2.31)

The accuracy of this equation, which is based on the additivity of small corrections, was confirmed many times and, the example given in Table 2.15 shows that the loss of accuracy is quite small. The MAD of the residuals is 0.02 pm for the bond lengths and 0.02° for the angles. For still larger molecules, the structure may be optimized at the CCSD(T)_fc/VTZ level and the effects of basis set enlargement (VTZ → VQZ) and core correlation are then estimated at a lower level. In this case, only the valence electrons are correlated (the so-called frozen core approximation). The estimation of the core correlation is discussed in Sect. 2.7.2. The calculation of the correction (AVTZ → AVQZ) is presented in Table 2.16 (when an electronegative atom is present, the addition of diffuse functions—augmented basis sets—may be necessary). Inspection of Table 2.16 shows that the extension of the basis set can be accurately calculated at the CCSD level of theory. The MP2 method is slightly less accurate; nevertheless, it allows us to estimate the extension with an acceptable accuracy. As the correction (AVQZ → AV5Z) is smaller, it is estimated with a better accuracy.

134.49

95.75

B–O

O–H

107.76

C4–H4

131.31

107.71

C3–H3

132.31

108.31

C2–H2

B–Fant i

155.09

N1–C4

B–Fsyn

133.62

C3–C4

108.09

C3–H

154.19

107.76

C2–H

C2–C3

144.76

C2–C3

128.82

154.55

N1–C3

N1–C2

125.42

N1=C2

wCVQZ

95.76

134.53

131.31

132.30

107.76

107.70

108.30

155.15

133.67

154.17

128.90

108.07

107.75

144.79

154.64

125.46

Equation (2.31)d

N1 C3 H

−0.03

C2 C3 H3

−0.06

OBFanti FBF

−0.04 −0.01 0.02

BOH

b Vogt

OBFsyn

0.00

0.01

0.00

0.01

H4 C4 C3

C3 C2 H2

−0.05 0.01

C2 C3 C4

118.37

119.39

113.14

122.24

139.106

138.596

136.633

84.559

118.38

119.34

112.98

122.28

139.111

138.572

136.638

84.584

96.017

−80.628

96.013

−80.606

H2 C2 C3 H3 N1 C2 C3

116.621

120.412

115.849

116.594

120.419

115.868

151.546

139.071

Equation (2.31)d

HC3 H

0.02

−0.08

0.02

C2 C3 H

C3 C2 H

−0.09 0.01

139.062

N1 C2 H

−0.04 151.577

wCVQZ

Res.

et al. (2015) et al. (2015) c Median absolute deviation d r BO = r (CCSD(T)_ae/wCVTZ) + r (MP2_ae/wCVQZ) − r (MP2_ae/wCVTZ) e e e e

a Császár

MADc

BF2

OHb

Azetea

Azirinea

Table 2.15 Comparison of the CCSD(T)_ae/wCVQZ results with those of (2.31), (distances in pm, angles in degrees)

0.019

−0.01

0.05

0.16

−0.04

−0.005

0.024

−0.005

−0.025

−0.004

0.022

−0.027

0.007

0.019

0.031

−0.009

Res.

30 2 Computational Methods

2.12 Lower Level Ab Initio Methods

31

Table 2.16 Calculation of the correction on the bond lengths due to the extension of the basis set from AVTZ to AVQZ (all values in pm) HF

N2

F2

CO

CCSD(T)

−0.33

−0.35

−0.51

−0.42

CCSD

−0.33

−0.35

−0.53

−0.41

MP2

−0.32

−0.34

−0.39

−0.38

Source Ruden et al. (2004)

2.12 Lower Level Ab Initio Methods As shown in Table 2.1, the accuracy of the CCSD(T) method is about 0.1 pm (mean error) for a molecule without heavy atom (Z < 18) and with a small non-dynamical correlation. However, for large molecules, the CCSD(T) method is still too expensive. It is therefore interesting to check the accuracy of lower level ab initio calculations which are more readily accessible. The two main sources of errors are the basis set convergence error and the electronic structure method error. However, the errors are not additive. The bond length normally decreases with the size of the basis set, whereas it increases with the level of theory. It is thus possible to make use of the concept of balanced calculation for which there is a near cancelation of the errors. The MP2 method has been shown to perform rather well, see Table 2.1. It also appeared that the remaining errors are generally mainly systematic and correction factors, or “offsets” can be derived empirically in order to predict molecular structures with an accuracy which may be competitive with the best experimental methods (i.e., a few tenths of pm). An alternative name for this method is template (TM) approach where the template is a structurally similar molecule whose equilibrium structure is accurately known (Piccardo et al. 2015). The equilibrium structure of the molecule is calculated with re = reMP2 + r (TM)

(2.32)

r (TM) = re (TM) − reMP2 (TM)

(2.33)

where

Obviously, in these equations, MP2 may be replaced by another method such as CCSD or another method known to give reliable results (see Sect. 2.13). This method is particularly excellent for the single bonds C–H, C(sp3 )–C(sp3 ), and C–N as well as for the bond angles because the offset r is quite small at the MP2/VTZ level of theory. Note that for the bond angles, either the MP2/VQZ level or the cheaper MP2/6-311 + G(3df,2pd) gives slightly more accurate results. One obvious difficulty is that the offset values are basis set dependent. Moreover, for a given basis set, the offset is not always constant, but may vary as the true equilibrium distance varies. In addition, the offset is a function of substituent’s effects.

32

2 Computational Methods

For instance, for the MP2/VTZ value of r(CC) in benzene: r = r e − r(calc.) = –0.13 pm, whereas the offset at the same level of theory is r = –0.65 pm for NC–CN. This large difference cannot be explained by the variation of the r e value because they are quite close. Constancy of the offset value in a given type of bond implies that similar errors occur in the calculation of that value, i.e., the finite basis set creates the same error, the partial neglect of electron correlation has the same effect on the calculated bond length, etc. Consequently, it is not surprising that the magnitude of the offset value is at least somewhat responsive to environmental perturbations from the surroundings of the bond. The conclusion of this discussion is that the offset method is useful but has to be used with caution. Nevertheless, in such a case, it is advisable to verify that the bonding in the two molecules is similar by using, for instance, the Atom In Molecule (AIM) theory; see Sect. 2.18. It is possible to improve this method by using the linear regression approach re = a · reMP2 + b

(2.34)

where the coefficients a and b determined by least-squares fit. For instance, the determination of the experimental structure of a fluorine derivative is a difficult problem because there is only one stable isotope for fluorine, making studies of isotopic species impossible. The calculation of a reliable ab initio structure is further complicated by the fact that fluorine is a highly electronegative atom which requires very large basis sets and highly correlated methods. On the other hand, a least-squares fit of the C(sp3 )-F single bond length of 15 molecules with the MP2/6-311 + G(3df,2pd) method gives a = 0.99827(20) and b = 0 with a standard deviation of 0.11 pm (Juanes et al. 2017). Likewise, it is possible to use this equation to predict the length of CC bonds in phenyl rings using the MP2/VTZ method. For 30 molecules, it gives a = 0.998414(73) and b = 0 with a standard deviation of 0.055 pm (Demaison et al. 2019). Similar results were obtained for the CO bond length. For a sample of 47 molecules with single and double bonds, the MP2/VQZ level of theory gives a = 1.0215(30) and b = −3.03(27) with a standard deviation of 0.2 pm (Demaison and Császár 2012).

2.13 Density Functional Theory (DFT) The Density Functional Theory (DFT) is a way to treat electron correlation in a much cheaper way than the correlated wavefunction methods like MP2, CCSD, or CCSD(T). The basic idea is to use the electron density ρ(r) which only depends on the three coordinates x, y, and z instead of the many-electron wavefunction which depends on many variables (3n neglecting the spin, n being the number of electrons). The theoretical basis is the Hohenberg and Kohn (1964) theorem that states that the density of any system determines all the ground state properties of the system, in particular the energy. The energy being variational with respect to the density, the minimum of the energy defines the electron density.

2.13 Density Functional Theory (DFT)

33

The problem is to find this density functional (a functional is a function of functions). It was solved by Kohn and Sham (1965). They modified the standard HF equations by introducing the local exchange-correlation term E XC that accounts for the exchange phenomenon and the dynamic correlation in the motion of the individual electrons. The energy is partitioned in several terms E = E T + E V + E XC

(2.35)

E T is the kinetic energy of the non-interacting electrons; E V is the electrostatic Coulomb interaction between two charges densities (nucleus/electron, nucleus/nucleus, and electron/electron). E T + E V corresponds to the classical energy of the charge distribution ρ. The problem is to find an approximation of E XC which is usually divided into two parts: exchange (X) and correlation (C) parts E XC = E X + E C

(2.36)

2.13.1 Local Density Approximation (LDA) The functional depends only on the (local) density at a given point. The model is a uniform electron gas. Thomas and Fermi studied this homogeneous electron gas in the early 1920. Their theory permits to calculate E X . E C is obtained by fitting an analytical form to the results obtained for the homogeneous electron gas. The errors due to the exchange and correlation parts tend to compensate each other approximately. However, the LDA has an obvious weakness in describing real, inhomogeneous systems.

2.13.2 Generalized Gradient Approximation (GGA) The solution to improve the LDA is to use not only the local density but also its gradient. Many gradient-corrected functionals have been developed, in particular by Becke (1993).

2.13.3 Meta-GGA A still better description of the inhomogeneity of the electron density is obtained by adding the Laplacian (second derivative) of the density.

34

2 Computational Methods

2.13.4 Hybrid DFT Another improvement is to mix a portion of exact exchange from Hartree–Fock theory (HF with the rest of the exchange-correlation energy from other DFT exchanges defining E XC as X XC + cDFT E DFT E XC = cHF E HF

(2.37)

The parameters determining the weight of each individual functional are typically specified by fitting the functional’s predictions to experimental or accurately calculated thermochemical data. The most popular example is B3LYP which stands for Becke, 3-parameter, Lee– Yang–Parr (Lee et al. 1988).

2.13.5 Double-Hybrid Density Functionals (DHDFT) Such functionals as B2-PLYP (sometimes also called B2PLYP) expand the DFT exchange-correlation energies into four terms: X X C C + (1 − cx )E DFT + (1 − ac )E DFT + ac E MP2 E XC = cx E HF

(2.38)

The first two terms describe the exchange energy as a mix of terms derived from GGA functionals and exact exchange HF. Likewise, the correlation energy is a sum C calculated of terms derived from GGA functionals and the correlation energy E MP2 with second-order perturbation theory (MP2). It is important to recognize that all four terms are derived from the same Kohn–Sham orbitals. The first DHDFT method of general applicability B2PLYP was proposed by Grimme (2006a). These functionals are better to take into account van der Waals forces. The convergence of the double hybrid can be more difficult than with simpler functionals. The running time is roughly twice of a B3LYP job. Furthermore, the double hybrids need bigger basis sets. One problem common to all these functionals is the treatment of dispersion interactions. The easiest and most popular solution is the D3 model of Grimme (2006b) which uses precomputed reference data for dispersion coefficients, and the concept of “fractional coordination number” to mimic the environment of an atom in a molecule. The functional B3LYP is at present one of the most popular approximations in chemistry and it is used in a considerable number of papers. There are several reasons for this success. First, it accounts for electron correlation, and yet the computational cost is of the same order as the HF method (i.e., proportional to n4 , n being the size of the system, whereas, for the MP2 method, the cost is proportional to n5 ). One further advantage is that the basis set convergence is much faster than in conventional correlated methods: A basis set of triple-zeta quality is sufficient in most cases and,

2.13 Density Functional Theory (DFT)

35

Table 2.17 Mean, standard deviation (St. dev.), and mean absolute error (MAE) for the B2PLYP/VTZ and B3LYP/SNSD deviations from CCSD(T) semiexperimental equilibrium structures (distances in pm and angles in degree) Bond

n#a

B2PLYP/VTZ Mean

B3LYP/SNSD St. dev.

MAE

All bonds

74

0.00

0.05

0.04

−0.01

0.09

0.07

CH

30

−0.02

0.04

0.03

−0.05

0.05

0.06

CC

21

0.02

0.05

0.04

0.05

0.09

0.09

CO Angles

St. dev.

MAE

Mean

7

0.02

0.02

0.02

0.03

0.04

0.05

46

0.00

0.05

0.03

0.00

0.07

0.05

Source Penocchio et al. (2015) a Number of data in the sample

as shown by Piccardo et al. (2015), a double-zeta basis set such as SNSD (downloadable on http://dreamslab.sns.it) already gives satisfactory results; see Table 2.17. Finally, B3LYP is able to give reliable geometries as well as force fields (Bauschlicher et al. 1997). B3LYP slightly overestimates single bond lengths and underestimates multiple bond lengths. Nevertheless, it is possible to correct B3LYP results (as for the MP2 ones, see Sect. 2.12) and obtain mean absolute errors of about 0.02 pm for all types of bonds. For instance, for the CO bond, using the 6-311 + G(3df,2pd) basis set, the median value of residuals is +0.018 pm for multiple bonds and −0.021 pm for single bonds (Demaison and Császár 2012). However, the differences are not constant, but there is a linear relationship between the residuals and the B3LYP values. Still for the CO bond, a linear fit gives re (CO) = 0.9780(29)r [B3LYP/6 − 311 + G(3df, 2pd)] + 0.0283(37)

(2.39)

with a correlation coefficient of ρ = 0.9996 and a standard deviation of σ = 0.21 pm. This is of quality comparable to the MP2/VTZ level of theory. Table 2.18 compares the results of the structures optimized with the B3LYP/SNSD or B2PLYP/VTZ levels of theory with accurate semiexperimental equilibrium structures. The agreement is satisfactory.

2.14 Calculation of the Force Field (Császár 2012) The concept of potential energy (hyper)surface (PES) is fundamental in chemistry. The equilibrium structure corresponds to its minimum. The PES governs the vibrations of the atoms. Its knowledge is also important for a better understanding of reaction kinetics. It is also a useful model for the study of local mode behavior. For most applications, it is enough to know the PES near its minimum. Therefore, it is

36

2 Computational Methods

Table 2.18 Mean, standard deviation (St. dev.), root-mean-square error (RMSE) and standard error of estimate (SEE) for the B2PLYP/VTZ and B3LYP/SNSD deviations from B2PLYP/VTZ semiexperimental equilibrium structures (distances in pm and angles in degree) CH

CC

CO

CN

∠(HCH)

∠(CCC)

na

82

45

30

14

29

22

B2PLYP Mean

0.02

0.01

0.27

0.015

St. dev. 0.11

0.28

0.15

0.018

RMSE 0.11

0.28

0.31

0.023

B3LYP

-.-.0.35

0.24

0.018715

−0.001425

Ab

−0.074795 −0.015977 −0.004834 0.006962

Bb

0.080798

0.022342

0.003455

R2 b,c

0.990739

0.999063

0.999746

0.999714

0.998167

0.999912

σ b,d

0.08

0.025

0.015

0.017

0.19

0.19

Mean

−0.010544 −1.819612 0.018197

0.63

0.48

0.49

0.043

-.-

St. dev. 0.13

0.35

0.29

0.022

-.-

RMSE 0.65

0.59

0.57

0.048

0.53

0.47

Ab

−0.093027 −0.023958 −0.015281 −0.003991 0.020715

Bb

0.095014

R2 b,c

0.988016

0.998736

0.999250

0.999539

0.992041

0.999644

σ b,d

0.09

0.029

0.025

0.021

0.40

0.39

0.028995

0.014730

0.000901

−0.003057

−1.993258 0.089102

Source Penocchio et al. (2015) a Number of bonds b Linear regression: A = 1 − slope; B = intercept c Square of the correlation coefficient d Standard deviation of the estimate  Mean e¯ = n1 ei  2 ei RMSE n  ¯2 (ei −e) Std σ = n−1

convenient to develop the PES in Taylor series as a function of the nuclear displacement coordinates around its minimum. In such a case, the first-order terms are zero. The coefficients of this expansion are called the force field. They are usually divided into two parts: the harmonic (or quadratic) force field which is the most important term in the expansion and the anharmonic force field. The expansion is written in the following form V =

1 1  1 f i j Ri R j + f i jk Ri R j Rk + f i jk Ri R j Rk Rl + · · · 2 ij 6 i jk 24 i jkl

(2.40)

where R denotes a set of nuclear displacement coordinates (internal coordinates, Cartesian coordinates, …). Unfortunately, different systems of coordinates have to be used and the transformation of the force constants to another coordinate system

2.14 Calculation of the Force Field (Császár 2012)

37

is rather complex. For the usual spectroscopic applications, it is enough to limit the development up to fourth order. Most computer packages compute analytically second (f ij ) derivatives for most methods. There are two ways to obtain all force constants (i.e., the cubic and quartic ones, and, eventually the quadratic ones): a least-squares fitting of the energy as described in Sect. 2.10 or by numerical differentiation. The second method is easier to use and often more accurate. For instance, when analytic second-order derivatives are available, the diagonal and semidiagonal cubic constants are given by  fi j j =

∂2V ∂ R 2j

 Ri =

−



2

∂2V ∂ R 2j

 Ri =−

(2.41)

where Δ is the displacement along the internal coordinate i. The numerical precision of the computed force constants depends on many factors (accuracy of the reference geometry, choice of the displacement size, , truncation errors, …), and it is expected to be worse for higher derivatives. However, the cubic and quartic force constants can be calculated with a higher accuracy than the quadratic ones. Indeed, the energy is composed of two terms opposite in sign but of similar magnitude: the electronic and the nuclear–nuclear repulsion (V NN ) contributions. V NN and its derivatives can be calculated exactly which is not the case for the electronic contribution, and for the anharmonic force constants, the V NN derivatives become increasingly dominant.

2.15 Semi-empirical Methods For very large molecules, the full Hartree–Fock method is still too expensive. In such cases, semi-empirical methods are used. They are based on the Hartree–Fock formalism but many approximations are made. Mainly, only valence electrons are taken into account and some two-electron integrals are neglected and other ones approximated empirically. This parameterization of the integrals is chosen in order to give results in agreement either with experimental data or with ab initio results, see for instance Steward (2013). It further allows us to take into account some electron correlation. The number of omitted integrals and the kind of parameterization define the different methods. These methods are now of very limited utility.

38

2 Computational Methods

2.16 Molecular Mechanics (MM) 2.16.1 Introduction The aim of molecular mechanics is to calculate the lowest energy of a molecular structure using the principle of classical physics and assuming that the contributions to energy are additive. Its main advantage, beyond its simplicity, is that it can be used for systems of thousands to millions of atoms. The objective of molecular mechanics is to find a force field that can be transferred from molecule to molecule in order to predict some properties of the molecule such as its structure. The basic assumption is that, for given bonds, their length and their angle do not vary much. For instance, The C–H bond lengths are between 106 and 110 pm for most molecules, with stretching frequencies between 2900 and 3300 cm−1 , the C=C bond lengths are close to 133 pm with stretching frequencies between 1620 an 1680 cm−1 , and the C–C bond length in alkanes is close to 153 pm and the ∠(CCC) bond angle is close to 109–114°. The structure of a given molecule, being close to the reference structure, is simply obtained by minimizing the potential energy. The parameterization of the force field and the reference structure is made using results from (mainly) small molecules whose properties (experimental or ab initio) are accurately known.

2.16.2 Calculation of the Energy One assumes that the molecule is made of point masses (atoms) mainly held together by strings obeying Hooke’s law. It is a first-order linear approximation according to which the force to compress or extend an elastic body is proportional to the deformation; see also (3.1) in Chap. 3. The steric energy E is the difference in energy between the real molecule and a hypothetical molecule where all the structural parameters are exactly at their reference values. It is given by Westheimer equation (Westheimer and Meyer 1946) E = E s + E b + E t + E nb + . . .

(2.42)

E s is the energy of the bond being stretched or compressed from its reference bond length ri0 , (not to be confused with r 0 from the effective structure), it is given by Es =

 ks  2 i ri − ri0 2 bonds

(2.43)

2.16 Molecular Mechanics (MM)

39

r i is the actual value of the bond length. Hooke’s law slightly overestimates E s . One possible solution is to add a higher-order term. E b is the energy of bending bond angles from their reference values. It is given by Eb =



kibj  2 θi j − θi0j 2 bond angles

(2.44)

θ ij is the angle between bonds i and j. Instead of the angle as variable, it is possible to use its cosine. E τ is the torsional energy. For a set of four bonded atoms A–B–C–D, the torsional angle τ is defined as the angle measured about the B–C axis from the ABC plane to the BCD plane. The expression of the energy is Eτ =

1  {V1 (1 − cos(τ − τ0 )) + V2 (1 − cos 2(τ − τ0 )) 2 dihedral

+V3 (1 − cos 3(τ − τ0 )) + · · ·}

(2.45)

Some terms in this equation may be zero. For instance, for the torsion of a XY3 group (typically a methyl group), V 1 = V 2 = 0. E nb is the energy of nonbonded interactions. There are two terms. The first one takes into account the attraction between two nonbonded atoms A and B due to London dispersion forces as well as the van der Waals repulsion. The Lennard-Jones 12–6 potential is often used because it is simple (Lennard-Jones 1924). It may be written (see Appendix 2.19.5) E LJ =

 nonbonded pairs



A B − 6 12 rAB rAB

 (2.46)

r AB is the distance between atoms A and B. The r −12 term is a repulsive term describing Pauli repulsion at short range. The r −6 term is an attractive long-range term. The second nonbonded term takes into account of the electronegativity differences between nonbonded atoms A and B. It is EQ =



1 QA QB 4π ε0 rAB nonbonded pairs

(2.47)

QA and QB are the charges on atoms A and B, respectively. ε0 is the permittivity. The estimation of the values of the partial charges QA and QB is not straightforward. For more details, see Appendix 8.7 of Chap. 8.

40

2 Computational Methods

2.16.3 Accuracy and Limitations The best achievable accuracy is 1–3 pm for the bond lengths, 1°–3° for the bond angles and up to 8° for the dihedral angles. Another important limitation is that electrons are ignored in molecular mechanics force fields; therefore, processes that involve electronic rearrangements, such as chemical reactions, cannot be described at the MM level. Finally, their applicability is restricted to those classes of compounds and bonding situations the force field was parameterized for.

2.17 Combined Quantum/Classical (QM/MM) Methods (Senn and Thiel 2009; Brunk and Rothlisberger 2015; Groenhof 2013) When the ab initio methods cannot be used because the molecule is too large, a solution is the combination of quantum mechanics and molecular mechanics (QM/MM). The reacting part of the molecule (abbreviated as Q) is treated with a quantum mechanical method and its environment (i.e., the rest of the molecule, abbreviated as M) with simpler molecular mechanical methods. This hybrid QM/MM strategy was originally introduced by Warshel and Levitt (1976) who studied enzymatic reactions. The main difficulty is to correctly describe the interactions between the two systems (Q and M). There are mainly two methods. In the subtractive method, the energy E of whole molecule is calculated at the MM level, E MM (M + Q). Then, the energy of the Q subsystem is calculated at the QM and MM levels giving E QM (Q) and E MM (Q). Finally, the correct energy is E QM/MM (Q + M) = E MM (M + Q) + E QM (Q) − E MM (Q)

(2.48)

The subscripts indicate the method. The subtractive method is for instance used in the ONIOM method implemented in GAUSSIAN. In the additive method, the energy is calculated in the following way E QM/MM (Q + M) = E QM (Q) + E MM (M) + E QM−MM (Q + M)

(2.49)

In this equation, the interaction between the two subsystems, E QM-MM (Q + M), is treated explicitly. The main difficulty is to evaluate this coupling term that may be written as the sum of three terms b vdW el + E QM−MM + E QM−MM E QM−MM (Q + M) = E QM−MM

(2.50)

2.17 Combined Quantum/Classical (QM/MM) Methods

41

The first term corresponds to the bonded interactions, the second one to the van der Waals interactions and the third one, to the electrostatic interactions between the two subsystems Q and M. The treatment of the first two terms is easy because they can be handled at the MM level. The case of the electrostatic coupling is more difficult and several different levels of sophistication may be used. The simplest method is called mechanical embedding. It treats the electrostatic interactions at the MM level. It is simple but has severe shortcomings. At present, one of the best methods is the electrostatic embedding: The MM charges are inserted in the QM Hamiltonian as oneelectron operators. The polarization of the Q system by the electrostatic interactions with the M system is accounted for. If the two subsystems are connected by a chemical bond, the bond is cut and the dangling bond of the Q system must be capped by a link atom (for instance, H) bound to Q. This link atom is only used for the QM calculation, the bond Q-M being described at the MM level. Even though QM/MM methods are often very efficient, they are still rather tricky to handle. The Q region has to be carefully selected. Furthermore, the potential energy surface generally has many local minima.

2.18 Quantum Theory of Atoms in Molecules (QTAIM or AIM) (Bader 1990; Gillespie and Popelier 2001) Ab initio methods can deliver accurate interatomic distances but do not give direct information on the bonds and their properties. Nevertheless, this information can be obtained by analyzing either the wavefunction or the electronic density, ρ. However, it is much easier to use the electronic density because ρ is a function of only three variables (x, y, and z) and has a direct experimental relevance (it can also be obtained by X-ray diffraction). The AIM method, developed by Richard F. W. Bader, uses the electronic density, ρ, derived from the wavefunction as starting point and provides a simple quantum definition for an atom in a molecule as well as for a bond. The AIM is also a method for analyzing the electron density distribution that determines all the properties of a molecule. The maximum of ρ corresponds to a nuclear position. Along the internuclear axis, ρ has a minimum value, and in a direction perpendicular to the internuclear axis, the density at this critical point is a maximum, and it is a saddle point called bond critical point. The magnitude of the electron density at this bond critical point ρ b serves as a parameter that can be used in evaluating the corresponding bond order (Cioslowski and Mixon 1991). It also gives the amount of electron density shared between the two bonded atoms. It is roughly proportional to the bond length. The strength of a bond increases and its length decreases as the bond critical point density ρ b increases, and the increasing charges on the bonded atoms have the same effect. There is still another factor affecting the bond lengths: They increase with

42

2 Computational Methods

the increasing coordination number of the central atom because of the ligand–ligand repulsion as will be further explained in Sect. 8.4. It is also possible to define the following useful parameters • The bonding radius r b is the distance from the bond critical point to the nucleus. It is identical to the covalent radius for homonuclear diatomic molecules and it is a well-defined property, contrary to the covalent radius see also Sect. 8.2.1. • The atomic charge q. It is simply the charge of the nucleus, Z, minus the electron population. The latter quantity is obtained by integrating ρ over the atomic basin (zero-flux surface in the gradient vector field of ρ). Note, however, that Badercharge values exaggerate the atomic charges (Maslen and Spackman 1985). A critical review of the different methods used to estimate atomic partial charges may be found in Meister and Schwarz (1994). The problem of deriving atomic charges from the results of ab initio calculations has been studied by many authors. A recent example is by Wiberg and Rablen (2018). See also Sect. 8.6.7. • The bond ellipticity ε provides a measure of the extent to which the charge is preferentially accumulated at different angles in a given plane perpendicular to the bond path and, for this reason, is a measure of the π-character of bond. It is defined from the eigenvalues, λ1 < λ2 < λ3 , of the Hessian of ρ at the bond critical point (the Hessian is the 3×3 matrix of second-order partial derivatives, ∂ 2 ρ/∂x∂y, …)

ε = (λ1 /λ2 ) − 1

(2.51)

ε = 0 indicates a circularly symmetric electron density found in linear molecules. The eigenvalues λ1 , λ2 , and λ3 are used to classify the different critical points. The number of nonzero eigenvalues, r, of a critical point is the rank, and the sum of the signs of the eigenvalues, s, is the signature. A critical point is denoted by (r, s). Important critical points are: • (r, s) = (3, −3) corresponds to a maximum, i.e., a nuclear position. • (r, s) = (3, −1) is a saddle point (maximum in two dimensions and minimum in the third one) i.e., a bond critical point (BCP). • (r, s) = (3, +1) is a ring critical point (RCP) found at the center of cyclic molecules (e.g., cyclopropane). • The Laplacian ∇ of the electron density shows where the field is locally concentrated (∇ < 0) or depleted (∇ > 0). When ∇ < 0, the concentration of charges is between the atoms forming the bonds, it is typical of covalent bonds. In the other hand, when ∇ > 0, the charges are away from the internuclear region. It is the case for ionic, hydrogen, or van der Waals bonds. Examples of the usefulness of the AIM methods are given below in Sects. 8.2.4 and 8.4.

2.19 Appendices

43

2.19 Appendices 2.19.1 Ratio of the Magnitudes of the Gravitational and Electrostatic Forces Between a Proton and an Electron (All Values in SI Units) Electrostatic force:   −19 2 1 e2 9 1.6 × 10 Fel = = 8.99 × 10 4π ε0 r 2 r2

(2.52)

Gravitational force:    −31 1.672 × 10−27 m e Mp −11 9.11 × 10 Fg = G 2 = 6.67 × 10 r r2 ⇒

Fel = 2.3 × 1039 Fg

(2.53) (2.54)

2.19.2 Nuclear Size The nucleus does not have definite boundaries because it is a quantum object. However, from scattering experiments, it is possible to define a mean radius called root-mean-square (rms) charge radius. The first determination was made by Geiger and Marsden in the laboratory of Rutherford (1911). They studied the scattering of alpha particles by gold nuclei. The work done to bring the alpha particle to rest is equal to its initial kinetic energy. If the distance of the particle to the nucleus (of charge +Ze) is d, the electric potential energy is Ze 1 =T · 2e · 4π ε0 d

(2.55)

If the kinetic energy of the alpha particle is T = 7.7 MeV and Z = 79 (for Au), it gives: d ≈ 3 × 10−14 m to be compared with the diameter of the atom, ∅ = 3 × 10−10 m. More generally, the majority of atomic nuclei are approximately spherical in shape and the radius is given approximately by r = r0 A1/3

44

2 Computational Methods

where r 0 ≈ 1.2 fm and A is the mass number. Of course, it is only an approximation, there is not a sharp cut off with a finite density of nucleus inside and zero density outside. Furthermore, some nuclei do not have a spherical symmetry. The deviation may be pointed out by the value of the quadrupole moment Q of the nucleus. The rms radius is often measured by scattering of electrons. Another method is to use optical isotopic shifts. When the s electrons are inside the nucleus, they are submitted to a potential different from the Coulomb potential. This potential depends on the volume of the nucleus, i.e., its radius. It is also possible to use muonic atoms where an electron is replaced by a muon. As the muon is more massive than the electron, the Bohr orbits are closer to nucleus permitting more precise measurements. The K α X-ray isotope shift may also be used. The rms radii and the radii changes in isotopic sequences are tabulated in Angeli and Marinova (2013) where the methods of measurements are also given. See also Sect. 3.9.3 where the effect of the nuclear size on the structure is discussed.

2.19.3 The Product of Two Gaussians is Another Gaussian Assume that the function ga = e−αa ra is on atom A and function gb = e−αb rb on atom B. We choose the axis Ox such that A and B lie on it. We may write 2

2

ga = e−αa [(x−xa )

2

+y 2 +z 2 ]

(2.56)

and the same for gb. Then ga gb = e−(αα xa +αb xb ) e2(αa xa +αb xb )x e−(αa +αb )r 2

If we choose the origin O such that

2

xa xb

2

(2.57)

= − ααab we obtain another Gaussian

ga gb = ce−(αa +αb )r with the coefficient c = e−(αα xa +αb xb ) 2

2

2

(2.58)

2.19.4 Relativistic Correction Due to the Dependence of the Electron Mass on Velocity For a Hg atom (Z = 80), the velocity of the 1 s electrons is 58% of the speed of light, its mass is then m = 1.23m0 , m0 being the rest mass. The nonrelativistic energy is

2.19 Appendices

45

E NR =

p2 +U 2m

(2.59)

The relativistic energy is  E R = c m 20 c2 + p 2 − m 0 c2 + U If v  c ⇒ Then

p2 m 20 c2

(2.60)

0 Higher-order potential constant [dimensionless]

(3.14, 3.73a–3.73d)

gJ

Rotational g factor [dimensionless]

(3.55)

h

Planck constant

k

Harmonic force constant [dyn cm–1 = 10−3 N m−1 ]

mA

Atomic mass of atom A [kg or u]

me

Electron rest mass

Mp

Proton rest mass

r

Bond length [pm or Å]

(3.1)

re

Equilibrium value calculated from Y 01

(3.56)

reBO

Isotopically independent Born–Oppenheimer bond length

(3.83)

rz

Zero-point average structure

(3.68)

r0

Effective structure

(3.58)

rs

Substitution structure

(3.64)

u

Unified atomic mass unit = 1.660 538 782(83) × 10−27 kg

υ

Vibrational quantum number: 0, 1, 2, …

(3.10)

A lk

Correction parameter for the Born–Oppenheimer approximation

(3.78, 3.79, 3.82)

αe

Rotation–vibration interaction constant [MHz or cm−1 ]

(3.40, 3.49)

γe

Higher-order rotation–vibration interaction constant [kHz]

(3.49)

μ

Reduced mass

(3.8)

μD

Electric dipole moment [C m or D, 1 D = 3.335 64 × 10−30

C m] (3.80) (continued)

3.1 Introduction

55

Table 3.1 (continued) Symbol Quantity [unit]

Equations

ν

Vibrational or rotational frequency

(3.24)

ωe

Harmonic vibrational frequency [cm−1 ]

(3.7, 3.9, 3.49)

ωe x e

Anharmonicity constant [cm−1 ]

(3.16, 3.49, 3.50)

ω e ye

Higher-order anharmonicity constant [cm−1 ]

(3.49)

ξ

Expansion parameter for the development of the potential

(3.15)

References books for this section are: Barrow (1962), Brown and Carrington (2003), Gordy and Cook (1984), Herzberg (1950), Le Roy (2011), and Tiemann (1992). The constants and parameters used in this chapter are listed in Table 3.1.

3.2 The Vibrational Energy 3.2.1 Harmonic Oscillator The variation of the electronic energy as a function of the bond length gives a curve called potential function whose minimum is the equilibrium structure. As a first approximation, it is possible to assume that this curve is a parabola 1 UV = − D˜ e + k(r − re )2 = − D˜ e + V (r ) 2

(3.1)

where D˜ e is the dissociation energy (not to be confused with the quartic centrifugal distortion constant, De , introduced in Sect. 3.5), k the quadratic (or harmonic) force constant, and r e the equilibrium bond length. We will assume that the molecule is formed of two atoms A of mass mA at the distance r A of the center of mass, and B of mass mB at the distance r B of the center of mass. We will call x A (t) the displacement of atom A and x B (t) the displacement of atom B. The kinetic energy may be written as T =

 1 m A x˙A2 + m B x˙B2 2

(3.2)

Taking into account the potential V, we have two equations of motion 

m A x¨A = − ∂∂xVA = −k(xA − xB ) m B x¨B = − ∂∂xVB = −k(xB − xA )

(3.3)

56

3 Diatomic Molecules

One solution is 

xA = xA0 eiωt xB = xB0 eiωt

(3.4)

(k − m A ω2 )xA − kxB = 0 −kxA + (k − m B ω2 )xB = 0

(3.5)

leading to 

To obtain a non-trivial solution, the determinant of this system of two equations must be zero. Developing it, gives   ω2 m A m B ω2 − k(m A + m B ) = 0

(3.6)

The first solution, ω = 0, corresponds to a translation motion: x A = x B . The second root is a vibrational motion whose angular frequency (in s−1 ) is  ωe =

k(m A + m B ) = mAmB

 k μ

(3.7)

where μ is the reduced mass μ=

mAmB mA + mB

(3.8)

It is common to express the vibrational frequency in unit of Hz (or multiple or cm−1 , 1 cm−1 = 29,979.2458 MHz)  1 ωe = 2π

k μ

(3.9)

In classical mechanics, the diatomic molecule behaves as a harmonic oscillator of mass μ and displacement coordinate x A − x B . The same situation is true in the quantum mechanical problem. The solution of the Schrödinger equation is well known and similar to that of the square-well problem. It follows that the vibrational energy is quantified and may be written as

1 hωe EV = υ + 2

(3.10)

υ is a positive integer called vibrational quantum number. It has to be noted that the first level, called vibrational ground state, with υ = 0 has an energy E 0 = hωe /2; i.e.,

3.2 The Vibrational Energy

57

it is above the bottom of the potential. It is a consequence of the uncertainty principle of Heisenberg. Example: the harmonic vibrational frequency of H35 Cl The atomic masses are usually given in atomic units, they have to be converted using mu = 1.6605388×10−27 kg, the force constant is given in dyn cm−1 , and it has to be converted in N m−1 . For H35 Cl, k = 4.84×105 dyn cm−1 = 484 N m−1 . It gives 1 ωe = 2π

 1.007825×34.9688 1.007825+34.9688 13

484 × 1.6605 × 10−27

= 8.66713 × 10 Hz = 2891 cm−1

(3.11)

It is useful to estimate the order of magnitude of the displacement. For the ground state of HCl, the displacement may be calculated from hωe 1 k(r − re )2 = 2 2

(3.12)

Equation (3.12) gives r − r e = 11 pm compared to the equilibrium value, r e = 127.46 pm. In other words, the displacement is small compared to the bond length. In the harmonic oscillator approximation, transitions are allowed only if υ = υ  – υ  = ±1. In absorption spectroscopy, which is the common case, it is υ = +1. Here, υ’ is the quantum number of the upper level and υ" the quantum number of the lower level. The spectrum is generally observed in the infrared range. However, vibrational spectra may also be measured by Raman spectroscopy.

3.2.2 Anharmonic Oscillator Equation 3.1, is only an approximation called harmonic approximation. Experimentally, one finds transitions with υ > +1; see Fig. 3.1. Furthermore, the overtone absorptions (with υ" > 0) are not equally spaced. Thus, a better approximation for the potential function is needed, for instance the Morse equation (Morse 1929)  2 V (r ) = D˜ e 1 − e−a(r −re )

(3.13)

D˜ e is the energy of dissociation measured from the bottom of the potential well. The problem is that this equation is not very convenient for calculations. It is better to use a series expansion called Dunham expansion (Dunham 1932). It is justified by the fact that the displacement is only a small fraction of the bond length.   V (ξ ) = a0 ξ 2 1 + a1 ξ + a2 ξ 2 + · · ·

(3.14)

58

3 Diatomic Molecules

Fig. 3.1 Anharmonic oscillator and corresponding vibrational spectrum

with the expansion parameter ξ=

r − re re

(3.15)

Note that k = 2 D˜ e a 2 = 2a0 re−2 ; a0 = D˜ e a 2 re2 = ωe2 /4Be ; and a1 = − ar e . Selected values of the ai parameters are given in Table 3.2, and their determination is discussed in Sect. 3.9. When one is interested in the equilibrium structure of diatomic molecules, i.e., the bottom of the potential, Dunham’s expansion, (3.14) is almost universally used. However, as seen in Table 3.2, its convergence is slow. For this reason, different expansion parameters have been proposed. In particular, Simon, Parr, and Finlan (SPF) (1973) suggested to use as expansion parameter (r − r e )/r instead of (r − r e )/r e . The SPF expansion parameter is superior in terms of both rate of convergence and region of convergence. As it has a proper behavior at large r, Table 3.2 Some typical valuesa of the Dunham ai potential constantsb defined by (3.14) a0 [cm−1 ]

a1

a2

a3

a4

a5

a6

LiF

154,087

−2.701384

5.05394

−7.65576

9.6986

−10.343

10.453

HBr

207,277

−2.43554

3.831169

−5.03908

5.863

−7.015

11.431

HCl

211,129

−2.36425

3.66281

−4.70624

5.2131

−5.5269

AgBr

240,184

−3.33655

7.39

−13.6

AlF

291,289

−3.183496

6.81574

−11.579

16.63

−22.51

29.89

BaO

358,703

−2.59066

4.0852

−6.888

15.80

−40.3

77

CCl+

434,564

−3.0005

5.7000

−8.400

9.47

a Source

Tiemann (1982, 1992); Hübner (1998) b Dimensionless, except a which is in cm−1 0

−7.75

8.3664

11.72

3.2 The Vibrational Energy

59

Table 3.3 Some typical valuesa of the Dunham constants Y 01 ≈ Be /MHz

Y 02 = − De /kHz

Y 11 = − α e /MHz

Y 10 ≈ ωe /cm−1

Y 20 = − ωe x e /cm−1

H35 Cl

317,580.97

−15945.8

−9208.96

2990.9664

−52.8364

H79 Br

253,850.6

−10381.5

−6997.58

2649.624

−45.444

12 C16 O

57,898.3404

−183.5202

−524.7536

2169.812593

−13.28794

2740.01367

−1.1294530

−11.927006

284.71102

−0.86123

205 Tl35 Cl a Source

Tiemann (1982, 1992); Hübner (1998)

it is mainly used for the expansion of the stretching potential of bonds in polyatomic molecules. Other expansion parameters are discussed in Le Roy (2011). The higher-order terms in the series expansion can be treated by a perturbation calculation, and the energy can be written as



1 1 2 − ωe xe υ + E V = ωe υ + 2 2

(3.16)

In this equation, all the parameters are in cm−1 , the unit normally used in infrared spectroscopy. The coefficient ωe x e is called anharmonicity constant. Higher-order terms may have to be introduced in (3.16); see (3.49). ωe x e is always positive and much smaller than ωe . For instance, for HCl, ωe = 2991 cm−1 and ωe x e = 52.8 cm−1 . Some typical values are given in Table 3.3.

3.3 The Rigid Rotor The rotation of any system is conveniently treated using the angular velocity Ω and the moment of inertia I. The distance between the two atoms, assumed to be point masses, is the bond length r = r A + r B . From the definition of the center of mass: mA r A = mB r B . The moment of inertia may be written as I = m ArA2 + m BrB2 =

mAmB 2 r = μr 2 mA + mB

(3.17)

In classical mechanics, the Hamiltonian is written as H=

P2 with P = I Ω 2I

(3.18)

Ω is the angular velocity not to be confused with the harmonic vibrational frequency ωe. In quantum mechanics, the Hamiltonian has the same form

60

3 Diatomic Molecules

H=

P2 2I

(3.19)

The eigenvalues of H are the rotational energy level of the molecule. It is possible to show that the angular momentum P is quantized with |P| =

h  J (J + 1) 2π

(3.20)

where J is a positive integer called rotational quantum number. The energy may be written in energy unit h2 J (J + 1) 8π 2 I

(3.21)

h J (J + 1) = B J (J + 1) 8π 2 I

(3.22)

ER = or in frequency unit ER =

B is called the rotational constant B=

h 505,379.07 or B (MHz) = 2 8π I I (uÅ2 )

(3.23)

The selection rule for J is J = J – J = 1 where J is the quantum number of the upper level and J the quantum number of the lower level. The frequency of a transition is then ν(J + 1 ← J ) = 2B(J + 1)

(3.24)

The rotational spectrum of a rigid diatomic molecule should be a series of equally spaced lines, the intensity of each line being proportional to the population of the lower energy level. According to Boltzmann distribution expression, it is Intensity ∝ μ2D ν 3 (J  + 1)e−E J " /kT

(3.25)

where μD is the electric dipole moment and ν the rotational frequency, (3.24). The rotational spectrum is usually observed in the microwave range, but for very light molecules, the rotational transitions are found in the infrared range. A molecule without dipole moment does not have any microwave spectrum. However, its rotational constant may be determined by Raman or infrared spectrum, thanks to different selection rules. See Chap. 5.

3.4 Vibrating Rotor

61

3.4 Vibrating Rotor A rough approximation is to assume that vibration and rotation do not interact. The total Hamiltonian is HT = HV + HR

(3.26)

ET = EV + ER

(3.27)

ψT = ψV ψR

(3.28)

the total energy

and the total wavefunction

Actually, when one compares a theoretical spectrum with the experimental spectrum of a single isotopologue, two main discrepancies appear: 1. Where one expects a single line, one finds a series of nearly equidistant lines whose intensity increases with the frequency. The less intense lines may be attributed to the excited states of vibration (υ  > 0). 2. When one considers the most intense lines of each series which are assumed to belong to the ground vibrational state (υ = 0), one observes that they are not equally spaced but that there is a deviation increasing with (J + 1)3 . It is explained by the fact that the molecule is not rigid. Going back to classical mechanics, it is obvious that the centrifugal force increases with the angular velocity and, therefore, the bond length is expected to increase with J. As the vibration motion is much faster than the rotation motion, one has to take into account the mean value of the bond length n r V =



ψV∗ r n ψV dτ

HT ψV ψR = HV ψV ψR +

(3.29)

hP2 1 ψV ψR 8π 2 μ r 2

= E T ψV ψR

(3.30)

Multiplying on the left by ψ V gives E V ψR +

1 hP2 ψV | 2 |ψV  = E T ψR 8π 2 μ r    HReff

(3.31)

62

3 Diatomic Molecules

To calculate the energy, one can use the virial theorem, which says that, for a quadratic potential T  = V  where T is the kinetic energy. Then, ET = T  + V  = 2V 

(3.32)

and V  =

1 r 2k k (r − re )2 υ = e ξ 2 υ = a0 ξ 2 υ 2 2

(3.33)

Equation (3.33) gives



2 1 hωe 1 2Be = υ + ξ υ= υ+ 2 kre2 2 ωe

(3.34)

In the harmonic approximation, the potential V is an even function; then ξ υ = 0. Using a series expansion gives 

1 r2



 1 1 = 2 = 2 (1 − 2ξ υ + 3 ξ 2 υ + · · · 2 re (1 + ξ ) υ r 

e  1 1 6Be = 2 1+ υ + re 2 ωe 

υ

The rotational constant in a vibrational state υ is then    

1 6Be 1 h Bυ = 1+ υ + = B e 8π 2 μ r 2 υ 2 ωe

(3.35)

(3.36)

One sees that, even in the vibrational ground state (υ = 0), the effective rotational constant B0 is different from the equilibrium rotational constant Be . This equation also explains the existence of vibrational satellites, each vibrational state having a different rotational constant. It is still necessary to take into account the anharmonicity. For that goal, it is possible to use Ehrenfest’s theorem which is the equivalent of Newton’s equation in quantum mechanics m

  ∂V d2 ξ υ = − = 2a0 ξ υ + 3a0 a1 ξ 2 υ + · · · = 0 2 dt ∂ξ υ

(3.37)

because is independent of time. Finally, 3 3 Be ξ υ = − a1 ξ 2 υ = − a1 2 2 ωe

(3.38)

3.4 Vibrating Rotor

63

The rotational constant in a vibrational state υ may be written as

1 Bυ = Be − αe υ + 2

(3.39)

with αe = −

6Be2 (1 + a1 ) ωe

(3.40)

The Dunham coefficient a1 is dimensionless, and its value is between −2 and −3; see Table 3.2. The consequence is that α e is always positive and that the anharmonicity is prevailing.

3.5 Centrifugal Distortion The rotational transitions for a given vibrational state υ should occur at intervals of 2Bυ . However, observation of the experimental spectrum shows that the intervals are smaller than 2Bυ and further decrease with increasing J. It is due to the centrifugal force whose effect is a slight increase of the bond length r. We will assume that the molecule is not vibrating but only rotating, and we will again suppose that the two atoms are bound by a spring-like bond which obeys Hooke’s law; see (3.1). The centrifugal force should be equal to the restoring force μr Ω 2 = k(r − re )

(3.41)

P = I Ω = μr 2 Ω

(3.42)

The angular momentum is

Then, k(r − re ) =

P2 P2 = μr 3 Ir

(3.43)

It is possible to express k as a function of ωe ; see (3.9) r − re =

J (J + 1) P2 h2 =r 2 2 2 4π μr I ωe ωe2 4π I

where use has been made of P2 = h2 J(J + 1) Equation (3.44) gives

(3.44)

64

3 Diatomic Molecules

r − re ξ= ≈ re



2Be ωe

2 J (J + 1)

(3.45)

Example: H35 Cl With Be = 317587 MHz and ωe = 2891 cm−1 , r – r e = 0.7 pm for J = 10 and 5.95 pm for J = 30. The Hamiltonian of the non-vibrating molecule is H=

P2 P2 + V = (1 − 2ξ ) + V 2μr 2 2μre2

(3.46)

Using the expression of r – r e , 3.44, the potential energy may be written in unit of frequency (i.e., dropping h) V =

B3 1 k(r − re )2 ≈ 4 e2 J 2 (J + 1)2 = De J 2 (J + 1)2 2 ωe

(3.47)

De is called quartic centrifugal distortion constant. Finally, the rotational energy, without vibration, is E R = Be J (J + 1) − De J 2 (J + 1)2

(3.48)

For the centrifugal distortion, the harmonic vibration is a good approximation in most cases.

3.6 Total Energy The total energy may finally be written (in frequency unit) using (3.10, 3.16, 3.22, 3.39, and 3.48)



E 1 1 2 1 3 = υ+ ωe − υ + ωe xe + υ + ωe ye + · · · h 2 2 2 + Be J (J + 1) − De J 2 (J + 1)2 + He J 3 (J + 1)3 + · · ·



1 1 2 J (J + 1) + γe υ + J (J + 1) + · · · − αe υ + 2 2

(3.49)

The energy is mainly defined by five parameters (higher-order parameters are also given in (3.49): ωe ye , H e , and γ e ): the harmonic vibrational frequency ωe , (3.7), the equilibrium rotational frequency, Be , (3.23), the quartic centrifugal distortion

3.6 Total Energy

65

constant, De , (3.48), the vibration–rotation interaction constant, α e , (3.40), and the anharmonicity constant, ωe x e , which may be expressed as a function of the other constants ωe xe = Be

αe ωe +1 6Be2

2 (3.50)

The typical order of magnitude is: ωe ≈ 104 GHz, Be ≈ 10 GHz, De ≈ 40 kHz, α e ≈ 200 MHz, and x e ≈ 0.02; see also Table 3.3. The five parameters defining the rovibrational energy may be calculated using the reduced mass μ which is a priori known and only three unknown parameters: the equilibrium bond length, r e , the harmonic force constant k, and the Dunham anharmonicity constant a1 . It is important to know how the parameters α e and De depend on μ, k, and r e . From (3.40), one get αe ∝ r −4 μ−3/2 k −1/2 or

αe 1 ∝√ Be kμ

(3.51)

De 1 ∝ Be kμ

(3.52)

and from (3.48) De = r −6 μ−2 k −1 or

When the mass of the molecule increases, α e and De decrease faster than Be and they decrease too when the force constant k increases. It is also possible to show that γ e /α e ∝ Be /ωe . It is also useful to check whether these parameters are independent. Indeed, the product kr 2e is approximately constant for a wide class of molecules and Badger (1934) found a more accurate relationship in which the equilibrium bond length is a simple function of the force constant k as well as two other constants depending on the row of the Mendeleev classification. A similar empirical formula was also found for the ai constants of the Dunham expansion. These empirical relations are not very useful for a diatomic molecule, but they would be very useful for a polyatomic molecule if it were possible to accurately determine force constants characteristic of the bond, which is rarely the case. However, there are a few exceptions, in particular for the CH bond. Because the mass of the hydrogen atom is much smaller than the masses of the other atoms, the frequency of the stretching vibration of the CH bond is much higher than the other vibrations and the coupling between this vibration and other vibrations is often negligible. In conclusion, when there is only one hydrogen atom in a molecule, there is an empirical relationship between the frequency ν(CHstretch ), called isolated stretching frequency, and r e (CH). This relationship may be used to determine r e (CH) with an accuracy of 0.2 pm, which is excellent for a polyatomic molecule; see Sect. 8.7.1. When there are several hydrogen atoms in the molecule, it is enough to replace them but one by deuterium. A similar relationship was also used to determine the Au–Au and Ag-Ag bond lengths (Perreault 1992).

66

3 Diatomic Molecules

More generally, when a vibrational mode has a characteristic frequency far from the others, it may be considered as isolated. It applies in particular to the OH bond. For more details, see Sect. 8.7.1.

3.7 Electronic Correction Before going further, it is now necessary to discuss the approximation that we have made: The center of mass of the electrons coincides with the nucleus. This is a very good approximation for most molecules. However, a small correction for unequal sharing of the electrons by the atoms and for non-spherical distribution of the electronic clouds around the atoms is sometimes non-negligible and has to be taken into account. This electronic correction can indeed be important for light molecules. We will only give here a brief summary, the full treatment being given in Sect. 4.10 for the polyatomic molecule. J the total angular momentum of a molecule may be written as the sum of N, the angular momentum due to the rotation of the nuclei, and L, the angular momentum of the electrons. The rotational Hamiltonian may be written as 1 (J − L)2 1 N2 + He = + He 2 I 2 I 1 J2 JL 1 L2 = + + He − I 2 I 2 I   

H=

H0 =HR +He

(3.53)

H

Since L is very small, the third term can be neglected and H can be treated as a perturbation of H0 . Equation (3.53) leads us to define an effective moment of inertia 1 2  |n|L|0|2 1 = − 2 Ieff I I n =0 E n − E 0

(3.54)

where I on the right is calculated using the nuclear masses. This effective moment of inertia can be expressed as a function of the molecular rotational g factor. The effective rotational constant Beff (obtained from the analysis of the rotational spectrum) is therefore Beff = B +

me gJ B n Mp

(3.55)

where B is the rotational constant calculated with atomic masses, Bn the rotational constant calculated with nuclear masses, me the mass of the electron, and M p the mass of the proton.

3.7 Electronic Correction Table 3.4 Electronic correction to the rotational constantsa (B0 and B in MHz)

67 ΔB = B − Bbeff

Molecule

B0

gJ

LiH

229,965.07

−0.6584

82.47

CO

57,635.97

−0.2691

8.45

HCl

312,989.3

PbS

3480.6

+0.45935

−78.31

−0.0644

0.12

a Source b See

Tiemann (1982, 1992); Hübner (1998) (3.55)

The gJ factor can be obtained experimentally from the analysis of the Zeeman effect on the rotational spectrum (Gordy and Cook 1984). gJ can also be calculated ab initio (Gauss et al. 1996). A few typical results are given in Table 3.4. As expected, the correction is the largest for very light molecules (as LiH) and it rapidly decreases when the mass of the molecule increases.

3.8 Definition of the Different Structures 3.8.1 Experimental Equilibrium Structure, re The rotational spectrum in its vibrational ground state gives B0 = Be − α e /2, and in the first excited vibrational state, it gives B1 = Be − 3α e /2 permitting to calculate Be = (3B0 – B1 )/2. The value of the equilibrium bond length is 

 re =

Ie = μ

h 8π 2 Be μ

(3.56)

3.8.2 Effective Structure, r0 The bond length is calculated using B0 , i.e., neglecting α e . As B0 = Be − α e /2, one gets for the ground-state moment of inertia I 0 as a function of the equilibrium moment of inertia I e

αe h h   = Ie 1 + (3.57) I0 = = 8π 2 B0 2Be 8π 2 Be − α2e because α e is much smaller than Be

68

3 Diatomic Molecules

Table 3.5 Structure of HCla r 0 (exp) − r 0 (calc)

H

Cl

B0 /MHz

r 0 /pm

1

35

312,989.2551

128.387028

1

37

312,519.084

128.386308

0.000015

2

35

161,656.313

128.124347

−0.000097

2

37

161,183.063

128.123405

−0.000059

3

35

111,075.84

128.010150

0.000113

3

37

110,601.62

128.008849

−0.000004

a Source

0.000032

Tiemann (1982, 1992); Hübner (1998)



 r0 =

I0 = μ



αe αe Ie = re 1 + 1+ μ 2Be 4Be

(3.58)

From (3.51), Sect. 3.6, it appears that the error decreases with the mass of the molecule. Replacing α e by its expression, (3.40), one finds that  3(1 + a1 ) −1/2 μ r0 = re − √ 4re k 

(3.59)

The term in brackets is a constant for a given molecule. It is then possible to rewrite this equation (Laurie 1958) A r0 = re + √ μ

(3.60)

where A is a positive constant for a given molecule. When r 0 has been determined for two different isotopologues, this equation permits to deduce r e . See the example of HCl in Table 3.5. The fit of the r 0 to (3.60) gives r e = 127.45858(16) pm. This value is close to the isotope-independent reBO value, 127.4606 pm calculated below with the help of (3.78); see below, Sect. 3.9.2. The small difference is mainly due to the fact that in the fit, contrary to (3.78), the breakdown of the Born–Oppenheimer approximation, was not taken into account.

3.8.3 Substitution Structure, rs (Costain 1958) Instead of using the ground-state moment of inertia I 0 , one uses the difference between the ground-state moments of inertia of two different isotopologues of rotational constants B0 and B0 ’ in the hope that it will eliminate most of the rovibrational correction.

3.8 Definition of the Different Structures

h I0 = 8π 2



1 1 − B0 B0



69



h αe αe Be2 = Ie + −1 16π 2 Be2 αe Be2

(3.61)

But as from (3.51) αe B 2 ω = e2  = αe Be ω



Be Be

3/2 (3.62)

Then h αe I0 = Ie + 16π 2 Be2



Be Be

1/2





αe − 1 = Ie 1 + 4Be

(3.63)

and the r s value is  rs =



αe I0 = re 1 + μ 8Be

(3.64)

re = 2rs − r0

(3.65)

re < rs < r0

(3.66)

which leads to (3.65)

and, as α e > 0

3.8.4 Zero-Point Structure rz (or rα0 ) This structure is the distance between average nuclear positions in the ground vibrational state at 0 K. This r z structure is interesting for two reasons: (i) It has a welldefined physical meaning, contrary to the r 0 or r s structures; (ii) it is also determinable by electron diffraction by conversion of the r g parameters taking into account the harmonic vibrational effects. The spectroscopic definition was given by Herschbach and Laurie (Herschbach and Laurie 1961). The B0 rotational constant is transformed into a Bz constant Bz = B0 +

αeharm α anharm B2 = Be − e with αeharm = −6 e 2 2 ωe

(3.67)

To calculate this structure, it is enough to know the harmonic force field. The relation between r e and r z is

70

3 Diatomic Molecules



3 Be = re + r r z = r e 1 − a1 2 ωe

(3.68)

r z > r0 > rs > re

(3.69)

As a1 < –1

One disadvantage of this structure is that it is not constant upon isotopic substitution. If we have two isotopologues 1 and 2, from (3.68), we get r2 Be (2) ωe (1) = = r1 Be (1) ωe (2)



μ1 μ2

(3.70)

When the isotopologue becomes heavier, the r z bond length becomes shorter.

3.9 Higher-Order Effects 3.9.1 Dunham Expansion Using the series expansion of the potential function, (3.14), Dunham (1932) has shown that the rovibrational energy of the molecule in a vibrational state υ and a rotational state J may be written as

 1 l k 1 E(υ, J ) = Ylk υ + J (J + 1)k h 2 l,k

(3.71)

With, for isotopologue α Ylkα = Ulk μ−(l+2k)/2 α

(3.72)

U lk is the Dunham isotope-independent parameter and μα the reduced mass of the isotopologue α, defined in (3.8). The first four potential constants of (3.14) can be evaluated from the Dunham constants ωe2 B2 Y2 = e ≈ − 01 4Be De Y02

(3.73a)

Y11 Y10 − 1 = −re a(Morse) 2 6Y01

(3.73b)

a0 = a1 =

3.9 Higher-Order Effects

a3 = −

71

7 Y12 Y01 1/2 9 19 = (are )2 a2 = + a1 (2 + a1 ) + 3 6 −Y02 8 8 12

(3.73c)

2 Y21 a2 a1 (are )3 + (3 + 13a1 ) − [4 + 3a1 (1 + a1 )] − 1 = − 15 Y02 5 2 4

(3.73d)

with the help of Y10 = 2

Be3 −Y02

1/2

Y20 = 1.5Be (a2 − 1.25a12 )

(3.74a) (3.74b)

With the exception of Y 01 , the Dunham constants are equivalent to the constants used in (3.49). The relationships are Y02 ≈ −De , Y11 ≈ −ae , Y10 ≈ ωe , Y20 ≈ −ωe xe

(3.75)

Some typical values are given in Table 3.3. The first Dunham coefficient Y 01 is approximately equal to the equilibrium rotational constant Be , the exact relation being (D) = Y01 = Be + Y01

 (D) + Y01 4π μre2

(3.76)

(D) Y01 is a small correction called Dunham correction (D) Y01



 Be3 21  2 3 15 + 14a1 − 9a2 + 15a3 − 23a1 a2 + a + a1 = 2ωe2 2 1

(3.77)

(D) This correction is rather small compared to Y 01 , for 12 C16 O: Y01 = −0.071 MHz (D) 35 and Y 01 = 57,898.34224 MHz and for H Cl: Y01 = −5.3 MHz and Y 01 = 317580.97 MHz; see also Table 3.6.

3.9.2 Breakdown of the Born–Oppenheimer Approximation Within the Born–Oppenheimer approximation, all the isotopologues of a molecule have the same molecular potential, which results in a single bond distance. Actually, the bond distance is found to be slightly dependent on the isotopologues. For instance, the equilibrium bond length of CO is 112.8336346(25) pm from 12 C16 O and 112.8327673(40) pm from 13 C18 O, i.e., a significant difference of 0.0008673(47) pm (Watson 1973). This variation was noted early, and following the pioneering work

72

3 Diatomic Molecules

Table 3.6 Derived parameters for the breakdown of the Born–Oppenheimer approximation (μg J )B MP

(01 )ad

r e /pm

−0.01526(6)

−1.8484(36)

−0.197(35)

112.82291(14)

−0.01526(6)

−1.8072(36)

−0.296(50)

−2.596(49)

−0.0148

−2.4758

−0.105(55)

CS/S

−2.223(98)

−0.0148

−1.9446

−0.264(110)

SiS/Si

−1.392(59)

−0.0106

−1.176

−0.205(69)

SiS/S

−1.870(65)

−0.0106

−1.5494

−0.310(75)

GeS/Ge

−1.463(70)

0.0008

−1.2244

−0.239(70)

GeS/S

−1.871(45)

0.0008

−1.6384

−0.233(45)

SnS/Sn

−1.76(19)

0.0058

−1.0608

−0.70(19)

SnS/S

−1.821(65)

0.0058

−1.6602

−0.167(25)

01

μY01 m e Be

CO/C

−2.061(34)

CO/O

−2.118(47)

CS/C

(D)

153.48224(23) 192.92639(19) 201.20431(10) 220.89829(22)

Source Tiemann et al.(1982a)

of Herman and Asgharian (1966), a coherent theory was proposed by Watson (1973, 1980). The correction to the Born–Oppenheimer approximation yields slight modification to the molecular potential, which becomes dependent on the nuclear masses and Y lk is now expressed as Ylk =

Ulk μ(l+2k)/2



me A me B 1+ lk +  mA m B lk

 (3.78)

where the U lk are mass-independent Dunham parameters, me is the electron mass, and mA and mB the masses of atom A and B. The ikl are Born–Oppenheimer breakdown parameters, of which only the i01 are significant. They may be expanded A A ad 01 = (01 ) +

(D) (μg J )B μY01 + MP m e Be

(3.79)

A ad A ) is the pure adiabatic part of 01 (01 Mp is the mass of the proton, (D) Y01 is defined in (3.77), and (μg J )B is the isotopically independent value of μgJ referred to the nucleus B as the origin (see Sect. 3.7)

(μg J )B = μg J + 2

Mp μD mA ere m A + m B

(3.80)

where μD is the signed electric dipole moment, e the electric charge of the electron, and r e the equilibrium bond length. When the structure is calculated using (3.78) but neglecting the adiabatic correction (01 )ad [i.e., taking into account the Dunham

3.9 Higher-Order Effects

73

correction, (3.77), and the electronic correction, (3.55)], it gives the adiabatic bond length read which differs for different isotopologues. As shown by Le Roy (1999), it is advantageous to choose one isotopologue as the reference (α = 1) to which the Dunham parameters of all other species are related. The Dunham parameters for (α = 1) are given by  Ylk(α)

=

Ylk(1)

+

m (α) A

A δlk m (α) A

+

m (α) B

B δlk m (α) B



μ1 μα

k+l/2 (3.81)

(α) (α) (1) where m (α) X is the mass of atom X(A, B) and m X = m X − m X and μα is the X X is reduced mass of isotopologue α. The relationship between δlk and lk

X X lk = −δlk

 −1 m (1) X A B Ylk(1) + δlk + δlk me

(3.82)

3.9.3 Effect of the Size of the Nuclei In addition to the mass variation by isotopic substitution also, the nuclear size will vary slightly giving rise to small changes in the Coulomb interaction between the electrons and the nucleus (Tiemann et al. 1982b). When at least one of the atoms is heavy, its nucleus can no longer be considered as a point charge and the charge is distributed over a volume to produce the so-called field-shift contribution. This may B be pointed out by the large difference found for A 01 and 01 , which are expected to be approximately equal (Watson 1980). This isotope effect which is called field shift in the theory of atomic spectra slightly modifies (3.79) where the mean square nuclear charge radius A,B is used as expansion parameter and the new molecular A,B is introduced parameter V01 Y01 =

U¯ 01 μ(l+2k)/2



1 + me

A B 01 + 01 mA mB

+

A V01 δ

2 B r AA + V01 δ r 2 BB

 (3.83)

with   U¯ 01 = U01 1 + V01A r 2 A + V01B r 2 B

(3.84)

A,B depends mainly on the electron density and its The field-shift parameter V01 derivatives with respect to the internuclear distance at the nucleus A or B. It is isotopically independent. The difference δ r 2 AA is the mean square nuclear charge radius on isotopic substitution A → A . These differences are tabulated as well as the values of (Angeli and Marinova 2013). This finite nuclear size correction is only significant when the accuracy of the measurements is extremely high and when

74 Table 3.7 Derived correction parameters of the rotational constant Y 01 for the breakdown of the Born–Oppenheimer approximation and for the field shifta

3 Diatomic Molecules AB

A 01

B 01

V A (104 Å−2 )b

208 Pb32 S

−12.94(141)

−1.997(71)

0

−1.333

−1.988(70)

2.45(19)

−11.86(92)

−2.120(76)

0

−1.520

−2.094(72)

2.21(19)

208 Pb130 Te

−11.98(21)

−1.794(110)

0

−1.405

−1.84(11)

2.12(16)

205 Tl35 Cl

−18.96(200)

−1.243(49)

0

−0.500

−1.257(73)

4.09(55)

208 Pb80 Se

a Source bV

B

Schlembach and Tiemann (1982) is fixed at zero

the mass number of the atom is not too small (>40). The field-shift parameter can be determined experimentally or calculated ab initio (Cooke et al. 2004) using A V01 =



Z A e2 dρel A 3ε0 kre dr re

(3.85)

Z A is the atomic number for atom A, ε0 the permittivity of free space, k the harmonic force constant, and ρ el the electronic density. Cl As a typical example, for TlCl, a fit to (3.78) gives Tl 01 = −21.5(64) and 01 = Tl Cl −1.30(19). Using (3.83), the results are 01 = −0.500 and 01 = −1.14(6) with Tl Cl = 4.09(55) × 10−4 Å−2 and V01 assumed to be zero; see Table 3.7. (Note that, V01 205 for Tl, the root-mean-square nuclear charge radius is 1/2 = 5.4759 fm and that δ205→203 = −0.0978 fm2 .) Actually, it is difficult to determine all the parameters of (3.83) by a least-squares fit, the system of normal equations being usually ill-conditioned (Giuliano et al. 2008); see Sects. 9.4.1 and 9.7. Finally, using U01 = μBeBO , the equilibrium bond length may be determined  reBO

=

h 8π 2 U01

(3.86)

3.10 Direct Potential Fit (DPF) Most equilibrium structures published up to now were obtained with either (3.78) or (3.83). However, there is a more sophisticated method permitting to determine the equilibrium structure. It is based on the fact that the Hamiltonian is one-dimensional and can be solved efficiently using standard numerical methods. The experimental

3.10 Direct Potential Fit (DPF)

75

data are directly fitted to the parameters defining an analytic potential energy function. It allows us to use more sophisticated potential energy functions, which extrapolate realistically at both large and small distances. The difficulty is that the fit is non-linear. This method has been recently reviewed by Le Roy (2011), and a typical example is the analysis of the rovibrational spectra of hydrogen halides by Coxon and Hajigeorgiou (2015).

3.11 Conclusion Spectroscopy allows us to determine the Born–Oppenheimer equilibrium structure of diatomic molecules as light as LiH and as heavy as PbTe with an accuracy better than 10−3 pm using (3.78) (or (3.83) when a heavy atom is present); see Tables 3.6 and 3.8. This is much better than can be achieved by ab initio methods. However, a word of caution is needed: The given uncertainties correspond to statistical errors. Among others, they do not take into account the systematic errors. For this reason, the true uncertainty is generally one order of magnitude larger; see, for instance, Table 3.8 Born–Oppenheimer bond lengths (in pm) for some diatomic molecules Molecule

recorr a

r be

References

ArD+

128.0375(7)

128.0349

Laughlin et al. (1987)

LiH

159.490811(16)

159.5595

Bellini et al. (1995)

LiCl

202.0700(8)

202.6914

Watson (1973)

BF

126.2672(7)

126.2762

Cazzoli et al. (1989)

CO

112.82427(6)

112.8336

Watson (1973)

112.82428(6)

Le Floch (1991)

112.8230(1) CS HF HCl

153.48192(12) 91.683897(4) 127.46149(9)

Authier et al. (1993) 153.4943 91.68942 127.4572

Coxon and Hajigeorgiou (2015) Watson (1973)

127.460400(108)

Ogilvie (1994)

127.460651(2)

Odashima (2006)

127.454677(6) HBr

Bogey et al. (1982)

141.4426(5)

Coxon and Hajigeorgiou (2015) 141.4465

141.44843(2)

Coxon and Hajigeorgiou (1991) Odashima (2006)

141.44292(1)

Coxon and Hajigeorgiou (2015)

PtS

203.8553(4)

203.9822

Cooke and Gerry (2004)

PbTe

259.4975987(11)

259.4977

Giuliano et al. (2008)

from U 01 , (3.78), or U¯ 01 , (3.83) b Calculated from Y , (3.56; 3.76) 01 a Calculated

76

3 Diatomic Molecules

the examples of HCl and CO in Table 3.8. In this table, the r e structure calculated from Y 01 , (3.56), (3.76), i.e., neglecting the breakdown of the Born–Oppenheimer approximation is also given. Although the effect of the breakdown is not constant, it shows that it is not sensible to determine a structure with a precision higher than about 0.007 pm (median absolute deviation) when this breakdown is neglected. For the sake of completeness, it has to be remembered that the accuracy of the final structure also depends on the accuracy of the Planck constant and of the atomic masses. However, these constants are now highly accurate and the effect of their uncertainty is generally negligible. Nevertheless, tiny differences may be observed in some cases. For instance, the equilibrium bond length of 194 Pt32 S is 203.98221 pm when calculated with (3.56) using the constants tabulated in Gordy and Cook (1984) whereas it is 203.98284 pm when calculated using the most recent constants (Cook and Gerry 2004).

References Angeli I, Marinova KP (2013) Table of experimental nuclear ground state charge radii: an update. At Data Nucl Data Tables 99:69–95 Authier N, Bagland N, Le Floch A (1993) The 1992 evaluation of mass-independent Dunham parameters for the ground state of CO. J Mol Spectrosc 160:590–592 Badger M (1934) A Relation between internuclear distances and bond force constants. J Chem Phys 2:128–131 Barrow GM (1962) Introduction to molecular spectroscopy. McGraw-Hill, New York Bellini M, De Natale P, Inguscio M, Varberg TD, Brown JM (1995) Precise experimental test of models for the breakdown of the Born–Oppenheimer separation: the rotational spectra of isotopic variants of lithium hydride. Phys Rev A 52:1954–1960 Bogey M, Demuynck C, Destombes JL (1982) Millimeter and submillimeter wave spectrum of CS 1  in high vibrational states: Isotopic dependence of Dunham coefficients. J Mol Spectrosc 95:35–42 Brown JM, Carrington A (2003) Rotational spectroscopy of diatomic molecules. Cambridge University Press Cazzoli G, Cludi L, Degli Esposti C, Dore L (1989) The millimeter and submillimeter-wave spectrum of boron monofluoride: equilibrium structure. J Mol Spectrosc 134:159–167 Cooke SA, Gerry MCL (2004) Internuclear distance and effects of born-oppenheimer breakdown for PtS, determined from its pure rotational spectrum. J Chem Phys 121:3486–3494 Cooke SA, Gerry MCL, Chong DP (2004) The calculation of field shift effects in the rotational spectra of heavy metal-containing diatomic molecules using density functional theory: comparison with experiment for the Tl-halides and Pb-chalcogenides. Chem Phys 298:205–212 Costain CC (1958) Determination of molecular structures from ground state rotational constants. J Chem Phys 29:864–874 Coxon JA, Hajigeorgiou PG (1991) Isotopic dependence of Born-Oppenheimer breakdown effects in diatomic hydrides: the X1 + states of HI/DI and HBr/DBr. J Mol Spectrosc 150:1–27 Coxon JA, Hajigeorgiou PG (2015) Improved direct potential fit analyses for the ground electronic states of the hydrogen halides: HF/DF/TF, HCl/DCl/TCl, HBr/DBr/TBr and HI/DI/TI. J Quant Spectrosc Radiat Transf 151:133–154 Dunham JL (1932) The energy levels of a rotating vibrator. Phys Rev 41:721–731 Gauss J, Ruud K, Helgaker T (1996) Perturbation-dependent atomic orbitals for the calculation of spin-rotation constants and rotational g tensors. J Chem Phys 105:2804–2812

References

77

Giuliano BM, Bizzocchi L, Cooke S, Banser D, Hess M, Fritzsche J, Grabow J-U (2008) Pure rotational spectra of PbSe and PbTe: potential function, Born-Oppenheimer breakdown, field shift parameter and magnetic shielding. Phys Chem Chem Phys 10:2078–2088 Gordy W, Cook RL (1984) Microwave molecular spectra. Wiley, New York Herman RM, Asgharian A (1966) Theory of energy shifts associated with deviations from BornOppenheimer behavior in 1 -state diatomic molecules. J Mol Spectrosc 19:305–324 Herschbach DR, Laurie VW (1961) Anharmonic potential constants and their dependence upon bond length. J Chem Phys 35:458–464 Herzberg G (1950) Spectra of diatomic molecules. Van Nostrand, New York Hübner H (1998) Diatomic molecules. In: Molecular constants mostly from microwave, molecular beam and sub-Doppler laser spectroscopy. In: Hüttner W (ed) Landolt-Börnstein, vol 24. Springer, Heidelberg Laughlin KB, Blake GA, Cohen RC, Hovde DE, Saykally RJ (1987) Determination of the dipole moment of ArH+ from the rotational Zeeman effect by tunable far infrared laser spectroscopy. Phys Rev Lett 58:996–999 Laurie VW (1958) Note on the determination of molecular structure from spectroscopic data. J Chem Phys 28:704–706 Le Floch A (1991) Revised molecular constants for the ground state of CO. Mol Phys 72:133–144 Le Roy RJ (1999) Improved parametrization for combined isotopomer analysis of diatomic spectra and its application to HF and DF. J Mol Spectrosc 194:189–196 Le Roy RJ (2011) Determining equilibrium structures and potential energy functions for diatomic molecules. In: Demaison, J, Boggs JE, Császár AG (eds) Equilibrium molecular structures. CRC Press, Boca Raton, pp 159–203 Morse PM (1929) Diatomic molecules according to the wave mechanics. II. Vibrational Levels. Phys Rev 34:57–64 Odashima H (2006) Isotopically invariant analysis of vibration-rotation transitions of HBr and its isotopologues. J Mol Spectrosc 240:69–74 Ogilvie JF (1994) Quantitative analysis of adiabatic and non-adiabatic effects in vibration-rotational spectra of diatomic molecules. J Phys B 27:47–61 Perreault D, Drouin M, Michel A, Miskowski VM, Schaefer WP, Harvey PD (1992) Silver and gold dimers. Crystal and molecular structure of Ag2(dmpm)2Br 2 and [Au2(dmpm)2](PF6)2 and relation between metal-metal force constants and metal-metal separations. Inorg Chem 31:695– 702 Schlembach J, Tiemann E (1982) Isotopic field shift of the rotational energy of the Pb-chalcogenides and Tl-halides. Chem Phys 68:21–28 Simons G, Parr RG, Finlan JM (1973) New alternative to the Dunham potential for diatomic molecules. J Chem Phys 59:3229–3234 Tiemann E (1982) Diatomic molecules. In: Hellwege K-H, Hellwege AM (eds) Molecular constants mostly from microwave, molecular beam and electron resonance spectroscopy. Landolt-Börnstein, vol 14a. Springer, Heidelberg Tiemann E (1992) Diatomic molecules. In: Hüttner W (ed) Molecular constants mostly from microwave, molecular beam and sub-Doppler laser spectroscopy. Landolt-Börnstein , vol 19a. Springer, Heidelberg Tiemann E, Arnst H, Stieda WU, Törring T, Hoeft J (1982a) Observed adiabatic corrections to the Born-Oppenheimer approximation for diatomic molecules with ten valence electrons. Chem Phys 67:133–138 Tiemann E, Knöckel H, Schlembach J (1982b) Influence of the finite nuclear size on the electronic and rotational energy of diatomic molecules. Ber Bunsenges Phys Chem 86:821–824 Watson JKG (1973) The isotope dependence of the equilibrium rotational constants in 1 states of diatomic molecules. J Mol Spectrosc 45:99–113 Watson JKG (1980) The isotope dependence of diatomic Dunham coefficients. J Mol Spectrosc 80:411–421

Chapter 4

Rotation of the Polyatomic Molecule

Abstract The rotational spectroscopy of the polyatomic molecule is reviewed. The determination of the rotational constants for the different types of rotors is discussed. The rovibrational correction and the electronic correction to the rotational constants are also described. Finally, the experimental methods are briefly presented.

4.1 Introduction This chapter is not intended to be a comprehensive review of rotational spectroscopy because there are several reference books: Bunker and Jensen (1998), Gordy and Cook (1984), Kroto (1992), Papousek and Aliev (1982), Perrin et al. (2011), Quack and Merkt (2011). Only the notions essential to understand how experimental data and structural information will be given. We will initially consider a molecule of N atoms assumed to be point masses before examining the quantum mechanics of molecular rotation in Sects. 4.4–4.8. We will furthermore first assume that this molecule is rigid: The distances between the atoms remain constants, in other words, there is no vibration. The experimental observables of moment of inertia and of rotational constant will be introduced. These parameters play an essential part in a structure determination. From the study of the diatomic molecule, it is known that the non-rigidity effects are not negligible. Centrifugal distortion will be discussed succinctly as well as the vibrational dependence of the rotational constants. The rovibrational correction to the rotational constants will be discussed in more detail in Sects. 5.5, 5.6, and 6.2. In particular, the Coriolis interaction, which is sometimes important for the analysis of the spectra and for the determination of rotational constants in vibrationally excited states, will be introduced in Sect. 5.5.

© Springer Nature Switzerland AG 2020 J. Demaison and N. Vogt, Accurate Structure Determination of Free Molecules, Lecture Notes in Chemistry 105, https://doi.org/10.1007/978-3-030-60492-9_4

79

80

4 Rotation of the Polyatomic Molecule

4.2 Classical Kinetic Energy of the Rigid Rotor We will use two Cartesian coordinate systems to describe free rotation under no external potential: 1. Space-fixed, XYZ, origin O, the “observer’s” axis system. This axis system is used to describe the position of the center of mass of the molecule. 2. Rotating, xyz, molecule fixed axis system, origin G: center of mass. The orientation of the axes x, y, z with respect to the space—fixed system is determined by three Euler angles θ, φ, and ψ. θ = zOZ and φ are the usual polar coordinates of the z axis in the XYZ system, and ψ is an angle in the xy plane measuring the rotation anticlockwise about the z axis, see Fig. 4.1 We consider an atom P in the molecule. We have OP = OG + GP = R + r

(4.1)

If there is an infinitesimal motion dP of P dOP = dR + dϕ × r

(4.2)

˙ +ω×r V=R

(4.3)

If we divide by dt

˙ is the velocity of translation (translation will be neglected in the following) where R and ω the radial velocity, it is a vector whose direction coincides with the axis of Fig. 4.1 Coordinate axes and Eulerian angles. Source modified version of the figure coming from https:// commons.m.wikimedia.org/ wiki/File:Euler_angles_zxz_ ext.png and licensed under the Creative CommonsAttribution-Share Alike 4.0 International

4.2 Classical Kinetic Energy of the Rigid Rotor

81

rotation. The kinetic energy T of the molecule may be written 2T =

N 

m α Vα2

N 

=

α=1

˙2 2T = R

N 

(4.4)

α=1

mα +

α=1

 2 ˙ m α R+ω × rα

N 

˙ ·ω m α (ω × rα )2 + 2R

α=1

N 

m α rα

(4.5)

α=1

N is the number of atoms, the first term is the energy  N of translation, and the last term is null because the origin is the center of mass: a=1 m a ra = 0 The kinetic energy of rotation is 2TR =

N 

m a (ω × ra )2 =

a=1

N 

  m a ω2 ra2 − (ω · ra )2

(4.6)

a=1

Or 2TR = ωx2

N 

N N         m a ya2 + z a2 + ω2y m a xa2 + z a2 + ωz2 m a ya2 + xa2

a=1

− 2ωx ω y

a=1 N 

a=1

m a xa ya − 2ω y ωz

a=1

N 

m a ya z a − 2ωx ωz

a=1

N 

m a xa z a

(4.7)

a=1

This equation may be rewritten using the moment of inertia tensor I which is a 3 × 3 symmetric matrix ⎞ Ix x Ix y Ix z I = ⎝ Ix y I yy I yz ⎠ Ix z I yz Izz ⎛

(4.8)

Its diagonal elements are (with g, g , g = x, y, z by cyclic permutation) Igg =

N 

  m a ga2 + ga2

(4.9)

a=1

and the non-diagonal elements Igg = −

N  a=1

Finally,

m a ga ga

(4.10)

82

4 Rotation of the Polyatomic Molecule

2TR = ωx2 Ix x + ω2y I yy + ωz2 Izz + 2ωx ω y Ix y + 2ω y ωz I yz + 2ωx ωz Ix z

(4.11)

or in matrix notation ˜ 2TR = ωIω

(4.12)

  ˜ = ωx ω y ωz is the transpose of the vector ω. where ω It is always possible to rotate the molecule fixed axis system, so that the offdiagonal terms of I vanish. The new axes are called principal inertial axes, and the three elements of the diagonal tensor are called principal moments of inertia (eigenvalues of the inertia matrix). By convention, they are noted I a ≤ I b ≤ I c . • When the three moments of inertia are identical, the molecule is called spherical top • When the three moments of inertia are different, the molecule is called asymmetric top • When two moments of inertia are different, the molecule is called symmetric top – If I b = I c , the molecule is called prolate top – If I a = I b , the molecule is called oblate top. A particular case is the linear molecule. If z is the axis of the molecule, x a = ya = 0 and Ix = I y =

N 

m a z a2 and Iz = 0

(4.13)

a=1

In this case, rotations are possible around the x and y axes and there are only two degrees of freedom.

4.3 Angular Momentum The angular momentum P may be written P=

N  a=1

m a ra × va =

N 

m a ra × (ω × ra )

(4.14)

a=1

In the principal inertial axis system, the expression of the angular momentum is a diagonal 3 × 3 matrix with Pa = I a ωa , etc. For a molecule in free rotation, E = T R = constant and |P| = constant. The rotation is defined by two equations

4.3 Angular Momentum

83

2E = Ia ω2 + Ib ω2 + Ic ω2 a c b P 2 = I 2 ω2 + I 2 ω2 + I 2 ω2 a a c c b b

(4.15)

P2 P2 2E = 2H = Iaa + Ibb + 2 P = Pa2 + Pb2 + Pc2

(4.16)

or Pc2 Ic

4.4 Rotational Hamiltonian of the Rigid Rotor In quantum mechanics, (4.16) remains valid, but the physical magnitudes

are replaced by operators. The operators Pg contain the Planck’s constant  = h 2π . In spectroscopy, it is common to define the Hamiltonian in units of frequency (i.e., divided by h), so the transformed Hamiltonian may be written   Pb2 h Pa2 Pc2 = APa2 + BPb2 + CPc2 H= + + 8π 2 Ia Ib Ic

(4.17)

with the rotational constants A=

h h h ≥B= ≥C = 8π 2 Ia 8π 2 Ib 8π 2 Ic

(4.18)

The conversion factor may be calculated h = Ia (uÅ2 ) · A(MHz) = 505,379.005(50) 8π 2

(4.19)

The matrix elements of the angular momentum operators may be found in the courses of quantum mechanics. For a symmetric top rotor with z as axis of symmetry, the diagonal elements are (Gordy and Cook 1984) J K M|Pz2 |J K M = K 2 2 J K M|P2x |J K M = J K M|P2y |J K M =

 2  J (J + 1) − K 2 2

(4.20a)

(4.20b)

and, using P2 = P2x + P2y + Pz2 J K M|P2 |J K M = 2 J (J + 1) and the off-diagonal elements are

(4.21a)

84

4 Rotation of the Polyatomic Molecule

2 [J (J + 1) − K (K ± 1)]1/2 4 [J (J + 1) − (K ± 1)(K ± 2)]1/2

(4.21b)

J K M|P2x |J K ± 2M = −J K M|P2y |J K ± 2M

(4.21c)

J K M|P2y |J K ± 2M =

In these expressions, J, K, and M are angular momentum quantum numbers with J = 0, 1, 2, . . . K = J, J −1, J −2, . . . , −J (projection along the internal symmetry axis) M = K = J, J −1, J −2, . . . , −J (projection along the space-fixed axis) The third quantum number, M, is relevant only in the presence of an electric or a magnetic field. In the following, we will drop it.

4.5 Symmetric Top 4.5.1 Rigid Rotor Approximation Assuming that z is the symmetry axis, I x = I y and the Hamiltonian may be written P2 + H= 2Ix



 1 1 Pz2 − 2Iz 2Ix

(4.22)

The rotational energy may be written E J K = J K |H|J K  =

   2 J (J + 1) 1 1 K2 + − 2 Ix Iz Ix

(4.23)

If the molecule is a prolate top, A=

h h h >B= =C = 2 2 8π Iz 8π Ix 8π 2 I y

(4.24)

The energy in frequency unit is EJK =

E J K = (A − B)K 2 + B J (J + 1) h

If the molecule is an oblate top,

(4.25)

4.5 Symmetric Top

85

A=

h h h =B= >C = 2 2 8π I y 8π Ix 8π 2 Iz

(4.26)

and the corresponding rotational energy in frequency unit is EJK =

E J K = (C − B)K 2 + B J (J + 1) h

(4.27)

Provided that the molecule has a permanent electric dipole moment, the selection rules for absorption or emission of electromagnetic radiation are J = 0, ±1 and K = 0 for K = 0

(4.28)

J = ±1 and K = 0 for K = 0

(4.29)

and

The absorption frequencies are obtained as ν = 2B(J + 1)

(4.30)

It has to be noted that all transitions of same J but different K have the same value. Actually, the inclusion of centrifugal distortion may resolve this degeneracy, see next section.

4.5.2 Centrifugal Distortion From the discussion of the diatomic molecule, Sect. 3.5, it appears that the first term of the rotational energy is quadratic in J (or, more exactly, is a function of P2 ), whereas the next term, the centrifugal distortion term, is quartic in J (function of P4 ). We may anticipate that the same behavior will apply to the symmetric top. In other words, we will have a term in J 2 (J + 1)2 , J(J + 1)K 2 , and K 4 . The rotational energy taking into account the centrifugal distortion may be written E J K = (A − B)K 2 + B J (J + 1) − D J J 2 (J + 1)2 h − D J K J (J + 1)K 2 − D K K 4

EJK =

(4.31)

This equation is valid for a prolate top. For an oblate top, it is enough to replace A by C. Taking into account the selection rules, the expression for the rotational frequencies is

86

4 Rotation of the Polyatomic Molecule

ν = 2B(J + 1) − 4D J (J + 1)3 −2D J K (J + 1)K 2

(4.32)

4.5.3 Determination of the Axial Rotational Constant Due to the selection rule K = 0, the axial rotational constant (A for a prolate top, C for an oblate top) cannot be obtained from the rotational spectrum. However, there are two powerful methods that permit to determine it from the infrared spectrum. The first one uses perturbation-allowed transitions. These transitions are normally forbidden but, thanks to a resonance between two nearby levels, the forbidden transition borrows some intensity to an allowed transition and becomes observable and is called perturbation allowed transition, see Sect. 21 of Papousek and Aliev (1982). The frequency difference between allowed transitions and “forbidden” transitions permit to obtain the axial rotational constants. With the other method, called loop method, energy differences in the ground state with two different K values are derived through the analysis of a fundamental degenerate band ν t  , a hot band (ν t + ν t  )—ν t  and the corresponding combination band (ν t + ν t  ) (For the definition of the bands, see Sect. 5.3). See an example in Fig. 4.2. For more details, see Graner and Bürger (1997).

Fig. 4.2 Scheme of levels in propyne, CH3 C≡CH used to obtain ground-state combination difference between K and K + 3 (K = |k|). Source Graner et al. (1988)

4.6 Linear Molecule

87

4.6 Linear Molecule The linear molecule is a special case of the symmetric molecule with I z = 0 and I x = I y . Thus, the rotational energy is E J = B J (J + 1)−D J J 2 (J + 1)2 + · · ·

(4.33)

as for the diatomic molecule.

4.7 Rotational Spectra of Linear and Symmetric Tops in Excited Vibrational States A linear molecule with N atoms has 3 N−5 modes of vibration, but the bending modes are always doubly degenerate because the deformation of the angle may occur in two orthogonal planes. For instance, in the case of a non-symmetric linear molecule XYZ, there are two non-degenerate parallel modes ν1 and ν3 which do not affect the linearity and correspond to the stretching vibrations of the bonds XY and YZ, and there is a doubly degenerate bending mode ν2 which corresponds to the deformation of the angle XYZ. The rotational spectra of the non-degenerate states behave in the same way as a ground state. On the other hand, as the molecule is no more linear in the bending state, there is an angular momentum pz =  along the symmetry axis with the rotational quantum numbers  = ±υ, ±(υ − 2), . . . ± 1 or 0.  is analogous to the K quantum number of the symmetric top. The energy in a degenerate state may be written E υ = Bυ J (J + 1) + (Aυ −Bυ )2 + · · ·

(4.34)

The term Aυ 2 is treated as vibrational energy, and the rotational energy is E υ = Bυ [J (J + 1) −2 ]−Dυ [J (J + 1)−2 ]2 + · · ·

(4.35)

With the selection rule J = 1 and  = 0. Finally, the rotational frequency is ν = 2Bυ (J + 1)−4Dυ (J + 1)3 + · · ·

(4.36)

Equation (4.36) is identical to the equation used for the ground vibrational state but with the difference that J = ||, || + 1, || + 2, . . . In other words, for an excited bending state with || = 1, the lowest value of J is 1. States with  = 0 are labeled

states, those with  = ±1, ±2, ±3 are called , ,  states, respectively. In a symmetric top molecule, there are also degenerate vibrations, in particular those involving motions of the atoms which are off-axis. For a prolate top, the term

88

4 Rotation of the Polyatomic Molecule

−2(Aυ ζ )K  should be added to (4.34) and for an oblate top A should be replaced by C.

4.8 Asymmetric Molecule 4.8.1 Rigid Rotor The matrix elements of the rigid rotor Hamiltonian may be easily found in the basis of the symmetric top eigenfunctions |J K M, but the resulting matrix which is a (2J + 1)× (2J + 1) symmetric matrix will not be diagonal in K because of the presence of terms J K M|H|J K ± 2M. The eigenvalues of this matrix are the rotational energy. J remains a good quantum number but not K because of the presence of non-diagonal elements in K. The energy levels of a slightly asymmetric rotor differ from the limiting symmetric top one because the levels −K and +K are separated, whereas they are degenerated in the symmetric rotor. By connecting the levels of the limiting prolate top with those of the symmetric oblate top, it is possible to label the levels: E(JK a K c ). The first subscript K a represents the K value of the limiting prolate top, and K c represents the K value of the limiting oblate top. K a and K c are called pseudo-quantum numbers. The selection rule is unchanged for J, i.e., J = 0, ± 1, but there are also restrictions for the pseudo-quantum numbers, which depend of the orientation of the permanent electric dipole moment μ. The rules are summarized in the table below for the possible variations of K a K c where e means even and o odd. μa = 0

μb = 0

μc = 0

ee⇔eo

ee⇔oo

ee⇔oe

oe⇔oo

oe⇔eo

eo⇔oo

4.8.2 Centrifugal Distortion The rotational Hamiltonian may be expanded as Hrot =

 β=a,b,c

B β Pβ2 +



  T βγ Pβ2 + Pγ2 + · · ·

(4.37)

β,γ

where T βγ are the quartic centrifugal distortion terms. Watson (1977) has shown that only five combinations of these terms are determinable experimentally and that there are two ways to reduce the Hamiltonian: the A-reduction and the S-reduction. The

4.8 Asymmetric Molecule

89

consequence is that the experimental rotational constants are slightly affected by the reduction. The following linear combinations can be determined from the analysis of the spectra: Bz = Bz(A) + 2 J = Bz(S) + 2D J + 6d2

(4.38a)

Bx = Bx(A) + 2 J + J K − 2δ J − 2δ K = Bx(S) + 2D J + D J K + 2d1 + 4d2

(4.38b)

B y = B y(A) + 2 J + J K + 2δ J + 2δ K = B y(S) + 2D J + D J K − 2d1 + 4d2 .

(4.38c)

Bξ(A) are the experimental constants in the so-called A-reduction, Bξ(S) are the experimental constants in the so-called S-reduction, and Bξ are the determinable constants (where ξ = x, y, z). However, these latter constants are still contaminated by the centrifugal distortion. As shown by Watson (1968a, b) and Kivelson and Wilson (1952), the rigid rotor constants Bξ are given by Bx = Bx +

 1 1 τ yyzz + τx yx y + τx zx z + τ yzyz 2 4

(4.39a)

where B y and B z are obtained by cyclic permutation of x, y, and z. The problem is that, experimentally, the τ constants are approximately determinable only for a planar molecule by means of so-called planarity relations, see Appendix 1. For a non-planar molecule, they can be calculated from the harmonic force field. Compared to the other corrections, the centrifugal distortion correction is generally quite small except for very light molecules. Furthermore, this correction is different from zero only for asymmetric top molecules, but in this case, it generally remains much larger than the experimental accuracy of the ground state rotational constants. In the calculations of semiexperimental equilibrium structures (see Chap. 6), this correction is generally ignored because it is often smaller than the uncertainty of the final equilibrium rotational constants. A typical example is given in Table 4.1 (see also Table 6.8). Nevertheless, it is worth noting that this correction is easily computed from the (ab initio) quadratic force field. Thus, there is no difficulty to take it into account, if necessary. However, there are additional terms to (4.39a). Watson (1968b) has shown that there is a mass-dependent contribution to the potential energy. This contribution has the effect of displacing the equilibrium configuration of a particular isotope slightly from the isotopic invariant configuration at the minimum of the potential. This shift of origin will show up in the values of the rotational constants whose equilibrium values are Bx = Bx −

 1 τx x x x + τx x yy + τx x zz 8

(4.39b)

90

4 Rotation of the Polyatomic Molecule

Table 4.1 Corrections to the ground-state rotational constants of trans–formic acid, HCOOH, ketene, H2 C=C=O, and furane, c-C4 H4 O (all values in MHz) Molecule

X

Experimental

Distortiona

Electronica

Vibrationala,b

HCOOHc

A

77512.2354(11)

−0.0443

11.81

403.53

B

12055.1065(2)

−0.1402

0.59

88.03

C

10416.1151(2)

0.1606

0.15

89.97

A

282101.19(41)

−0.5517

66.407

1482.83

B

10293.3212(8)

−0.0325

0.203

23.623

C

9915.9055(8)

0.3090

0.136

37.781

H2

CCOd

Furanee

A

9447.12291(17)

−0.0039

0.4727

79.9622

c-C4 H4 O

B

9246.74363(16)

−0.0037

0.4633

70.1748

C

4670.82538(21)

0.0057

−0.1311

39.8993

aX

corrected –X exp b From the MP2/VTZ

anharmonic force field et al. (2007) d Guarnieri et al. (2010) e Demaison et al. (2011) c Demaison

This correction can be important for light molecules but there are still additional terms whose detailed form has not been discussed so far. For this reason, this last correction, (4.39b), is generally neglected.

4.9 Rovibrational Correction The analysis of the spectra gives the rotational constants Bυξ for a given vibrational state υ. In a perturbational treatment, the rotational constant Bυξ is given by (Mills 1972)   dk ξ αk υk + 2 k      ξ dj di ξ υj + + γi j υi + γli l j i  j + · · · + 2 2 i≥ j i≥ j

Bυξ = Beξ −



(4.40)

where ξ = a, b, c. The summations are over all vibrational states, each characterized by a quantum number υ i and a degeneracy d i . Beξ is the equilibrium rotational constant, and αiξ and γiξj are the vibration–rotation interaction constants of different order. The last term γlξi l j is different of zero only for degenerate modes. The convergence ξ

of this expansion and the determination of the αi are discussed in Sect. 6.2. The

4.9 Rovibrational Correction

91 ξ

perturbations affecting the αi are described in Sect. 5.5 for the Coriolis interactions and in Sect. 5.6 for the anharmonic resonances.

4.10 Electronic Correction (Gordy and Cook 1984; Sutter and Flygare 1976) The largest correction to the ground-state rotational constants is the rovibrational correction, see Sect. 4.9 and Table 4.1 (see also Sect. 6.2). However, for some molecules. The electronic correction is not negligible. This correction has already been discussed in Sect. 3.7 of Chap. 3 for the particular case of a diatomic molecule. The generalization to polyatomic molecules is straightforward. Atomic masses are used to calculate the rotational constants. However, a small correction for unequal sharing of the electrons by the atoms and for non-spherical distribution of the electronic clouds around the atoms is sometimes non-negligible and has to be taken into account. The discussion will be limited to molecules of zero spin and zero electronic angular momentum in the ground electronic state. It is the case of a great majority of molecules. A more rigorous derivation may be found in Sutter and Flygare (1976). The total angular momentum J of a molecule may be written as the sum of N, the angular momentum due to the rotation of the nuclei, and L, the angular momentum of the electrons. The rotational Hamiltonian for the nuclear system plus the Hamiltonian for the unperturbed electronic energies may be written as  2 1  Jξ − Lξ 1  Nξ2 + He = + He H= 2 ξ Iξ 2 ξ Iξ =

 Jξ Lξ 1  Lξ2 1  Jξ2 + He − + 2 ξ Iξ Iξ 2 ξ Iξ ξ       H0 =HR +He

(4.41)

H

Since Lξ is very small, the third term can be neglected, and H’ can be treated as a perturbation of H0 . We now assume that the molecule is not in a pure 1 state ψ0(0) (L = 0) but in a perturbed state ψ0(1) , which has some electronic momentum. The correct effective rotational Hamiltonian is then   (4.42) Heff = ψ0(1) H R + H ψ0(1) A perturbation calculation up to second order gives

92

4 Rotation of the Polyatomic Molecule

Heff

⎛ 2 ⎞ 1  2⎝ 1 2  n|Lξ |0 ⎠ = J − 2 2 ξ ξ Iξ Iξ n=0 E n − E 0

(4.43)

Equation (4.43) leads to a definition of an effective moment of inertia (I ξ )eff 2 1 1 2  n|Lξ |0 = − 2 (Iξ )eff Iξ Iξ n=0 E n − E 0

(4.44)

where I ξ on the right is calculated using the nuclear masses. This effective moment of inertia can be expressed as a function of the molecular rotational g factor in the principal axis system, whose definition is gx x =

 2Mp  |n|Lx |0|2 Mp   2 Z i yi + z i2 − Ix i m Ix n=0 E n − E 0

(4.45)

and gyy and gzz are obtained by cyclic permutation. In this equation, M p is the mass of the proton and m the mass of the electron. The effective rotational constant Beff (obtained from the analysis of the rotational spectrum) is therefore 

Bξ

 eff

= Bξ +

  m gξ ξ B ξ n Mp

(4.46)

where ξ = a, b, c, and Bξ is the rotational constant calculated with atomic masses, and (Bξ )n is the rotational constant calculated with nuclear masses. The g factor can be obtained experimentally from the analysis of the Zeeman effect on the rotational spectrum (Sutter and Flygare 1976), but it is now much easier to calculate the g factor ab initio (Gauss et al. 1996). A few typical results are given in Table 4.1. As expected, the correction is the largest for very light molecules (as LiH) and rapidly decreases when the mass of the molecule increases. There are, however, a few exceptions. As the expression of g shows, see (4.45), g may become large when an electronic excited state is close to the ground state, (because the denominator E n −E 0 is small). This is the case for ozone (O3 ), where the electronic correction is extremely large: gaa = −2.9877(9) (28) leads to a huge electronic correction of −173 MHz for the A rotational constant.

4.11 Determination of the Rotational Constants The geometry of a molecule is defined by a set of internal coordinates: bond lengths (usually but not necessarily) between bonded atoms, bond angles between adjacent bonds, and dihedral angles. A set of non-redundant internal coordinates is equal to the

4.11 Determination of the Rotational Constants

93

number of normal modes (see Chap. 5). To determine a structure from the rotational constants, it is essential to use a non-redundant set which may require some expertise (Mendolicchio et al. 2017). On the other hand, to calculate rotational constants from the geometry, any convenient set may be used. Except for very simple molecules, it is difficult to express the moments of inertia as a function of the internal coordinates. For instance, it is easy to show that the moment of inertia of a linear molecule X 1 , X 2 , … X n may be written I =

n 1  m i m j ri2j M i> j

(4.47)

where M is the mass of the molecule, mi the mass of atom i, and r ij the distance between atoms i and j. See Appendix 2 for the expression of some moments of inertia. More generally, the moments of inertia of a molecule are calculated in two steps: First the internal coordinates are converted into Cartesian coordinates. Thompson (1967) proposed an elegant method for this conversion. Then, it is easy to calculate the moments of inertia using (4.9) and (4.10). To obtain the rotational constants from an experimental spectrum, once the spectrum is assigned, it is necessary to fit the rotational and centrifugal distortion constants using (4.32) to (4.37) with the help of dedicated computer programs (see Sect. 4.13). As the rotational constants are mainly used to obtain the geometrical structure of molecules, their practical determination will be discussed in Chap. 6, Sect. 6.3, together with the calculation of the structures.

4.12 Experimental Techniques There are many reviews on this subject. A recent one on the technical aspects is Grabow and Caminati (2009). Two other reviews report recent applications of microwave spectroscopy (Caminati and Grabow 2009; 2018). Microwave spectroscopy covers a huge frequency scale (5–8 orders of magnitude larger than linewidths), so it is not possible to cover the full range with a single technique. Typically, spectroscopy in the cm-, mm- and sub-mm wavelengths is considered separately. Presently time-domain techniques are dominant, with frequency-domain techniques mostly used in the mmw and sub-mmw ranges.

4.12.1 Frequency-Domain Microwave Spectroscopy Microwave spectroscopy started with a paper by Cleeton and Williams (1934) in which they describe the observation of the 1.26 cm inversion line of NH3 , made with a magnetron. However, the real birth of microwave spectroscopy started after

94

4 Rotation of the Polyatomic Molecule

the Second World War thanks to availability of microwave hardware developed for radars (microwave sources–mainly klystrons and detectors). The simplest spectrometer consists of a coherent source (historically a klystron), a cell containing the gas at low pressure and a detector (crystal diode) measuring the microwave power. The cell is generally a rectangular waveguide of 1–3 m length sealed at each end with a microwave transmitting window and connected to a vacuum line. The use of such a spectrometer is extremely limited (except in the millimeterwave range where the absorption is stronger) because: (i) the absorption of the gas is weak (ii) there are strong spurious cell absorptions iii) the noise due to the detector is important. The sensitivity is improved by applying a modulation to the source. A tuned amplifier with a finite pass band permits to reduce the noise by two orders of magnitude. The most used spectrometer up to the eighties was the Stark-modulated spectrometer introduced by Hughes and Wilson (1947). An on–off square-wave electric field is applied to the gas. When the field is on, the rotational energy levels are split by the Stark effect, and the Stark components are at a different frequency of the zero-field line. The effect is the same as the instantaneous appearance and disappearance of the gas. The signals are thus modulated at the frequency of the square-wave and are detected using a narrow-band amplifier followed by a phase-sensitive detection technique, which allows the signal components, which are phase-coherent with the reference voltage to reach the output. Thanks to the Stark spectrometer thousands of spectra were recorded. However, this kind of spectroscopy has several weaknesses: It lacks of resolution, its sensitivity is sometimes not good enough (in particular the measure the spectra of isotopologues in natural abundance), and for large molecules, the recorded spectra are extremely crowded and difficult to assign. To improve the resolution (and marginally the sensitivity), a superheterodynedetection bridge-spectrometer with double phase-demodulation was proposed (Rudolph and Schwoch 1971) with no success. To improve the sensitivity, thanks to the advent of minicomputers (and later microcomputers), computer-controlled Stark-effect spectrometers were used which allowed adding up scans, thus improving greatly the signal/noise ratio (Gwinn et al. 1968). Actually, the best method to increase at the same time the resolution and the sensitivity is to use the Fourier transform (FT) technique as will be described below in Sect. 4.12.2. There are two main ways to simplify the spectra. The first one is to use a gas jet into which the sample is seeded. Adiabatic cooling of the rotational and vibrational degrees of freedom in a supersonic jet considerably simplifies the rotational spectrum as only low-J transitions in the ground vibrational state are observed. The second method is double resonance (Woods et al. 1966; Baker 1979; Jones 1979). Consider three energy levels |1 > , |2 > , and |3 > with respective populations N 1 , N 2 , and N 3 . Assume that transitions are allowed between |1 > and |2 > , and |2 > and |3 > , see Fig. 4.3. The sample is irradiated simultaneously at two different frequencies: The pump at the frequency of the transition ν 12 = |2 > ← |1 > is used to saturate it, while the second source (signal) is at the frequency of the transition

4.12 Experimental Techniques

95

Fig. 4.3 A three-level system used to illustrate double resonance

ν 23 = |3 > ← |2 > . One thus observes the changes in population produced by the pump radiation by means of the lower power signal source. The pump increases the population N 2, whereas the absorption coefficient of the signal is proportional to the difference N 2 −N 3 . If the pump is modulated, only one transition appears when the pump frequency is scanned, whereas the signal frequency is kept at the value ν 23 . Conversely, the pump can be kept at the frequency ν 12, whereas the signal is scanned. The pump and the signal can be microwave or infrared sources. Another advantage of this method is that it permits to observe weak transitions.

4.12.2 Time-Domain Microwave Spectroscopy The development of Fourier transform microwave spectroscopy (FTMW) coupled to static samples or supersonic jet expansion was a big step forward because of the large increase in resolution and the observation of transient species such as weakly bound molecular species. Last but not least, thanks to FTMW, it is less difficult to study low-population species like isotopic species in natural abundance. A special mention should be given to the chirped-technique, which will replace most experiments in a few years, even in the millimeterwave range. In frequency-scanning microwave spectroscopy, each spectral element is studied one at a time. However, a considerable improvement in S/N and resolution can be obtained if all the spectral elements are studied simultaneously. This is the multiplex mode of operation. This method is well known and was first used in NMR spectroscopy and, later, in infrared spectroscopy. The gas is excited by a series of regularly spaced microwave pulses using fast-switching microwave diodes. After each pulse, the molecules transiently relax by spontaneous emission. Since the detection of the signal takes place in the absence of any microwave power, the S/N is unaffected by any source noise. Furthermore, there is no power nor modulation broadening increasing the resolution. The signal is averaged for signal-to-noise improvement and treated by fast Fourier transform to give the spectrum. A hundreds of MHz bandwith can be covered by a single-pulse train. The first FTMW spectrometer was described by Ekkers and Flygare (1976). A considerable improvement was brought by (Balle et al. 1980) (see also Balle and Flygare 1981) by combining the techniques of FTMW, a Fabry–Perot cavity, and

96

4 Rotation of the Polyatomic Molecule

a pulsed nozzle source of molecules synchronized with the microwave pulse. Thanks to the jet the molecules are cooled considerably simplifying the spectra and is well adapted to the study of weakly bound complexes. The Fabry–Perot cavity brings a significant enhancement in sensitivity and resolution. The weak point is the bandwidth of individual measurements that is severely restricted (about 1 MHz). However, the scanning is usually automated, so that the spectrum over a frequency range of several GHz can be obtained without user intervention. However, this scanning is still time consuming. Figure 4.4 shows the block diagram of such a spectrometer with laser ablation. This last difficulty was solved by Pate et al. (Brown et al. 2008; Shipman and Pate 2011) by using a high-speed (>4 G samples/s) arbitrary waveform generator to generate a “chirped” microwave pulse that sweeps up to 12 GHz in less than one μs combined with the detection of the full frequency band in a single experimental event. In this way, broadband operation (typically 10 GHz) can be achieved routinely. This spectrometer has a sensitivity per pulse smaller than the Balle–Flygare spectrometer, but the large reduction in measurement time allows for increased sensitivity over long scans. In the last years, the use of chirped-pulse techniques has become standard and will probably replace other MW techniques. Chirped-pulse techniques can be combined with double-triple resonance experiments and extended to the mmw-range. Very recently (Patterson et al. 2013; Lobsiger et al. 2015), chirped-pulse methods

Fig. 4.4 Block diagram of a Fourier transform microwave (FTMW) spectrometer with a supersonic jet and laser ablation

4.12 Experimental Techniques

97

were used to detect molecular chirality, opening new perspectives for this technique. For reviews on this subject, see Grabow (2013) and Pate et al. (2018).

4.12.3 Millimeterwave and Submillimeterwave Spectroscopies With the exception of some backward-wave oscillators which can operate up to 1 THz (Winnewisser et al. 1994; Petkie et al. 1997), the conventional sources used in the microwave range do not operate well at short wavelengths. One of the oldest methods to obtain millimeterwaves and submillimeterwaves is the frequency multiplication of a microwave source (King and Gordy 1953, 1954). In the early days, this method required a lot of skill and was used by very few groups. Now, thanks to commercial multipliers, this technique is much easier to use up to about 3 THz (Drouin et al. 2005; Pearson et al. 2011). A more recent technique consists in producing a sideband source by photomixing two lasers (Pine et al. 1996) or by mixing a far-infrared laser with a microwave source (Verhoeve et al. 1990). A third technique is to use a conventional Fourier transform infrared spectrometer. However, thermal sources give little power in the millimeter and submillimeter region. This difficulty was recently solved by the use of a synchrotron radiation (SR) as source. The application of SR to this field was pioneered in the mid-1990s by Bengt Nelander at the MAX I storage ring in Lund, Sweden, and has become more widespread since the opening of new dedicated facilities. A recent spectrometer is described by Barros et al. (2015). Finally, free-electron lasers such as FELIX (Oepts et al. 1995) can also be used in the far-infrared range.

4.13 Software • The analysis of rovibrational data has benefited from the introduction by Pickett (1991) of a general code based on tensor algebra, allowing for flexible introduction of custom Hamiltonian terms. The use of the SPFIT and SPCALC programs is now standard. They calculate energies and intensities for asymmetric rotors and linear molecules with up to 999 vibrational states and up to 9 spins. SPFIT is used for fitting transitions and term values. SPCALC is used for predicting line positions and strengths. • JB95 from David F. Plusquellic is a graphical user interface program written in the C programming language to aid in the analysis of complex molecular spectra. Resources are provided for the deconvolution of multiple overlapping rotational

98

4 Rotation of the Polyatomic Molecule

bands from different conformations of a molecule and for the analysis of molecular spectra when internal rotation, centrifugal distortion, nuclear quadrupole coupling interactions, large-amplitude motions, and inertial-frame reorientation effects are resolved. (https://www.nist.gov/services-resources/software/jb95-spe ctral-fitting-program) • AABS is a program for the assignment and analysis of broadband spectra (Kisiel et al. 2005). • PGOPHER is a program used for the automatic assignment and fitting of spectra (Western and Billinghurst 2019). A collection of programs for various aspects of the rotational spectroscopy problem are available free of charge from http://www.ifpan.edu.pl/~kisiel/prospe. htm.

Appendix 1 Centrifugal distortion correction for a planar molecule. From the experimental quartic centrifugal distortion constants, it is possible to determine a set of five determinable combinations (Watson 1977) Tx x = − J − 2δ J = −D J + 2d1 + 2d2

(4.48a)

Tyy = − J + 2δ J = −D J − 2d1 + 2d2

(4.48b)

Tzz = − J − J K − K = −D J − D J K − D K

(4.48c)

T1 = −3 J − J K = −3D J − D J K − 6d2

(4.48d)

1 T2 = −(Bx + B y + Bz ) J − (Bx + By ) JK + (Bx − By )(δJ + δK ) 2 1 1 = −(Bx + B y + Bz )D J − (Bx + B y )D J K − (Bx − B y )d1 − 6Bz d2 2 2 (4.48e) where the notation of Watson (1977) has been used Tαα =

1 ταααα 4

(4.49a)

Tαβ =

1 τααββ 4

(4.49b)

Appendix 1

99

There are seven quartic centrifugal distortion constants for a planar asymmetric top

∗ = τabab 4 for which the symmetry plane is ab: T aa , T bb , T cc , T ab , T bc , T ca , and Tab (Note that τ caca = τ bcbc = 0). The constants T 1 and T 2 are linear combinations of the last four constants of the former set ∗ T1 = Tab + Tbc + Tca + 2Tab

(4.50)

∗ T2 = ATbc + BTca + C Tab + 2C Tab

(4.51)

To obtain all the constants, the planarity relations have to be used (Dowling 1961)   Taa 1 2 2 Tbb Tcc Tbc = B C − 4 + 4 + 4 2 A B C   Taa 1 Tbb Tcc Tca = C 2 A2 − + 2 A4 B4 C4

(4.52) (4.53)

and Tab = And

  Taa 1 2 2 Tbb Tcc A B − 4 − 4 + 4 2 A B C

∗ Tab may be obtained from Eq. (4.50)

(4.54)

(4.55)

∗ , and they are discussed by Actually, there are different ways to determine Tab Yamada and Winnewisser (1976).

Appendix 2 Expression for the moments of inertia of some simple molecules (the symmetry group and one example are given in parentheses.

1 M i≥ j

n 

m Xi m X j rX2 i X j

(D3h , PF5 )

XY5

c

(D4h , XeF4 )

planar, XY4

(C3v , NH3 ) θ 2

  θ θ mXmY 2 + rXY 3 − 4 sin2 2 M 2

2 2 I x = I y = 3m Y req + 2m Y rax

2 Iz = 3m Y req

2 Iz = 4m Y rXY

Ix = I y = Iz 2

2 sin2 Iz = 4m Y rXY

2 I x = I y = 2m Y rXY sin2

I x = I y = Iz /2

(D3h , BF3 )

pyramidal, XYb3

2 Iz = 3m Y rXY

2 I = 4m Y rXY

2 I = 83 m Y rXY

I =

Moments of inertiaa

planar, XY3

Symmetric

(Oh , SF6 )

XY6

(Td , CH4 )

XY4

Spherical

(C∞υ , HC ≡ CF)

X1 X2 · · · XaN

Linear

Molecule

(continued)

100 4 Rotation of the Polyatomic Molecule

2 Iz = 4m Y rXY sin2

θ 2

  θ θ 4 3m Y 2 2 I x = I y = 2m Y rXY sin2 + 1 − sin2 (m X + m Z )rXY 2 M 3 2    θ 4 m Z rXZ + (3m Y + m X )rXZ + 6m Y rXY 1 − sin2 M 3 2

Moments of inertiaa

d The

atoms i and j, mi is the mass of atom i, M is the total mass of the molecule, and z the figure axis (principal symmetry axis)

bipyramidal molecule, r eq and r ax are the equatorial and axial bond length, respectively y-axis is perpendicular to the xz plane, with z the C 2 axis. I y = I z + I x

c Trigonal

ij is the distance between b θ is the YXY angle

ar

θ 2 Iz = 2m Y rXY sin2 2   1 θ θ 2 2 + 2m Y (m X + m Z )rXY cos2 + 4m Y m Z rXZ rXY cos Ix = [m Z (2m Y + m X )rXZ M 2 2

(C2v , H2 CO)

planar, ZXY2 b,d

Asymmetric

(C3v , CH3 F)

axial, ZXY3 b

Molecule

(continued)

Appendix 2 101

102

4 Rotation of the Polyatomic Molecule

References Baker JG (1979) Microwave-microwave double resonance. In: Chantry GW (ed) Modern aspects of microwave spectroscopy. Academic Press, London, pp 65–122 Balle TJ, Flygare WH (1981) Fabry-Perot cavity pulsed Fourier transform microwave spectrometer with a pulsed nozzle particle source. Rev Sci Instrum 52:33–45 Balle TJ, Campbell EJ, Keenan MR, Flygare WH (1980) A new method for observing the rotational spectra of weak molecular complexes: KrHCl. J Chem Phys 72:922–932 Barros J, Evain C, Roussel E, Manceron L, Brubach J-B, Tordeux M-A, Couprie M-E, Bielawski S, Szwaj C, Labat M, Roy P (2015) Characteristics and development of the coherent synchrotron radiation sources for THz spectroscopy. J Mol Spectrosc 315:3–9 Brown GG, Dian BC, Douglass KO, Geyer SM, Shipman ST, Pate BH (2008) A broadband Fourier transform microwave spectrometer based on chirped pulse excitation. Rev Sci Inst 79:053103 Bunker PR, Jensen P (1998) Molecular symmetry and spectroscopy. NRC Research Press, Ottawa Caminati W, Grabow J-U (2009) Microwave molecular systems. In: Laane J (ed) Frontiers of molecular spectroscopy. Elsevier, Amsterdam, pp 455–552 Caminati W, Grabow J-U (2018) Advancements in microwave spectroscopy. In: Laane J (ed) Frontiers and advances in molecular spectroscopy. Elsevier, Amsterdam, pp 569–598 Cleeton CE, Williams NH (1934) Electromagnetic waves of 1.1 cm wave-Length and the absorption spectrum of ammonia. Phys Rev 45:234–237 Demaison J, Herman M, Liévin J (2007) Anharmonic force field of cis- and trans-formic acid from high-level ab initio calculations, and analysis of resonance polyads. J Chem Phys 126:164305 Demaison J, Császár AG, Margulès L, Rudolph HD (2011) Equilibrium structures of heterocyclic molecules with large principal axis rotations upon isotopic substitution. J Phys Chem A 115:14078–14091 Dowling G (1961) Centrifugal distortion in planar molecules. J Mol Spectrosc 6:550–553 Drouin BJ, Maiwald FW, Pearson JC (2005) Application of cascaded frequency multiplication to molecular spectroscopy. Rev Sci Instrum 76:093113 Ekkers J, Flygare WH (1976) Pulsed microwave Fourier transform spectrometer. Rev Sci Inst 47:448–454 Gauss J, Ruud K, Helgaker T (1996) Perturbation-dependent atomic orbitals for the calculation of spin-rotation constants and rotational g tensors. J Chem Phys 105:2804–2812 Gordy W, Cook RL (1984) Microwave molecular spectra. Wiley, New York Grabow J-U (2013) Fourier transform microwave spectroscopy: Handedness caught by rotational coherence. Angew Chem Int Ed 52:11698–11700 Grabow J-U, Caminati W (2009) Microwave spectroscopy: experimental techniques. In: Laane J (ed) Frontiers of molecular spectroscopy. Elsevier, Amsterdam, pp 383–454 Graner G, Bürger H (1997) Hot bands in infrared spectra of symmetric tops and some other molecules. A useful tool to reach hidden information. In: Papousek D (ed) Vibration-rotational spectroscopy and molecular dynamics. World Scientific, Singapore, pp 239–297 Graner G, Demaison J, Wlodarczak G, Anttila R, Hillman JJ, Jennings DE (1988) A preliminary determination of the A0 rotational constant of propyne. Mol Phys 64:921–932 Guarnieri A, Demaison J, Rudolph HD (2010) Structure of ketene-revisited r e and r m structures. J Mol Struct 969:1–8 Gwinn WD, Luntz AC, Sederholm CH, Millikan R (1968) On-line control, data collection, and reduction for chemical experiments at Berkeley. J Comput Phys 2:439–464 Hughes RH, Wilson EB (1947) A microwave spectrograph. Phys Rev 71:562–563 Jones H (1979) Infrared-microwave double resonance techniques. In: Chantry GW (ed) Modern aspects of microwave spectroscopy. Academic Press, London, pp 123–216 King WC, Gordy W (1953) One to two millimeterwave spectroscopy. I Phys Rev 90:319–320 King WC, Gordy W (1954) One to two millimeterwave spectroscopy IV. Experimental methods and results for OCS, CH3 F, and H2 O. Phys Rev 93:407–412

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Kisiel Z, Pszczolkowski L, Medvedev IR, Winnewisser M, De Lucia FC, Herbst E (2005) Rotational spectrum of trans–trans diethyl ether in the ground and three excited vibrational states. J Mol Spectrosc 233:231–243 Kivelson D, Wilson EB (1952) Approximate treatment of the effect of centrifugal distortion on the rotational energy levels of asymmetric-rotor molecules. J Chem Phys 20:1575–1579 Kroto HW (1992) Molecular rotation spectra. Dover, New York Lobsiger S, Perez C, Evangelisti L, Lehmann KK, Pate BH (2015) Molecular structure and chirality detection by Fourier transformmicrowavespectroscopy. J Phys Chem Lett 6:196–200 Mendolicchio M, Penocchio E, Licari D, Tasinato N, Barone V (2017) Development and implementation of advanced fitting methods for the calculation of accurate molecular structures. J Chem Theory Comput 13:3060–3075 Mills IM (1972) Vibration-rotation structure in asymmetric and symmetric molecules. In: Rao KN, Mathews CW (eds) Molecular spectroscopy: modern research, vol 1. Academic Press, New York, pp 115–140 Oepts D, van der Meer AFG, van Amersfoort PW (1995) The free-electron-laser user facility FELIX. Infrared Phys Technol 36:297–308 Papousek D, Aliev MR (1982) Molecular vibrational-rotational spectra. Elsevier, Amsterdam Pate BH, Evangelisti L, Caminati W, Xu Y, Thomas J, Patterson D, Pérez C, Schnell M (2018) Quantitative chiral analysis by molecular rotational spectroscopy. In: Polavarapu PL (ed) Chiral analysis: advances in spectroscopy, chromatography and emerging methods, 2nd edn. Elsevier, Amsterdam, Chapter 17, pp 679–729 Patterson D, Schnell M, Doyle JM (2013) Enantiomer-specific detection of chiral molecules via microwave spectroscopy. Nature 497:475–478 Pearson JC, Drouin BJ, Maestrini A, Mehdi I, Ward, J, Lin RH, Yu S, Gill JJ, Thomas B, Lee C, Chattopadhyay G, Schlecht E, Maiwald FW, Goldsmith PF, Siegel P (2011) Demonstration of a room temperature 2.48–2.75 THz coherent spectroscopy source. Rev Sci Instrum 82:093105 Perrin A, Demaison J, Flaud J-M, Lafferty WJ, Sarka K (2011) Spectroscopy of polyatomic molecules: determination of the rotational constants. In: Demaison J, Boggs JE, Császár AG (eds) Equilibrium molecular structures. CRC Press, Boca Raton, pp 89–124 Petkie DT, Goyette TM, Belov SP, Bettens RPA, Albert S, Helminger P, De Lucia FC (1997) A fast scan submillimeter spectroscopic technique. Rev Sci Instrum 68:1675–1683 Pickett HM (1991) The fitting and prediction of vibration-rotation spectra with spin interactions. J Mol Spectrosc 148:371–377 Pine AS, Suenram RD, Brown ER, McIntosch KA (1996) A terahertz photomixing spectrometer: application to SO2 self broadening. J Mol Spectrosc 175:37–47 Quack M, Merkt FJ (eds) (2011) Handbook of high resolution spectroscopy. Wiley, New York Rudolph HD, Schwoch D (1971) A High-resolution bridge-type microwave spectrometer for molecular rotational spectroscopy. Z Angew Phys 31:197–204 Shipman ST, Pate BH (2011) New techniques in microwave spectroscopy. In: Quack M, Merkt FJ (eds) Handbook of high resolution spectroscopy. Wiley, New York, pp 801–828 Sutter DH, Flygare WH (1976) The molecular Zeeman effect. In: Boscchke FL (ed) Topics in current chemistry, vol 63, Springer, Berlin, pp 91–196 Thompson HB (1967) Calculation of Cartesian coordinates and their derivatives from internal molecular coordinates. J Chem Phys 47:3407–3410 Verhoeve P, Zwart E, Versluis M, Drabbels M, TerMeulen JJ, Meerts WL, Dymanus A, Mclay D (1990) A far infrared laser sideband spectrometer in the frequency region 550–2700 GHz. Rev Sci Instrum 61:1612–1625 Watson JKG (1968a) Determination of centrifugal distortion coefficients of asymmetric-top molecules. J Chem Phys 46:1935–1949 Watson JKG (1968b) Simplification of the molecular vibrational hamiltonian. Mol Phys 15:479–490 Watson JKG (1977) Aspect of quartic and sextic centrifugal effects on rotational energy levels. In: Durig JR (ed) Vibrational spectra and structure, vol 6. Elsevier, Amsterdam, pp 2–89

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Western CM, Billinghurst BE (2019) Automatic and semi-automatic assignment and fitting of spectra with PGOPHER. Phys Chem Chem Phys. 21: 13986–13999 Winnewisser G, Krupnov AF, Tretyakov MY, Liedtke M, Lewen F, Saleck AH, Schieder R, Shkaev AP, Volokhov SV (1994) Precision broadband spectroscopy in the terahertz region. J Mol Spectrosc 165:294–300 Woods RC, Ronn AM, Wilson EB (1966) Double resonance modulated microwave spectrometer. Rev Sci Instr 37:927–933 Yamada K, Winnewisser M (1976) Comments on quartic centrifugal distortion constants for asymmetric top molecules. Z Naturforsch A 31:131–138

Chapter 5

The Vibrations of Polyatomic Molecules

Abstract The vibrational spectroscopy of the semirigid (i.e., without largeamplitude motions) polyatomic molecule is reviewed. The normal modes are defined. The importance of molecular symmetry is pointed out. The Coriolis interaction and the anharmonic resonances are discussed.

5.1 Introduction This chapter is limited to the introduction of the notions permitting to understand how the vibrations affect the structure determination. Reference books are: Bunker and Jensen (1998), Califano (1976), Herman et al. (1999), Papousek and Aliev (1982), Perrin et al. (2011), Quack and Merkt (2011), and Wilson et al. (1955). We will consider a molecule of N atoms assumed to be point masses. A real molecule is not rigid, its atoms vibrate around their equilibrium position. One may distinguish three different kinds of motion: translation (t), rotation (r), and vibrations (v). Note that to vibrations belong internal rotation and ring puckering. The total kinetic energy T may be written T = Tt + Tr + Tv + Ttr + Ttv + Trv

(5.1)

The subscripts have the following meaning: tr = interaction between translation and rotation, tv = interaction between translation and vibration, and rv = interaction between rotation and vibration. One may choose the molecule fixed axis system (see Chap. 4) so that the coupling terms in (5.1) are minimal. This is done with the help of six equations called Eckart conditions, see (5.17). The T tr and T tv terms are strictly zero because the molecule is isolated. But the last term, T rv , is only minimized. It is called Coriolis energy and may be important in some cases. It will be discussed in Sect. 5.5. First, we will only consider the vibrations. The position of the atoms may be described by 3N Cartesian coordinates. One needs three coordinates to describe the motion of the center of mass of the molecule © Springer Nature Switzerland AG 2020 J. Demaison and N. Vogt, Accurate Structure Determination of Free Molecules, Lecture Notes in Chemistry 105, https://doi.org/10.1007/978-3-030-60492-9_5

105

106

5 The Vibrations of Polyatomic Molecules

and, for nonlinear molecules, three coordinates to describe the rotation of the molecule about its center of mass. There are 3N − 6 degrees of freedom left for the vibrations. If the molecule is linear, molecular rotations can only occur about the two axes that are perpendicular to the molecular axis. In that case, there are only 3N − 5 degrees of freedom.

5.2 Classical Kinetic Energy of the Rigid Rotor Let x α , yα , zα be the instantaneous coordinates of the αth atom and aα , bα , cα the values of these coordinates at equilibrium in the molecule fixed axis system. Displacement from equilibrium will be defined by δxα = xα −aα , δyα = yα −bα , and δz α = z α −cα . The kinetic energy T may be written (in the following, the vectors and the matrices will be noted in bold) 2T =

N 

+

m α vα2 = ξ˙ Mξ˙ = q˙ + q˙ =

α=1

3N 

qi2

(5.2)

i=1

where + means transpose, vα is the velocity of atom α, and M is the 3N × 3N diagonal matrix of the masses, ξ+ = (δx1 , δy1 , δz 1 , . . . , δxn , δyn , δz n )

(5.3)

q = M1/2 ξ

(5.4)

and

It is possible to use a series expansion for the potential energy (assuming that the displacements qi remain small) 2V = 2V0 + 2

 3N   ∂V i=1

∂qi

 3N   ∂2V qi + qi q j + · · · ∂qi ∂q j 0 0 i, j=1

(5.5)

Neglecting the higher-order terms gives the harmonic approximation that is satisfactory as a first approximation and that we will assume in this section. Assuming that the energy of the equilibrium configuration is zero implies V 0 = 0 as well as for the first derivative   ∂V =0 (5.6) ∂qi 0 so that

5.2 Classical Kinetic Energy of the Rigid Rotor

2V =

3N 

107

f i j qi q j = q+ fq

(5.7)

i, j=1

with  f i j = f ji =

∂2V ∂qi ∂q j

 (5.8) 0

The Newton’s equations of motion may be written d ∂T ∂V + =0 dt ∂ q˙i ∂qi

(5.9)

Using the expressions of T and V, (5.2) and (5.7) give q¨i +

3N 

f i j q j = 0,

j = 1, 2, . . . , 3N

(5.10)

j=1

A possible solution for this set of simultaneous differential equations is  qi = li cos( λi t + ϕi )

(5.11)

Substituting this solution in the differential equations gives a set of equations 3N  

 f i j − δi j λi li = 0,

j = 1, 2, . . . , 3N

(5.12a)

i=1

with ωi =



λi /2π

(5.12b)

In other words, the λi are the eigenvalues of the force constants matrix f, the li are the elements of the eigenvector matrix, and ωi the harmonic vibration wave number. There are 3N eigenvalues but there are six values for translation and rotation that are close to zero. In the following, we will assume that they are zero, i.e., we will only consider the vibration. Equation (5.12a) may also be written using matrices, it gives 

    l+  0 f l l tr = l+ 0 0 tr

which may be rewritten in the following way

(5.13)

108

5 The Vibrations of Polyatomic Molecules

l + f l = dim.(3N − 6) × (3N − 6)

(5.14a)

f l tr = 0 dim.(3N − 6)

(5.14b)

The orthonormality of (l l tr ) leads to the relations l + l = E dim.(3N − 6) × (3N − 6)

(5.15a)

l+ tr l = 0 dim.6 × (3N − 6)

(5.15b)

5.3 Normal Modes Equation (5.11) shows that each atom oscillates around its equilibrium position with the harmonic frequency ωi = λ1/2 i /2π, phase ϕ i , and amplitude l i . For one given solution λi , each coordinate vibrates with the same frequency and the same phase but the amplitude may be different. Each atom reaches the maximum and the minimum of the displacement at the same time. Such a mode of vibration is called normal mode of vibration and its frequency is known as normal or fundamental frequency. It is convenient to introduce a new set of coordinates, Qi , called normal coordinates, such that the potential is diagonal. These normal coordinates are defined in terms of the mass-weighted Cartesian displacement coordinates qi Q = l +q

(5.16)

Note that there are only 3N − 6 (3N − 5 for linear molecules) nonzero normal coordinates. It follows l+ tr q = 0

(5.17)

This matrix equation contains the six Eckart conditions. It further gives 2V = q+ fq = Q+ l + f lQ = Q+ Q =

3N −6 

λi Q i2

(5.18)

i=1

likewise for T ˙ +l+lQ ˙ = 2T = q˙ + q˙ = Q

3N −6  i=1

Q˙ i2

(5.19)

5.3 Normal Modes

109

It is possible to introduce the angular momentum Pi =

∂T = Q˙ i ∂ Q˙ i

(5.20)

The vibrational Hamiltonian may be written H =T+V=

3N −6 1 + 1 1  P P + Q+ Q = (Pi2 + λi Q i2 ) 2 2 2 i=1

(5.21)

In quantum mechanics, this equation remains valid but P is now an operator whose definition is Pi =

 ∂ i ∂ Q˙ i

(5.22)

with, as usual,  = h/2π . The Hamiltonian may be written 3N −6 3N −6 h 2  ∂ 2 ψ(Q k ) 1  − 2 + λk Q 2k ψ(Q k ) = W (k)ψ(Q k ) 8π k=1 2 k=1 ∂ Q 2k

(5.23)

W (k) are the eigenvalues of the Hamiltonian. The advantage of the normal coordinates is obvious, it is possible to separate the Hamiltonian in 3N − 6 independent equations, one for each Qk . The total wavefunction is ψ = ψ(Q 1 )ψ(Q 2 ) · · · ψ(Q 3N −6 )

(5.24)

and the total vibration energy is W = W (1) + W (2) + · · · + W (3N − 6)

(5.25)

 1 hωk , υk = 0, 1, 2, . . . W (k) = υk + 2

(5.26)

With 

υ k is an integer called vibrational quantum number. In (5.26), ωk , the harmonic vibration wave number, is in units of Hz. It is usually given in units of cm−1 , i.e., divided by the speed of light (c) in cm s−1 . The ground-state energy (also called zero-point energy) is

110

5 The Vibrations of Polyatomic Molecules

W0 =

3N −6 1  ωk h 2 k=1

(5.27)

Nomenclature When υ l = 1 and all other υ k = 0 with k = l, the level W (l) is called fundamental level W (υl = 1) = hωl + W0

(5.28)

When υ l > 1 and all other υ k = 0 with k = l, the level W (l) is called overtone level W (υl > 1) = υl hωl + W0

(5.29)

When several quantum number υ k are different from zero, the level is called combination level. All the solutions of (5.12) are not necessarily distinct. The corresponding frequencies are said to be degenerate. This degeneracy is a consequence of the molecular symmetry when the molecule has more than twofold axes. For instance, a linear polyatomic molecule consisting of N atoms should have (3N − 5) fundamental vibrations but it has (N − 1) non-degenerate stretching modes and (N − 2) two-dimensional bending vibrations (the bending vibrations may occur in two orthogonal planes). The quantum numbers associated with a degenerate vibration are υ t and t instead of υ b1 and υ b2 , see Sect. 4.7 (the index t is kept for degenerate modes).

5.4 Molecular Symmetry 5.4.1 Introduction When a molecule has some symmetry, the calculation of the normal modes is greatly simplified because a symmetry operation on a molecule must not change any of its physical properties. In particular, it cannot change the kinetic and potential energies. This may reduce the number of nonzero terms in the series expansion of the potential energy, see (5.5). The method and the notations used rely on the group theory. The symmetry also permits to determine the degeneracy of the frequencies as well as the selection rules. Furthermore, it allows us to determine whether the molecule has a permanent dipole moment as well as its direction. Only a short outline limited to the essential notions will be given here. A more detailed introduction may be found in the classic reference: Wilson et al. (1955). A more recent reference is: Bunker and Jensen (1998).

5.4 Molecular Symmetry

111

A symmetry operation is an action that leaves the molecule looking the same after it has been carried out, i.e., the bond lengths and the bond angles remain unchanged. Each symmetry operation has a corresponding symmetry element, which consists of all the points that stay at the same place when the symmetry operation is performed: axis, plane, or point. There are operations that keep the orientation of a trihedron (operations of first kind) and operations, which invert it (operations of second kind). The product of two operations of the first kind is still of the first kind, so that their set constitutes a subgroup (G1 ) of the whole group (G). The number of elements of the second kind is either zero or equal to those of G1 . It is possible to obtain them all by associating one of them with G1 . It is possible to characterize a group G by using a symbol naming G1 and, if necessary, a subscript naming the operation of second kind. To each group G belongs the identity operation E consisting of doing nothing.

5.4.2 Subgroup G1 In addition to E, the elements of G1 are rotations: • C n a n-fold axis of rotation with n an integer. Rotation by 2π /n leaves the molecule unchanged. • Dn in addition to C n , there are n two-fold rotations about axes perpendicular to the principal axis (the n-fold axis of rotation). • T contains the symmetry elements of a regular tetrahedron, i.e., 4 C 3 axes and 3 C 2 axes. • O contains the symmetry elements of a cube.

5.4.3 Operations of Second Kind It principally concerns a reflection in a plane of symmetry (usually called σ ) whose addition to G1 does not introduce new axes of rotations. • The plane of symmetry is orthogonal to the principal axis of G1 . It is marked by the index h (for horizontal): C nh , Dnh , T h , Oh . • The plane of symmetry includes the principal axis. For G1 = C n , one uses the index v (vertical). For G1 = Dn , when there is more than one horizontal symmetry plane, one uses the index d (diagonal), the symmetry plane is the plane bisecting two binary axes. • S n Rotation–reflection: rotation of 2π /n about the axis followed by a reflection in a plane perpendicular to the axis. S 1 is the same as reflection and S 2 is the same as inversion (through the center of symmetry), also called C i (i for inversion).

112

5 The Vibrations of Polyatomic Molecules

5.4.4 Molecular Point Groups We may group together molecules that possess the same symmetry elements. These groups of symmetry elements are called point groups. Some important point groups are given in Table 5.1 using the Schoenflies notation. As an example, we will discuss the case of the planar molecule XeF4 that belongs to the symmetry group D4h , see Fig. 5.1. It has a C 4 axis perpendicular to the plane called principal axis because it has the largest n. It has four different C 2 axes. It has two planes of symmetry parallel to the principal axis: σ v , two planes of symmetry parallel to the principal axis and bisecting the angle between two C 2 axes: σ d , and one plane of symmetry perpendicular to the principal axis: σ h . It also has an inversion center at the Xe atom. Finally, it has a C 2 and a S 4 axis (both coincident with C 4 ). Table 5.1 Some important point groups Point group

Symmetry operations

Examples

C1

E (no symmetry)

CHBrClF

Cs

E, σ h (mirror plane)

CH2 BrCl

Ci

E, i (inversion center)

CHBrCl-CHBrCl, anti

C2

E, C2 (twofold axis)

H2 O2

C 2v

E, C2 , 2σ v

H2 O, CH2 F2

C 3v

E, C3 , 3σ v

NH3 , CH3 Cl

C ∞v

E, C∞, ∞σ v (linear)

HCN, OCS

C 2h

E, C2 , σ h , i (planar with inversion center)

CHF=CHF, trans

D2d

E, 3C2 (mutually perp.), S4 (coincident with one C2 ), 2σd (through the S4 axis)

H2 C=C=CH2

D3d

E, C3 , 3C2 (perp. to C3 axis), S6 (coincident with C3 axis), i, 3σd

C6 H12 (cyclohexane)

D2h

E, 3C2 (mutually perp.), 3σ (mutually perp.), i (planar, with inversion center)

CH2 =CH2

D3h

E, C3 , 3C2 (perp. to C3 axis), 3σv , σh (trigonal)

BF3 (planar molecule)

D6h

E, C6 , 6C2 (perp. to C6 axis), 6σv , σh , C2 and C3 and S6 C6 H6 (benzene) (all coincident with the C6 axis), i (hexagonal)

D∞h

E, C∞ , ∞C2 (perp. to the C∞ axes), ∞σv , σh , i (linear with inversion center)

CO2 , HC≡CH

Td

E, 3C2 (mutually perp.), 4C3 , 6σ, 3S4 (coincident with the C2 axes) (tetrahedral)

CH4

Oh

E, 3C4 (mutually perp.), 4C3 , 3S4 and 3C2 (coincident with the C4 axes), 6C2 , 9σ, 4S6 (coincident with the C3 axes), i (octahedral or cubic)

SF6

5.4 Molecular Symmetry

113

Fig. 5.1 Molecular symmetry of the molecule XeF4

5.4.5 Classification of the Vibrational Modes It is possible to apply two symmetry operations in sequence but the order in which the two operations are applied is important, i.e., symmetry operations do not in general commute. However, for some groups, the operations commute, these groups are called Abelian. For molecules belonging to these groups, there is no degenerate vibration. The main Abelian groups are: C 2 , C 2v , C 2h , D2h , D2d , and S 2 . Molecules belonging to other groups generally have simple and degenerate vibrations. Every non-degenerate mode of vibration corresponds either to a symmetric (s) or to an antisymmetric (a) configuration for any symmetry operation. As an example, we will consider the molecule H2 O of C2v symmetry, see Fig. 5.2. The vibrational frequencies ωi are calculated using (5.12). As the symmetry operation acts simultaneously on all atom displacements, it can only either simultaneously change signs of all displacement coordinates, or remain them unchanged. Thus, a non-degenerate vibration can only be symmetric or antisymmetric with respect to any symmetry operation permitted by the symmetry of the molecule. For example, water, H2 O, has three fundamental modes whose experimental frequencies are: ν 1 = 3657 cm−1 , ν 2 = 1595 cm−1 , and ν 3 = 3756 cm−1 . This molecule has three symmetry elements: a twofold z-axis (which is the principal axis b) and two planes of symmetry: one is the plane xz (defined by the principal axes a and b) of the molecule itself, the other, yz (defined by the principal axes b and c) perpendicular to the plane of the molecule and passing through the oxygen and the midpoint of the line joining the hydrogen atoms. The operation C 2 leaves the normal modes ν 1 (symmetric stretching) and ν 3 (HOH bending) unchanged whereas it inverts all

114

5 The Vibrations of Polyatomic Molecules

Table 5.2 Character table of C2v group σ xz

σ yz

c

1

1

1

Tz

1

1

−1

−1

Rz

1

−1

1

−1

T x ; Ry

1

−1

−1

1

T y ; Rx

C2v

E

C 2z

A1

1

A2 B1 B2

b

a For

B1 and B2 the convention of Wilson et al. (1955) is used is the symmetry axis c T = translation in the g (= x, y, z) direction g Rg = rotation about the axis g bz

displacement vectors for the mode ν 3 (asymmetric stretching). The reflection through the xz and yz planes leaves ν 1 and ν 2 unchanged, whereas a reflection through the yz plane inverts all displacement vectors for the mode ν 3 . Both symmetric stretching and bending vibrations have A1 symmetry, whereas the asymmetric stretching vibration is of B1 symmetry, see Table 5.2. For non-degenerate modes, A indicates a symmetric mode and B an antisymmetric mode, in both cases, relatively to the principal symmetry axis. Subscripts 1 or 2 are used with A or B to designate the species which are symmetric or antisymmetric under one of the twofold rotations about an axis perpendicular to the principal axis in Dn or a vertical plane σv in a group like C2v . E indicates a doubly degenerate mode an F a triply degenerate mode. For molecules with an inversion center (i), the vibrations are either symmetric or antisymmetric, in this case, the notation is not s and a, but g (gerade = even) and u (ungerade = odd). When there is no inversion center but a rotation–reflection, s and a are replaced by primes ( ) and double primes ( ). This is the case for the groups Cnh and Dnh with n odd. The effect of applying two symmetry operations in sequence within a given point group is summarized in multiplication tables. The most important rules are: A × A = A, B × B = A, A × B = B, A × E = E, B × E = E, etc.

(5.30)

g × g = g, u × u = g, g × u = u, ’ × ’ = ’, ” × ” = ’, ’ × ” = ”

(5.31)

And for the subscripts on A or B: 1 × 1 = 1, 2 × 2 = 1, 1 × 2 = 2

(5.32)

To deduce the number and degeneracies of the normal modes, the theory of irreductible representations has to be used. It is beyond the scope of this book. Furthermore, most computer programs give you the correct answer. The vibrational modes are labeled by indices. The index is usually assigned to be increasing with descending wave number, symmetry species by symmetry species, starting with the totally symmetric species.

5.5 Coriolis Interaction

115

5.5 Coriolis Interaction Up to now, the cross-term T rv of (5.1) has been neglected. It is possible to show that it is due to a force F Cor acting on each atom i of mass mi and of velocity υ i in the molecule fixed axis system FCor = 2m i (υi × )

(5.33a)

 is the angular velocity of the molecule fixed axis system giving the acceleration γCor γCor = 2υi × 

(5.33b)

At this point, it is useful to study a simple example, a XY2 molecule of symmetry C 2v (such as H2 O). We consider the vibration ω3 whose symmetry is B1 , see Fig. 5.2. The symmetry axis of the molecule is z and the atoms are in the plane xz. When the molecule does not rotate, the motion of each atom is described by Atom X0

Atom Y1

Atom Y2

x 0 = a0 sin ω3 t

x 1 = −a1 sin ω3 t

x 2 = −a1 sin ω3 t

z0 = 0

z1 = −b1 sin ω3 t

z2 = b1 sin ω3 t

The velocity at t = 0 for each atom when it passes through its equilibrium position is Atom X0

Atom Y1

Atom Y2

x˙0 = a0 ω3

x˙1 = −a1 ω3

x˙2 = −a1 ω3

z˙ 0 = 0

z˙ 1 = −b1 ω3

z˙ 2 = b1 ω3

When the molecule is furthermore in rotation, the Coriolis interaction has to be taken into account and the Coriolis acceleration for a rotation about the y-axis perpendicular to the plane of the molecule ( = y ) γx = −2˙z γ y = 0 γz = 2x˙ Fig. 5.2 Vibrations of H2 O (z = b is the symmetry axis and xz = ba the symmetry plane)

(5.34)

116

5 The Vibrations of Polyatomic Molecules

for each atom Atom X0

Atom Y1

Atom Y2

(γx )0 = 0

(γx )1 = 2b1 ω3

(γx )2 = −2b1 ω3

(γz )0 = 2a0 ω3

(γz )1 = −2a1 ω3

(γz )2 = −2a1 ω3

It is interesting to compare this Coriolis interaction with the bending vibration ω2 of symmetry A1 . If we draw the vectors representing these Coriolis accelerations, we find that when the molecule vibrates with the frequency ω3 , the Coriolis acceleration induces the bending vibration Q2 but with the frequency ω3 . To determine whether a Coriolis interaction is possible, it is enough to take into account the fact that the term T rv must be totally symmetric. A simple calculation shows that  α Q r ζrαs Q˙ s (5.35) Trv = r,s

α

In this equation, ζrαs is a Coriolis coupling constant. α has the symmetry of the rotation Rα which can be found in the table of characters of the group, see Table 5.2. If the product of the symmetry species of two vibrational modes contains the species of rotation, Coriolis interaction takes place between these two modes (Jahn 1939). For instance, for H2 O, Q3 is of symmetry B1 and Q2 of symmetry A1 . The product of the symmetry species of these two modes is B1 ×A1 = B1 ; this is the symmetry c , which can be of a rotation around axis c and the Coriolis coupling constant ζ23 calculated from the harmonic force field, is different from zero as shown above. Another example of Coriolis interaction is found in formaldehyde, H2 CO, see Fig. 5.3. The υ 4 = 1 and υ 6 = 1 states are of symmetry B1 and B2 , respectively. As B1 × B2 = A2 , this product has the symmetry of a rotation around the symmetry axis a, see Table 5.2. Actually, it is not necessary to use the character table to identify Coriolis interactions. As an example, we will consider the vibration υ 6 = 1. For a rotation about the symmetry axis z (principal axis a), (5.33) shows that the Coriolis acceleration of the mode υ 4 = 1 describes the mode υ 6 = 1, see Fig. 5.3. When the frequencies of the interacting states are very different, the Coriolis interaction is taken into account by a perturbation calculation. However, when these two frequencies are close, the two interacting vibrational states have to be treated Fig. 5.3 Vibrations υ 4 = 1 and υ 6 = 1 for formaldehyde

5.5 Coriolis Interaction

117

by a direct diagonalization of the Hamiltonian. When there is a Coriolis interaction between two vibrational states υ k and υ l , there is a non-diagonal term in υ υk , υl |H|υk + 1, υl − 1 =

ξ 2i Beξ ζkl



ωl + ωk



ωk ωl



(υk + 1)υl Pξ + · · · 4 (5.36)

ξ

ζkl is the Coriolis-zeta constants which couples the vibrations υ k and υ l trough the ξaxis (a, b, c). The spectra are strongly perturbed and difficult to assign. Furthermore, the least-squares system of equations becomes strongly nonlinear and it is difficult to obtain accurate parameters. When the Coriolis interaction is not too strong, it is possible to take it into account by a second-order perturbation calculation. For the α-constants (see (6.2) of Chap. 6), it gives the following term   ξ 2  ξ 2  3ωk2 + ωl2 ζkl B ξ αk (Cor.) = −2 e ωk ωk2 − ωl2 l 

(5.37)

When the α-constants are summed to determine the equilibrium rotational constants (see Chap. 6 and (6.23)), the resonance contribution disappears  2 2 ξ ξ 1  ξ  Be ζkl (ωk − ωl ) α = 2 k k ωk ωl (ωk + ωl ) l>k

(5.38)

In theory, even if the Coriolis interaction is handled only approximately, the equilibrium rotational constants should not be affected. However, (5.38) is only a firstorder approximation and it fails in case of a strong Coriolis interaction. A typical example is found in trans formic acid, HCOOH of symmetry C s . Table 5.3 shows the difficulty to determine reliable values for the individual α-constants for the states υ 7 = 1 (OCO scissor mode at 626.17 cm−1 ) and υ 9 = 1 (COH torsion at 640.73 cm–1 ) whereas the sum α 7 + α 9 is correctly determined. These two states are coupled through strong A- and B-type Coriolis resonances. Table 5.4 gives the values of α6ξ + α8ξ for states υ 6 = 1 (in-plane C-O stretch at 1104.85 cm–1 ) and υ 8 = 1 (outof-plane C-H wag at 1033.47 cm–1 ) that are also coupled by strong A- and B-type Coriolis resonances and demonstrates the failure of (5.38). Although the Coriolis interaction is a harmonic phenomenon, its analysis can be quite difficult and it is often easier to calculate the rovibrational correction from an ab initio force field using (5.38).

118

5 The Vibrations of Polyatomic Molecules ξ

ξ

ξ

ξ

Table 5.3 Values of α7 , α9 and α7 + α9 for states υ 7 = 1 and υ 9 = 1 of trans HCOOH in Coriolis interaction (ξ = a, b, c) Experimental

CCSD(T)/VTZ

Axis

α7

α9

α7 + α9

Year

a

−416.9425(43)

0.6705(42)

−416.2721(60)

2006

b

6.61933(19)

61.88599(20)

68.50532(27)

c

21.36583(15)

11.04986(14)

32.41569(20)

a

−243.10(92)

−172.80(93)

−415.9(1.3)

b

20.370(64)

48.212(65)

68.582(91)

c

21.1419(45)

10.8344(37)

31.9762(58)

a

−279.1(9.6)

−136.0(9.6)

−415.0(13.6)

b

8.27(19)

60.46(18)

68.74(26)

c

21.284(18)

11.047(17)

32.331(24)

a

2030.5(2.6)

−2442.7(2.6)

−412.1(3.7)

b

63.3(1.9)

30.8(1.2)

94.1(2.2)

c

14.80(26)

−7.50(20)

7.31(32)

a

337.02

−769.68

−432.66

b

50.87

14.08

64.95

c

18.98

12.09

31.07

2002

1986

1978

2004

Source Demaison et al. (2007) ξ

ξ

Table 5.4 Values of α6 + α8 for states υ 6 = 1 and υ 8 = 1 of trans HCOOH in Coriolis interaction (ξ = a, b, c)

ξ

a

b

c

Exp.

447.1

105.7

60.1

446.8

105.4

60.6

CCSD(T)/VTZ

−823.7

30.4

−2.9

404.3

99.5

57.3

Source Demaison et al. (2007)

5.6 Anharmonicity The harmonic approximation, (5.5), permits to calculate the vibrational frequencies with an accuracy of about 5% (10% for the vibrations which involve a hydrogen atom). However, it does not allow us to predict the existence of the vibrational overtones and the anharmonic resonances. Furthermore, the cubic anharmonic constants play a leading role in the calculation of the rotation–vibration interaction constants. For these reasons, it is necessary to take into account the cubic and quartic terms in the potential energy

5.6 Anharmonicity

V (Q) =

119

1 1 1  λk Q 2k + kst Q k Q s Q t + kstu Q k Q s Q t Q u + · · · 2 k 6 kst 24 kstu (5.39)

It is practical to introduce a dimensionless normal coordinate qk (to be not confused with the mass-weighted Cartesian coordinates of Sect. 5.1, (5.4)) qk = V (q) =

√ γk Q k with γk =

√ 2π cωk λk =  

1 1 1  ωk qk2 + φkst qk qs qt + φkstu qk qs qt qu + · · · 2 k 6 kst 24 kstu

(5.40) (5.41)

The symmetry operations belonging to the group of the molecule do not change the potential energy. Therefore, the product of the normal coordinates must be total symmetric and only a few cubic, φ kst , and quartic terms, φ kstu are different from zero. For instance, for a triatomic molecule of C2v symmetry as H2 O, Q1 and Q2 are of symmetry A1 but Q3 is of symmetry B1 . Therefore, φ 111 and φ 222 are different from zero but only the terms involving Q23 are different from zero (because B1 × B1 = A1 ). As the cubic and quartic terms are small, a perturbation calculation is often accurate enough. W = W (0) + W (1) + W (2)

(5.42)



 W (1) ∝ φkktt qk2 qt2 + φkkkk qk4

(5.43)

With

W

(2)

∝

 υ

   υ|qq qs qt υ  υ  qq  qs  qt  |υ φkst φk  s  t  ωυ − ωυ 

(5.44)

The vibrational energy may be written G(υ) =

 k

 ωk

1 υk + 2

 +

 k≥s

   1 1 υs + xks υk + 2 2

(5.45)

The x ks are the anharmonicity constants with for instance xkk =

1 1  2 8ωk2 − 3ωs2   φkkkk − φ 16 16 s kks ωs 4ωk2 − ωs2

and similar expressions for the non-diagonal terms x ks

(5.46a)

120

5 The Vibrations of Polyatomic Molecules 2 ωt (ωt2 − ωk2 − ωs2 ) 1 1  φkkt φtss 1  φkst − φkkss − 4 16 t ωt 2 t kst

    2  2  2 ω ωs k (a) (b) (c) + Ae ζk,s + Be ζk,s + Ce ζk,s + ωs ωk

xks =

(5.46b)

where kst = (ωk + ωs + ωt )(ωk − ωs − ωt )(−ωk + ωs − ωt )(−ωk − ωs + ωt ) (5.47) The perturbation calculation is only valid when there is no degeneracy. For instance, inspection of (5.46) for x kk shows that the calculation fails when ωs ≈ 2ωk . This is called a Fermi resonance. Inspection of the denominator kst of (5.46b) for x ks shows that there are further Fermi resonances due to ωk ≈ ωs + ωt or ωk ≈ ωs − ωt

(5.48)

In the Fermi resonances, the cubic constant φ kst plays the leading role. Higherorder resonances involving the quartic force constants are also possible, for instance the Darling–Dennison resonance: 2ωk ≈ 2ωs . For instance, in H2 O, the frequencies ω1 (A1 ) and ω3 (B1 ) are close but there are not coupled by a harmonic constant because they do not belong to the same symmetry. On the other hand, for the overtones, an anharmonic coupling is possible through φ 1133 because the product Q21 Q23 is totally symmetric. In the simple case of Fermi resonance between two levels, the resonance itself has to be treated by the construction and diagonalization of the two coupled states. The off-diagonal Fermi elements F are  F = υr , υs , υt |HFermi |υr + 1, υs + 1, υt − 1 = φr st

(υr + 1)(υs + 1)υt 8

1/2 (5.49)

or F = υr , υs |HFermi |υr + 2, υs − 1 =

φrr s 2



(υr + 1)(υr + 2)υs 2

1/2 (5.50)

If E i(0) and E (0) j are the vibrational energies of the unperturbed vibrational states i and j, the perturbed energies are obtained by diagonalizing the 2 × 2 matrix 

E i(0) F F E (0) j

The perturbed energies are given by

 (5.51)

5.6 Anharmonicity

121

λ=

E i(0) + E (0) j 2

  δ 2 + F2 ± 2

(5.52)

with δ = E i(0) − E (0) j

(5.53)

The eigenfunctions may be written   ψ = cψ (0) + sψ (0)  i i j   ψ j = −sψi(0) + cψ (0) j

(5.54)

with √  4F 2 + δ 2 + δ and s = 1 − c2 c= √ 2 4F 2 + δ 2

(5.55)

When two vibrational states i and j are in Fermi resonance, the rotational energy may be written E R (Ai , Bi , Ci ) = ψi |HR |ψi              (0) = c2 ψi(0) HR ψi(0) + s 2 ψ (0) j HR ψ j   = E R (Ai , Bi , Ci ) + s 2 E R (A j , B j , C j ) − E R (Ai , Bi , Ci ) (5.56) where the prime ( ) indicates the rotational constants affected by the Fermi resonance. If the resonance is not too strong and assuming that the rotational energy is a linear function of the rotational constants (it is only a first-order approximation), the rotational constants may be written  ξ ξ ξ ξ Bi = Bi + s 2 B j − Bi

(5.57)

with ξ = a, b, c. A typical example is given by the Fermi resonance between the states υ 1 = 1 and υ 2 = 2 of F2 O for which the two states are separated by about 6.8 cm−1 , see Table 5.5. It is possible to estimate the rotational constants of the υ 2 = 2 state from those of the ground state and the υ 2 = 1 state ξ

ξ

B ξ (υ2 = 2) = B0 − 2α2

(5.58)

Using ξ

ξ

B ξ (υ2 = 2) = B ξ (υ2 = 2) + b2 (2α2 − α1 )

(5.59)

122

5 The Vibrations of Polyatomic Molecules

Table 5.5 Analysis of the Fermi resonance in F2 O (all values in MHz, except for s which is dimensionless) ξ

A

B

C

Experimental rotational constants υ=0

58,782.63

10,896.431

9167.41

υ1 = 1

59,213.58

10,824.35

9128.09

υ2 = 1

59,481.65

10,854.07

9114.13

υ2 = 2

59,711.17

10,818.01

9092.99

Experimental vibration–rotation interaction constants under Fermi resonance α1

−430.95

72.081

α2

−699.02

42.361

53.28

α 2 (from υ 2 = 2)

−928.54

78.421

74.42

39.32

Unperturbed constants (corrected for the Fermi resonance) ξ

ξ

B ξ (υ 2 = 2)

60,180.67

10,811.7092

9060.85

= B0 − 2α2 (υ2 = 1)

s2 (2α 2 - α 1 )

−469.50

6.301

32.14

= B ξ (υ2 = 2) − B ξ (υ2 = 2)

B ξ (υ 1 = 1)

58,744.08

10,830.651

9160.23

= B ξ (υ1 = 1) − s 2 (α1 − 2α2 )

α1

38.55

65.78

7.18

2α 2 − α 1

−1436.59

18.942

99.38

s2

0.327

0.333

0.323

ξ

ξ

=

ξ B0

ξ

− B ξ (υ1 = 1)

ξ

allows us to determine b2 (2α2 − α1 ) from B ξ (υ2 = 2) and B ξ (υ2 = 2). These corrections strongly affect the values of the α-constants. For instance, the value of α1A (in MHz) is −430.95 before correction whereas the unperturbed value (corrected for Fermi resonance) is 38.55 (Morino and Saito 1966; Taubmann et al. 1986). This is further discussed in Sect. 6.6 and some typical examples are given in Table 6.7. The occurrence of anharmonic resonances is common and, for this reason, it is difficult to determine reliable values for the experimental α-constants, whereas the computed ab initio constants are not affected. This is one of the reasons why the semiexperimental structure is often more accurate than the experimental one (and easier to determine).

5.7 Internal Coordinates Up to now, two kinds of coordinates have been introduced: the Cartesian and the normal coordinates. Although both kinds are useful, they have some disadvantages: their chemical meaning is not obvious and the force field is not transferable from one molecule to another one. The use of internal coordinates circumvents both difficulties.

5.7 Internal Coordinates

123

Furthermore, with well-chosen internal coordinates, the force constants matrix is nearly diagonal. Assuming that the displacements are infinitesimal, the relationship between internal coordinates, R, and Cartesian ones, x, is linear R = Bx

(5.60)

If n is the number of atoms, B is a 3n × (3n − 6) matrix, which is a function of the structure of the molecule. The inverse relation is x = AR

(5.61)

The relationship with normal coordinates is R = LQ

(5.62)

Note that the matrix L in (5.62) is not identical to the matrix l of (5.13). And the force field may be written 2V = xT f x x = RT f R R = QT Q

(5.63)

f R = AT f x A and f x = BT f R B

(5.64)

LT f R L = 

(5.65)

With

and

More relations are discussed in Winnewisser and Watson (2001). Changes interatomic distances or in angles between the bonds are normally used as internal coordinates. These coordinates are not affected by translation or rotations of the molecule and they provide a physically significant set to describe the potential energy of the molecule. Two main kinds of force fields are used: the central force field and the valence-bond force field. In the central force field, the potential energy is expressed in terms of squares of interatomic distances. For instance, for H2 O, it gives 2V = kHO (δrHO )2 + kHO (δrH O )2 + kHH (δrHH )2

(5.66)

In the valence-bond force field, changes in lengths of bonds and changes in angles between bonds are preferred. For H2 O, it gives 2V = kHO (δrHO )2 + kHO (δrH O )2 + kα (δrα )2

(5.67)

124

5 The Vibrations of Polyatomic Molecules

where δr α is the change in the angle between the two OH bonds. To improve the force field, it may be helpful to add some central force terms between non-bonded atoms. It is equivalent to take into account the van der Waals forces between non-bonded atoms. This modified force field often called Urey– Bradley force field. One might also add a bond stretching angle bending interaction constant. These additional terms ameliorate the force field but there are more force constants to determine which complicates the problem.

5.8 Conclusion One of the main difficulties of the vibrational analysis is to correctly take into account the interactions between the vibrational states as explained in Sects. 5.5, 5.6, and 6.6. If the molecule is quite small, the variational method can be used (Bowman et al. 2008; Császár et al. 2012; Tennyson 2016). However, the usual way is the band-byband analysis because it is faster and easier. Although it is widely used, it is not a good method because the determined parameters are effective ones, i.e., the interactions are empirically and partially taken into account. The so-called global analysis is much better. The first step is to analyze the ground state, then the first lowest excited state, and so on, climbing the ladder. At each step, if an interaction is possible, it is taken into account and the interesting parts of the spectra permitting to determine it are analyzed. The difficulty is obvious: the infrared spectra are needed in the whole range. Furthermore, the rotational spectra have to be measured from the centimeterwaves to the submillimeterwaves. Finally, the help of an ab initio anharmonic force field is very useful. But, at the end, a collection of reliable parameters is obtained. A typical example is the global rovibrational analysis of carbonyl sulfide, OCS (Lahaye et al. 1987).

References Bowman JM, Carrington T, Meyer H-D (2008) Variational quantum approaches for computing vibrational energies of polyatomic molecules. Mol Phys 106:2145–2182 Bunker PR, Jensen P (1998) Molecular symmetry and spectroscopy. NRC Research Press, Ottawa Califano S (1976) Vibrational states. Wiley, New York Császár AG, Fábri C, Szidarovszky T, Mátyus E, Furtenbacher T, Czakó G (2012) Fourth age of quantum chemistry: molecules in motion. Phys Chem Chem Phys 14:1085–1106 Demaison J, Herman M, Liévin J (2007) Anharmonic force field of cis- and trans-formic acid from high-level ab initio calculations, and analysis of resonance polyads. J Chem Phys 126:164305 Herman M, Liévin J, Vander Auwera J, Campague A (1999) Advances in chemical physics: global and accurate vibration Hamiltonians from high-resolution molecular spectroscopy, vol 108. Wiley, Hoboken, NJ Jahn HA (1939) Note on the Coriolis coupling terms in polyatomic molecules. Phys Rev 56:680–683

References

125

Lahaye J-G, Vandenhaute R, Fayt A (1987) CO2 laser saturation Stark spectra and global rovibrational analysis of the main isotopic species of carbonyl sulfide (OC34 S, O13 CS, and 18 OCS). J Mol Spectrosc 123:48–83 Morino Y, Saito S (1966) Microwave spectrum of oxygen difluoride in vibrationally excited states; ν1 − 2ν2 Fermi resonance. J Mol Spectrosc 19:435–453 Papousek D, Aliev MR (1982) Molecular vibrational-rotational spectra. Elsevier, Amsterdam Perrin A, Demaison J, Flaud J-M, Lafferty WJ, Sarka K (2011) Spectroscopy of polyatomic molecules: determination of the rotational constants. In: Demaison J, Boggs JE, Császár AG (eds) Equilibrium molecular structures. CRC Press, Boca Raton, pp 89–124 Quack M, Merkt FJ (eds) (2011) Handbook of high resolution spectroscopy. Wiley, New York Taubmann G, Jones H, Rudolph HD, Takami M (1986) Diode laser spectroscopy of the ν1 /2ν2 Fermi diad of OF2. J Mol Spectrosc 120:90–100 Tennyson J (2016) Perspective: accurate ro-vibrational calculations on small molecules. J Chem Phys 145:120901 Wilson EB, Decius JC, Cross PC (1955) Molecular vibrations: the theory of infrared and Raman vibrational spectra. McGraw Hill, New York Winnewisser BP, Watson JKG (2001) The A matrix in molecular vibration–rotation theory. J Mol Spectrosc 205:227–231

Chapter 6

Equilibrium Structures from Spectroscopy

Abstract The determination of the geometry of a molecule from its rotational constants is reviewed in detail. First, the different empirical methods are critically examined. Then, the different ways to determine an equilibrium structure are described. A particular emphasis is put on the semiexperimental method that combines ground-state experimental rotational constants and computed ab initio rovibrational corrections. The accuracy of the different methods is discussed.

6.1 Introduction As shown in Chaps. 3 and 4, the structural information is contained in the equilibrium moments of inertia that are reciprocals of the equilibrium rotational constants. Spectroscopic methods are extremely accurate and permit obtaining rotational constants with eight or more significant digits and, in most cases, a theory exists that allows one to fit the spectra to within the uncertainties of the most accurate observations. One might think that the equilibrium structure of a molecule can be obtained with a precision comparable to that of the rotational constants. However, there are two difficulties: (i) even in its ground vibrational state, a molecule undergoes zero-point vibrational motions that induce vibration–rotation interactions making the extraction of an accurate equilibrium structure from experimental rotational constants a difficult task; (ii) a molecule has at most three different rotational constants (two for a symmetric molecule and only one for a linear molecule) and the number of independent structural parameters is often much larger, unless the molecule has the formula XYn and sufficiently high symmetry, as for the linear molecules XY2 (e.g., CO2 ) or the asymmetric molecules XY2 (e.g., SO2 ). However, based on the Born–Oppenheimer approximation (see Sect. 2.3), the equilibrium structure is isotopically invariant and additional information may be supplied by the rotational constants of isotopologues. In theory, rotational constants of isotopologues solve the problem. However, in practice, it is not easy because there are experimental as well as computational difficulties. Sections 6.2 and 6.3 explain how to obtain equilibrium rotational constants (see also Chap. 4). The next Sects. 6.4 and 6.5, describe the different ways to determine © Springer Nature Switzerland AG 2020 J. Demaison and N. Vogt, Accurate Structure Determination of Free Molecules, Lecture Notes in Chemistry 105, https://doi.org/10.1007/978-3-030-60492-9_6

127

128

6 Equilibrium Structures from Spectroscopy

an empirical structure using only the ground-state rotational constants. Section 6.6 is devoted to the experimental equilibrium structure. Sections 6.7 and 6.8 discuss the combined use of experimental ground-state rotational constants and ab initio calculations to derive an equilibrium structure (called semiexperimental). Finally, the Sect. 6.9 is devoted to the particular case of weakly bound complexes. Recent references on this subject are: Demaison (2007); Gordy and Cook (1984); Groner (2000); Puzzarini et al. (2010); Rudolph and Demaison (2011); Vázquez and Stanton (2011).

6.2 Vibrational Dependence of the Rotational Constants; See also Sect. 4.9 The difference between the equilibrium rotational constants and the experimental constants is called the rovibrational contribution and may be written as a series expansion. The effective rotational constant about the ξ-axis (ξ = a, b, c) in a vibrational state characterized by the vibrational quantum numbers υ = (υ 1 , υ 2 , …) with degeneracies d 1 , d 2 , … (for linear and symmetric molecules; for an asymmetric molecule d i = 1) is given by Bυξ

=

Beξ

−



ξ αk

k

       dj dk di ξ ξ υk + + υj + + γi j υi + γli l j i  j + · · · 2 2 2 i≥ j i≥ j (6.1) ξ

ξ

Beξ is the equilibrium rotational constant, and αi and γi j are the vibration-rotation interaction constants of different orders. The summation is over all the normal modes. ξ The last term, γli l j , is different from zero only for degenerate modes. The convergence

of the series expansion is usually fast, αiξ being about two orders of magnitude smaller than Beξ and γiξj two orders of magnitude smaller than αiξ ; see Table 6.1. For this reason, the γ -terms are generally neglected, except for light molecules. The first-order vibration–rotation interaction constants (also called α-constants) can be derived by standard perturbation theory (Mills 1972) ⎧⎡  2  2  ξ 2 ⎤  ⎪ 2 ⎨  3 akξ γ ξ 3ωk + ωl2 ζkl  B ⎢ ⎥ ξ + αk = −2 e ⎣ ⎦ γ 2 2 ωk ⎪ 4I ω − ω e ⎩ γ =a,b,c k l l 

 +π

c ξ ξ ωk φkkl al 3/2 h l ωl



(6.2)

6.2 Vibrational Dependence of the Rotational …

129

Table 6.1 Vibrational contribution to the rotational constants (in MHz)a   Molecule Be C = αi di /2 C/B (%) D = γi j di d j /4 HCN

44511.620

198.137

0.45

FCN

10586.782

2.395

D/C (%) 1.21

32.604

0.31

– 0.248

– 0.76

ClCN

5982.8975

12.0644

0.20

– 0.0207

– 0.17

BrCN

4126.5059

6.2838

0.15

0.0596

0.95

3.5084

0.11

– 0.0049

– 0.14

– 0.45

1.91

– 0.70

0.40

0.448

1.1

0.51

0.227

0.50

ICN

3329.0568

SO2 , Ab

60502.45

– 274.194

SO2 , B

10359.234

41.295

SO2 , C

8844.869

45.166

a Source:

Demaison (2007) b Morino and Tanimoto (1994)

In this equation, ωk is the harmonic frequency of the normal mode k, Ieξ is the    ξγ equilibrium moment of inertia, ak = ∂ I ξ γ ∂ Q k e is the derivative of the (ξ, γ ) element of the inertia tensor with respect to the normal coordinate Qk at equilibrium, ξ ξ ζkl = −ζlk is a Coriolis coupling constant, and φ kkl is a cubic force constant in the dimensionless normal coordinate representation. c is the speed of light, and h is Planck’s constant. The last term is the anharmonic contribution, which is far from negligible, and the term in brackets is the harmonic contribution. Note the presence in this term of a component with a denominator ωk2 − ωl2 , which is the Coriolis contribution, introduced in Sect. 5.5. This term will be discussed further in Sect. 6.6. Equation (6.2) is appropriate when the rotational constants are in units of cm−1 ; it has to be multiplied by the speed of light in cm·s−1 to obtain it in units of Hz. Several quantum chemistry programs calculate the force field and the α-constants, see Sect. 6.11.1.

6.3 Determination of the Rotational Constants 6.3.1 Ground-State Constants The first step in a structure determination is to obtain experimental ground-state rotational constants. Tremendous progress has been achieved in the last forty years, and it is now easy to determine ground-state rotational constants with a very high precision. When the molecule has a permanent dipole moment (even a tiny dipole moment is enough for instance induced by isotopic substitution as in H2 C = CD2 ), microwave spectroscopy in the 1–1000 GHz range (i.e., between 30 and 0.03 cm) is the method of choice. A pulsed-jet supersonic expansion Fourier transform microwave spectrometer (FTMW) is now widely used because it is fast, extremely precise (a fraction of a kHz) and highly sensitive (see Sect. 4.12).

130

6 Equilibrium Structures from Spectroscopy

The FTMW often permits measuring the rotational spectra of several isotopic species in natural abundance such as 13 C, 15 N, 34 S. It is much more difficult to measure the spectra of rare isotopic species such as 18 O or deuterium. In such a case, the synthesis of an enriched sample is required, which can be tedious and expensive. Another drawback is that only low J lines can be recorded (because of the low temperature of the jet). For this reason, conventional Stark spectroscopy is still used, but, now millimeterwave spectroscopy is frequently preferred because it has access to many high J lines in a large frequency range although it is less sensitive. See Sect. 4.12. High-resolution infrared and Raman spectroscopies also allow us to determine rotational constants with a precision almost as good as microwave spectroscopy. They are particularly useful when the molecule has no dipole moment. When the infrared spectrum is measured in the gas phase at low pressure and when the resolution of the spectrometer is high enough, a fine structure may appear. This structure occurs because rotational transitions are occurring at the same time as vibrational transitions. The rovibrational energy of the molecule may be written E υr = E υ (v) + Er (J, . . .)

(6.3)

where v is the vector of the vibrational quantum numbers (υ 1 , υ 2 , …) and J the rotational quantum number. If a transition occurs between two levels of which the upper is denoted by a single prime and the lower by a double prime, the frequency ν of the rovibrational transition is given by     hν = [E υ v −E υ (v )] + [Er J  −Er (J  )]

(6.4)

where the selection rule for υ i is the same as in low-resolution vibrational spectroscopy and for J is the same as in pure rotational spectroscopy, i.e., J = 0, ±1. From the analysis of a high-resolution rovibrational spectrum, it is possible to obtain the ground-state rotational constants by the method of ground-state combination differences (GSCD) (Blass and Edwards 1967). The principle of this method is quite simple: when two transitions arrive at the same upper level, their frequency difference only depends on the ground-state rotational constants. A minor complication is that the values of the experimental rotational constants of an asymmetric top depend on the reduced Hamiltonian used. For this reason, as suggested by Watson (1977), the determinable combinations, which do not depend on the reduction, should be used. However, these determinable combinations Bξdet contain a small contribution of the distortion constants; see Sect. 4.8.2 rigid

Bξ

= Bξdet − 2Tξ  ξ 

(6.5)

with ξ, ξ ‘, ξ “ = a, b, c and where T ξ  ξ  is a quartic centrifugal distortion constant as defined by Watson, see (4.38) and (4.39). These distortion constants are small, and they are easily calculated from the harmonic force field. It is therefore possible to take them into account. Several computational chemistry programs provide the different

6.3 Determination of the Rotational Constants

131

kinds of rotational constants, see Sect. 6.11.1. A recent determination of the structure of sulfine, CH2 =SO, confirms that taking into account this small correction improves the fit (Demaison et al. 2020). Neglecting the centrifugal distortion correction in the case of SO2 decreases the bond length by 0.00003 Å (0.003 pm) and increases the bond angle by 0.005°. There is another correction that is small but often larger than the centrifugal correction: the electronic correction. This effect is discussed in Sect. 4.10.

6.3.2 Rotational Constants in a Vibrationally Excited State Rotational constants in a vibrationally excited state are determined in the same way as the ground-state constants. However, doing so is much more difficult. The very efficient pulsed-jet supersonic expansion Fourier transform spectroscopy cannot be used because the very low temperature of the jet depopulates the excited vibrational states. The less-sensitive Stark or millimeterwave spectroscopies can only access the low-lying vibrational states because the intensity of a rotational transition is proportional to the population of its lower state, which is given by the Boltzmann law exp(–E’’/kT ). If the energy of the excited vibrational state is high, above about 1000 cm−1 , the population of its rotational levels will be small and the rotational transitions between them will be too weak to observe. In this rather common case, only infrared spectroscopy may be used to determine the α-constants. Of course, it is still much more difficult for the isotopologues measured in natural abundance. Furthermore, the rotational spectra in excited states are often complicated by Coriolis interactions or/and Fermi resonances; see Sects. 5.5, 5.6, and 6.6.

6.4 Empirical Structures 6.4.1 Introduction Although much progress has recently been made in the field of equilibrium structure determinations, it is still a time-consuming task and is furthermore limited to rather small molecules. For this reason, empirical methods are still currently used. Most of them only use the ground-state rotational constants. They are much simpler, but their accuracy may be rather poor.

132

6 Equilibrium Structures from Spectroscopy

6.4.2 Effective Structures (r0 ) Historically, the effective structure method (r 0 ) is the oldest method. The structural parameters are directly fitted to the ground-state moments of inertia. In other words, the rovibrational corrections εξ = I0ξ − Ieξ are neglected. These corrections are indeed quite small compared to the values of the moments of inertia. They are less than 1% in most cases (Demaison and Nemes 1979); see Table 6.2. Thus, neglecting these corrections may seem to be a good approximation. However, the least-squares system is often ill-conditioned; see Sects. 9.4.1 and 9.7. As a consequence, the solution may be quite inaccurate. The accuracy is about 1% for diatomic molecules; see Sect. 3.8.2. In favorable cases, the bond lengths in polyatomic molecules are determined with an accuracy of about 0.2 pm and, for the angles, it is about 0.2–0.3°. For instance, in the case of SO2 , which is a very favorable case, the r 0 structure derived from the I a and I b moments of inertia is: r 0 (SO) = 143.22 pm and ∠0 (SOS) = 119.535°. These values are rather close to the equilibrium structure: r e (SO) = 143.0793(4) pm and ∠e (SOS) = 119.3290(2)° (Lafferty et al. 2009). This satisfactory agreement is obtained for two reasons: i) the structure can be obtained from only one isotopic species, which avoids the problem of ill-conditioning; ii) the molecule is planar, hence the equilibrium moments of inertia are not independent (Ice = Iae + Ibe ). However, because of the inertial defect, the relation does not hold for the ground-state moments of inertia. To determine the structure, it is possible to use either I a and I b , or I b and I c , or I c and I a . The best results are obtained with the two smallest moments of inertia (I a and I b ) because the rovibrational correction increases with the value of the moment of inertia. A striking counterexample is the r 0 structure of the planar molecule phosgene, OCCl2 . In this case, the rotational constants of at least two isotopologues are required because there are three geometrical parameters to determine. A fit of the ground-state moments of inertia of a full set of isotopologues gives r 0 (C = O) = 116.6(10) pm; r 0 (C–Cl) = 174.6(10) pm and ∠0 (ClCCl) = 111.3(10) pm (Demaison et al. 1997) to Table 6.2 Variation of the vibrational correction ε = I 0 − I e as a function of the moments of inertia I 0 (uÅ2 ) Molecule

Number of isotopologues

I0

ε

ε/I 0 (%)

Mean

Range

HCN

11

11.707

0.049(1)

0.007

0.4

N2 O

12

40.232

0.206(4)

0.015

0.5

OCS

12

83.101

0.250(6)

0.018

0.3

OCSe

27

125.019

0.349(9)

0.026

0.3

Experimental, semiexperimental and ab initio equilibrium structures; Demaison J; Molecular Physics; 105:3109-3138; Dec 10, 2007, reprinted by permission of the publisher Taylor&Francis Ltd, http://www.tandfonline.com a Demaison (2007)

6.4 Empirical Structures

133

be compared with the equilibrium structure: r e (C = O) = 117.59(4) pm, r e (C–Cl) = 173.75(2) pm and ∠e (ClCCl) = 111.85(2) pm (Demaison and Császár 2012). A comparison of 45 equilibrium and effective angles has been made (Rudolph and Demaison 2011). The median absolute deviation (MAD) [for definition, see (9.35)] is only 0.2°, corresponding to a standard deviation of 0.3° but with a maximum deviation of 1.7°. This outcome is the reason that the effective angles sometimes seem more reliable than the distances. The comparison of r 0 bond lengths to r e values for 55 bonds indicates that the MAD is 0.3 pm with a maximum deviation of 1.4 pm (Rudolph and Demaison 2011). When hydrogen atoms are present, the determined bond lengths are affected by a mostly systematic error of about 0.5(2) pm but with r e – r 0 > 0 for the ≡C(sp)– H bond and r e − r 0 < 0 when an sp2 or sp3 carbon is involved (Demaison and Wlodarczak 1994). Finally, as the rovibrational contributions to the inertial moments are predominantly positive, an r 0 structure may appear slightly expanded compared to the equilibrium structure. Indeed, in many cases, r 0 > r e . In conclusion, the r 0 method is useful for a first estimate of the structure, but it cannot be considered as reliable, except in the cases when the least-squares equations are well-conditioned.

6.4.3 Substitution-Like Structures 6.4.3.1

Kraitchman’s Equations

A better assumption than the neglect of the rovibrational correction is to assume that it remains constant upon isotopic substitution. Indeed, in many cases, the range of the εξ values is small compared to the mean values. For instance, in the particular case of OCS, the mean value of ε = I 0 − I e is 0.250 uÅ2 for twelve isotopologues whereas the range is only 0.018 uÅ2 ; see Table 6.2. However, there are at least two cases where this assumption is not valid: (i) when a hydrogen atom is substituted by a deuterium atom because the change of mass is large; (ii) when there is a large rotation of axes upon isotopic substitution. Following a suggestion by Costain (1958), analytical equations derived by Kraitchman (1953) have been widely used to determine the Cartesian coordinates of the substituted atoms; it is the so-called r s method. For a linear molecule, the calculation is quite simple. If we assume that z is the axis of the molecule, and if the Cartesian coordinates are measured from the center-of-mass of the parent molecule, the equilibrium moment of inertia of the parent molecule is Iex = Iey =



m i z i2

(6.6)

where the sum is over all atoms, mi is the mass of atom i and zi its Cartesian coordinate. If one atom is substituted, its mass becomes m + m and the equilibrium moment of inertia of the isotopic species is

134

6 Equilibrium Structures from Spectroscopy 

Iex =



m i z i2 + μz 2 with μ =

Mm M + m

(6.7)

where M is the total mass of the parent molecule. The difference of the two moments of inertia gives the value of the Cartesian coordinate of the substituted atom   1/2 1 x x |z s | = I − I0 μ 0

(6.8)

If the rovibrational correction is neglected, the substitution coordinate is equal to the equilibrium coordinate. For a general asymmetric top molecule, the equations are significantly more complicated, and their derivation is given in the Appendix 6.13.1 of this chapter. The Cartesian coordinates of the substituted atom are given by  P  − Pa 1  P  − Pa Pa − Pa b × c μ Pb − Pa Pc − Pa    Pc Pb Pa 1+ 1+ = μ Ia − Ib Ia − Ic

a2 =

(6.9)

The planar moments of inertia are defined by Pa = Paa =

 1 m i ai2 (−Ia + Ib + Ic ) = 2

(6.10)

The primed quantities refer to the daughter (substituted) isotopologue. The squared coordinates b2 and c2 are obtained by cyclic permutation. In the bottom part of (6.9), the last two factors are near unity because the numerators Pg (g = a, b, c) are expected to be much smaller than the denominators I g – I g . In conclusion, the coordinate g depends mainly on the isotopic difference Pg . Note that the sign of the coordinates is unknown, which can sometimes be a problem. When the substituted atom lies in the ab symmetry plane, c = 0 and ΔPc = 0. Likewise, when the substituted atom lies on the a symmetry axis, b = c = 0 and ΔPb = ΔPc = 0. These equalities are exact only with the equilibrium planar moments of inertia. Chutjian (1964) derived expressions for molecules in which symmetrically equivalent atoms are simultaneously determined. For the double substitution of a pair of atoms, see Appendix 6.13.2. Pierce (1959) proposed the double substitution method (where two atoms are substituted at the same time) to locate atoms near a principal axis by taking second differences of moments of inertia. This method has not found widespread application because of the necessity of very accurate data for many isotopologues. Furthermore, it was shown that the accuracy of the Pierce method is not satisfactory (Demaison et al. 1990; Le Guennec et al. 1991, 1993).

6.4 Empirical Structures

6.4.3.2

135

Accuracy (Demaison and Rudolph 2002)

The drawback of the substitution method is that Kraitchman’s equations are numerically unstable for small coordinates and may give inaccurate coordinates. From (6.8), it is seen that z s2 =

I0 − I0 ε Ie + ε = = z e2 + μ μ μ

(6.11)

or for the error δz = |z s − z e | =

1 ε 2 μ|z e |

(6.12)

Costain (1966) assumed that ε/μ is approximately constant and proposed the following empirical rule to estimate the uncertainty (in pm), now called Costain’s rule δz =

K |z|

(6.13)

Originally, Costain proposed K = 12, but, later, a slightly larger value was adopted: K = 15 (van Eijck 1982). The error is large when z is small (small coordinates are frequent in large molecules). But, as shown by (6.11), the error is also large when ε is large, i.e., when a hydrogen atom is substituted by deuterium or when a large rotation of axes occurs upon isotopic substitution (usual for oblate top molecules). But, ε can also be large in many other cases. For instance, in OCSe, although O is the farthest atom from the center of mass, its substitution coordinate is the least accurate in contradiction with (6.13): zs (O) = 225.06 pm to be compared with ze (O) = 224.86 pm whereas zs (C) = 109.42 pm to be compared with ze (C) = 109.54 pm. It is simply explained by the fact the ε(18 O) − ε(16 O) = 0.0182 uÅ2 , whereas ε(13 C) − ε(12 C) = −0.0025 uÅ2 is much smaller (Le Guennec et al. 1993). Furthermore, ε increases with the mass of the molecule; see Fig. 6.1, which shows the variation of ε upon bromine isotopic substitution (79 Br → 81 Br) for seventeen diatomic molecules going from the light DBr (I 0 = 3.9 uÅ2 ) to the heavy CsBr (I 0 = 476.4 uÅ2 ). A typical example is the heavy linear molecule SiC6 for which I 0 = 826.8 uÅ2 , the error on the C3 coordinate is |ze − zs | = 1.2 pm whereas the coordinate is not small: ze = 73.5 pm. Another difficulty is that, when one Cartesian coordinate is small (i.e., one atom is close to one principal axis), (6.8 or 6.9) may deliver an imaginary value. Still, the most worrying aspect is that the r s method is not able to correctly predict small changes in bond lengths. A typical example is the equatorial conformer of fluorocyclohexane, for which the r s values are (in pm): C1C2 = 152.2(4); C2C3 = 151.4(7); C3C4 = 153(2), whereas the corresponding rese values are 151.22(4); 153.12(7); and 152.55(3) (Juanes

136

6 Equilibrium Structures from Spectroscopy

Fig. 6.1 Plot of the variation of ε as a function of I 0 with isotopic substitution (79 Br → 81 Br) for seventeen diatomic molecules going from the light DBr (I 0 = 3.9 uÅ2 ) to the heavy CsBr (I 0 = 476.4 uÅ2 ), all values in uÅ2 . Reproduced from Journal of Molecular Spectroscopy, 215. Demaison J, Rudolph HD. When is the substitution structure not reliable? 78–84. Copyright 2002, with permission from Elsevier

et al. 2017). The Cartesian and internal coordinates of equatorial fluorocyclohexane are also compared in Table 6.3 whose inspection confirms the poor accuracy of the r s structure. Table 6.3 Comparison of the r s structure and the semiexperimental equilibrium structure for equatorial fluorocyclohexanea Cartesian coordinatesb (Å) as

ae

bs

be

cs

ce

C1

−0.9270(16) −0.92971(27) ic

C2

−0.2839(63) −0.25583(69) 1.2579(12) 1.25371(13) −0.1784(84) −0.18171(66)

0

C3

1.2239(12)

1.22310(19) 1.2615(12) 1.25710(11)

C4

1.9336(8)

1.92679(14) ic

0

0.3308(45)

0.32898(37)

0.2124(71)

0.21498(41)

−0.2885(52) −0.28680(51)

Internal coordinates (distances in pm and angles in degree) rs

re

C1C2 152.16(44)

rs

re 151.22(4)

C1C2C3

110.64(36)

110.17(3)

C2C3 151.42(68)

153.12(7)

C2C3C4

111.18(29)

111.01(2)

C3C4 153.17(31)

152.55(3)

C1C2C3C4

55.89(70)

55.99(5)

Reproduced from Physical Chemistry, Chemical Physics; Juanes M, Vogt N, Demaison J, León I, Lesarri A, Rudolph HD; Axial–equatorial isomerism and semiexperimental equilibrium structures of fluorocyclohexane (2017) 19: 29162–29169, with permission from the PCCP Owner Societies a Juanes et al. (2017) b The uncertainty is calculated with Costain’s rule, δz = 0.0015/|z| c Imaginary coordinate

6.4 Empirical Structures

137

Nevertheless, the r s method has the advantage of permitting a partial structure determination when only a few isotopologues have been studied. However, a comparison of r 0 and r s bond lengths to r e values for 55 bonds indicates that the maximum deviation of the residuals is much larger for r s − r e than for r 0 − r e , 2.1 pm instead of 1.4 pm, respectively (Rudolph and Demaison 2011). When possible (i.e., when the rotational constants of a sufficiently large set of isotopologues is available), it is better to use the least-squares fitting method where, in addition to the internal coordinates, the three rovibrational corrections εξ (ξ = a, b, c) are fitted. This method is sometimes called rIε , (Rudolph 1991). It avoids the problem of imaginary coordinates, and the first-moment equations (mi zi = 0) are automatically satisfied. Its obvious inconvenience is that this method introduces three supplementary parameters, which is often enough to undermine the quality of the fit. For this reason, the method described in Sect. 6.5 is preferred.

6.4.3.3

rc Method

For molecules with few atoms, it is possible to considerably improve the accuracy of the substitution method. It can be shown that (Watson 1973; Smith and Watson 1978) 2Iξs − Iξ0 = Iξe +

n ∂ 2 (Mεξ ) 1  m i m i + ··· M i=1 ∂m i2

(6.14)

where ξ = a, b, c and n is the number of atoms. The substitution moment of inertia, Iξs , must be calculated using the squares of all atomic position coordinates as individually obtained by the r s method, even if the square is negative. Nakata et al. (1980, 1981) suggested the use of complementary sets of isotopologues in order to eliminate the first term in the summation of (6.14). In other words, substitutions with positive and negative mi have to be used. This method gives accurate results; see for instance Table 8 of Demaison et al. (1997). The large number of isotopologues required limits its application to very small molecules.

6.5 Mass-Dependent Structures It was observed that the rovibrational correction ε is approximately proportional to the square root of the moment of inertia. It was first found empirically (Demaison and Nemes 1979) and, then, justified theoretically by Watson (1973) who confirmed that ε is a homogeneous function of atomic masses of degree ½. Therefore, the ground-state moment of inertia may be approximated by ξ

I0 = Imξ + cξ



ξ

Im

(6.15)

138

6 Equilibrium Structures from Spectroscopy

where the I m are calculated from the structure as if the molecule were a rigid rotor. When a structure is calculated with this approximation, it gives the rm(1) structure (Watson et al. 1999). This method was called m because it is mass-dependent as shown by (6.15) and “1” because one extra parameter per axis is added. It does not require more parameters than the r s -fit (or rIε ) method and, often, gives better results. There are, however, two cases where this method fails: (i) when a hydrogen atom is substituted by deuterium; (ii) when the substituted atom is close to the center of mass. Indeed, a plot of ε as a function of I 0 in the case of N2 O shows that there is not one line but two: the top one corresponding to the isotopic species x N14 Ny O (x = 14, 15 and y = 16, 17, 18) and the bottom one to the species x N15 Ny O, see Fig. 6.2. This behavior is general for atoms close to the center of mass as shown in Table 6.4. Generally, ε increases with I 0 but for an isotopic substitution close the center of mass, ε decreases, leading to negative values for ε. An analysis of several triatomic linear XYZ molecules where the coordinate of the central atom Y is small showed that ε varies as mX mZ /M where mX and mZ are the masses of the atoms X and Z, respectively (Le Guennec et al. 1993) Watson et al. (1999) generalized this method by proposing to include an additional empirical term, giving the rm(2) structure ξ I0

=

Imξ

 + cξ

ξ

Im + dξ

 m m · · · m 1/(2n−2) 1 2 n M

(6.16)

Fig. 6.2 Plot of ε = I 0 − I e versus I 0 for N2 O. All values in uÅ2 (Le Guennec et al. 1993)

6.5 Mass-Dependent Structures Table 6.4 Variation of the rovibrational correction ε = I 0 − I e (in uÅ2 ) as a function of the isotopic substitution

139 Molecule

Isotope

ε

COa2

16.12.16

0.1580

16.12.18

0.1642

0.0062

16.13.18

0.1620

0.0040

18.12.18

0.1709

0.0129

16.13.16

0.1560

– 0.0020

18.13.18

0.1686

0.0106

16.12.32

0.2454

16.12.34

0.2494

0.0040

16.13.32

0.2428

– 0.0026

18.12.32

0.2563

0.0109

16.12.80

0.3440

16.12.82

0.3454

0.0014

16.13.80

0.3415

−0.0025

18.12.80

0.3622

0.0182

14.14.16

0.2018

14.14.18

0.2103

0.0085

14.15.16

0.1996

– 0.0022

15.14.16

0.2060

0.0042

OCSb

OCSec

N2

Od

ε

a Graner

et al. (1986) b Lahaye et al. (1987) c Le Guennec et al. (1993) d Teffo and Chédin (1989)

where n is the number of atoms, and cξ and d ξ are fitting parameters (one per axis). To correctly take into account the substitution H → D in a bond XH, Watson et al. (1999) defined an effective XH distance  rmeff (XH)

= rm (XH) + δH

M m H (M − m H )

1/2 (6.17)

δ H is an additional fitting parameter, and the expression in parentheses is the inverse of the reduced mass of the H atom vibrating against the rest of the molecule. Although this equation gives satisfactory results for very small molecules, it does not give reliable results for large molecules. Actually, as will be shown in Sect. 6.7, it is better not to mix hydrogenated and deuterated molecules and to use instead the mixed estimation method; see Sects. 6.7 and 9.7. Watson also generalized the method to take into account a large rotation of axes upon isotopic substitution. However, instead of three unknown cξ parameters (one for each principal axis), six parameter are now necessary, cξ η (ξ, η = a, b, c). This complication significantly reduces the quality of the fit. Furthermore, a large rotation

140

6 Equilibrium Structures from Spectroscopy

Table 6.5 Different structures of OCSe and SO2 (distances in pm, angles in deg.) SOb2

OCSea r0

r(OC)

r(CSe)

κc

r(SO)

∠(OSO)

κc

115.364(59)

171.299(44)

102

143.358(17)

119.420(30)

2.98

rm

115.532(14)

170.816(22)

268

143.084(24)

119.442(32)

343

(2) rm

115.373(2)

170.950(2)

899

143.068(12)

119.340(20)

1486

re

115.327(2)

170.981(2)

131

143.0782(15)

119.3297(30)

8

(1)

a Unit

weighted fit of 27 isotopologues, except for r e where only 8 isotopologues were used weighted fit of 17 isotopologues, except for r e where only one isotopologue was used c Condition number b Unit

axis is exceptional (it mainly concerns oblate top molecules, which are much less common than prolate top molecules). The different structures of two small molecules, the linear OCSe and the triatomic SO2 , are compared in Table 6.5. The improvement, when going from r 0 to rm(2) via rm(1) , is shown by the decrease of the standard deviation of the fitted parameters, but there is a parallel increase of the condition number κ (a large condition number is an indicator of ill-conditioning; see Sect. 9.4.1). The rm(2) structure is within a few tenths of pm of the r e structure, which may be considered as satisfactory. However, in the case of OCSe, the rm(1) (OC) bond length is still far from the r e value. Because of the small value of the Cartesian coordinate of the central carbon atom, the rm(2) approximation is required. The weak points of this method are that there are more parameters to fit: ca , cb , cc , d a , d b , d c , and δH and that (6.16) is only an approximation. When the leastsquares system is ill-conditioned, which is the rule when the number of parameters to determine is large, the small remaining errors due to the approximate model are considerably amplified and the derived parameters are inaccurate. For this reason, the general rm(2) method was rarely used with success on a large molecule. However, a trivial amendment improves it significantly: using the method of mixed estimation (see Sect. 9.7), the structural parameters are fitted concurrently to predicate parameters and moments of inertia, associated with appropriate uncertainties. The predicate parameters are usually obtained by high-level quantum-chemical computations; see Chap. 2. With this modification, it was possible to determine accurate structures for molecules as large as fructose, C6 H12 O6 , (24 atoms, 66 independent structural parameters) (Vogt et al. 2016). As an example, the rm(2) and the semiexperimental equilibrium structures of the equatorial conformer of ethynylcyclohexane, C8 H12 , are compared in Table 6.6 (Vogt et al. 2018). The agreement is very satisfactory.

6.6 Experimental Equilibrium Structure Table 6.6 Comparison of the r m and r e equilibrium structures for the equatorial conformer of ethynylcyclohexane (distances in pm and angles in degree)a

141 rese

(2)

rm

rs

C7C8

120.53(11)

120.50(13)

120.9(4)

C1C7

146.33(13)

146.30(15)

147.4(4)

C1C2

153.38(11)

153.44(13)

151.5(8)

C2C3

152.51(11)

152.61(15)

156.9(16)

C3C4

152.48(10)

152.68(14)

153.2(4)

C1C7C8

178.39(21)

178.41(22)

178.3(16)

C7C1C2

111.060(66)

111.126(85)

109.0(39)

C1C2C3

110.929(98)

110.913(96)

109.6(6)

C2C3C4

111.460(89)

111.35(10)

111.7(3)

C3C4C5

110.940(84)

111.02(12)

110.9(3)

C2C1C6

110.56(10)

110.65(11)

113.3(7)

C1C2C3C4

−56.09(10)

−56.07(15)

−55.1(9)

Reprinted from Journal of Chemical Physics; Vogt N, Demaison J, Rudolph HD, Juanes M, Fernández J, Lesarri A; Semiexperimental and mass-dependent structures by the mixed regression method: Accurate equilibrium structure and failure of the Kraitchman method for ethynylcyclohexane (2018) 148: 064306, with permission from AIP Publishing a Vogt et al. (2018)

6.6 Experimental Equilibrium Structure The α-constants may be determined experimentally. In principle, it is enough to measure the rotational spectra for each vibrational fundamental in an excited state. From (6.1), we have ξ

αk = Bk(0) − Bk(υk =1)

(6.18)

where Bk(0) is a ground-state rotational constant and Bk(υk =1) the rotational constant in the excited state υ k = 1. However, as explained in Sect. 6.3.2, it is not easy to determine experimentally a full set of rotational constants for excited states of vibrational fundamentals. Moreover, this approach is often complicated because at least some excited states are not fully isolated but are in resonance either by Coriolis interaction or anharmonic (Fermi, Darling–Dennison, etc.) resonance; see Sects. 5.5 and 5.6. Unfortunately, such complications are frequently the case for a polyatomic molecule. It is then necessary to analyze the interactions between the excited states, which is not easy even for small molecules. However, when there are only two interacting vibrational states, it is not too complicated. For the Coriolis interaction, it is obvious from the denominator, that the Coriolis term, (6.2), becomes very large when the two frequencies ωk and ωl are close. In

142

6 Equilibrium Structures from Spectroscopy

such a case, the perturbation calculation breaks down, and the resonance should be treated by construction and diagonalization of a matrix of the rotational states of the coupled vibrations. The case of the anharmonic resonances, even weak, is particularly important because they are not taken into account in (6.2). The experimental rotational constants have to be corrected before using them for a structure determination. When the interaction is between two vibrational levels i and j, the perturbed rotational energy may be written using the results of Sect. 5.6 E R (Ai , Bi , Ci ) = ψi |HR |ψi              (0) = a 2 ψi(0) HR ψi(0) + b2 ψ (0) H ψ R j j   2 = E R (Ai , Bi , Ci ) + b E R (A j , B j , C j ) − E R (Ai , Bi , Ci ) (6.19) (Note that a2 + b2 = 1). When the resonance is not too large, it is possible to write 

ξ

Bi



ξ

ξ

ξ

= Bi + b2 (Bi − B j )

(6.20)

  + B ξj = Biξ + B ξj

(6.21)

It gives 

Biξ



When more than two levels are interacting, the situation is much more complicated and it is difficult to obtain unperturbed rotational constants. In some particular cases, it is possible to avoid this correction. For instance, there is a Fermi resonance between the levels υ 3 = 1 and υ 2 = 20 of many linear XYZ molecules. To cancel the effect of this resonance, the equilibrium rotational constants are usually calculated with Be = [5B000 − B100 − B001 − B O22 0 /2]

(6.22)

making use of (6.18). For instance, in the case of OCS, α 1 varies from 18.13 MHz before the Fermi correction to 20.14 MHz after the correction. Taking into account the Fermi resonance increases the C=O bond length by almost 0.2 pm and decreases the C=S bond length by the same value (Morino and Matsumura 1967; Lahaye et al. 1987). Unfortunately, this simplification is no longer valid when more than two states interact. In Table 6.7, the effect of neglecting the anharmonic interactions is illustrated for a few molecules. In conclusion, the determination of a purely experimental equilibrium structure is not easy because it is extremely difficult to gather the necessary experimental information and to correct for the resonances. For these reasons, up to now, accurate purely experimental equilibrium structures are only known for very small molecules (mainly up three independent structural parameters).

6.6 Experimental Equilibrium Structure

143

Table 6.7 Comparison of an accurate equilibrium structure, r e , (in pm) and the structure, r e (α), determined using the experimental α constants and neglecting anharmonic interactions r e − r e (α)

Molecule

Parameter

re

r e (α)

OCSa

r(C=O)

115.62(1)

115.45(2)

r(C=S)

156.14(1)

156.30

−0.16

r(C –H)

109.19(9)

109.725(4)

−0.53(9)

r(C=O)

110.55(3)

110.474(2)

0.08(3)

r(C –H)

107.02(10)

106.60(1)

r(C≡P)

153.99(2)

154.045(2)

−0.06(2)

r(C –F)

127.61(7)

128.4

−0.8

r(C≡P)

154.45(6)

153.8

0.6

HCO+b HCPc FCPd

0.17(2)

0.42(10)

a Lahaye

et al. (1987) (1988) and Puzzarini et al. (1996) c Puzzarini et al. (1996b) d Dréan et al. (1996) b Woods

As a typical example, details of the corrections used to calculate the experimental rotational constants of 32 S16 O2 are given in Table 6.8. The first conclusion is that at least one order of magnitude in precision is lost when going from the ground-state constants to the equilibrium ones. However, the final standard deviations are too optimistic because they are calculated using the law of propagation of errors that does not take into account the systematic errors. As expected, the main correction comes from the α-constants; see (6.1). The contribution of the γ -constants is not negligible,  although it is difficult to evaluate accurately. Flaud and Lafferty (1993) give γ /4 = 2.70 MHz, whereas Morino and Tanimoto (1994) estimate it at 1.9 MHz. It is easy to understand why: whereas it is relatively easy to obtain accurate rotational constants for the fundamental vibrational states, it is much more difficult to achieve the same goal for the combination and overtone bands because most of them are perturbed thereby significantly affecting the γ -constants. This accuracy problem is Table 6.8 Experimental equilibrium rotational constants of 32 S16 O2 (all values in MHz)a X

A

B

C

X0  α/2  γ /4

60778.5526(39)

10318.07332(60)

8799.70375(54)

−273.5665

41.1766

45.2091

γ 222 /8

0.0713

0.0000

0.0000

−276.1977

41.1604

45.4009

2.7025

Xrovib

0.0161

Xelec

19.9137

0.6564

Xcd

−0.1014

−0.2081

60522.17(52)

10359.682(74)

Xe a Source

of experimental data: Flaud and Lafferty (1993)

−0.1918

0.4271 0.2570 8845.7887(77)

144

6 Equilibrium Structures from Spectroscopy

confirmed by the value of the equilibrium inertial defect, Δe = −0.0014(6) uÅ2 , which is different from zero although two orders of magnitude smaller than the ground-state value, 0 = 0.163 uÅ2 . The electronic correction is also not negligible. The centrifugal distortion correction is much larger than the uncertainty of the ground-state rotational constants, but it has a small effect on the accuracy of the structure: there is a tiny increase of 0.001 pm for the SO bond length. Thus, except in very accurate works, it may be neglected. Neglecting the electronic correction, decreases the r(SO) bond length by 0.01 pm and increases the ∠(OSO) bond angle by 0.007°. These deviations may be considered as negligible for most works.  This conclusion also applies to the smaller γ /4 correction. Using the equilibrium moments of inertia derived from Table 6.8 allows us to obtain the equilibrium structure: r e (SO) = 143.0782(15) pm and ∠(OSO) = 119.3297(30)°. The estimated uncertainties are quite small, but they do not take into account the systematic errors. The true uncertainties may be one order of magnitude larger. See also Sects. 6.10 and 9.11. The high accuracy achieved for SO2 is mainly because the structure of this molecule can be determined from the moments of inertia of a single isotopologue. Hence, the equations are well-conditioned; see Sect. 9.4.1. When more than one isotopologue (the most frequent case) has to be used, the situation is less favorable because it is difficult to avoid the problem of ill-conditioning. A typical example is given by the structure of the formyl cation HCO+. Its structure was determined using the experimental equilibrium rotational constants of H12 C16 O+ , H13 C16 O+ , and H12 C18 O+ : r e (CH) = 109.72 pm and r e (CO) = 110.47 pm (Woods 1988). The results disagree with several accurate ab initio calculations; in particular for the CH bond, which seems much too long (Puzzarini et al. 1996a). An obvious explanation is that the fit was not well-conditioned and the two bond lengths were fully correlated [see Chap. 9 and Demaison et al. (1997)] because the rotational constant of DCO+ is missing. The rotational constant for this species was measured later, and its inclusion in the fit considerably improved the conditioning. However, statistical diagnostics indicate that this constant is not compatible with the other three. A careful analysis of the rovibrational spectra shows that the experimental rovibrational constant α 1 of D12 C16 O+ is heavily perturbed because the state υ 1 = 1 is in strong interaction with a nearby state. The unperturbed value of α 1 (obtained from the ab initio force field) is 339.79 MHz whereas the experimental value is more than 100 MHz smaller, 227.45 MHz (Puzzarini et al. 1996a). The υ 1 state at 2585.91 cm−1 may interact with two nearby states, either υ 2 = 40 at 2621.36 cm−1 or the combination state 011 0 at 2574.66 cm−1 . It was first assumed that the resonance was with the υ 2 = 40 state. However, soon after, an analysis of the rotational spectra of different υ 2 states indicated that the υ 2 = 40 is not significantly perturbed (in MHz): α 2 (υ 2 = 1) = −98.540; α 2 (υ 2 = 2) = −98.665; α 2 (υ 2 = 3) = −98.856; and α 2 (υ 2 = 4) = −99.092. The correct explanation is that a Coriolis interaction couples the e component of the 111 0  state to the 100  state. The solution is to determine α 1 from the combination B(100 0) + 2B(011 1) where the Coriolis contributions to each perturbed state

6.6 Experimental Equilibrium Structure

145

cancel out. This treatment gives α 1 = 339.791(24) MHz. With the corrected rotational constants, the experimental equilibrium structure has r e (CH) = 109.204 pm and r e (CO) = 110.558 pm in perfect agreement with the ab initio calculations and a recent semiexperimental structure: r e (CH) = 109.19(9) pm and r e (CO) = 110.55(3) pm (Dore et al. 2003). This example clearly shows the difficulties in obtaining a reliable purely experimental equilibrium structure, even for a small molecule. This case explains why such structures are only available for very small molecules: Diatomics, triatomics, tetratomics (NX3 , PX3 , AsX3 , SbH3 , BiH3 with X = H, F), and pentatomics (CH3 X, SiH3 X, GeH3 X with X = F, Cl, Br, I), see Chap. 10.

6.7 Semiexperimental Equilibrium (se) Structure The semiexperimental method has been employed in many studies and by many authors (see Vázquez and Stanton (2011) and Mendolicchio et al. (2017) for details), but it is worth pointing out here that the fundamentals of this technique, yielding an rese equilibrium structure, were laid down in 1978 by Pulay et al. (1978). Although they used a very modest Hartree–Fock (HF) level of theory, they could determine an accurate SE structure of methane, CH4 , with r e = 108.62(5) pm, which was later improved and confirmed by the almost perfect agreement of the SE structure and the one computed at a high level of ab initio theory [r e = 108.595(30) pm] (Stanton 1999). The pioneering work of Allen et al. (1990, 1992), Clabo et al. (1988) must be quoted because they were among the first workers to determine accurate semiexperimental structures, for instance for HNCO (East et al. 1993) and ketene, CH2 =C=O, (East and Allen 1995). As the determination of an experimental equilibrium structure is extremely complicated and time-consuming, it is attractive to calculate the rovibrational correction from the anharmonic force field. This method avoids the problem of measuring rotational constants of excited vibrational states of all needed isotopologues. Furthermore, the difficulty originating from the anharmonic resonances disappears. Finally, the summation of the α-constants eliminates the (possibly small) denominator in the Coriolis term, (6.2). The rovibrational correction may be written ξ

ξ

 ξ dk αk 2 k ⎧⎡ ⎫ 2  2 ⎤  ξ  ξξ ⎪ ⎨  3 akξ γ ⎬  2 ⎪ (ωl − ωk )2 ζkl   φ a c kkl l ⎢ ⎥ ξ = Be + π − ⎣ ⎦ γ 3/2 ⎪ ⎪ ωk ωl (ωk + ωl ) h ⎩ kγ 4ωk Ie ⎭ ωl k