Complementary Bonding Analysis 9783110660067

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Simon Grabowsky (Ed.) Complementary Bonding Analysis

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Complementary Bonding Analysis |

Edited by Simon Grabowsky

Editor PD Dr. Simon Grabowsky University of Bern Department of Chemistry, Biochemistry and Pharmaceutical Sciences Freiestrasse 3 CH - 3012 Bern Switzerland [email protected]

ISBN 978-3-11-066006-7 e-ISBN (PDF) 978-3-11-066007-4 e-ISBN (EPUB) 978-3-11-066027-2 Library of Congress Control Number: 2020948397 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Cover image: alice-photo / iStock / Getty Images Plus Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface Chemical bonding has always been my main fascination in science. I learned about the experimental access to it via electron-density determination from high-resolution single-crystal X-ray diffraction experiments during my PhD work in Professor Peter Luger’s group at the Free University of Berlin. During my postdoctoral stay at the University of Western Australia, Professor Dylan Jayatilaka and Professor Mark Spackman opened the theoretical access to me. During my research stay at the University of Bremen, the German Research Foundation generously sponsored a workshop, for which I invited the leading method and software developers from both worlds (quantum chemistry and quantum crystallography) to teach a diverse portfolio of methods and software to graduate students, PhD students, postdocs and early career researchers. This was something I knew from my higher University education was missing. Both the support by the teachers and the reception by the participants was overwhelming, so that the “Workshop Tools for Chemical Bonding 2019” (Bremen, Germany, July 14– 19, 2019) was a success and the collaborative project complementary bonding analysis was born. It is thanks to my colleague in Bremen, Professor Jens Beckmann, who had the initial idea of recording and publishing the lectures and software sessions of the workshop, and to the initiative of Ms. Berber-Nerlinger from the deGruyter publishing house, that this book “Complementary Bonding Analysis” has become reality. But mostly, I owe a debt of gratitude to all the 29 authors of the 14 chapters of this book, most of which, but not all of which, were also teachers at the workshop. I have enjoyed the collaboration and the contact with them over nearly 2 years. Finally, I thank my former coworker, Dr. Malte Fugel, and my former PhD colleague, Dr. Stefan Mebs, for their practical work on the comparisons of huge varieties of bonding descriptors which pushed me into the direction of formalizing the concept of complementary bonding analysis as a principle in research and teaching. I hope that our readers find the presented smorgasbord of methods and software as useful as I do. Bern, October 1, 2020

https://doi.org/10.1515/9783110660074-201

Simon Grabowsky

Contents Preface | V

Part I: The importance of chemical bonding concepts Simon Grabowsky 1 Introduction to complementary bonding analysis | 3 Dietmar Stalke 2 Chemical concepts of bonding and current research problems, or: Why should we bother to engage in chemical bonding analysis? | 9

Part II: Bonding descriptors from quantum chemistry Ángel Martín Pendás and Carlo Gatti 3 Quantum theory of atoms in molecules and the AIMAll software | 43 Miroslav Kohout 4 Electron localizability indicator and bonding analysis with DGrid | 75 Jean Christophe Tremblay 5 Is there a unique way of localizing molecular orbitals, and why not | 113 Eric D. Glendening, Clark R. Landis, and Frank Weinhold 6 Natural bond orbital theory: Discovering chemistry with NBO7 | 129 Avital Shurki, Benoît Braïda, and Wei Wu 7 Valence bond theory with XMVB | 157 Trevor A. Hamlin, Pascal Vermeeren, Célia Fonseca Guerra, and F. Matthias Bickelhaupt 8 Energy decomposition analysis in the context of quantitative molecular orbital theory | 199

Part III: Bonding descriptors from quantum crystallography Piero Macchi 9 Introduction to quantum crystallography | 215

VIII | Contents Benoît Guillot, Christian Jelsch, and Piero Macchi 10 Multipole modeling with MoPro and XD | 235 Alessandro Genoni and Dylan Jayatilaka 11 X-ray constrained wavefunction analysis with Tonto | 269 Dana Nachtigallová and Pavel Hobza 12 Introduction to noncovalent interactions | 309 Mark A. Spackman, Peter R. Spackman, and Sajesh P. Thomas 13 Beyond Hirshfeld surface analysis: Interaction energies, energy frameworks and lattice energies with CrystalExplorer | 329 Rubén Laplaza, Francesca Peccati, David Arias-Olivares, and Julia Contreras-García 14 Visualizing non-covalent interactions with NCIPLOT | 353 Appendix | 379 Index | 383 E1

Erratum to: Chapter 2 Chemical concepts of bonding and current research problems, or: Why should we bother to engage in chemical bonding analysis? By Dietmar Stalke | 395

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Part I: The importance of chemical bonding concepts

Simon Grabowsky

1 Introduction to complementary bonding analysis The bond is arguably the most important concept in chemistry. The design of drugs or materials very often depends on the chemist’s understanding or notion, perception and intuition of the strengths, characters and properties of bonds. In fact, all developments in chemistry, materials science, life science and other disciplines are directly or indirectly based on an understanding of chemical bonding and reactivity. However, bonds are not observables, there is no proof that they exist at all. Coulson stated in 1955: “A chemical bond is not a real thing: it does not exist, no one has ever seen it, no one ever can. It is a figment of our imagination” [1]. Despite this inconvenient fact, chemists use concepts related to the nature of chemical bonds to explain reality. Our understanding of properties and reactivity is based on our notion of chemical bonding in all its facets, although Coulson adds: “Concepts like hybridization, resonance, covalent, and ionic structures do not appear to correspond to anything directly measurable” [2]. Whenever we discuss chemical bonding in this textbook, but also in any other scientific context, we must keep in mind that there is no rigorous or general definition of a chemical bond or of chemical bonding [3, 4]. Why is the concept of the chemical bond so useful if even its very existence is in doubt? Hoffmann et al. pointed out: “Chemistry has done more than well in creating a universe of structure [...] with just this imperfectly defined concept of a chemical bond. Or maybe it has done so well precisely because the concept is flexible and fuzzy” [5]. Chemical bonds are universal tools that can flexibly be adapted to construct models of chemical structure that are related to observations. These models have largely been based on the concept of directed two-center-two-electron bonds drawn in Lewis structures for the last 100 years [6]. This notion corresponds to a pair of electrons shared between two atoms and is therefore synonymous with covalent bonding [7]. It is indicated by a solid line connecting these two atoms (e. g., H–H). However, Lewis’ electron-pair bond was purely based on empirical findings, and, at that time, it was lacking a physical foundation. Soon after the introduction of electron-pair bonding in 1916 [6], Lewis realized that quantum mechanics is necessary for a physically sound description of chemical bonding [8]. In 1927, Heitler and London published their quantum mechanical description of the H2 molecule, which is characterized by the introduction of ionic resonance forms (H+ H− and H− H+ in addition to H–H) resulting in an energetic stabilization of the wavefunction of the H2 molecule [9]. Pauling’s work unified Lewis’ empirical concept and the description provided by quantum mechanics, which is the reason for the continuing success of Lewis structures [10]: Even today, Simon Grabowsky, University of Bern, Department of Chemistry, Biochemistry and Pharmaceutical Sciences, Freiestrasse 3, 3012 Bern, Switzerland, e-mail: [email protected] https://doi.org/10.1515/9783110660074-001

4 | S. Grabowsky solid lines are used to represent bonds. However, the question arises if every solid line is, in fact, an electron pair shared between two atoms [11, 12]. Many more detailed bonding concepts have been developed on top of or beyond Lewis structures, many of which will be covered in this book. There are the concepts of resonance [13] and multicenter interactions [14], hybridization [15, 16], covalency vs. ionicity [17], charge-shift bonding [18], hydrogen bonding [19], van-der-Waals-type interactions such as London dispersion [20, 21], etc., all of which ascribe a certain character or nature to the chemical bond. Hence, a huge variety of different bonding descriptors that qualitatively represent or quantify certain aspects of the character of a chemical bond have been developed. An (incomplete) selection of these descriptors, and how they are obtained and interpreted with respect to the myriad of existing bonding concepts, is the topic of this textbook. We focus on descriptors from the frameworks of the Quantum Theory of Atoms in Molecules (QTAIM, Chapter 3), the Electron Localizability Indicator (ELI, Chapter 4), Natural Bond Orbital Theory (NBO, Chapter 6), Valence Bond Theory (VB, Chapter 7), and Energy Decomposition Analysis (EDA, Chapter 8). Chapters 2 and 5 bring these methods into perspective in the light of electron densities and molecular orbitals. One aspect of the complementarity of bonding descriptors in general lies in the fact that bonding can be viewed from within different realms or spaces, which we simplify as (i) real space, (ii) orbital space, (iii) and an energetic viewpoint, which might emerge from (ii) or augment both (i) and (ii) (Figure 1.1). Although a classification is ambiguous, QTAIM and ELI rather fit into (i), NBO and VB into (ii) and EDA, if used standalone, into (iii). This means that all these bonding descriptors accentuate different aspects of the character of bonds. This, in turn, might lead to conflicting or opposing conclusions about bonding although based on the very same wavefunction, which can be regarded as a dilemma [22, 23]. This is, of course, due to the absence of a unified or rigorous definition of what a bond is, as discussed above. But the prob-

Figure 1.1: The double bond in ethylene depicted with bonding descriptors from different realms: A localization domain of the ELI-D in the real space picture, the π-NBO in the orbital (or Hilbert) space picture, the total energy density color coded with itself adding an energetic perspective. The green spheres represent bond critical points of QTAIM.

1 Introduction to complementary bonding analysis |

5

lem goes even deeper than that; quantum mechanics does not uniquely define atoms in molecules [24]. And many bonding descriptors, such as bond indices or population analyses, directly depend on the definition of an atom in a molecule [17, 25]. Therefore, the dilemma cannot be solved at a fundamental level. Naturally, the literature is full of controversies around the interpretation of chemical bonding based on bonding descriptors arising from different frameworks, e. g., discussions on differences between topological real-space approaches and energy partitioning analysis [26, 27], on the meaning of bond paths in QTAIM [28–30], or on the use of arrows in extended Lewis structures [31–33]. Although such debates are instructive and educational, they reflect a certain notion of chemical bonding theories to be all-encompassing and exclusive. With this book, we wish to contribute to an end of “either/or” and “better/worse” discussions around chemical bonding. We want to convey how to obtain and interpret various bonding descriptors from various frameworks in a neutral mindset, so that employing the most useful selection of them based on the individual research question will be more easily feasible in the future. This is the philosophy behind complementary bonding analysis, which is a comparative and inclusive approach to chemical bonding [34]. Rahm states within a discussion about EDA methods that “However, rather than risking extinction, I suspect that EDA methods subjected to comparative studies will thrive. Comparisons will bring out complementarities in different approaches and ultimately allow us to get a better overall grasp of electronic structure and chemical bonding” [35]. In this book, the experts who develop the respective methods teach ways of interpreting selected bonding descriptors arising from their bonding theory framework. However, for many interested users, the first hurdle is to know how to derive these descriptors. Therefore, a central focus of this book is the software underlying the discussed methods. Although software changes over time and some input options might be different or not available anymore in a few years, the compilation that encompasses this book can always serve as a first clue for any interested user on what range of possibilities there are for the analysis of the research question at hand. Ultimately, discussions on the nature and character of chemical bonding should not be solely in the hands of experts, but should broadly underlie any interesting compound synthesized or analyzed in the chemical laboratory. This creates a broader knowledge base, especially in the light of the ambiguity that surrounds chemical bonding analysis. The complementarity of bonding analysis schemes featured in this book has another aspect to it: the complementarity between experiment and theory. Calculating a wavefunction and optimizing the geometry is only one way of obtaining reliable bonding properties. Another way described here is the analysis of the single-crystal X-ray diffraction experiment. Nowadays, both crystal structure determinations and isolatedmolecule calculations are a sine qua non for publications on novel compounds, materials, drugs or synthetic routes. However, often researchers do not realize that even a routine in-house X-ray structure determination experiment offers more than only accurate atomic coordinates. The measured X-ray structure factors carry all the informa-

6 | S. Grabowsky tion about the electronic structure of the compound under investigation, limited only by the resolution of the experiment and the experimental errors. Extraction of that chemical bonding information and the related molecular and solid-state properties is at the heart of the field of quantum crystallography (Figure 1.2) [36–38].

Figure 1.2: Visualization of the concept of quantum crystallography as an amalgamation of diffraction experiments and wavefunction/ orbital/ electron density models to obtain insight into chemical bonding; here an ELI-D localization domain (blue) and basin (orange) representation of a N. . . Si periinteraction in a naphthalene framework.

Consequently, complementary bonding analysis also deals with the complementarity between quantum chemistry and quantum crystallography. Whereas quantum chemistry could be viewed as a computational approach to chemistry, quantum crystallography consists of the interplay between experiment and theory. In that sense, it is a natural science in its own right (see Chapter 9). Two methods for deriving chemical bonding information from the X-ray diffraction structure factors are described in detail in this book: Multipole modeling (Chapter 10) and X-ray constrained wavefunction fitting (Chapter 11). Subsequent to the former, an experimentally-based QTAIM analysis is possible; subsequent to the latter, all the bonding analysis schemes that are based on wavefunctions are accessible, restrained to the experimental observations. Since quantum crystallography mostly deals with the crystalline solid state, intermolecular interactions are a class of bonds easily accessible to this method. They are introduced in Chapter 12, and two different ways of analyzing them are discussed in detail together with the relevant software: Hirshfeld-surface and interaction-energy based CrystalExplorer analysis (Chapter 13) as well as the noncovalent interaction (NCI) index (Chapter 14). Chemical bonding is not simple or easily understood. Therefore, we believe that complementary bonding analyses are needed. This textbook offers a practical introduction to complex questions of chemical bonding that lie at the very heart of chemistry.

1 Introduction to complementary bonding analysis | 7

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Coulson CA. The contributions of wave mechanics to chemistry. J Chem Soc. 1955;2069–84. Coulson CA. The Spirit of Applied Mathematics, an Inaugural Lecture Delivered Before the University of Oxford on 28 October 1952. Oxford: Clarendon Press; 1953. Shaik S, Rzepa HS, Hoffmann R. One molecule, two atoms, three views, four bonds? Angew Chem Int Ed. 2013;52:3020–33. Ball P. Beyond the bond. Nature. 2011;469:26–8. Alvarez S, Hoffmann R, Mealli C. A bonding quandary—or—a demonstration of the fact that scientists are not born with logic. Chem Eur J. 2009;15:8358–73. Lewis GN. The atom and the molecule. J Am Chem Soc. 1916;38:762–85. Langmuir I. J Am Chem Soc. 1919;41:1543–59. Lewis GN. Valence and the Structure of Atoms and Molecules, vol. 14. Chemical Catalog Company,Incorporated; 1923. Heitler W, London F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z Phys. 1927;44:455–72. Pauling L. The Nature of the Chemical Bond and the Structure of Molecules and Crystals. Ithaca, NY: Cornell University Press; 1939. Frenking G, Shaik S. The Chemical Bond: Fundamental Aspects of Chemical Bonding. Weinheim: Wiley-VCH; 2014. Zhao L, Schwarz WHE, Frenking G. The Lewis electron-pair bonding model: the physical background, one century later. Nat Rev Chem. 2019;3:35–47. Hückel E. Quantum-theoretical contributions to the benzene problem. I. The electron configuration of benzene and related compounds. Z Phys. 1931;70:204–86. Coulson CA. The nature of the bonding in xenon fluorides and related molecules. J Chem Soc. 1964;1442–54. Pauling L. The nature of the chemical bond. Application of results obtained from the quantum mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J Am Chem Soc. 1931;53(4):1367–400. Alabugin IV, Bresch S, dos Passos Gomes G. Orbital hybridization: a key electronic factor in control of structure and reactivity. J Phys Org Chem. 2015;28:147–62. Fugel M, Hesse MF, Pal R, Beckmann J, Jayatilaka D, Turner MJ, Karton A, Bultinck P, Chandler GS, Grabowsky S. Covalency and ionicity do not oppose each other —relationship between Si-O bond character and basicity of siloxanes. Chem Eur J. 2018;24:15275–86. Shaik S, Danovich D, Morrisson Galbraith J, Braïda B, Wu W, Hiberty PC. Charge-shift bonding: a new and unique form of bonding. Angew Chem Int Ed. 2020;59:984–1001. Steiner T. The hydrogen bond in the solid state. Angew Chem Int Ed. 2002;41:48–76. Liptrot D, Power P. London dispersion forces in sterically crowded inorganic and organometallic molecules. Nat Rev Chem. 2017;1:0004. Wagner JP, Schreiner PR. London dispersion in molecular chemistry—reconsidering steric effects. Angew Chem Int Ed. 2015;54:12274–96. Henn J, Leusser D, Stalke D. Chemical interpretation of molecular electron density distributions. J Comput Chem. 2007;28:2317–24. Jacobsen H. Chemical bonding in view of electron charge density and kinetic energy density descriptors. J Comput Chem. 2009;30:1093–102. Parr RG, Ayers PW, Nalewajski RF. J Phys Chem A. 2005;109:3957–9. Bultinck P, Popelier P. Atoms in Molecules and Population Analysis. London: Taylor and Francis; 2009. p. 215–27.

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[26] Kovács A, Esterhuysen C, Frenking G. The nature of the chemical bond revisited: an energy-partitioning analysis of nonpolar bonds. Chem Eur J. 2005;11:1813–25. [27] Bader RFW. Comment on the comparative use of the electron density and its Laplacian. Chem Eur J. 2006;12:7769–72. [28] Poater J, Solà M, Bickelhaupt FM. Hydrogen–hydrogen bonding in planar biphenyl, predicted by atoms-in-molecules theory, does not exist. Chem Eur J. 2006;12:2889–95. [29] Grimme S, Mück-Lichtenfeld C, Erker G, Kehr G, Wang H, Beckers H, When WH. Do interacting atoms form a chemical bond? Spectroscopic measurements and theoretical analyses of dideuteriophenanthrene. Angew Chem Int Ed. 2009;48:2592–5. [30] Bader RFW. Bond paths are not chemical bonds. J Phys Chem A. 2009;113:10391–6. [31] Himmel D, Krossing I, Schnepf A. Dative bonds in main-group compounds: a case for fewer arrows! Angew Chem Int Ed. 2014;53:370–4. [32] Frenking G. Dative bonds in main-group compounds: a case for more arrows! Angew Chem Int Ed. 2014;53:6040–6. [33] Himmel D, Krossing I, Schnepf A. Dative or not dative? Angew Chem Int Ed. 2014;53:6047–8. [34] Fugel M, Beckmann J, Jayatilaka D, Gibbs GV, Grabowsky S. A variety of bond analysis methods, one answer? An investigation of the element–oxygen bond of hydroxides Hn XOH. Chem Eur J. 2018;24:6248–61. [35] Andrés J, Ayers PW, Boto RA, Carbó-Dorca B, Chermette H, Cioslowski J, Contreras-García J, Cooper DL, Frenking G, Gatti C, Heidar-Zadeh F, Joubert L, Pendás AM, Matito E, Mayer I, Misquitta AJ, Mo Y, Pilmé J, Popelier PLA, Rahm M, Ramos-Cordoba E, Salvador P, Schwarz WHE, Shahbazian D, Silvi B, Solà M, Szalewicz K, Tognetti V, Weinhold F, Zins EL. Nine questions on energy decomposition analysis. J Comput Chem. 2019;40:2248–83. [36] Grabowsky S, Genoni A, Bürgi HB. Quantum crystallography. Chem Sci. 2017;8:4159–76. [37] Genoni A, Bučinský L, Claiser N, Contreras-García J, Dittrich B, Dominiak PM, Espinosa E, Gatti C, Giannozzi P, Gillet JM, Jayatilaka D, Macchi P, Madsen AØ, Massa L, Matta CF, Merz KM, Nakashima PNH, Ott H, Ryde U, Schwarz K, Sierka M, Grabowsky S. Quantum crystallography: current developments and future perspectives. Chem Eur J. 2018;24:10881–905. [38] Macchi P. Quantum Crystallography – Fundamentals and Applications. DeGruyter Textbook, deGruyter Oldenbourg; 2021.

Dietmar Stalke

2 Chemical concepts of bonding and current research problems, or: Why should we bother to engage in chemical bonding analysis? 2.1 Introduction Today’s pressing challenges like sustainable energy, climate change or even land use and poverty are all closely connected to chemistry and its products. To address and partly meet them, we have to find ways to produce energy- and atom-economically materials with the desired properties like energy storing, light-conversing or healing properties. This can only be achieved by synthetic strategies to selectively modify and fine-tune atomic connectivities. Unfortunately, the one-to-one causality between structural changes and the modification of function is still unknown in any natural science. It would be a major step forward to be able to ab initio reliably predict which alterations in chemical bonding would cause what effects. Currently, however, the syntheses and findings of new promising target molecules heavily rely on the expertise and the chemical intuition of a skilled preparative chemist and frequently even on serendipity. To facilitate rational design, we need to understand the relation between structural changes (i. e., changes in the atomic topology) and the variation of the functionality, both in materials- and life-science. The electron density (ED) is the common observable and calculable denominator, which needs to be analyzed and rationalized to understand chemical bonding and eventually function [1–4]. The real-space ED can be investigated both by the theoretical wave function and experimentally by the X-ray diffraction experiment. Unfortunately, the experiment does not provide the direct answer and still needs to be modeled because only the position and intensity of the Bragg reflection can be determined while the phase information is lost. That needs to be estimated to recover the ED from the diffraction The original version of this chapter has been revised by the author. An Erratum is available at DOI: https://doi.org/ 10.1515/9783110660074-016

Acknowledgement: This work was supported by the Deutsche Forschungsgemeinschaft within the Priority Program 1178 “Experimental charge density as the key to understand chemical interactions,” the DNRF-funded Center for Materials Crystallography, and the PhD program CaSuS, Catalysis for Sustainable Synthesis, funded by the Land Niedersachsen and the Volkswagenstiftung. The author is particularly indebted to many capable students for providing the results that form the basis of this article. Dietmar Stalke, Universität Göttingen, Institut für Anorganische Chemie, Tammannstraße 4, 37077 Göttingen, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110660074-002

10 | D. Stalke pattern. As soon as the sort and position of the atoms in the crystal lattice is assumed the related structure factors can be calculated from the parameterized model to give the calculated structure factors Fcalc . They are compared to the observed ones (Fobs ). In an iterative least-squares refinement process of the model, the parameters Fcalc are converged to Fobs . If the figure of merit “R-value’’ is better than 5 %, it means that the model describes 95 % or more of the experiment. Classically, this is achieved by the Independent Atom Model (IAM), describing the x,y,z positional coordinates of the atom type in the unit cell and the six parameters of the anisotropic displacement (frequently misnamed the “thermal ellipsoid’’). The atom-centered multipole model (MM) by Hansen and Coppens [5] allows a much more adapted model to the experimental ED because it accounts, e. g., for aspherical valence density and charge transfer. Partitioning the total atomic density ρ(r) into three components produces a much more precise image of the crystal’s content: (i) The spherical core density ρc (r) mostly stays unmodified by any chemical interactions. (ii) The spherical valence density ρv (κr) mirrors, e. g., the oxidation state of the atom. (iii) The aspherical valence density ρd (κ′ r) is deformed by the bonding partner and even by the next neighbors in the lattice. κ and κ’ represent radial scaling parameters that allow for an expansion or a contraction of the related density. The aspherical valence shell density ρd (κ′ r) is modeled by spherical harmonics, the multipoles. They are reminiscent of the orbitals from theory and provide the most flexible way to designate the measured density. Although the core density ρc (r) stands for the inner electron energy levels of an atom, normally untouched by chemical reactions, from very accurate experiments, it is clear that even those regions of ED close to nuclei may be affected. This phenomenon is called core polarization [6, 7]. The spherical valence density ρv (κr) and especially kappa account for the oxidation state of the refined element. The higher the oxidation state the smaller the radius. Hence, the oxidation state is an experimentally directly accessible feature from the ED, i. e., the Fe(II) in a porphyrin system can independently be distinguished from the Fe(III) and not just indirectly by looking at the Fe–N distances. The aspherical valence shell density ρd (κ ′ r) mimics the polarization effects of the bonding partners and models even the nonbonding regions, inaccessible by mere geometrical parameters from atom positions. The ED model generated that way is then analyzed according to Bader’s Quantum Theory of Atoms in Molecules (QTAIM) [8]. The main descriptors and tools for chemical bonding in this framework are introduced and explained in Chapter 3 by Ángel Martín Pendás and Carlo Gatti. This demanding ED analysis goes beyond simple 1,2- and 1,3-distance considerations and needs to address leading concepts in chemistry, should challenge commonly held perceptions and should help to develop more detailed and resilient models to improve materials and processes. To my estimate, the currently most rewarding areas are covalent versus dative bonding and catalysis. I am going to present these two in more detail in this chapter.

2 Chemical concepts of bonding and current research problems | 11

2.2 Covalent, dative and aromatic bonding in organosilicon chemistry 2.2.1 Si(II) and Si(0) compounds In this chapter, I primarily resume results from [23, 49–51]. The first topic will be sketched on the field of contemporary silicon(II) compounds. After they have been synthesized only recently, they are employed in small molecule activation on their way to sustainably back-up expensive, rare and toxic transition metal catalysts [9–14]. Current industrial-scale silicon chemistry, i. e. predominantly polymeric silicones, still rests on Si(IV) conversion, but low-valent silicon species contribute more and more to industrial-scale catalytic application [15]. While the lower oxidation state +II gradually gets more stable while descending Group 14, the +IV oxidation state still is favored by silicon. This way, it behaves as a role model for all of the other Group 14 elements, leaving carbon exceptional. Bonds to silicon cover a wide range in inorganic and organometallic chemistry [16], as far as the bonding type and the variety of bonding partners are concerned [17]. Bonding in silicon(II) needs to be understood as it would be most rewarding to stabilize silylenes with silicon in oxidation state +II and revamp this abundant, nontoxic and cheap element to a catalyst. In Si(II), the HOMO-LUMO gap is as small and advantageous for catalyses as in rare and expensive transition metal catalysts. Characteristic is the high reactivity, because silylenes are analogues of carbenes [18–20]. We modified the original synthesis of monomeric chlorosilylene L1 SiCl (1, L1 = PhC(NtBu)2 in Scheme 2.1) [21] employing L1 SiHCl2 and bis-trimethyl silyl lithium amide to give 1 in much higher yields [22]. N-heterocyclic carbene stabilized L2 SiCl2 (2, L2 = 1,3-bis(2,6-iPr2 C6 H3 )imidazol-2-ylidene, N-heterocyclic carbene, NHC) was prepared by reducing HSiCl3 in the presence of L2 (NHC) without using any alkali metals [23]. Silylene 1 as well as 2 and their derivatives are widely activating small molecules and functioning as ligands for transition metals [24]. The bis-silylene (L1 Si-SiL1 (3) [25]) also received remarkable attention and reactions with alkynes [26], ketones [27, 28], N2 O [29], aluminium hydride [30], carbodiimide [31] and others are reported. In 2008, Robinson et al. published the synthesis of NHC-stabilized bis(monochlorosilylene) (4a) and a stable diatomic silicon molecule (5) via the reduction of L2 SiCl3 or L2 SiCl4 with KC8 [32]. Subsequently, Filippou et al. obtained a series of L2 Si(X)-Si(X)L2 (X = Cl, Br, I) compounds (4) from the reaction of 1,2-C2 H4 X2 and 5 via elimination of a C2 H4 molecule [33]. The NHC ligand is superior in stabilizing low oxidation states in main group as well as in transition metal chemistry. It has to be regarded a neutral carbene donor ligand rather than a carbanion. Thus, the Si–NHC bond seems not to be a classical twocenter two-electron covalent bond but to be a hybrid of the classical double bond in R2 C=CR2 olefins and the dual donor-acceptor bond in R2 Si󴀘󴀯SiR2 disilylenes. Hence, it

12 | D. Stalke

Scheme 2.1: Silylenes stabilized by NHCs and amidinate.

was most rewarding to look at (NHC)SiCl2 (2 in Scheme 2.1) by means of charge density determination [23]. In the structure of 2, the three substituents at the central silicon atom are arranged in an almost rectangular fashion (Cl1–Si–Cl2 97.25, C–Si–Cl2 94.66, C–Si–Cl1 98.80°). Immediately, this indicates a compelling case of predominantly p-orbital bonding at silicon and very little s-orbital mixing. Silicon shows no hybridization from the geometrical parameters. The same is valid for the topology from QTAIM analysis, which witnesses s-character of the lone-pair (Figure 2.1(a)). The second derivative of the ED, the Laplacian, i. e., the energy density, clearly displays the spherical arrangement of the valence-shell charge concentration (VSCC) in the nonbonding region of the silicon atom (Figure 2.1(b)). A stereochemically active lone pair from a spn hybrid orbital at silicon would have given a much more concentrated and directed VSCC. The Si–C bond of 1.99 Å is 0.10 Å longer than the average Si–Caryl single bond but shows significant ellipticity ε, normally associated with double bond character in non-polar C=C bonds. Obviously, we are facing a different scenario in 2 (Figure 2.1(c)). Qualitatively, the NHC carbon atom accommodates the singlet lone pair in the directed sp2 hybrid orbital, donating to the vacant Si p-orbital, orthogonal to the SiCl2 plane. Simultaneously, the silicon s-orbital singlet lone pair donates back to the unoccupied p-orbital of the carbene carbon atom. Therefore, the arrangement of the NHC ligand relative to the SiCl2 plane has to be wider than 90° (96° in 2 and 97° in (NHC)SiBr2 [34]), because otherwise the s-/p-orbital overlap integral vanishes. Consequently, this bonding mimics the dual donor-acceptor bond known from transition metal complexes rather than from the classical double bond. The ellipticity has to be attributed to the s/p-(back)donation. Although the π*-orbital is not available for the back donation it adds to the

2 Chemical concepts of bonding and current research problems | 13

Figure 2.1: (a) ED determination in (NHC)SiCl2 (2), (b) singlet lone pair character in the s-orbital at silicon and (c) dual donor acceptor bond in 2 (a reproduced from [23] with permission of John Wiley & Sons, Inc.)

delocalised density of the heteroaromatic ring. The central carbene atom searches for that additional interaction. From various charge density investigations [35, 36], it is known that most of the ring density is located at the nitrogen atoms, leaving the carbon atom quite depleted. Even more capable ligands than the NHCs are the cyclic alkyl(amino) carbenes (cAACs). Their syntheses were reported in 2005 by Bertrand et al. [37]. In the cAACs, one σ-withdrawing and π-donating nitrogen atom in the NHCs is substituted by an σ-donating quaternary carbon atom yielding a lower lying LUMO. Consequently, cAACs are superior ligands for the stabilization of various unstable chemical species [38], radicals [39–46] and elements in different oxidation states [47, 48] because of their stronger π-accepting properties. It was only with this ligand that we could synthesize and study a silylone, a single silicon(0) atom coordinated to two dual donor acceptor cAACs [49]. The abovementioned NHC coordinated dichlorosilylene (NHC)SiCl2 (2) is the starting material in that synthesis. Three equivalents cAAC need to be added to 2. The first replaces the NHC, proving to be the better donor ligand. Therefore, a second cAAC coordinates to the Si(II) atom. The remaining equivalent C–C couples with the NHC. This side product can be filtered off. The target molecule (cAAC)2 SiCl2 (7) contains a tetrahedrally coordinated central Si(II) atom, bound with considerably shorter distances than in 2 to two chlorine atoms (2.07 vs. 2.17 Å) and to two cAACs (1.85 vs. 1.99 Å), respectively (Scheme 2.2). A different bonding is already indicated by these geometric parameters. In addition, EPR spectroscopy indicates an unpaired biradical character of 7 with the related hyperfine structured signal, indicating that the unpaired electrons are located at the carbene carbon atoms. The singlet-triplet excitation energy required for cAAC (209 kJ/mol) is much lower than for NHC (372 kJ/mol). Hence, the calculated Si–C bond dissociation energy of 227 kJ/mol in 7 is sufficiently compensating for the required singlet-triplet excitation energy in the SiCl2 fragment. The much higher singlet-triplet gap in NHC explains thermodynamically why the singlet state in (NHC)SiCl2 (2) is maintained. Different to the long dual donor-acceptor bond in 2 two

14 | D. Stalke

Scheme 2.2: Some features of (NHC)SiCl2 (2) in comparison to (cAAC)2 SiCl2 (7).

short classical Si–C 2-center 2-electron bonds are formed, remarkably even against the higher steric strain in 7 compared to 2. In this bonding, the two cAAC radical substituents serve as simple one-electron donating organic residues R each and the single dash from the Lewis diagram matches the findings quite well. In (NHC)SiCl2 (2), however, the double arrow, visualizing the dual donor-acceptor bond, is much closer to reality (Scheme 2.2). The silicon(II) atom in (cAAC)2 SiCl2 (7) can be dehalogenated, and hence reduced, twice by two equivalents of potassium graphite to give Si(cAAC)2 (8, Figure 2.2) [50]. Formally, 8 now contains a silicon(0) central atom provided the Si–C contacts convert from the starting short classical Si–C 2-center 2-electron single bonds in 7 to longer dual donor-acceptor bonds like in 2. However, the two Si–C distances in 8 of 1.85 and 1.86 Å, respectively, compare much better to 7 than to 2 and suggest 2-center 2-electron bonds. But if they were like in (cAAC)2 SiCl2 (7), a diradical is expected with two unpaired electrons at the carbene carbon atom. In fact, 8 is EPR silent. This issue was rewarding to be clarified by a charge density investigation [51]. Here, a look at the Laplacian, the second derivative of the ED, clarifies the changes. A negative Laplacian ∇2 ρ(rBCP ) [eÅ−5 ] shows local concentration of the ED. A negative Laplacian at the bond critical point (BCP), accompanied by a high ED, indicates the covalent character of a bond. On the other hand, a distinct positive Laplacian and low ED at the BCP are associated with closed-shell interactions. Even if this categorization cannot be applied strictly to very polar bonds (the electronegativity of carbon is 2.55 and that of silicon only 1.90) it remains a feasible way to compare bonds. Further arguments provided to the discussion come from the eigenvalues (λi ) of the Hessian matrix. The elliptic-

2 Chemical concepts of bonding and current research problems | 15

Figure 2.2: Synthesis (top) and some charge density features of Si(cAAC)2 (8): (a) three-dimensional Laplacian ∇2 ρ(r), (b) ∇2 ρ(r) contour map orthogonal and (c) in the C2 Si plane. Contour lines are drawn at ±0.5,1.0, 1.5, … eÅ−5 (b) and ±2, 4, 6, … eÅ−5 (c) interval level; blue lines: charge concentration, red lines: charge depletion (reproduced from [51] with permission of John Wiley & Sons, Inc.)

ity ε(rBCP ) = λ1 /λ2 − 1 quantifies the deviation from a cylindrical shape indicating, for example, a higher π-contribution in nonpolar C=C double bonds. η = |λ1 |/λ3 provides reasoning concerning the bond type. ηBCP below unity indicates closed shell interactions and increases with increasing covalent character. (3,-3) critical points in the second derivative, so-called valence-shell charge concentrations (VSCCs), indicate bonding or nonbonding electrons [52], although their presence in nonbonding regions should not be taken as a one-to-one representation of a Lewis lone pair [53]. We found two distinct VSCCs of −2.82 and −2.80 eÅ−5 in the nonbonding region of the central silicon atom in 8 at the positions where lone pairs would be expected (Figures 2.2(a) and (b)). They are not involved in any chemical bonding which renders them potential lone-pair indicators [54]. The Laplacians along the Si1–C1 and Si1–C24 bond paths feature a very similar shape (Figure 2.2(c)). At the BCP, they have a slightly positive value and reach their minimum at about −30 eÅ−5 , close to the carbene carbon atoms. The rather low ED at the BCP and the charge concentration and depletion along the different directions at the Si1–C1 and Si1–C24 BCPs support the very polar character of both Si–C bonds. It cannot be a totally closed shell interaction because the covalent character is documented by, e. g., a NMR spectroscopic 1 JSi,C coupling. Hence, the QTAIM topology would not provide an ultimately decisive set of criteria either. In both cases, η is less than unity and even smaller than in other previously reported cases for various analyzed Si–C, Si–O and Si–N bonds [55–60]. From this data, both Si–C bonds in 8 have

16 | D. Stalke to be regarded as very polar bonds with a slight covalent contribution. However, by analyzing their ellipticity ε along the bond paths, a significant difference in the Si1–C1 (exp. 0.56; theo. 0.29 at the BCP) and Si1–C24 (exp. 0.13; theo. 0.21 at the BCP) bonds is recognized, possibly indicating different π-contributions (Figure 2.2(c)). Most probably, this results from the different torsion angles between the five-membered ring and the silicon atom (C5–N1–C1–Si1 18.7°, C28–N2–C24–Si1 11.5°). They even cause a difference in ρ(rBCP ) of 0.726 for the Si1–C1 and 0.741 eÅ−3 for the Si1–C24 bond and in the Bader charges of −0.51 for C1 and −0.37e for C24. The bent structure 8 (C1–Si1–C24 angle of 119.10°), even against the steric hindrance of the two bulky cAAC ligands, makes it difficult to interpret 8 as a silaallene with two orthogonal C=Si=C double bonds. Then the molecule should be straight like allenes (a in Scheme 2.3). The cAAC substituents, however, make a difference. If the silicon bound substituents are carbenes, then the molecule should rather be called a siladicarbene or a silylone (d in Scheme 2.3). In the case of carbon as central element, hence as carbones, they have been predicted by Tonner and Frenking in 2007 from theory to be bent (C–C–C ≈ 135°) and show two remarkable proton affinities (292 and 155 kcal/mol) [61]. A year later, those molecules were synthesized and employed in metal coordination by Bertrand et al. [62] and Fürstner et al. [63]. Remarkably, even the trisilaallene synthesized by Kira et al. shows a bent structure of 136° at the central silicon atom (c in Scheme 2.3) [64]. These investigations made it very clear that from a distance alone the knowledge concerning the actual bonding remains very limited. The differences of ligands (e. g., NHC versus cAAC) and the various electronic features of the resulting compounds can only be accessed by charge density investigations, both from quantum chemical cal-

Scheme 2.3: (a) In the straight allenes, two or more C=C σ/π double bonds are cumulated orthogonally between the terminally sp2 - and the inner sp-hybridized carbon atoms, (b) in the carbones, the partial dual-donor-acceptor bonds maintain the lone-pair character at the central carbon atom and support the bending, (c) the silaallene from Kira et al. is also bent to 136.5°, and the silylone in (d) shows similar bonding as the carbone in b.

2 Chemical concepts of bonding and current research problems | 17

culations and diffraction experiments. In conclusion, studying the ED in the bonds of (NHC)SiCl2 (2), (cAAC)2 SiCl2 (7) and Si(cAAC)2 (8) revealed new perspectives of old problems. From the abovementioned findings, it makes sense to transfer the concept of dative bonding to main group metals, established by Haaland in the late 80s [65], to contemporary silicon chemistry and possibly even to nonmetal chemistry as a whole. It fills the continuum between the cornerstones of a covalent 2-center 2-electron σ-bond and a 2-center 4-electron σ-and π-bond in carbon chemistry. Certainly, that concept should not be abused just to create hype [66], but on the other side it should not be bluntly condemned either [67, 68]. The dual donor-acceptor bond [M]󴀘󴀯CO between an organometallic d-block metal residue [M] and, e. g., a donor ligand such as carbon monoxide CO is well established in carbonyl chemistry, maybe it can help to explain the quasi-linear structure of carbon suboxide C3 O2 as well (O=C=C=C=O vs. OC󴀘󴀯C󴀗󴀰CO) [69]. In addition, the Dewar–Chatt–Duncanson model might not be limited to d-orbitals either and work with a considerably small HOMO-LUMO gap in main group elements as well [70]. Charge density investigations will provide us with further insight beyond barriers and free from stereotypical concepts that are valid only to subsections of chemistry.

2.2.2 Hexasilabenzene Si6 R6 In this chapter, I primarily resume results from [77]. Ever since its discovery, the benzene molecule C6 H6 has fascinated the imagination and creativity of many researchers. This is because of its special aesthetic symmetry, physical and chemical properties and vast applications. This, of course, can be traced back to the special aromatic bonding in the C6 perimeter. σ-bonding is accompanied by π-bonding, which gives the aromatic system stability upon cyclic conjuga⃛ bonds are of the same length (1.39 Å), even shorter than halfway between tion. All C−C a standard C–C single (1.54 Å) and a C=C double (1.34 Å) bond. They are amazingly inert and difficult to substitute. Aromatic carbon compounds underline the exceptional character of that element to form multiple bonds. All of the other heavier Group 14 elements are much more reluctant to do so. It even turned out to be difficult to switch one or two CR ring members to SiR [71–74]. Only in 2010, Scheschkewitz, Rzepa et al. succeeded in the synthesis and characterization of hexasilabenzene Si6 R6 (9) [75]. They reacted the lithiated disilylene R3 Si2 Li precursor with an excess of tetrachlorosilane SiCl4 to give an unsymmetrically substituted trichlorocyclotrisilane R3 Si3 Cl3 . Total dehalogenation with lithium naphthalene resulted in dimerization and formation of Si6 R6 (Scheme 2.4). However, this tempting simple gross formula is misleading, because the single crystal X-ray structure analysis showed the six bulky substituents R not evenly distributed among the six silicon atoms: two carry two substituents, two only one and two even none. This gives oxidation state +II for the first silicon atom

18 | D. Stalke

Scheme 2.4: Synthesis (top) and resonance forms of the ring isomer 9 of hexasilabenzene Si6 R6.

(SiR2 ), +I for the second (SiR) and 0 for the third (Si). Si6 R6 (9) forms a puckered tricyclic structure rather than a planar monocyclic structure like aromatic C6 R6 . Two three membered rings are fused to opposite sides of a central rhombohedral planar Si4 four membered ring, arranged at an angle of 113° relative to its plane. All Si–Si bond distances are almost identical. Within the four membered ring, they measure 2.309 Å at the side and 2.325 Å transannular, so that the Si3 rings are almost ideal equilateral triangles with the remaining distances being 2.326 Å long. From the equidistant bonding point of view, there might be aromaticity, even against the nonplanarity. And indeed, theory found aromaticity, witnessed by, e. g., the nucleus-independent chemical shift (NICS(0)) [75]. The authors suggested the term dismutational aromaticity, facilitated by the resonance of the double bond, the transannular bond and the lone pair. Most interestingly, the ring isomer 9 converts to the cage isomer 10 upon heating or irradiation (Figure 2.3, top), reminiscent to the paddle-wheel structure of C5 H6 [1.1.1] propellane, where two paddles are tip-connected by an additional SiR2 unit. 10 marks a minimum of the potential energy surface and is the thermodynamic silicon counterpart to benzene [76]. Both isomers are most challenging but rewarding touchstones for experimental charge density determination, so we embarked to study them [77]. From the multipole model based on high resolution X-ray data, we found all bond paths and related BCPs in 9 (Figure 2.3(A)), already predicted from theory [76]. The Bader charges from integration of the atomic basins echo the formal oxidation states: +1.6e for SiR2 , +0.6e for SiR and –0.3e for Si. The two polar Si–C bonds (ηBCP Si–Si ≥ 1; Si–C ≤ 0.5) deplete the Si(II) atom most while the Si(0) atom even shows some electron accumulation. The electron density σ BCP at the Si–SiR2 BCPs is similar (0.537 and 0.545 eÅ–3 ), followed by the amount of the partially Si=SiR double bonds (0.595 eÅ–3 ). The least density is found at the BCP of the transannular Si–SiR bond (0.466 eÅ–3 ). The lowest Laplacian

2 Chemical concepts of bonding and current research problems | 19

Figure 2.3: Topological properties from experimental charge density investigations of the ring isomer 9 and the cage isomer 10 of hexasilabenzene Si6 R6 with R = 2,4,6 triisopropylphenyl (reproduced from [77] with permission of John Wiley & Sons, Inc.)

∇2 ρ(rBCP ) at the Si=SiR BCP (–2.285 eÅ−5 ) and the highest at the transannular Si–SiR (–1.164 eÅ−5 ) support the fact that the first bond at the outside of the four-membered ring with partial π-contribution is the strongest (colored in red in Figure 2.3(C)) and

20 | D. Stalke the transannular is the weakest covalent bond. In the nonbonding region of the Si(0), we found a distinct VSCC of –2.08 eÅ−5 , indicative of a lone pair. Similarly, there are two VSCCs present at the transannular bond, witnessing its presence in addition to the bond path. The accumulation of the static deformation density in the plane of the R2 SiSi2 three membered ring (colored in blue in Figure 2.3(E)) further supports that fact: Between Si2 and Si3, density is accumulated. This figure impressively shows the ring strain in the three-membered ring. All the density maxima lay outside the straight lines of the triangle, supporting the considerably bent bonds. The transformation of the ring isomer 9 to the cage isomer 10 requires no substituent scrambling but only a slight rearrangement of the Si–Si bonds and a twist of the central four membered ring. This preserves the substitution pattern and various oxidation states of the silicon atom. Like in [1.1.1] propellanes, 10 shows two bridgehead Si(0) atoms at the hub of the paddle wheel. In a carbon derivative, Luger et al. experimentally found a bond path in the inside of the cage along the hub and a BCP with a positive value for the Laplacian [78]. In the sila-propellane reported by Klopper, Breher et al., the distance between the unsubstituted bridgehead silicon atoms is reported as 2.636 Å [79], similar to that in 10 (2.642 Å). Different to the all-carbon propellane and the transannular bond in 9, we could not find a bond path and a BCP along the hub in 10 (Figure 2.3(B)). The distinct Laplacian in the apical non-bonding region of the Si(0) atoms (–4.00 eÅ−5 ) rather supports the biradicaloid character of the molecule (Figure 2.3(D)) [80]. Different to 9, the static deformation density in the plane of the paddle shows no accumulation at the hub, but also indicates bent bonds. Again, the ED maxima are located outside the straight atom connection lines (Figure 2.3(F)). 9 and 10 are very similar as far as the Laplacians at the Si–Si BCPs are concerned, indicating the bond strengths to be on par. The only exception is the weakest transannular Si–SiR bond in 9. Although the bond distances in the propellane core of 10 indicate single bonds and the tip-connecting silicon atom shows the most pronounced σ-bonding, the generally higher ellipticities ε indicate a higher amount of delocalized ED in 10, compared to 9.

2.3 Sulfur-nitrogen chemistry This chapter resumes primarily results from [92, 95, 96, 101, 106]. Sulfur nitrogen chemistry is distinctly rich [81] because both elements are in many respects complementary. The diagonal relationship in the Periodic Table of the Elements shows them to be similar but also sufficiently different. Both elements span a wide range of stable oxidation states [82–86]. Nitrogen as one of the second-row elements tends to form double bonds and sulfur with the larger radius can easily expand the coordination number. The electronegativity difference between S and N is only 0.4; hence, much smaller than, e. g., between the two homologous elements from the last

2 Chemical concepts of bonding and current research problems | 21

chapter, silicon and carbon of 0.7. Nevertheless, by this it is already clear that any S–N bonding is also substantially Sδ+ –Nδ− polarized. Sulfur nitrogen chemistry can readily be understood from the settled sulfur oxygen chemistry, although different in important aspects. The isovalencelectronical replacement of oxygen in SOn m− molecules and ions by NR imido groups yields the polyimido sulfur species S(NR)n m− , n = 2, 3, 4 and m = 0, 2. This substitution gives the bent sulfur diimides S(NR)2 from sulfur dioxide SO2 (1B to 1C in Figure 2.4), or the trigonal planar sulfur triimides S(NR)3 from the sulfur trioxide SO3 (2B to 2C in Figure 2.4). Single oxygen atoms might even isovalencelectronically be replaced by CR2 groups to get the polyimido sulfur ylides (3B to 3C in Figure 2.4) [87]. Columns 1 and 3 in Figure 2.4 contain compounds with sulfur in oxidation state +IV and columns 2 and 4 of +VI. Moving from row B to C in Figure 2.4 marks the replacement of oxygen by imido groups. This shifts the chemistry from the solidstate domain of, e. g., the sulfate and sulfite minerals to the molecular array, because the organic periphery promotes solubility in hydrocarbon solvents and prevents electrostatic aggregation. Nevertheless, the electron-rich nitrogen atoms with their bent S–N–R environment and directed lone pair(s) of electrons facilitate metal coordination in contact ion pairs with plenty of interesting features. For the utilization of polyimido sulfur species in targeted syntheses, it is vital to understand the S–N bonding. Their redox and even radical chemistry is much richer than that of the oxo analogues [88]. If the S–O bonding introduced by Pauling [89] was only carbon-copied to the S–N bonding the difference in reactivity could not be rationalized. He suggested as many S=O double bonds as possible to avoid formal charges (line A in Figure 2.4). To rationalize the identical S–O and S=O bond distances, resonance like in organic aromatics was introduced. The price was high: justification of the short distances by σ- and π-bonding required valence expansion and d-orbital participation at sulfur, a concept that later turned out to be wrong [90]. d-Orbitals for the pblock elements are energetically totally and utterly out of reach. Short bond distances can also be achieved by covalent bonding, reinforced by electrostatic interaction, formally introduced by the move from line A to line B in Figure 2.4. This re-introduces formal charges and Lewis’ eight-electron-rule is obeyed. Experimental charge density investigations can shed light on this bonding issue and add arguments to this still vital debate. Experiments on SO2 [91] and SO4 2– [92] identified the S–O bond as predominately ionic. In the first, e. g., the ellipticity ε of 0.09 shows the bond to be between a single and a 1.5-fold bond. All other properties also support the Lewis structure with the formal charges (1B in Figure 2.4) to be the more appropriate description. In SO4 2– , the bonds have to be described as covalent single bonds with only minute double bond contribution. The ellipticity ε ranges from 0.014 to 0.076 in the experiment and is only 0.004 from theory. These make the hypervalent description of the sulfur atom for sulfate superfluous (4B in Figure 2.4) [93]. Also, the Lewis structures with formal charges from line C in Figure 2.4 are definitely much more appropriate to describe the polyimido compounds adequately. They are more prone to redox chemistry than their oxo analogues, indicating that the S–N

22 | D. Stalke

Figure 2.4: Canonical forms of some SOx - (lines A and B) and S(NR)x -species (line C), according to Pauling (line A) and Lewis (lines B and C) as well as the Laplacians from experimental charge density studies (line D) (1, 2, 3D reproduced from [96, 101], Copyright 2021 American Chemical Society, and 4D from [103, 106] under the CC BY-NC license).

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bond is much easier cleaved and that sulfur is much more oxophilic. Polyimido sulfur compounds form N-centered radicals upon oxygen exposure [88, 94, 95]. In nonpolar hydrocarbon solvents, they tend to be more stable than in polar media. Latter promote reduction by cleavage of the Sδ+ –Nδ− bond, accompanied by RN=NR diimide formation. Charge density investigations substantiate these impressions from reactivity. In S(NtBu)2 (11), both the experimental and theoretical Laplacians show only a single, non-deconvoluted lone pair VSCC at both nitrogen atoms and the sulfur atom in the plane of the molecule (1D in Figure 2.4), indicating their sp2 hybridization [96, 97]. The ED ρ(r) at the BCPs (1.93 and 2.24 eÅ−3 ), the Laplacian ∇2 ρ(r) (−9.44 and −9.38 eÅ−5 ) and the ellipticity ε (0.07 at both BCPs) characterize those bonds as 3-center-4-electron covalent bonds with a polarized σ- and π-contribution. This concept was already introduced by Rundle in 1947 [98–100]. The same applies to the sulfur triimide S(NtBu)3 (12). The zero flux surface at ∇2 ρ(r) = 0 (2D in Figure 2.4) depicts charge accumulation on top and underneath the trigonal planar molecule. This is the reason why the sterically nonhindered molecule is relatively unattractive to bulky nucleophiles like carbanions this way. Although the central sulfur atom is electronically depleted it can only be reached through the holes in the zero flux surface in the plane along the N–S–N bisection. This requires small or slim nucleophiles. S(NtBu)3 reacts smoothly with MeLi, PhCCLi, SC4 H3 Li or SC4 H2 Li2 but not with nBuLi or tBuLi. Against the steric strain the S–N bond distances in 12 are shorter than those in 11, because of the higher oxidation state of +VI compared to +IV. The ED ρ(r) at the BCP of the non-crystallographically C3h symmetric molecule (2.27 eÅ−3 ), the Laplacian ∇2 ρ(r) (−10.56 eÅ−5 ) and the ellipticity ε (0.22 at both BCPs) show an extended π-system in 4-center-6-electron covalent bonding. However, neither in 11 nor in 12 there is need to formulate more than one double bond and to assume valence expansion at the sulfur atom. In the polyimido sulfur ylide (RN)2 S(CH2 )2– (13, 3C in Figure 2.4), there is one oxygen from the sulfite SO3 2– replaced by one methylene substituent and two by NR substituents each [101]. In this cap-shaped dianion, there is neither any indication for double bonding nor for any valence expansion. The diaza(carba)sulfite 13 is embedded in a dimeric [(thf)Li2 {H2 CS(NtBu)2 }]2 structure. Apart from their S,C-substituents, all nitrogen atoms are coordinated to two lithium atoms each, addressing them by their VSCCs at the bond path. The ylidic carbon atom develops a single lone-pair VSCC in the nonbonding region, pointing to the midpoint of a Li3 triangle, a structural motif well known from organolithiums [102]. The ED at the BCP for both S–N bonds is 1.670 and 1.547 eÅ–3 , respectively. The Laplacians are considerably low, indicating covalent bonding (–11.518 and –9.774 eÅ–5 ). The high ellipticity at one S–N BCP (0.07 and 0.30) is not indicating S=N double bonding but stems from the leaning over of the easy-to-polarize lone pair VSCC at the sulfur atom to the electronegative nitrogen atom. The question to what extent the S–C bond is a ylenic S=C or ylidic S+ –C– bond is decided from the topology. The electron density (1.369 eÅ–3 ) and the Laplacian at the

24 | D. Stalke BCP (–5.214 eÅ–5 ) indicate a standard single bond. The BCP is only shifted 0.03 Å from the nonpolar midpoint of the bond to the carbon atom, prompting a not very polar bond either. The integration of the atomic basin of the sulfur atom gives a charge of +0.87e. Overall, the Lewis structure with the positive formal charge at the S-, a negative charge at the ylidic C- and at the imido N-atoms each fits much better the reality than the electroneutral form with the resonating double bond. Reliable comparability of formal S=N double and S–N single bonds, however, is only provided inside the same molecule [103]. A most simple, and thus convincing molecule is tetraimido sulfuric acid H2 S(NtBu)4 (14, 4C in Figure 2.4). The imido analogue of sulfuric acid H2 SO4 is better written as (tBuN)2 S(HNtBu)2 to emphasize the fixed hydrogen positions and the different substituents. Sulfuric acid has been known for almost 1200 years [104, 105] and is the most important bulk starting material in today’s chemical industry to give, e. g., fertilizers. The imido analogue could only be made and investigated recently (Scheme 2.5) [106]. Surprisingly, the hydrogen atoms in the molecule are not disordered and their positions establish and separate the amido from the imido substituents. The amido S–N(H) bond paths are longer than the imido S–N bond paths (1.649 vs. 1.543 Å). Both are considerably curved, indicative of polar bonds. The mere distances might suggest S–N(H) single and S=N double bond character, but the topology tells another story: the BCP in the amido bond is shifted slightly less to the electropositive sulfur atom, indicating that this bond is less polarized than the latter. ρ(r) (1.81 vs. 2.17 eÅ–3 ) and ∇2 ρ(r) (–14.8 vs. –21.2 eÅ–5 ) at the BCPs charac-

Scheme 2.5: From sulfuric acid to tetraimido sulfuric acid by gradual O/NR replacement (reproduced from [106] under the CC BY-NC license).

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terise them both as shared covalent interactions. Not even the high ellipticity ε accounts for double bond character. Counterintuitively, it is even higher at the amido than at the imido bond (0.43 vs. 0.32), demonstrating that it would not stem from double bond character like in nonpolar C=C bonds. Here, the lone-pair densities at each nitrogen atom (one VSCC at the N(H)s, two at each imido nitrogen atom) bent over to the electropositive sulfur atom to compensate for the high positive charge (4D in Figure 2.4). Obviously, the two lone-pair densities at the imido nitrogen atoms are more attractive to the electropositive sulfur atom, shortening the S–N distance. This leaping into the bonding region causes the deformation detected by ε. Thus, the differences in bond lengths depend on the strength of the polarization, i. e., the more polarized the shorter, and not on π-contributions in double bonding. All S–N bonds in 13 and 14 are polarized single bonds and the concept of valence expansion or hypervalency is obsolete. Most interestingly for applications, the sum of all four Sδ+ –Nδ− bond distances in tetraimido anions [S(NtBu)4 ]2– in various transition and main group metal complexes spans only the narrow range of 6.34 to 6.40 Å (6.35 Å in 14). This documents the huge adaptability of polar S–N bonds, discovered from charge density studies. Their polarity provides the flexible geometry of the SN4 -moiety to suit the various Lewis acidity- and orbital-guided requirements of different metal cations. The sulfur atom in the middle serves as a redox electron sponge and can easily be shifted, not holding on to a fixed tetrahedral geometry. Recently, this feature turned out to be very advantageous in the area of single-molecule magnets. The paramagnetic metal cations in those complexes can be magnetized and keep that magnetization even when the external field is removed. It would be most rewarding to transfer that feature to industrial application, because that would revolutionize, e. g., high-density data storage and spintronics [109, 110]. In a first observation, we discovered that the acute N–S–N chelating angle in [Co{(NtBu)3 SMe}2 ] gives rise to zero-field slow magnetic relaxation and a hysteresis loop [111]. More recently, we found that the common perception that the distortion of the tetrahedral CoN4 coordination to D2d and C2v and further to linear C∞ would presumably further increase the magnetic anisotropy and thereby the energy barrier of spin reversal is not quite correct. In fact, the dx 2 −y2 and dxy orbitals degenerate at an optimal N–Co–N angle of about 77° and any smaller angle would not improve the single-molecule magnetic abilities. This angle can readily be adjusted by SN ligands (Figure 2.5(A)) [112]. For other reasons, lanthanides are most promising ions in potential single-molecule magnets. They show a genuine large unquenched orbital momentum because of their large number of unpaired electrons and strong spin-orbit coupling. First results of dysprosium and terbium metal polyimido sulfur complexes show promising results. The dimeric dysprosium complex [{(thf)2 Li(NtBu)2 S(tBuN)2 DyCl2 }2 •ClLi(thf)2 ] shows real single-molecule magnetic behavior of slow magnetic relaxation at zero field, while with the monomeric Tb and Dy complexes [{(thf)2 Li(NtBu)2 S(tBuN)2 LnCl2 (thf)2 ], Ln = Tb, Dy, this is induced only by an applied external field [113]. The same holds for [Dy{(NtBu)3 SMe}2 (μ-Cl2 )Li(THF)2 ] and

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Figure 2.5: A: Only the optimal N–Co–N angle of about 77°degenerates the dx 2 −y2 and dxy orbitals

and maximizes zero-field splitting D up to −171 cm−1 ; B: The acute N–Dy–N bite angles from 60 to 64° and the SLDyLS vector approaching linearity cause slow relaxation of magnetization at zero field in [Dy{(NtBu)3 SMe}2 (μ-Cl2 )Li(THF)2 ]. (Reprinted under the CC BY-NC license from [112] and with permission from [114], Copyright 2021 American Chemical Society).

on the other hand for [Dy{(NtBu)2 SCH2 PPh2 }2 (μ-Cl2 )Li(THF)2 ] and [Dy{N(tBu)2 SPh}2 (μ-Cl2 )Li(THF)2 ], respectively (Figure 2.5(B)) [114]. Apart from the nature of the S–N bond these ligands provide many levers to optimize single-molecule magnetic properties: the number of imido groups can be adjusted from one to four, the various organic substituents might provide further bulk, side-arm donation and, e. g., the mainstay to surface-fixation. Furthermore, the central sulfur atom is prone to redox chemistry and even to radical formation, providing even more unpaired electrons and ligand-to-metal-charge-transfer options beneficial to single-molecule magnets.

2.4 Catalysis In this chapter, primarily I resume results from [125]. The synthesis of today’s bulk chemicals heavily relies on catalysis, reducing energy consumption and preventing waste in atom-economically tailored reaction sequences [115–118]. This can be facilitated by either homogeneous and heterogeneous catalysis. In the first, the soluble molecular catalyst and the starting material share the same phase, which is, in most cases, the solution phase consisting of petro-chemically based organic solvents. In the latter, the solid-state catalyst is exposed to the starting material in solution or in the gaseous state. As a major drawback, it is still not always predictable, which alterations cause the desired effects. Hence, the benchmarking of working models along established concepts is required. A firm understanding of the structure–property relationship of a chemical system is vital for the improve-

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ment of catalysts. ED investigations provide an ideal tool for understanding these interactions [1–4, 119–124]. From the knowledge of atomic distances and their arrangement in the crystal, many properties, both at the molecular and macroscopic scale, can be deduced. However, as seen in the last chapters the most basic concepts, such as the chemical bond and reactivity, are still strongly disputable. Solid-state heterogeneous catalysis is the most important sustainable principle in energy-intensive chemical bulk production, even more important than homogeneous catalysis. It reduces energy consumption, minimizes waste and in many cases supersedes petrochemical organic solvents. To further promote these green processes, it is essential to learn more about their mode of action. Solid-state materials, containing cobalt and other transition metals, are prominent catalysts in organic transformations and are therefore in the focus of current research. This is why we studied catalytic abilities by experimental and theoretical charge density investigations [125]. A nickel-cobalt catalyst can, e. g., facilitate the oxidation of styrene [126] and nano-sized solid solutions of copper/cobalt molybdate and chromium phosphate promote the reduction of p-nitrophenol to p-aminophenol [127]. Cobalt-substituted calcium phosphate catalyses the oxidative dehydration and dehydrogenation of propane2-ol used as a benchmark substrate for characterizing acid and base properties of solid-state catalysts (Scheme 2.6) [128, 129]. The maximum catalytic effect is reached at a cobalt content of 0.32, the maximum solubility of Co in calcium phosphate. Later, the conversion of ethane, propane, butan-2-ol and propanol was studied [130].

Scheme 2.6: Cobalt phosphate catalyses the oxidative dehydration and dehydrogenation of propane-2-ol to give 2-butanone and 2-butene (Reprinted under the CC BY-NC license from [125]).

On this basis, we set out to investigate the catalytically active surfaces in cobalt phosphate Co3 (PO4 )2 and tried to quantify substrate occupancy by means of charge density analysis. An extensive characterization using topological analysis and theoretical methods for experimental and theoretical electron density distributions was performed (Figure 2.6) [125]. As starting point, the low-resolution structure of cobalt orthophosphate has been known for long. It crystallizes in the monoclinic space group P21 /n [131–133]. The unit cell contents is made up by two formula units of Co3 (PO4 )2 , giving two symmetry independent Co(II) atoms. One is six-fold coordinated in an almost ideal octahedral fashion; the five-fold coordination of the second gives a considerably distorted trigonal bipyramidal environment. While the first is O-corner-sharing

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Figure 2.6: In Co3 (PO4 )2 , the (011) face is identified to be best suited for catalysis. Five-fold coordinated Co ions (depicted in bronze, oxygen in red, carbon in black) in close proximity to advantageously oriented electron depletion sites (black dots) suit the reactant’s oxygen-lone pairs most for chemisorption and heterogeneous C–H activation (reprinted under the CC BY-NC license from [125]).

connected to six PO4 3– anions, the latter only to three and to one additional edgesharing phosphate ion. These different Co atoms will be termed Co[6o] and Co[5by] , respectively. Cobalt cation layers, interconnected by the phosphate anions, compose the complete solid-state structure (Figure 2.7(a)). The sinus curved metal lines in the first are generated by corner- and edge-sharing Co[6o] octahedra and Co[5by] trigonal bipyramids (Figure 2.7(b)). In addition to the arrangements of the atoms in the solid derived from the classical spherical IAM, we wanted insight in the electron distribution around the metals, because it is known that catalysis takes advantage of the vacant coordination sites of d-metals. Like the lone pairs in nonbonding regions in the abovementioned molecules, charge density investigations facilitate the quantification of these vacant coordination sites in the solid. To trigger catalysis, they not just have

Figure 2.7: An excerpt from the crystal structure of cobalt phosphate. The coordination polyhedra of phosphorus are colored in magenta, those of Co are bronze. a, b, c are lattice constants. (a) alternating layers of phosphates and cobalt atoms and (b) connectivity of Co-polyhedra to give the sinus curved metal lines (reprinted under the CC BY-NC license from [125]).

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to be holes in the right position to be settled by substrates, but certainly they have to have the right properties in addition, i. e., acidity/basicity, electron depletion/accumulation, positive/negative polarity, etc. Where they are and what they are can be quantified by the aspherical density in the multipole model (MM) formalism [5]. First, local coordinate systems are required. We used density functional theory (DFT) calculations, together with manual judgment, to set up the local coordinate systems, because the actual coordination polyhedra around the Co ions in Co3 (PO4 )2 are distorted from the ideal shape (Figure 2.8(a and b)). The so-derived ED can be investigated by a complete topological analysis according to QTAIM [8], based on the experimental diffraction data (MM-Expt) as well as the structure factors derived from DFT calculations (MM-DFT). First, guidance comes from the static deformation density. It visualizes the difference between the total MM ED and a spherical reference IAM ED. This already provides a very informative picture of the ED distribution around the metal atoms. Regions of negative charge concentrations are depicted in sky blue and those of negative charge depletion in orange (Figure 2.8(c and d)). Interestingly, the number of six charge concentration regions coincides in both metals, but not the number of depleted regions. Co[6o] shows only six minima, each directed to the negatively charged phosphate oxygen atom, already fueling the idea of an electrostatic Co–O interaction. In contrast, Co[5by] shows eight regions of negative deformation density arranged at the eight corners of a cube. Five of the vertices point to a coordinated phosphate oxygen atom each and the remaining three exposed minima are left uncoordinated, indicating free coordination sites, attractive to potential catalytically convertible substrates (Figures 2.8(e,f)). The indications of the difference between the Co atoms found in the deformation density are mirrored in the Laplacian. Each dark blue array in Figures 2.8(g–k) represents a VSCC. The red regions represent the negative charge depletions. For Co[5by] , the match between the features of the Laplacian from either the MM-Expt and the MM-DFT refinement is very good. At Co[6o] , on the other hand, differences, such as the position and number of depletions, are obvious. The MM-DFT data give eight negative charge depletion points arranged into an approximate cube (Figure 2.8(i)), and six VSCCs roughly in the center of the faces formed by the cube of charge depletions. The vertices of the charge depletion cube do not fully coincide with the ion ligand bonds. In contrast, from the experimental multipole refinement (Figure 2.8(g)), there are only six charge depletions that form an octahedron with vertices approximately along the six ion-ligand bonds. Considering the reaction of butan-2-ol with oxygen to butene and butanone (Scheme 2.6), at least one of the two reactants must be chemisorbed at the catalyst’s surface, and thus be activated. If the butanol is adsorbed over several atoms on the surface due to its size, the negatively polarized alcohol oxygen atom most likely coordinates via the two oxygen lone pairs to an electronically depleted site of the cobalt atoms and the positively polarized methyl hydrogen atoms to a nearby basic phosphate site. If we assume differences in the heterogeneous catalytic ability of the

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Figure 2.8: Left column Co[6o] and right column Co[5by] ; (a) and (b) depict the local coordinate systems; (c) and (d) show the deformation density with the iso-surface values of ± 0.65 eÅ−3 for negative charge accumulation in sky blue and positive in orange; (e) and (f) show the coordination sites; (g) and (h) depict the Laplacian with iso-surfaces of 240 eÅ−5 (red) and −1250 eÅ−5 (blue) for the experimental, (i) and (k) for the theoretical data with iso-surfaces of 260 eÅ−5 (red) and −1260 eÅ−5 (blue) (reprinted under the CC BY-NC license from [125]).

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solid-state faces in cobalt phosphate, we need to establish criteria to identify effective catalytic sites [134]. Since the crystal structure of cobalt phosphate shows two differently coordinated metal sites, the first question to be asked is whether both Co atoms are equally active, and if not, what causes the differences. If the plain coordination number of the cobalt atoms is considered, the saturated Co[6o] most likely is not tremendously active as it lacks a vacant site for the substrate to interact with. Figures 2.8(g and i) clearly show that there are no free sites at Co[6o] , whereas Co[5by] holds three of them. However, it has to be kept in mind that Co-atoms exposed at the surface inevitably render free coordination sites. Depending on how exposed they are on the surface, the number of free coordination sites also varies, which affects the probability of chemisorption. However, statistically speaking, Co[5by] would still have to have more free coordination sites than Co[6o] at the surface, because it has already three more vacant sites in the bulk. In addition, the present distortion of the geometry will promote catalysis [135]. When, e. g., C–C bonds are activated in a substrate by rhodium and nickel catalysts, it is proven that this is a structurally sensitive reaction that takes place on coordinatively unsaturated atom pairs of the d metals on corners or edges of minute crystallites [118]. Thinking in terms of concepts of homometallic cooperativity [136], short distances between the metal centers are needed. In the crystal structure of cobalt phosphate, there are such short distances between two five-fold coordinated edge-sharing Co-ions (3.032 Å) (highlighted in the black ellipses in Figure 2.9) and between a five-fold and a six-fold Co-ion (3.145 Å). Figure 2.9 shows the four most likely low-index crystal faces with Co… Co cooperative sites. The layer is chosen such that no covalent P–O bonds are broken and cobalt atoms appear on the surface. The positions of electronically depleted areas found from the multipole refined Laplacians are superimposed onto the surface of the cobalt atoms as black areas. At the (101) face, only a few pairs of Co[6o] and Co[5by] atoms are reachable for the alcohol. (10-1) is similarly unattractive, because it is almost fully occupied by acidic metal cations with the basic phosphate oxygen atoms unavailable. The (110) face looks more promising because there are two Co[5by] atoms in close proximity and phosphate oxygen atoms available for the C–H activation. The most promising site seems to be, however, the (011) face. The most Co[5by] atom pairs in close proximity of 3.032 Å are here found in an open groove with close-by phosphate oxygen atoms. Those metals expose three and four vacant coordination sites each. The pairs at the edges even have up to six vacant coordination sites per Co atom. They are perfectly arranged to serve as adsorption sites for the two oxygen lone pairs of the butanol molecules, which then μ-bridge two Co atoms (Figure 2.10). Hence, we conclude that the catalytically most active plane is the (011) face. Finally, we note that the above arguments were made based on the ED for the bulk crystal. In order to verify that the predicted adsorption of, e. g., butan-2-ol onto Co[5by] on the (011) surface is favorable, we used DFT to calculate the adsorption energy of butan-2-ol onto several different adsorption sites for two different surface ter-

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Figure 2.9: Space filled view of the most stable crystal planes of cobalt phosphate. Red: oxygen, bronze: cobalt, phosphorus is hidden inside the O4 tetrahedra (reprinted under the CC BY-NC license from [125]).

Figure 2.10: Space filling representation of the (011) face of cobalt phosphate with a chemisorbed butan-2-ol molecule (reprinted under the CC BYNC license from [125]).

minations of the (011) surface and found the predicted adsorption site to be indeed the most favorable as expected, confirming that the bulk ED can be used as a tool to qualitatively predict adsorption behavior at the surface. Nevertheless, the surface structure of a real catalyst may deviate from bulk-truncated cuts, and so based on this analysis we cannot fully exclude the presence of additional relevant active sites and surface facets. However, insofar as the catalyst surface structure resembles any of the

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low-index surfaces, the (011) surface has the right structural features and ED for catalytic conversion. Related investigations can help to identify most catalytically active sites and inspire catalyst design by accumulating those sites. Precious transition metal fragments might be arranged economically in the right way on an abundant material as, e. g., silica or corundum.

2.5 Conclusions I really hope that this chapter has gotten some points across: – Charge density investigations provide physically meaningful bond criteria beyond mere distances. The electron density ρ(r), its second derivative the Laplacian ∇2 ρ(r), the position of the bond critical point, etc. along the bond path and other QTAIM properties establish firm descriptors to judge the nature of the bond and hence form the basis for decisions how to derivatize the material. – Chemical concepts can be evaluated and (re)assessed. The dative bond is a fundamental support to the Lewis diagram. Although established in d-block coordination chemistry it opens up the view to phenomena in main group chemistry and helps to take off our specialist’s blinkers. – The method might even help preventing us from taking the wrong scientific perceptions. Not all phenomena ask for new concepts but might be explained by existing ones. It facilitates unifying concepts rather than building and maintaining Potemkin villages. In this spirit, let us pick it up and develop this ground-breaking method further, because it is quite clear that it is worthwhile to analyse the chemical bond. The polarity of the bond decides upon how easily it can be formed and cleaved in what media, guiding the way to new synthetic protocols. Encoding the nature of the chemical bond and the vacant nonbonding regions provide a low barrier access to new beneficial materials. Although still a predominantly experimental science, bond analysis helps chemistry to promote rational design of target compounds.

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Part II: Bonding descriptors from quantum chemistry

Ángel Martín Pendás and Carlo Gatti

3 Quantum theory of atoms in molecules and the AIMAll software 3.1 From electron densities to atoms in molecules Chemical bonding in the twenty-first century is inexorably linked to the application of quantum mechanics to chemistry. Today, the electronic structure of systems of moderate complexity can be reliably obtained even with desktop computers. The latter spit out intricate wavefunctions that need be analyzed if chemical insight, not only energies or energy differences, are needed. However, the large dimensionality of Ψ poses a severe problem to the rationalization of chemical phenomena. As already noticed by Van Vleck [1], the amount of information stored in a wavefunction when the number of electrons of a system exceeds N ≃ 1000 is so large that it escapes human comprehension, so that an information compression technique is needed. Historically, two basic, in a sense opposite, techniques have been used to tackle this problem. On the one hand, physicists and chemists have taken profit of the models and approximations used to solve Schrödinger’s equation in order to rationalize the computed Ψ’s [2]. For instance, within the Hartree–Fock approximation, a wavefunction is written as a single determinant constructed from one-electron functions or orbitals. In this mean-field approach, electrons move in the average potential created by the rest of the particles of the system. Knowledge and manipulation of the N 3D orbital (spinorbital) functions of a system is much easier than that of the full wavefunction. However, this simplification comes at the expense of several problems and caveats. First, orbitals are not unique, as they may be subjected to an infinitely many number of transformations that leave the total Ψ invariant. Unfortunately, orbital-based chemical interpretations are not immune to this freedom. Since orbitals come in a typical fashion (i. e., localized, delocalized) depending on the method used to obtain the wavefunction, e. g., molecular orbital theory (MO) or valence bond theory (VB); orbital interpretations are method dependent. Second, as the mean-field is abandoned and electron correlation is taken into account, the pristine single determinant orbital concept vanishes. In multiconfiguration approaches, for instance, many more than N partially occupied functions appear, and the simple MO machinery ceases to be welldefined. Ángel Martín Pendás, Universidad de Oviedo, Avda. Julián Clavería 8, 33006, Oviedo, Spain, e-mail: [email protected] Carlo Gatti, CNR-SCITEC, Istituto di Scienze e Tecnologie Chimiche ”Giulio Natta”, sezione di via Golgi, c/o Dipartimento di Chimica, Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110660074-003

44 | Á. Martín Pendás and C. Gatti The second category of techniques compresses the information contained in the wavefunction by using its probabilistic interpretation. If we forget about the behavior of all electrons but one or two (i. e., we integrate out the others’ coordinates), we get the density of finding electrons or electron pairs at given positions of space. These reduced densities, as they are known, are quantum mechanical observables and can be experimentally accessed in principle. They are manifestly invariant under orbital transformations, so they do not depend on models of computational methods, and have clear-cut interpretations [3]. Since they live in the standard 3D space in which chemical phenomena are interpreted, they provide first-hand insights. Notice that if we take the ⃗ as an example, constructing a theory of chemical bonding as deelectron density, ρ(r), tailed and predictive as MO theory through the analysis of just one scalar field instead of through the exam of N orbitals should be a not easy enterprise. Over the years, it has been found that subtle details of the reduced densities, are hidden in their derivatives and overall topographies, so that these real space techniques are commonly known as Quantum Chemical Topology (QCT), a term coined by Paul Popelier [4]. Given that reduced densities, are dominated by the nuclear positions, it was clear from the beginning that a theory of chemical bonding based on electron densities should focus on density differences, on how electrons redistribute in chemical processes. The easiest way to do that is to build a density difference map. If we have an initial and a final electron density, ρi and ρf , we build Δρ = ρf −ρi . Then we try to locate regions where charge is concentrated or depleted. However, soon it became clear that the choice of the reference density, ρi , could completely alter the interpretation, as it happens when spherically averaged or valence-specific atomic densities are used to build the reference density in difluorine. An interesting solution to this problem is to focus on the analysis of the density itself. This has lead to examining its topology, since as Paul Mezey noted, if we want to study the behavior of a scalar function without recourse to an external reference, only its value and that of its derivatives are available.

3.1.1 Topological analyses The topological analysis of the electron density, as a means to obtain chemical information from ρ was pioneered by Richard F. W. Bader, from McMaster University in Canada [5]. It is a rather intuitive and elegant procedure which can be grasped and mastered rather easily. Today it has been generalized and applied to many other scalar fields like the electron localization function (ELF) of Becke and Edgecombe [6], the electrostaic potential, the reduced energy gradient behind the noncovalent interactions index (NCI) [7], etc. A number of software packages, both open-source and proprietary exist that have been interfaced with many standard electronic structure packages like GAUSSIAN [8], GAMESS [9], ORCA [10] or MOLCAS [11], to name just a few. In this chapter, we will describe how to use the AIMAll suite, written by Todd A. Keith [12]. Before doing that, we need to examine briefly the mathematical tools that

3 Quantum theory of atoms in molecules and the AIMAll software

| 45

we will use, as well as to present in a more formal way the Quantum Theory of Atoms in Molecules (QTAIM) [5]. Let us take a differentiable vector field, y.⃗ We can now introduce a fictitious time ⃗ coordinate, t, and consider the system of equations given by dr/dt = y.⃗ This trick transforms the field into a dynamical system. The solution of the above equation defines the ⃗ trajectories or field lines of r(t), and the theory of dynamical systems [13], allows us to fully classify their properties. If we now turn to ℝ3 , we can take a scalar function f : ℝ3 → ℝ

r ⃗ → f (r)⃗

(3.1) (3.2)

An ⃗ ⃗ explicit form for the field (or flux) lines of the gradient field after writing dr/dt = ∇f (r)⃗ is given by and define an associated vector field through its gradient, ∇f⃗ =

𝜕f ⃗ 𝜕f ⃗ 𝜕f ⃗ i + 𝜕y j + 𝜕z k. 𝜕x

t

⃗ = r(t ⃗ 0 ) + ∫ ∇f⃗ (r(s))ds, ⃗ r(t)

(3.3)

t0

which is simply interpreted as a temporal movement guided by the gradient field at the position reached at (fictitious) time t. It is easily proven that except at special points (the critical points of the field, see below), only one gradient lines passes through each point in space, that the gradient vector is tangent to the trajectory at every point, and that the trajectories are orthogonal to the isoscalar surfaces of the field at any point. Moreover, every line must begin or end at points where ∇f⃗ (r)⃗ = 0.⃗

These null gradient points are the so-called critical points (CPs) of the field. Their number and properties characterize its global structure (or topology). It is customary to classify them according to the linear approximation, which substitutes the full field in the proximity of a CP by its first-order Taylor expansion. Thus, in the vicinity of a CP rc⃗ , r ⃗̇ = f ⃗(r)⃗ ≃ f ⃗(rc⃗ ) + J(r ⃗ − rc⃗ ) = J(r ⃗ − rc⃗ ),

(3.4)

⃗

f (x,y,z) where J is the Jacobian matrix at rc⃗ , J = 𝜕𝜕(x,y,z) . In the special case of gradient fields, the Jacobian matrix at rc⃗ is simply the matrix of second derivatives of the field, i. e., the Hessian matrix of ρ : Hαβ = 𝜕2 ρ/𝜕xα 𝜕xβ , where α, β run over x, y, z. We can now translate our reference frame to the CP being studied, so that the flux lines of this firstorder approximation to the field will be the solution of

r ⃗̇ = H r.⃗

(3.5)

In this equation, the differential equations for the x, y, z coordinates are coupled by the nondiagonal elements of the Hessian, but a rotation of the coordinates so that

46 | Á. Martín Pendás and C. Gatti they coincide with the eigenvectors of H decouples them. Since the Hessian is symmetric, this is an orthogonal transformation that warrants that the new diagonal Hessian (eigen)values are real. If U is matrix that diagonalizes H at the critical point:

U t HU = diag(λi ), where the λi ’s are the eigenvalues of H, then the new coordinates η⃗ are defined by r ⃗ = U η,⃗ and the immediate solution of the differential equations pro-

vides ηi (t) = ηi (t0 )eλi (t−t0 ) , {i = 1, 3}. The behavior of the trajectories of the field along each eigendirection depends only on the sign of its associated λ value. Either they start or end at the CP when sgnλ is positive or negative, respectively. The case where one or more eigenvalue is exactly zero will not be examined here. These degenerate cases can be usually neglected in chemistry, since an infinitesimal change in the geometry of a system will in general change any vanishing eigenvalue to a nonzero value. With these provisos, there are clearly only four types of nondegenerate CPs for a gradient field. Ordering the eigenvalues such that λ1 ≤ λ2 ≤ λ3 we may have: (i) All the λi < 0. In this case, all the curvatures are negative, and the trajectories converge toward the CP, which is called a sink or attractor. (ii) λ1 , λ2 < 0, λ3 > 0. Then, in the plane defined by the η⃗ 1 and η⃗ 2 vectors the gradient lines approach the critical point; however, in the orthogonal η3 direction they escape from it. This is a saddle point of the first kind. (iii) λ2 , λ3 > 0 and λ1 < 0, so we have a symmetric situation, or saddle point of the second kind. (iv) Finally, if all λi > 0, then every field line escapes the CP in all directions, and the CP is known as a source or repellor of the field. Figure 3.1 explains graphically all these possibilities. The usual labeling of CPs in QCT uses two integer indices, written as (r, s). The rank, r, is defined as the number of nonzero eigenvalues of the Hessian at the CP, and the signature, s, as the difference between the number of positive and negative curvatures. In our case, the four nondegenerate cases described above thus correspond to (3, −3), (3, −1), (3, +1) and (3, +3) points, respectively. Notice that either the attraction basins of the (3, −3) CPs or the repulsion basins of the (3, +3) CPs, respectively, are 3D regions. If we take the former, for instance, the union of all of the (3, +3) attraction basins of the field is, except a null-measure set (formed by 2D surfaces or 1D lines which correspond to the attraction basins of the other CPs) the full ℝ3 space, so that the attraction (or repulsion) basins of a field induce a topology in ℝ3 , i. e., an exhaustive partition into disjoint regions: ℝ3 ≡ ⋃A ΩA , where A runs over all the attraction (repulsion) basins. The surfaces that separate the basins are called separatrices. At all points belonging to a separatrix, rs⃗ , the gradient of the field is parallel to the surface, so that if the unit exterior normal vector to the surface at point rs⃗ is denoted by n⃗ s , then ∇f (rs⃗ ) ⋅ n⃗ s = 0. Separatrices are local zero-flux surfaces of the gradient field. All local zero-flux surfaces are also global zero-flux objects, and ⃗ = 0. ∫S ∇f⃗ (r)⃗ ⋅ n(⃗ r)ds The number and type of the CPs of a given scalar field is limited through topological invariants, which depend only on the intrinsic properties of the space in which

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Figure 3.1: The four types of CPs of a 3D scalar field.

the field is defined. In ℝ3 , which is the space in which we embed finite molecules, the Euler–Poincarè or Morse invariant [14] reads: n3 −n2 +n1 −n0 = 1, where ni is the number of CPs with i negative curvatures. Similarly, if we deal with 3D periodic systems, like crystals, the appropriate space in which the unit cell is embedded is not ℝ3 , but a 3D torus, 𝕊3 , since each spatial direction is equivalent to a closed line or circumference. In this case, n3 − n2 + n1 − n0 = 0.

3.1.2 The topology of the electron density The Quantum Theory of Atoms in Molecules (QTAIM) [5] is based on the topology in⃗ In the Born-Oppenheimer approximaduced by the electron density scalar field, ρ(r). tion with fixed nuclei, the stationary ρ for a system of N electrons with spin spatial coordinates x⃗i is obtained by averaging out all the electrons but one from Born’s probability, ρ(r)⃗ = N ∑ ∫ dx⃗2 . . . dx⃗N Ψ∗ (x⃗1 , . . . , x⃗N )Ψ(x⃗1 , . . . , x⃗N .) s1

(3.6)

In this expression, all of the spin coordinates have been integrated, and we have used the fact that the N electrons are indistinguishable. Notice that ρ does not provide the

48 | Á. Martín Pendás and C. Gatti probability of finding one electron at r,⃗ but the probability density of finding electrons at r,⃗ being thus normalized to N. The electron density, is an observable that can be determined experimentally [15] by means of several techniques, like X-ray or electron diffraction, Compton scattering, etc. In an ideal X-ray diffraction setup, the total intensity of diffracted rays, supposed elastic and coherent dispersion becomes ⃗ ⃗ 2πik⋅r ⃗ dr|⃗ 2 , where k⃗ is the dispersion vector that bisects the angle between I(k)⃗ = | ∫ ρ(r)e the diffracted ray and the inverse of the incident ray, with module 2 sin(θ)/λ. The dispersion amplitude is thus related to the Fourier transform of ρ, which can be obtained by deconvolution. The density can also be accurately calculated through electronic structure determinations. The morphology of the ρ field is strongly determined by the nuclei [16, 17]. It can be shown that the nuclear Coulombic attraction potential forces a cusp (or singularity) of the density at the nucleus. Kato’s atomic theorem [18] establishes that the spherī cally averaged density at a nuclear position ρ(̄ r)⃗ satisfies −2Zα = 𝜕 ln ρ(r)/𝜕| r ⃗ − R⃗ α |, so that densities decay exponentially. It has also been shown that at large distance the density decays exponentially, but simple properties, like monotonicity, have never been proven, and others, like convexity, have been shown not to hold in general. In a number of cases, maxima of the electron density not associated to nuclear positions may appear. These so-called nonnuclear attractors [19–21], where originally thought as computational artifacts, but are now accepted and even associated to exotic behaviors in the case of electrides [22, 23]. Altogether, the electron density field can be envisioned as a an exponentially decaying map from cusps that coincide, in the vast majority of times, with nuclear positions. Although cusps are not differentiable, they are topologically equivalent to maxima, and they will be considered like that in the following. The topological partition induced by the attraction basins of ρ is, thus, an atomic partition. The Quantum Theory of Atoms in Molecules (QTAIM) starts from the computational and experimental finding that basins of atoms (or of groups of atoms) in chemically similar environments are extremely similar in shape and density, so that chemically close atoms do show transferable basins. A number of ways to look at basins have been used in the literature. Figure 3.2 shows a relief map of the density of the LiH molecule in a plane containing the nuclei and in an orthogonal plane passing through the critical point located in the internuclear axis. Maps where the gradient lines ending up or emanating from the different CPs are projected on a plane are also common. The relief map shows two clear nuclear cusps (the Li cusp has been cropped so that the H cusp is visible), and a saddle point in between the nuclei with λ > 0 along the internuclear axis. The orthogonal projection shows that the other two curvatures have negative signs, and that this is a (3,-1) CP. The gradient field projections allow us to grasp the Li and H atomic basins, as well as the separatrix or interatomic surface, which is the 2D attraction basin of the (3,-1) CP. Notice that two gradient lines

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Figure 3.2: Relief (left panel) and gradient line (right panel) representations of ρ in a 6-311G*//B3LYP LiH molecule. In each panel, the density is plotted in planes containing the internuclear axis and being orthogonal to it at the intermediate critical point.

(in green), originate at the (3,-1) CP ending up at the Li and H nuclei. Together they form its 1D repulsion basin. These first-order saddle points are known as bond critical points (BCPs), since they are commonly found among chemically bonded atoms, although this is a heuristic assignment. Plotting the repulsion basins of each BCP, i. e., their bond paths, is equivalent to drawing the chemical graph, so that the binary relationship among the basins that one obtains after inducing a topology in the real space through ρ provides a way to chemically structuring the space. The rest of the possible CPs are easily found in more complex molecules. Figure 3.3 displays the computed CPs as well as the molecular graphs in the water, diborane and cubane. For H2 O, two standard O-H BCPs are found, with straight-line bond paths. In diborane, BCPs are found between the B atoms and the hydrogens, but instead of a BCP, a (3,+1) CP is found between the borons, in agreement with the absence of a standard chemical link there. This is a minimum of ρ in the B2 H4 plane, and a maximum in the orthogonal direction, and it is called a ring critical point (RCP). The figure also plots the gradient lines connecting the B-H BCPs to the RCP. Notice how the bond paths (BPs) connecting the B atoms and the bridging hydrogens are strongly curved inwards. This is many times found in 3c,2e bonds. Outward curved BPs are usually

Figure 3.3: Critical points and molecular graphs obtained for the H2 O, B2 H6 and cubane molecules from 6-311G*//B3LYP calculations. CPs are classified by color: green, red and blue for bond, ring and cage points, respectively.

50 | Á. Martín Pendás and C. Gatti found in strained links, like in the C-C links in cubane. It has been suggested that, by doing so, the linked atoms improve their preferred angular arrangement (e. g., tetrahedral), and that the difference between the BP length and the interatomic distance may be taken as a measure of bond strain. In cubane, besides the C-H and C-C BCPs we find one RCP at the center of each face of the cube, and a final CP at its geometrical center. This is a (3,+3) minimum, or a cage critical point (CCP). The Euler–Poincaré relationship holds in all cases: n + b + r − c = 1. The set of CPs of a given density provides a continuum to discrete map in ℝ3 . Instead of examining infinitely many points, we can establish correlations between scalar or vector densities at the CPs and chemical behavior. This is the origin of the local stance in the QTAIM. A flavor of how this is done is now sketched. For instance, provided that chemical bonding has been traditionally related to electron density accumulation along the bonded internuclear axis, the value of the density itself at the BCP, ρb , can be used as a characteristic value for each type of bond: ρb is thus a bond strength descriptor. In this sense, the values of ρb for the C-C bonds in ethane, ethene and ethyne provide a calibration ruler for other cases. If they are linearly fitted to the standard bond orders one, two and three, then ρb for the C-C bond in benzene lies in between that of C2 H6 and C2 H4 , as expected. In classical topological analysis, there are two other properties computed at bcps that have spread widely: the ellipticity, ϵ and the bond radius, rb . If, as stated before, we order the eigenvalues of the Hessian matrix of the density (now at a BCP) in ascending order, then for a diatomic molecule in an axially symmetric electronic state, λ1 = λ2 , so that the orthogonal density accumulation toward the internuclear axis is axially isotropic. The ellipticity at a BCP is defined as the ratio ϵ = λ1 /λ2 − 1. This quantity thus measures the asymmetry in the perpendicular density accumulation, determining the easy and hard charge accumulation directions [24, 25]. Most texts on the QTAIM comment on the ellipticity of the C–C bond in C2 H4 , which turns out to be ϵ ≈ 0.30. This indicates an accumulation of charge in the plane orthogonal to the molecular plane, and standard interpretations relate it to the π cloud. Ellipticity is then lined to the π character of a bond, and the value ϵ = 0.23 displayed by the C–C bond in benzene matches a partial double bond. It is nonetheless important to warn about overinterpretations in quantum chemical topology. First, the π character of a chemical bond is not an observable quantity, but a quantity based on a given type of orbital description. Second, ϵ is actually sensing anisotropies in the density distribution, and does not include any direct information on orbital contributions to such anisotropy. This can be very easily understood by turning to the ethyne molecule, in which the axial symmetric distribution leads to ϵ = 0, and thus to no π character. The distance from the BCP to each of the two nuclei it binds to is called the bond or bonding radius of each of the nuclei. It correlates rather well with the notion of bonded size. Its evolution along a homologous series reflects very clearly how electronegativity differences affect the atomic size. For instance, as we run over the second period A-H

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diatomics, rb (H) evolves from 1.72 to 0.71 on passing from LiH to CH, ending up in 0.28 au in HF. If we simply take as a reference half the bond distance in H2 , rb = 0.70 au, we can clearly decide about the charge state of each system immediately.

3.1.3 The Laplacian of the electron density Any other chemically relevant scalar field can be studied with the topological method. Keeping an eye on the density, the next scalar whose CPs are not directly related to those of ρ itself is the Laplacian field [26]: ∇⃗ 2 ρ. It is related to local charge accumulations or depletions, since it is easy to show that the mean value of ρ on the surface of an infinitesimally small sphere centered at a given point r ⃗ is larger or smaller than ρ(r)⃗ depending on whether the sign of ∇⃗ 2 ρ(r)⃗ is positive or negative, respectively. At a point with ∇⃗ 2 ρ < 0, the electron density accumulates with respect to its nearbies, while the contrary is true if the Laplacian is positive. The topology of the Laplacian field can only be understood after it has been exρ󸀠 (r) amined in spherically symmetric atoms. Since ρ = ρ(r), then ∇⃗ 2 ρ(r) = ρ󸀠󸀠 (r) + 2 r , so the sign of the Laplacian is not fixed, depending on a subtle balance. Admitting an exponential behavior for the density, ρ(r) ≈ e−ζr , ∇2 ρ(r) ≈ e−ζr (ζ 2 − 2ζ /r), and a cutoff radius exists at which a sign change occurs. The Laplacian is negative (and large, divergent) at short distances, changes sign and approaches zero from above at large distances. Atomic densities can be well described as a set of exponential segments as we move away from the nucleus that are ascribed to the standard atomic shells. Their Laplacians show then a quick oscillatory behavior, with one negative (accumulation) and positive (depletion) region for each shell. The radii at which the several maxima or minima occur correlate well with estimations of the positions of the shells. However, it is important to not overshoot interpretations. Past the calcium atom the number of shells in the Laplacian fall short that expected from the number of electrons, although other scalar fields, like the Electron Localizability Indicator (ELI) proposed by M. Kohout [27] do not have this problem. Anyway, negative and positive regions of the Laplacian field in representative element molecules provide important information about the fine structure of the electron density. Regions with negative ∇⃗ 2 ρ are called charge concentration (CC) regions, while regions with positive Laplacian are usually known as charge depletions (CDs) [28]. As a molecule is formed, we expect the atomic shells of the different atoms to distort. Instead of spherical CCs, several minima compatible with the molecular symmetry will appear. Distortions in the inner atomic shells are usually small (although present), while those in the valence shell are clearly visible. Examining the structure of the valence CCs (VCCs) provides very useful chemical information. Unlike for the electron density field, the topology of the Laplacian is complex, with many CPs of all the possible kinds.

52 | Á. Martín Pendás and C. Gatti

Figure 3.4: Critical points of the Laplacian in 6-311G*//B3LYP Cl2 (left) and NaCl (right). BCPs of ρ are shown in green, and (3,+3), (3,+1), (3,-1) and (3,-3) CPs of the Laplacian in yellow, green, magenta and dark blue, respectively. Degenerate points are shown in cyan. Isolines (negative in red, positive in blue) are also shown on a plane containing the internuclear plane.

Figure 3.4 shows the CPs of ∇⃗ 2 ρ in the 6-311G*//B3LYP Cl2 and NaCl molecules, together with isolines projected on a plane containing the internuclear plane. In dichlorine the K, L and M CCs of each Cl atom are clearly visible. The first two maintain sphericity, while the valence one is very distorted, having fused with its equivalent on the other atom. At the BCP, the Laplacian is negative and we classify this as a shared-shell interaction. Degeneracy of the CPs is lifted on the internuclear axis, but not out of it, where axial symmetry forces annular degeneracy. We identify two valence charge concentrations (VCCs) on the axis close to the BCP, and a ring of CC on each atom that is slightly displaced to their rear part. Two maxima of the Laplacian are found just outside the valence shells on the axis. They are related to what we call a σ-hole. The situation in NaCl is very different. Now the K, L and M shells of Cl are visible, but the M shell in Na has disappeared. The valence shells are not fused, and ∇⃗ 2 ρ > 0 at the BCP. This is a closed-shell interaction. When we define atomic observables, we will check that the atomic population of the Na region is close to 10 electrons, so a very appealing image where the valence shell has been transferred from the Na atom to the chlorine appears. In this case, the ring of CC in the valence shell of the Cl atom is displaced toward the Na moiety, showing the polarization of the distribution. When symmetry is fully broken, the VCCs condense on discrete points in space. Figure 3.5 shows the CCs of ∇⃗ 2 ρ in the 6-311G*//B3LYP H2 O and ClF3 molecules as well as projected isolines. The two O-H bonds are shared-shell, as expected, and the valence shell of oxygen is now distorted forming a tetrahedron of CCs, two bonded, and two nonbonded. The latter are located where lone pairs are expected. They follow the VSEPR rules [29], and the angle between them and the oxygen atom is 138.6∘ , considerably larger than the one existing between the BCCs, 103.8∘ . The valence shell CC tetrahedron can be associated to the four valence pairs in oxygen. ClF3 poses a more interesting case, with five valence pairs in the central Cl atom. As it can be seen, all Cl-F interactions are closed-shell, limiting the meaning of octet expansion, but the

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Figure 3.5: Laplacian of a 6-311G*//B3LYP H2 O molecule (left) and ClF3 (right). Only the BCPs of ρ in green, and the (3,+3) CPs of the Laplacian in yellow are shown. Isolines in the molecular plane are also depicted.

usefulness of the VSEPR rules is clear. The T structure follows from applying the rules to the clear five VCCs, two of them nonbonded.

3.2 The quantum theory of atoms in molecules There are deeper theoretical reasons that justify using the topology induced by the electron density to partition the space into chemically relevant regions. This was a program started by R. F. W. Bader and coworkers [30–32], who tried to generalize quantum chemistry to subsystems or regions in ℝ3 . This faces several problems that are not easily overcome. One of the easiest to understand is the presence of surface terms that do not necessarily vanish upon integration when a domain has finite limits. Many tenets of quantum mechanics are shuttered by this. Among them, the hermiticity of its operators, particularly the momentum operator p⃗̂ if we use the position representation. Max Born’s probabilistic interpretation of the wavefunction is possible since the quantum fluid is incompressible. If we consider a fluid with mass density ρ and current density j ⃗ = ρv,⃗ then the rate of variation of the mass contained in a volume Ω with ⃗ S.⃗ Since the mass must equal m = ∫ ρ dr,⃗ the divergence = − ∮S j⋅d surface S = 𝜕Ω is dm dt Ω 𝜕ρ theorem allows us to write = −∇⃗ ⋅ j,⃗ the continuity equation of the fluid. Using 𝜕t

Born’s probability density ρ = Ψ∗ Ψ (for one spinless particle, the generalization is d ̂ = iℏΨ,̇ simple), then dt ∫Ω Ψ∗ Ψ dr ⃗ = ∫Ω 𝜕t𝜕 (Ψ∗ Ψ) dr.⃗ Using Scrhödinger’s evolution HΨ and after a few elementary steps we arrive at a continuity equation if we define the quantum mechanical current density as j⃗ = −

iℏ ⃗ − Ψ∇Ψ ⃗ ∗] . [Ψ∗ ∇Ψ 2m

(3.7)

̂ ∗ Ψ+Ψ∗ (HΨ)], ̂ If, on the other hand, we write 𝜕t𝜕 (Ψ∗ Ψ) = −i [−(HΨ) it is easy to find ℏ 𝜕 ∗ ̂ ∗ Ψ dr ⃗ = that for a system that extends to infinity, 𝜕t ∫ Ψ Ψ dr ⃗ = 0 if and only if ∫(HΨ)

54 | Á. Martín Pendás and C. Gatti ̂ dr.⃗ This is the standard Hermitian condition imposed on quantum mechan∫ Ψ∗ (HΨ) ical operators. For a subsystem, we can repeat the above argument and arrive at ̂ ∗ Ψ − Ψ∗ (HΨ)] ̂ dr ⃗ = ∮ j ⃗ ⋅ dS,⃗ ∫ [(HΨ)

(3.8)

S

Ω

which shows that operators that might be Hermitian in a description of the total system may lose this property for a general subsystem unless the surface term vanishes. Since the current density involves gradients of the wavefunction, it is important to p2 = study the behavior of the kinetic energy operator. Using for a single particle T̂ = 2m −ℏ2 2 ∇, 2m

we get

−ℏ2 ⃗ ⃗ ∗ −ℏ2 2 ∗ ∇ (Ψ Ψ) = ∇ ⋅ ∇(Ψ Ψ) 2m 2m −ℏ2 2 ∗ = [∇ (Ψ Ψ) + 2 (∇Ψ∗ ) (∇Ψ) + Ψ∗ ∇2 Ψ] , 2m

(3.9)

or, for a many electron system, −ℏ2 −ℏ2 ∑ ∇i2 (Ψ∗ Ψ) = ∑(Ψ∗ ∇i2 Ψ + Ψ∇i2 Ψ∗ + 2∇i Ψ∗ ∇i Ψ). 2m i 2m i

(3.10)

Integrating spins and all spatial coordinates but that of one electron, −ℏ2 ℏ2 ℏ2 ⃗ ⋅ dS.⃗ ∫(∇2󸀠 + ∇2 )ρ(r,⃗ r ⃗ 󸀠 )dr ⃗ = ∫ ∇󸀠 ∇ρ(r,⃗ r ⃗ 󸀠 )dr ⃗ − ∮ ∇ρ 4m 2m 4m Ω

(3.11)

S

Ω

From this, we can introduce two possible kinetic energy densities, which are usually ℏ2 ℏ2 󸀠 called K and G, K = − 4m (∇2󸀠 + ∇2 )ρ(r,⃗ r ⃗ 󸀠 ), G = 2m ∇ ∇ρ(r,⃗ r ⃗ 󸀠 ), as well as a difference 2

ℏ operator related to the Laplacian of ρ, L = − 4m ∇2 ρ. Now we can rewrite equation (3.11) as

⃗ r ⃗ = ∫ G(r)d ⃗ r ⃗ − ∫ ∇2 ρdr,⃗ ∫ K(r)d Ω

Ω

KΩ = GΩ + LΩ ,

Ω

(3.12)

so that for the subsystems of the QTAIM, the kinetic energy of a subsystem does not depend on whether we use the K or G densities. It can be shown [33] that there is an infinite family of kinetic energy densities (the Laplacian family) for which the QTAIM regions provide unique kinetic energies, although this does not exhaust all the possibilities. A large number of other integral expressions valid over subsystems can be found. These are usually called atomic theorems [34, 35]. Particularly important is the virial theorem, which lies historically at the origin of the QTAIM. Using simple arguments, it

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𝜕j ⃗ = −Ψ∗ Ψ∇⃗ V̂ + ∇⃗ ⋅ σ, where the left-hand side is the Newtoniancan be shown that m 𝜕t like force density acting at a given point of space and σ is Pauli’s stress tensor,

σ=

ℏ2 ⃗ + ∇⃗ ⊗ (∇Ψ ⃗ ∗ )Ψ − ∇Ψ ⃗ ∗ ⊗ ∇Ψ ⃗ − ∇Ψ ⃗ ⊗ ∇Ψ ⃗ ∗ }. {Ψ∗ ∇⃗ ⊗ (∇Ψ) 4m

(3.13)

In quantum mechanics, the force density contains a classical term coming from the gradient of the potential energy plus a nonclassical one determined by the divergence ⃗ of the stress tensor, which is called the Ehrenfest force, m 𝜕j = f ⃗ +f ⃗ . In a stationary 𝜕t

class

Ehr

system, the total force density vanishes. In classical physics, as well as in quantum ̂ = 2⟨T⟩. ̂ Using mr ⃗ ⋅ 𝜕j ⃗ = r ⃗ ⋅ ∇σ ⃗ t − Ψ∗ Ψr ⃗ ⋅ mechanics, the virial theorem states that ⟨r ⃗ ⋅ ∇⃗ V⟩ 𝜕t ∇⃗ V̂ = 0 for a stationary system, and after a few manipulations, we may arrive at ̂ Ω = ⟨r ⃗ ∇⃗ V⟩ ̂ Ω − ∫ rσ ⃗ t dS⃗ − 2⟨T⟩ S

ℏ2 ⃗ ⋅ dS.⃗ ∮ ∇ρ 4m

(3.14)

S

Since the last term vanishes for a QTAIM subsystem, the virial theorem holds for a subsystem. The first term of the right-hand side is known as the basin virial, 𝒱 b , and the second as the surface virial, 𝒱 s , which measures its flux across the surface.

3.2.1 Atomic observables Since almost all operator densities that include gradients and bear any chemical relevance are related to the kinetic energy, we will assume in the following that the operator densities of interest exist and are well-defined within QTAIM basins. These densities can be integrated over the basins to reconstruct the full expectation value. If o(̂ r)⃗ ̂ = ∑ ∫ o(r)⃗ dr ⃗ = ∑ O , is the spatial density of a one-electron operator O,̂ then ⟨O⟩ Ω Ω Ω Ω so that any expectation value is additively reconstructed from the atomic observables. It has been shown over the years that the latter are transferable if the atom (or group of atoms) lie in chemically similar environments. Similarly, if g(r1⃗ , r2⃗ ) is the density oper̂ = ∑ 󸀠 ∫ ∫ 󸀠 g(r ⃗ , r ⃗ ) dr ⃗ dr ⃗ = ∑ 󸀠 G 󸀠 , ator of a two-electron operator G,̂ then ⟨G⟩ 1 2 1 2 Ω,Ω Ω,Ω Ω Ω Ω,Ω and the expectation value is pairwise-additively reconstructed. A large number of one-electron densities can be used to define atomic observables. If we use the unit density, we will get a measure of the atomic size. Usually, atomic basins extend to infinity along some directions so that, rigorously, VΩ diverges. It is customary to cut the integration at a given electron density isosurface, like that with ρ = 0.001 au. These atomic volumes compare reasonably well with van der Waals ones. If the density integrated is the electron density itself, then we get a partition of the number of electrons. This provides atomic populations, NΩ , or charges, QΩ = Z − NΩ . Using pair densities the variances of these populations, which are not constants of motion, can also be determined, σ 2 (Ω) = ⟨N 2 ⟩Ω − NΩ2 .

56 | Á. Martín Pendás and C. Gatti Other important atomic properties are the moments of the atomic charge distribution [36]. Using spherical harmonics Slm instead of complex ones, we define the spher⃗ r.⃗ Since S00 = s, S11̄ = py , ical atomic multipole moments as NΩlm = ∫Ω r l Slm (θ, ϕ)ρ(r)d

S10 = pz , S11 = px , N 00 is the atomic population, the three (N 11 , N 11 , N 10 ) determines the atomic dipolar moment, μΩ , with large l contributions being quadrupole, octupole, etc. moments. Atomic moments are not invariant under translations, so care has to be taken. For instance, the reconstruction of the electronic molecular dipole moment from the atomic dipoles involves ̄

⃗ r ⃗ = ∑ ∫ rρd ⃗ r ⃗ = ∑ ∫ (r ⃗ − R⃗ A )ρdr ⃗ + ∑ ∫ R⃗ A ρdr ⃗ μ⃗ = ∫ rρd A Ω A

A Ω A

A Ω A

= ∑ μ⃗ A + ∑ NA R⃗ A = μ⃗ pol + μ⃗ CT , A

(3.15)

A

so that on top of the sum of the atomic dipoles (the polarization term) we need to take into account a charge transfer contribution. Energetic quantities follow similar rules. Of course, the kinetic energy is the sum of the atomic kinetic energies, T = ∑A TΩA . The electron-nucleus attraction can be also en partitioned, and VAB = −ZB ∫Ω

A

ρ(r)⃗ dr ⃗ is ⃗ R⃗ B | |r− en . ∑A,B VAB

the attraction of electrons in basin A with

nucleus B. It follows that V en = Partitioning the electron repulsion, which is a two-electron observable, requires the consideration of the pair density, ρ2 (r1⃗ , r2⃗ ), the probability density of finding a pair of electrons at points r1 and r2 , V ee =

1 −1 ρ2 (r1⃗ , r2⃗ ), ∬ dr1⃗ dr2⃗ r12 2

(3.16)

ee −1 ee so that if we define VAB = ∫Ω dr1⃗ ∫Ω dr2⃗ r12 ρ2 (r1⃗ , r2⃗ ), V ee = ∑A,B VAB A B Finally, forces can also be partitioned. The Ehrenfest force can be integrated over atomic basins, and given that the gradient of the potential contains electron-electron and electron-nuclear terms, a pairwise additive decomposition also exists. Defining ρ(r)⃗ en ee −3 = ∫Ω dr1⃗ ∫Ω dr2⃗ r12 (r1⃗ − r2⃗ )ρ2 (r1⃗ , r2⃗ ), we arrive at fAB = −ZB ∫Ω ⃗ ⃗ 3 (r ⃗ − R⃗ B )dr,⃗ and fAB A

|r−RB |

A

B

⃗en + ∑ (f ⃗en + f ⃗ee ). The Hellmann–Feynman force acting upon a nucleus, F⃗ , fA⃗ = fAA A B=A ̸ AB AB ⃗ne = − ∑ f ⃗en . can be similarly partitioned, F⃗A = ∑B fAB B BA

3.2.2 Electron localization and delocalization measures The pair density introduced above can be also subjected to a topological analysis [37, 38], although being it a six-dimensional object this enterprise has not proceeded too far. However, its study within QTAIM atomic basins has provided extremely interesting insights. The second-order density matrix is defined as ρ2 (x⃗1󸀠 , x⃗2󸀠 ; x⃗1 , x⃗2 ) = N(N − 1) ∫ dx⃗3 . . . dx⃗N Ψ∗ (x⃗1󸀠 , x⃗2󸀠 , . . . , x⃗N )Ψ(x⃗1 , x2 , . . . , x⃗N ),

(3.17)

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integrating to N(N − 1) ordered pairs. Its spinless diagonal is the pair density used above. If electrons had no correlated motions, it could be obtained from the first-order N−1 density, ρind 2 (x⃗1 , x⃗2 ) = N ρ(x⃗1 )ρ(x⃗2 ). The (N − 1) factor accounts for the normalization of the density of the second electron, which should exclude the first one. Normally, the statistical dependence is taken into account via conditional probabilities. Thus, when a reference electron is known to be located at x⃗1 , the (conditional) probability to find a second one at x⃗2 is ρ(x⃗2 |x⃗1 ) = ρ2 (x⃗1 , x⃗2 )/ρ(x⃗1 ). From it, the exchange-correlation hole hxc , that measures the difference between the conditional and the standard density is introduced as hxc (x⃗2 |x⃗1 ) = ρ(x⃗2 |x⃗1 ) − ρ(x⃗2 ). The exchange-correlation hole excludes one electron from the electron count, ∫ dx⃗2 hxc (x⃗2 |x⃗1 ) = −1, and has the property hxc (x⃗2 → x⃗1 |x⃗1 ) = −ρ(x⃗1 ). Now let us introduce the (ordered) pair population of a region A, DAA = ∫A dx⃗1 ∫A dx⃗2 ρ2 (x⃗1 , x⃗2 ). It can be shown that the variance of the electron population in A can be written as σ 2 (NA ) = DAA − NA (NA − 1) [39–41]. We say that when the pair population is equal to NA (NA − 1) (and this can be shown to occur only when NA is an integer number) the electrons are localized in region A. After simple rearrangements, σ 2 (NA ) = NA + ∫ dx⃗1 ρ(x⃗1 ) ∫ dx⃗2 hxc (x⃗2 |x⃗1 ) = NA − λ(A), A

(3.18)

A

where λ(A), the localization index of basin A is interpreted as the number of localized electrons in a basin. When hxc is completely contained in region A, hxc (x⃗2 ∉ Ω|x1 ∈ Ω) = 0, then σ 2 (Ω) = 0, so that complete localization of the exchange-correlation hole is equivalent to electron localization. If the holes are localized in two regions A and B, then for the A ∪ B union it follows that σ 2 (A ∪ B) = −2 ∫A dx⃗1 ρ(x⃗1 ) ∫B dx⃗2 hxc (x⃗2 |x⃗1 ) = δ(A, B) = 0 We call δ the delocalization index between the regions. It vanishes if electrons are not shared (delocalized) among them. It is clear that a sum rule exists, and that λ(A) +

1 ∑ δ(A, B) = NA , 2 B=A̸

(3.19)

so that the average atomic population NA can be decomposed into localized and delocalized contributions. The delocalization index (DI) [42, 43] between two regions may also be understood in terms of the covariance of their electron populations, δ(A, B) = −2cov(NA , NB ), or as the number of shared pairs of electrons. It is thus a measure of the covalent bond order. In fact, it can be shown that it smoothly tends to the Wiberg– Mayer bond order [44], if the QTAIM projection is substituted by a Mulliken’s population analysis. In a Hartree–Fock description of the H2 molecule, the delocalization index is exactly equal to one. We call this an ideal covalent bond. Each new independent ideal link added to a given pair of atoms adds one to δ. In this way, the Hartree– Fock descriptions of ethane, ethylene and acetylene provide DIs between the CH3 -, CH2 - and CH-groups very close to one, two and three, respectively.

58 | Á. Martín Pendás and C. Gatti

3.3 The interacting quantum atoms energy decomposition As shown, all the operators in the standard Coulomb hamiltonian under the Born– Oppenheimer approximation can be partitioned using the QTAIM spatial decomposition. This allows us to provide an energy decomposition analysis (EDA) based on the QTAIM. Initial approaches, escaping from the complexity of constructing the pair density, used only the first-order density matrix, ρ(r,⃗ r 󸀠⃗ ), since the kinetic energy can ̂ = ∫(−1/2)∇2 ρ(r,⃗ r 󸀠⃗ )dr|⃗ r ⃗ → r 󸀠⃗ . At stationary points of the potential be written as ⟨T⟩ r⃗ ̂ = −E is fulfilled, we can use the QTAIM energy surface, where the virial theorem ⟨T⟩ partition to write ∑A TA = −E, so that we can define a virial atomic energy EA = −TA which reconstruct additively the molecular energy [5]. Unfortunately, this partition is only meaningful at equilibrium (or transition state) geometries, providing quantities that are difficult to interpret chemically. We can circumvent this problem by using the pair density, ρ2 . Doing that, we note that the energy can always be written as ̂ r ⃗ , r 󸀠⃗ ) 󸀠⃗ ⃗ dr ⃗ + 1 ∫ ∫ ρ (r ⃗ , r ⃗ )r −1 dr ⃗ dr ⃗ E = h + Vee + Vnn = ∫ hρ( 1 1 |r1 →r1 1 2 1 2 12 1 2 2 ZA ZB 1 , + ∑∑ 2 A B=A̸ RAB

(3.20)

where ĥ = t ̂ − ∑A ZA /rA is the one-electron hamiltonian. All these integrals can now be partitioned by decomposing the space into QTAIM regions, leaving terms that have already been defined: en nn ee E = ∑ TA + ∑ VAB + ∑ VAB + ∑ VAB . A

A,B

A>B

A>B

(3.21)

nn In this expression, VAB = ZA ZB /RAB is the only newly defined contribution. This is the interacting quantum atoms (IQA) partition [45–48], which is valid at all molecular geometries, be they stationary points on the potential energy surface or not. If all intra and interatomic terms are gathered appropriately we separate the so-called atomic A self-energy from the interatomic interaction energy. The first, EAself , is defined as Eself = en ee int en en ee ee TA + VAA + VAA , while the second as EAB = VAB + VBA + VAB + VAB . If we separate the atoms forming a molecule fully, their atomic self-energies will be equal to their free atomic energies, and all the mutual interatomic interaction energies will vanish. As the atoms approach, their self-energies will be modified upon interaction. IQA provides an EDA familiar from atomistic simulations, int E = ∑ EAself + ∑ EAB . A

A>B

(3.22)

Notice that we can join several QTAIM atoms together to form a QTAIM group. In this case, the self-energy of the group is found by adding all the atomic self-energies of

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their atoms together with all their mutual interaction energies. Similarly, the interaction energy between two groups is the sum of all the interatomic interaction energies between atoms belonging to different groups. If we chose a reference state for each atom (or group), e. g., the isolated atom, then the change in self-energy as measured from this reference, usually positive since upon interaction the free atom is no longer variational in the molecule, is called the atomic deformation energy: EAdef = EAself −EAref . With respect to the chosen reference, the molecular binding energy is thus int Ebind = ∑ EAdef + ∑ EAB . A

A>B

(3.23)

In IQA, at variance with other EDAs, a reference needs not be introduced, providing a reference-independent energetic decomposition. The IQA interaction energies are usually decomposed by taking into account that the electron repulsion can always be decomposed into a Coulomb and a nonCoulombic (or exchange-correlation) contribution, purely quantum mechanical in nature. This is done after writing ρ2 (r1⃗ , r2⃗ ) = ρ(r1⃗ )ρ(r2⃗ ) + ρxc (r1⃗ , r2⃗ ), where ρxc , the exchange-correlation density is intimately linked to the exchange-correlation hole introduced above. Using this decomposition, the electron repulsion can be written as ee,cl ee xc VAB = VAB + VAB . Gathering all classical terms in the interaction energy expression, int cl xc EAB = VAB + VAB .

(3.24)

It has been shown over the years [49] that the classical interaction energy measures the ionic component of a given interaction, while the exchange-correlation −1 one, which can be understood as a kind of r12 scaled delocalization index, for xc δ(A, B) = 2 ∫A ∫B ρxc (r1⃗ , r2⃗ )dr1⃗ dr2⃗ while VAB = − ∫A ∫B ρxc (r1⃗ , r2⃗ )/(r12 )dr1⃗ dr2⃗ , measures the energy due to electron sharing, thus covalency. Taylor expansions [50] have also shown that the ionic interaction is controlled cl by the atomic charges of the atoms which interact, VAB ≈ QA QB /RAB , and that the xc covalent component depends on the covalent bond order, VAB ≈ (−1/2)δ(A, B)/RAB .

3.3.1 The Source Function (SF) analysis Sections 3.1 and 3.2 of this chapter provide examples of the study of chemically relevant scalar fields, such as the electron density and the Laplacian of the electron density, with the topological method. The Source function (SF) tool introduces another interesting perspective in a study of a scalar field f (r)⃗ [51–53]. When combined with a suitable space partitioning, such as that provided by QTAIM, SF analysis shows how the various atoms or groups of atoms in a system contribute to determine the value of f (r)⃗ at arbitrary, but conveniently selected positions [52]. It does so within a causeeffect framework, where the value of f (r)⃗ at a given point of R3 is not retrieved in terms

60 | Á. Martín Pendás and C. Gatti of separate contributions from nuclear-centered basis functions. Rather, it is determined by features of the overall distribution of the scalar within the various atomic basins, locally weighted by the inverse of the distance to the scalar reconstruction point r ⃗ (hereinafter called the reference point, RP). For, instance, by taking f = ρ, it has been shown [51, 52] that ρ(r)⃗ may be envisaged as determined by contributions from a local source LS(r,⃗ r 󸀠⃗ ), operating at all other points r 󸀠⃗ in R3 , ρ(r)⃗ = ∫ LS(r,⃗ r 󸀠⃗ )dr 󸀠⃗

(3.25)

LS(r,⃗ r 󸀠⃗ ) = −(4π|r ⃗ − r 󸀠⃗ |)−1 ∇2 ρ(r 󸀠⃗ ),

(3.26)

and given by:

where (4π|r ⃗ − r 󸀠⃗ |)−1 is an influence function [54] measuring how effective is ∇2 ρ(r 󸀠⃗ )dr 󸀠⃗ ⃗ The concentration or dilution of the electron density, as in determining the effect ρ(r). expressed by its Laplacian at all other points r 󸀠⃗ , is the cause for the effect, the value of the electron density at the RP r.⃗ If the integration of the local source (LS) over R3 (equation (3.25)) is replaced by a sum of disjoint integrations over the QTAIM atomic basins Ω, ρ(r)⃗ = ∫ LS(r,⃗ r 󸀠⃗ )dr 󸀠⃗ = ∑ ∫ LS(r,⃗ r 󸀠⃗ )dr 󸀠⃗ = ∑ SF(r,⃗ Ω), Ω Ω

Ω

(3.27)

the electron density at the RP r ⃗ may be written as a sum of atomic contributions, SF(r,⃗ Ω), each of them called the Source Function from atom to the electron density at r ⃗ [51]. Equation (3.27) expresses the well-known fact that ρ(r)⃗ is only apparently a local quantity. In reality, it is determined by all atoms of the system. The degree of participation of an atom in building the density at a point is just expressed by its own SF value, compared to those from the other atoms in the system. From the tenets of density functional theory, we know that the electron density is uniquely mapped to the external potential arising by the position and charge of all nuclei in the system (and by any externally-applied field, if present). The relevant feature of the SF tool is that of enabling us to express quantitatively such nonlocal dependence using the language of chemistry, that is, in terms of the single atom or group of atoms influence in determining ρ(r)⃗ [51–55]. Being completely independent from the way ρ(r)⃗ is expressed and evaluated, say through an atomic nuclear-centered basis set, or through a multipolar model expansion, or even given numerically on a grid, SF values are fully model-independent for given equivalent ρ(r)⃗ and for a given space partitioning scheme. As such, and as any QCT approach, the SF tool represents a natural choice for comparing, on the same grounds, experimentally and theoretically derived electron densities [52]. In particular when f = ρ, SF values are amenable to experimental determination provided an accurate ∇2 ρ distribution derived from high-quality singlecrystal X-ray diffraction intensity data is available [52, 56, 57].

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The SF is customarily applied to f = ρ, but recently other chemically relevant scalar choices, such as the electron spin density s and the molecular electrostatic potential Vel , including the nuclear and the electron distribution contributions, have been explored [58–60]. In such cases, SF reconstruction is performed through mathematical expressions formally resembling equations (3.25 and 3.27), but where the cause ∇2 ρ is replaced by the causes ∇2 s and ∇2 Vel , respectively [58–60]. Within SF analyses, QTAIM space partitioning is normally adopted so as to identify each basin SF contribution with one due to a chemically meaningful object having well-defined expectation values for its observables (see Section 3.2.1). Yet, any other disjoint or fuzzy boundaries space exhaustive partition is also mathematically possible (Ω being defined accordingly in equation (3.27)) [52]. In order to manifest the capability of an atom (or group of atoms) Ω to determine the scalar f at the RP r ⃗ relative to that of all other atoms or groups in the system, atomic (atomic group) SF contributions are often analyzed in terms of Source Function per⃗ (r)⃗ × 100) [52, 55]. SF % values may centage contributions, SF %, (SF % = SF(Ω, r)/f be portrayed using a ball and stick molecular model where each ball has a volume proportional to its associated SF % value and where the location of the chosen RP is denoted by a dot (Figure 3.6). Bond critical points (bcps) are customarily adopted as RPs when SF electron density reconstructions are analyzed to discuss chemical bonding features [51–53, 55]. Assuming the bcp as the most representative electron density location for a given chemical interaction, the pattern of the atomic SF contributions provides a visible representation of the more or less delocalized nature of the interaction [52, 55]. (Figure 3.6). The SF analysis may be applied to any chemical interaction. It has been found to be of particular interest where conventional covalent bonding— characterized by dominant sources from the pair of bonded atoms at BCP—does not describe a system appropriately or where chemical bonding varies widely in nature, depending on some well-identified chemical variables [52, 53, 55]. This is, for instance, the case of hydrogen bonds (Figure 3.6) [55, 61]. Despite the prominent role of bcps as reference points in the SF analysis, many other choices are possible and in some cases even more convenient [52, 55–60, 62–64]. The ball and stick models here considered are not available in the present version of AIMAll. In particular, bcp positions are often irrelevant when f = s since s almost vanishes at these points [58, 59, 65], while charge concentrations associated to the unpaired electrons distribution appear particularly meaningful in such a case [58, 59, 65]. For f = Vel , maxima and minima of Vel on chemically significant electron density isosurfaces have been selected as suitable RPs [60]. In general, the electron spin density SF provides insights on how the spin density due to a paramagnetic center transmits to the neighboring atoms and on which are the atomic or the atomic region sources for the regions of negative spin polarization [58, 59, 65]. SF analysis of Vel has also interesting applications [60, 66]. For instance, it enables one to determine the SF atomic group contributions to the positive Vel regions associated to the so called σ- and π-holes

62 | Á. Martín Pendás and C. Gatti

Figure 3.6: Ball-and-stick representation of the Source Function percentage (SF %) atomic contributions relative to carbon-carbon, metal-metal and hydrogen-bond linkages. Each “atomic” ball has a volume proportional to the corresponding SF % value. The location of the reference point (a bond critical point in all displayed cases) is denoted by a dot. Blue and yellow balls denote positive and negative SF % contributions, respectively. Description of single cases: (a) SF % contributions at the carbon-carbon bcp in ethane, ethene and ethine. The SF % values show the dominance of the C atom contributions as well as their increasing weight with increasing bond order along the series (adapted from Figure 3, Ref. 63, with permission from the Royal Society of Chemistry); (b) SF % contributions at the metal-metal bcp in the [M2 (formamidinate)4 , M= Mo, Pd] isostructural binuclear transition metal complexes. The SF % patterns clearly distinguish the SF contributions in a complex with a metal-metal formal bond order of 4 (M = Mo) from those in a complex with a formal bond order of 0 (M = Pd) (adapted from Figure 8, Ref. 63, with permission from the Royal Society of Chemistry); (c) SF % contributions at the hydrogen bond bcp as a function of the donor to acceptor distance in the H2 N-H⋅ ⋅ ⋅ H2 O hydrogen-bonded model complexes and (d) in the Adenine:Thymine (AT) DNA base pair complex (Adapted with permission from Figures 2 and 3 of Ref. 61. Copyright 2018 Wiley-VCH Verlag 15530 GmbH&Co. KGaA).

which are deemed to play an active role in molecular recognition (halogen, chalcogen, pnicogen bonds, etc.). The percentage values of such contributions have been shown to serve as a valuable tool to design the separation of enantiomers through HPLC techniques [60, 66]. Recently, the interesting concept of SF reconstructed partial electron densities has also been proposed [59, 61]. These densities, which are defined as ρ{Ω, subset}(r)⃗ =

∑

∫ LS(r,⃗ r 󸀠⃗ )dr 󸀠⃗ =

Ω,subset Ω

∑

Ω,subset

SF(r,⃗ Ω),

(3.28)

represent hypothetical electron distributions determined only by a given subset of atoms in the system. When compared against the full electron density, e. g., through the corresponding 2D contour plots maps, they visibly highlight (Figure 3.7) the major or minor role a given subset of atoms plays in determining the electron density in the various system’s regions [59]. At variance with standard partial densities, the SF

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Figure 3.7: Source Function (SF) reconstructed partial electron and electron spin densities: (a) SF electron density reconstruction in the least squares molecular plane of the Adenine:Thymine DNA base pair complex, using the SF contributions from all atoms in the complex (top left map), only those belonging to ring 3 (R3) and ring 4 (R4) (top right map), only those belonging to R3 (bottom left map) or to R4 (bottom right map). From the four maps it is clear that the basic features of the central N27-H28⋅ ⋅ ⋅ N11 hydrogen bond may be reasonably well described either by using only the R3 atoms or only the R4 atoms SF contributions, while the longer and weaker C12-H13 ⋅ ⋅ ⋅ O30 hydrogen bond requires that the remaining rings atomic sources be also included for its proper description (adapted from Figure 5 of Ref. 61 with permission of the International Union of Crystallography). Hydrogen bond distances in the complex are shown in Figure 3.6; (b) SF electron spin density reconstruction in an end-on ferromagnetic azido-bridged di-copper molecular complex. The partial reconstructed spin density for two wavefunction models [CASSCF(6,6) and DFT-UB3LYP] and various subset of atoms is shown in the least squares molecular plane of the four ligand N atoms around each Cu atom (N and Cu are drawn as dark blue and azure balls, respectively, in the molecular drawing at left). The following atomic subsets are considered (from left to right panels): the two Cu, all N, all C of the four pyridinic units; the two Cu, all N; the two Cu; all N; all azidic N. The first subset reproduces almost exactly the total spin density (not shown) in the selected plane, while, except for the subset of the Cu atoms, the sources of the remaining individual subset of atoms are unable to explain, even qualitatively, the electron spin distribution around them (adapted with permission from Figure 5 of Ref. 59. Copyright 2018 Wiley-VCH Verlag 15530 GmbH&Co. KGa). Partial spin density reconstructions for the various atomic subsets largely depend on the wave-function method, which reveals the effect of the chosen method on both spin delocalization and polarization mechanisms (for more details, see Ref. 59 and Ref. 65).

64 | Á. Martín Pendás and C. Gatti partial densities represent a rigorous cause-effect decomposition of ρ that is fully independent from the adopted set of functions expressing ρ. Partial SF reconstructions may be clearly defined for any other scalar field f and have been found to be particularly useful for f = s [59, 65]. They immediately reveal whether the SF contribution from a given atom is capable or not to account for the qualitative features of its own basin electron spin distribution (Figure 3.7).

3.4 The AIMAll software There are several software packages that perform QTAIM or QTAIM-like analyses in molecules that in one way or another owe their structure to AIMPAC [67], a set of codes from the laboratory of late Professor R. F. W. Bader. Some of them have not been updated for many years, like the AIM2000 inititative [68], while others are development versions coded in research groups for their everyday work (e. g., the PROMOLDEN code [69]). Only a couple of them can be recommended for general purposes: MULTIWFN [70], and AIMAll [12]. Due to its versatility and cross-platform character, only the AIMAll code will be summarized in this chapter. We should notice that there are many other codes that are able to offer QTAIM properties. For instance, basic BCP densities and Laplacian properties, as well as their natural atomic orbital (NAO) counterparts, are available (through the NBCP keyword) in the natural bond orbital (NBO) program. The electron density SF analysis for molecular systems is implemented in the MULTIWFN [70] and AIMALL [12] general purpose packages. The XD-2006 [71] and CRITIC2 [72] (linked to Quantum Expresso) packages perform electron density SF analysis for periodic systems, using X-ray experimentally derived and first principle electron densities, respectively. Early versions of the TOPOND code [73] perform SF analysis of the electron density for molecules, polymers, slabs and crystals, but this analysis has not yet been implemented in the most recent TOPOND version fully embedded in CRYSTAL 17 code [74]. A complete in vacuo SF analysis for the electron density, the electron spin density and the molecular electrostatic potential, including also the evaluation of partial SF reconstructions and the gathering of local SF contributions due to any selected atomic subset is performed by a set of codes [75] originating from the Richard Bader’s AIMPAC code and written as development versions by C. Gatti for his everyday work. The basic problem which any QTAIM code faces is two-fold: on the one hand finding the derivatives of the electron density or of any other selected scalar field whose topology is needed and, on the other, integrating operator densities over atomic domains: 3D integrals in the case of one-electron operators and 6D integrals when twoelectron densities are selected. In molecules, the aim of this chapter, most electronic structure calculations are done with localized Gaussian basis sets, so that all necessary derivatives can be obtained analytically. This is not always the case in condensed

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matter, where specialized numerical grid techniques to access the appropriate derivatives have been devised [76, 77]. It is therefore integrating over basins that gives rise to difficulties since, unfortunately, the zero-flux condition is a nonanalytical constraint. No local expression allows us to evaluate whether a point in space lies at a separatrix of a gradient field or not. This leads to expensive numerical integrations, the cornerstone of any QTAIM integration. We will not delve into the specific algorithms that have been devised to tackle this problem. As a rule of thumb, if only 3D integrations are needed a QTAIM job can be usually done for a standard size molecule in a desktop computer. When 6D integrations are necessary, large molecules may need supercomputer facilities. AIMAll has both a GUI that allows the user to run interactively a small calculation, as well as command line options for noninteractive sessions. Both commercial and trial versions exist, available at http://aim.tkgristmill.com/.

3.4.1 AIMAll components and capabilities AIMAll is a cross-plattform code (Linux, Windows, OSX) code to perform QTAIM analysis from electronic structure calculations performed with Gaussian basis primitives. It is efficient and fast, mostly automatic, with simultaneous calculation and visualization capabilities of the topology of the electron density and several other scalar fields. It is fully parallelized in a shared memory environment and computes a large number of properties, both local, at critical points, and integrated. It is composed of a number of independent codes, fully integrated among themselves: – AIMQB: The primary AIMAll program. It automatically runs, with various options, a full QTAIM analysis using the programs AIMEXT, AIMINT and AIMSUM. The user can launch AIMQB directly or from a shortcut or from the AIMStudio graphical interface, or dragging-and-dropping a .wfn, .wfx or fchk file onto it. Its commandline usage is: aimqb [options] [wfnfile | wfxfile | fchkfile] – AIMUTIL: Not used directly – AIMEXT: Runs the topological analysis (and builds 2D and 3D grid files for futher use). Command-line usage: aimext [-progress] [-wsp] [wfnfile | wfxfile] [-input ...] – AIMINT: Computes atomic properties. Command-line usage: aimint [-nproc=...] [-wstat] [inpfile wfnfile | wfxfile] – AIMSUM: Generates summary file. – AIMSTUDIO: The graphical suite. AIMAll reads descriptions of the electronic structure (nuclear coordinates, basis functions, molecular orbitals, etc.) from .wfn, .wfx or fchk files, which are produced by a variety of general purpose programs. The GAUSSIAN [8] suite invokes the writing a .wfn or .wfx file by using the keyword output=wfn(wfx), or by reading directly the .fch

66 | Á. Martín Pendás and C. Gatti or .fchk checkpoint file. In the GAMESS code [9], a .wfn file is written with the keyword aimpac=.t., and in ORCA [10] by using !AIM.

3.4.2 Running a basic AIMAll calculation A standard, interactive AIMAll session is run by invoking the AIMstudio code. One or several wavefunction (.wfn, .wfx) files are then selected to be computed through the Run AIMQB dialog, as shown in Figure 3.8. Computational conditions and integration options are also selected at this step. Default values exist in all cases, which have to be modified in difficult or particular cases. As shown in the figure, one can select the basin integration method and the fineness of the integration grid, how many atoms are integrated, whether atomic Ehrenfest or Hellmann–Feynman forces are evaluated, or whether interatomic surfaces are computed. The user decides also at this preliminary step the complexity of the energetic analysis to be performed. As explained, a full IQA calculation might be rather time consuming. All properties are identified through acronyms that are easy to decode: for instance, T(A) is the atomic kinetic energy. There is an online extensive documentation with a description of the meaning of all these acronyms. We will explore here how to compute and visualize the most common topological tasks in AIMAll. Once the AIMQB code completes its calculations, a number of text files (.sum, .sumviz) with the results of the calculations are generated that can be read within AIMStudio in order to manipulate the information there stored graphically, as indi-

Figure 3.8: Running AIMQB and selecting a wavefunction file as well as determining the general computation conditions in AIMAll.

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cated in the scheme below. In the case examined, a Hartree–Fock calculation on cyclopropanone (available in the AIMAll test folder) is loaded.

This loads the molecule in the GUI. From this moment on, the user can manipulate easily, navigating through the different dialogs. Notice that only the properties which have been actually computed through AIMQB will be available. Using arrow keys, the mouse to zoom in and out or to rotate the molecule, and the shift+mouse combination to translate adequately the molecule, the user selects the desired view, as shown below.

The computed bond paths as well as the CPs of the electron density are shown by default. Their presence and aspect can be fully controlled by selecting options in the

68 | Á. Martín Pendás and C. Gatti BCPs, RCPs and CCPs menus. They are, by default shown in green, red and blue, respectively. As shown, the C-C BCPs in cyclopropanone are relatively strained, differing considerably from straight lines. A large number of properties can be examined at CPs by selecting the properties dialog from the specific CP menu, as illustrated in the adamantane molecule.

If computation of the CPs of the Laplacian has been selected in AIMQB, its analysis is available through the LAPCPs menu. To that end, the user needs to load in the current window the .agpviz files that can be found in the atomicfiles folder that will have been created. Similarly, depending on the integration options activated when running AIMQB, several atomic properties will be available to be shown using the atoms menu. In the picture below, this is done for the water molecule. Atomic charges and atomic kinetic energies are shown. Notice, for instance, that the charges add up to about zero. A measure of the quality of the numerical integrations can be guessed by examining the integrated values of the Laplacian, which should ideally be zero for zero-flux surrounded regions.

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Localization and delocalization indices can also be depicted by activating the populations, pairs and localization subdialog. An example, again in the H2 O, shows how the total atomic population can be partitioned into its localized and delocalized parts. It is to be highlighted that the covalent bond order in a polar link like that in the O-H pair is considerably smaller than in a pure homopolar bond, tending to zero in the case of total charge transfer. Notice that the current version of the code does not compute pair densities beyond the single determinant approximation.

3.4.3 Loading vector maps, relief maps, contours and surfaces The AIMAll code can easily superimpose vector maps, contours of many scalar fields projected on planes, or even isosurfaces of those fields on top of the molecular graph. This is done by selecting the Contours, Relief Maps, Vector Maps or Isosurface dialogs. If a Contour map of the electron density on top of the water molecule graph is needed, a new 2D grid is selected in the Contour dialog. Among the possibilities shown, the user selects the electron density and the plane in which to plot the contours. Three points define the plane. Usually, it is reasonable to make this plane coincide with that defined by the position of three nuclei. In that case, the geometry of each of them can be obtained by clicking on the nucleus, right-clicking with the mouse and copying to the clipboard the coordinates, which are then pasted. Once the isocontours are built, a .g2dviz file will be created to be loaded in the current window. This will make the isocontours appear on the GUI. Clicking on any isoline will pop up its isovalue.

70 | Á. Martín Pendás and C. Gatti

Plotting an isosurface implies building a 3D grid (through the Isosurface dialog), selecting the scalar field and the isovalue, and likely selecting another field that will be mapped with colors on the isosurface. Transparency options exist that allow to still observe the molecule through the surface. A map of the electrostatic potential on top of a density isosurface together with the interatomic surfaces constructed in a previous step for the water molecule shows that the electrophilic regions (blue/positive electrostatic potential) lie almost fully within the H atoms QTAIM basins.

Finally, if a full IQA calculation has been requested, the user may examine either the summary file directly, or explore intra and interatomic contributions through the Atoms dialog. Notice that not every functional is available for IQA partitioning in AIMAll. For instance, doing a M06-2X optimization of the LiH molecule with an augcc-pVDZ basis set, the topological charge of the Li basin is +0.907 au. In the picture below, E self (called E_IQA_Intra in AIMAll) is shown for the two atoms, as well as the electrostatic component in the right subplot. The exchange-correlation interaction

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energy is much smaller than the electrostatic one, and the localization indices of both atoms, very close to two, indicate that we have a clear Li+ H – ionic system.

3.4.4 Using AIMAll for production purposes Powerful as the GUI is, in real production runs the code has to be invoked on a command line and executed in a batch or queueing environment. There is an extensive description on how to do that, including how to specify the several options of AIMQB that can be selected through the GUI at http://aim.tkgristmill.com/manual/aimqb/ aimqb.html The command line version is invoked as: – Windows: c:\aimall\aimqb -nogui file.wfn – OSX: /AIMAll/AIMQB.app/Contents/MacOS/aimqb -nogui file.wfn – Linux: /usr/local/AIMAll/aimqb.ish -nogui file.wfn

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Frisch MJ et al. Gaussian 03, Revision C.02. Wallingford, CT: Gaussian, Inc.; 2004. Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su SJ, Windus TL, Dupuis M, Montgomery JA. J Comput Chem. 1993;14:1347. Neese F. Wiley Interdiscip Rev Comput Mol Sci. 2011;2:73–8. Karlström G, Lindh R, Malmqvist P-Å, Roos BO, Ryde U, Veryazov V, Widmark P-O, Cossi M, Schimmelpfennig B, Neogrady P, Seijo L. Comput Mater Sci. 2003;28:222–39. Keith TA. The AIMAll program. 2015; The code is avalaible at http://aim.tkgristmill.com. Hirsch MW, Smale S. Ecuaciones diferenciales, sistemas din’amicos y ’algebra lineal. Alianza: Madrid; 1974. Morse M. Trans Am Math Soc. 1931;33:72–91. Gatti C, Macchi P, editors. Modern Charge-density Analysis. Dordrecht: Springer; 2012. Bader RFW, Bedall PM. J Chem Phys. 1972;56:3320. Smith VH, Price PF, Absar I. Isr J Chem. 1977;16:187. Kato WA. Commun Pure Appl Math. 1957;10:151. Martín Pendás A, Blanco MA, Costales A, Sánchez PM, Luaña V. Phys Rev Lett. 1999;83:1930–3. Cao WL, Bader RFW, Gatti C, MacDougall PJ. Chem Phys Lett. 1987;141:380. Cioslowski J, Liu G. Chem Phys Lett. 1997;277:299–305. Dye JL. Science 1990;247:663–8. El Bakouri O, Postils V, Garcia-Borràs M, Duran M, Luis JM, Calvello S, Soncini A, Matito E, Feixas F, Solà M. Chemistry – A European J. 2018;24:9853–9. Bader RFW, Slee TS, Cremer D, Kraka E. J Am Chem Soc. 1983;105:5061–8. Cremer D, Kraka E, Slee TS, Bader RFW, Lau CDH, Dang TTN, MacDougall PJ. J Am Chem Soc. 1983;105:5069–75. Bader RFW, Essén H. J Chem Phys. 1984;80:1943–60. Kohout M. The Chemical Bond II. Springer International Publishing; 2015. p. 119–68. Bader RFW, Gillespie RJ, MacDougall PJ. J Am Chem Soc. 1988;110:7329–36. Gillespie RJ. Molecular Geometry. London: Van Nostrand Reinhold; 1974. Bader RFW, Srebrenik S, Nguyen-Dang TT. J Chem Phys. 1978;69:3680. Srebrenik S, Bader RFW. J Chem Phys. 1974;61:2536. Srebrenik S, Bader RFW. J Chem Phys. 1975;63:3945. Anderson JSM, Ayers PW, Hernandez JIR. J Phys Chem A. 2010;114:8884–95. Bader RFW, Popelier PLA. Int J Quant Chem. 1993;45:189–207. Bader RFW, Dang p, Adv TT. Quantum Chem. 1981;14:3663. Bader RFW, Larouche A, Gatti C, Carroll MT, MacDougall PJ, Wiberg KB. J Chem Phys. 1987;87:1142–52. Cioslowski J, Liu G. J Chem Phys. 1996;105:8187. Cioslowski J, Liu G. J Chem Phys. 1999;110:1882. Daudel R. The Fundamentals of Theoretical Chemistry. Oxford: Pergamon Press; 1968. Aslangui C, Constanciel R, Daudel R, Kottis P. Adv Quantum Chem. 1972;6:93. Daudel R, Bader RFW, Stephens ME, Borett DS. Can J Chem. 1974;52:1310. Daudel R, Bader RFW, Stephens ME, Borrett DS. Can J Chem. 1974;52:1310. Bader RFW, Stephens ME. J Am Chem Soc. 1975;97:7391. Huggins ML, Mayer JE. J Chem Phys. 1933;1:643–6. Martín Pendás A, Blanco MA, Francisco E. J Chem Phys. 2004;120:4581. Martín Pendás A, Francisco E, Blanco MA. J Comput Chem. 2005;26:344. Blanco MA, Martín Pendás A, Francisco E. J Chem Theory Comput. 2005;1:1096. Francisco E, Martín Pendás A, Blanco MA. J Chem Theory Comput. 2006;2:90. Otero de la Roza A, DiLabio GA. Non-Covalent Interactions in Quantum Chemistry and Physics: Theory and Applications. 2017. Martín Pendás A, Francisco E. Phys Chem Chem Phys. 2018;20:21368–80.

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Bader RF, Gatti C. Chem Phys Lett. 1998;287:233–8. Gatti C. Struct Bond. 2012;147:193–286. Gatti C. Phys Scr. 2013;87:048102. Arfken G. Mathematical Methods for Physicists. 3rd ed. San Diego, California: Academic Press; 1985. Gatti C, Cargnoni F, Bertini L. J Comput Chem. 2003;24:422–36. Gatti C, Saleh G, Lo Presti L. Acta Crystallogr, Sect B, Struct Sci Crystal Engrg Mater. 2016;72:180–93. Thomsen MK, Gatti C, Overgaard J. Chemistry – A European J. 2018;24:4973–81. Gatti C, Orlando AM, Lo Presti L. Chem Sci. 2015;6:3845–52. Gatti C, Macetti G, Lo Presti L. Acta Crystallographica Section B Structural Science, Crystal Engineering and Materials. 2017;73:565–83. Peluso P, Gatti C, Dessì A, Dallocchio R, Weiss R, Aubert E, Pale P, Cossu S, Mamane V. J Chromatogr A. 2018;1567:119–29. Gatti C, Macetti G, Boyd RJ, Matta CF. J Comput Chem. 2018;39:1112–28. Gatti C, Bertini L. Acta Crystallogr A, Found Crystallogr. 2004;60:438–49. Gatti C, Lasi D. Faraday Discuss. 2007;135:55–78. Monza E, Gatti C, Lo Presti L, Ortoleva E. J Phys Chem A. 2011;115:12864–78. Macetti G, Lo Presti L, Gatti C. J Comput Chem. 2018;39:587–603. Gatti C, Dessì A, Dallocchio R, Mamane V, Cossu S, Weiss R, Pale P, Aubert E, Peluso P. Molecules. 2020;25:4409 (19 pages). Keith TA, Laidig KE, Krug P, Cheeseman JR, Bone RGA, Biegler-König FW, Duke JA, Tang T, Bader RFW. The AIMPAC95 programs. 1995. The code is avalaible at http://www.chemistry.mcmaster. ca/aimpac. Biegler-König F, Schönbohm J. J Comput Chem. 2002;23:1489–94. Martín Pendás A, Francisco E. A QTAIM/IQA code (Available from the authors upon request). Lu T, Chen F. J Comput Chem. 2012;33:580–92. Volkov A, Macchi P, Farrugia LJ, Gatti C, Mallinson P, Richter T, Koritsansky T. XD2006 – A Computer Program Package for Multipole Refinement, Topological Analysis of Charge Densities and Evaluation of Intermolecular Energies from Experimental and Theoretical Structure Factors. University at Buffalo, State University of New York, NY USA. 2006. http://xd.chem.buffalo.edu. de-la Roza AO, Johnson ER, Luaña V. Comput Phys Commun. 2014;185:1007–18. Gatti C. Acta Crystallogr, Sec A. 1996;52:C555. Dovesi R, Erba A, Orlando R, Zicovich-Wilson CM, Civalleri B, Maschio L, Rérat M, Casassa S, Baima J, Salustro S, Kirtman B. Wiley Interdiscip Rev Comput Mol Sci. 2018;8:e1360. Gatti C. AIMPAC modified to evaluate SF contributions (unpublished, available on request: [email protected]), Milano, 2018. Kohout M. DGRID, version 4.6. Otero-de-la Roza A, Blanco MA, Martín Pendás A, Luaña V. Comput Phys Commun. 2009.

Miroslav Kohout

4 Electron localizability indicator and bonding analysis with DGrid The bonding in molecules and solids can be analyzed in two distinct, though often complementary, ways. The bonding analysis in Hilbert space is explicitly referring to the orbitals from which the wavefunction is built. Here, the orbitals, respectively different orbital dependent expectation values, are manipulated and evaluated to gain insight into the bonding situation. Such analyses are very powerful and widely used. Together with the corresponding programs they are described in detail in other chapters of this book. The following sections of this chapter are dedicated to another approach where the bonding analysis is performed in real position (and sometimes momentum) space. With real space the usual 3-dimensional Cartesian space R3 with position vectors r ⃗ is meant. The N-electron wave function Ψ(x⃗1 , . . . , x⃗N ), with the electronic space-spin coordinates x⃗i = (ri⃗ , σi ) is an imaginary 4N-dimensional object (omitting the nuclear coordinates which are fixed in the Born–Oppenheimer approximation). As a first step, the N-matrix ΓN is created (now even 8N-dimensional): ΓN (x⃗1󸀠 , . . . , x⃗N󸀠 ; x⃗1 , . . . , x⃗N ) = Ψ∗ (x⃗1󸀠 , . . . , x⃗N󸀠 )Ψ(x⃗1 , . . . , x⃗N ).

(4.1)

Setting the primed coordinates to the unprimed ones results in a 4N-dimensional real function ΓN (x⃗1 , . . . , x⃗N ), the diagonal part of the N-matrix, which is connected to the probability that each of the electrons with spin σi is located in the corresponding volume element dri⃗ . For chosen set of spin values, we arrive in the 3N-dimensional real space. The integral of ΓN (x⃗1 , . . . , x⃗N ) over all electronic positions x⃗i followed by the summation over the spin up/down-values is normalized to 1, which means that all the electrons must be somewhere in the space. If we allow for all but one electron to be located somewhere in space, we obtain the 1-electron reduced density matrix (1-RDM): Γ1 (x⃗1󸀠 ; x⃗1 ) = N ∫ Ψ∗ (x⃗1󸀠 , x⃗2 , . . . , x⃗N )Ψ(x⃗1 , x⃗2 , . . . , x⃗N )dx⃗2 . . . , dx⃗N .

(4.2)

⃗ The diagonal part of γσ (r 󸀠⃗ ; r)⃗ is the σ-spin Γ1 (x⃗1󸀠 ; x⃗1 ) is sometimes written as γσ (r 󸀠⃗ ; r). ⃗ It yields N-times the probability to locate a σ-spin electron in electron density ρσ (r). the volume element dr ⃗ with all remaining electrons somewhere in space. It should be mentioned that it describes neither the electron number 1, nor another specific electron. The electrons are indistinguishable; thus, the (antisymmetric) wave func-

Miroslav Kohout, Max-Planck-Institut für Chemische Physik fester Stoffe, Nöthnitzer Straße 40, 01187 Dresden, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110660074-004

76 | M. Kohout tion is given by Slater determinants, i. e., each electronic coordinate is successively attributed to each orbital. The electron density ρ(r)⃗ = ρα (r)⃗ + ρβ (r)⃗ is normalized to the total number of electrons N. If we allow all but two electrons to be located somewhere in space, the reduction analogous to equation (4.2) (but with the factor N(N−1) instead of N) yields 2 Γ2 (x⃗1󸀠 , x⃗2󸀠 ; x⃗1 , x⃗2 ), the 2-RDM of which the diagonal part Γ2 (x⃗1 , x⃗2 ) for the spin-pair αα gives the same-spin pair density ραα 2 (r1⃗ , r2⃗ ) (similarly for the opposite-spin pairs). The ββ αβ βα αα total pair density ρ2 = ρ2 + ρ2 + ρ2 + ρ2 is normalized to the number of electron N(N−1) N(N−1) pairs 2 . Again, it gives 2 times the probability to find an electron in volume element dr1⃗ and another one in dr2⃗ . But, it does not operate explicitly on electrons number 1 and 2. A real-space bonding analysis based on electron density only, can be evaluated as well using experimentally determined electron densities. But in general, going beyond the scope of electron density, the wave function of the system must be determined by an appropriate quantum mechanical (QM) package before a real space analysis begins. The necessary density matrices or densities need to be evaluated from the orbitals. In contrast to the bonding analysis methods based on a specific representation of orbitals, the methods based on density matrices are invariant with respect to the unitary transformation of orbitals. Thus, using real space indicators the explicit reference to orbitals is in principle not needed and only the (invariant) value at the examined position is taken into account. Of course, clever chosen combination of real space indicators and an orbital based approach can form powerful tool to gain insight into the bonding situation. In the following sections, the usage of the DGrid program [1] will be described with particular focus on the evaluation utilizing the wavefunction from the ADF package. A detailed description of all the possibilities offered by DGrid can be found in the latest DGrid manual. Bear in mind that DGrid does not perform any data visualization. For this, an external program is needed.

4.1 Wavefunction file To analyze given molecule or a solid-state system with the DGrid package [1], first the corresponding wavefunction must be available in a format suitable for DGrid. As the QM packages computing the wavefunction have different ways how to output the data, DGrid has an interface to several widely used programs, like Gaussian, GAMESS, Molpro, ADF or the solid-state code FHI-aims. The first three mentioned packages use Gauss-type of orbitals (GTOs) in the wavefunction, whereas ADF utilizes Slater-type of orbitals (STOs) and FHI-aims computes with numerical atomic orbitals (NAOs). Sometimes the wavefunction data are written out in specific package format, like is the case for the check-point file or TAPE21 produced by Gaussian and ADF, respectively,

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whereas GAMESS writes the wavefunction data directly to the output file. Molpro and other programs use the so-called Molden format. In case of the FHI-aims program, our group included the necessary output routines into the FHI-aims code. DGrid converts the QM package data and writes its own wavefunction file in a specific format. The format follows the same scheme for all the different wavefunction representations. It accommodates the usage of GTOs, STOs, NAOs, molecular or crystal orbitals or even data from a configuration interaction (CI) for the evaluation of highly correlated density matrices. As an example, let us analyze the Mn2 (CO)10 molecule. The relativistic ZORA calculation was performed with the ADF package employing the TZP basis set and the PBE functional. ADF produces the binary file TAPE21 which has to be converted to an ASCII file. This is done with the ADF utility dmpkf (for details cf. the ADF manual): dmpkf TAPE21 > Mn2CO10_TZP_PBE.kf Then the resulting file Mn2CO10_TZP_PBE.kf is converted to the DGrid wavefunction file: dgrid Mn2CO10_TZP_PBE.kf In the above command, it is assumed that the DGrid executable is know to the computer system under the name dgrid (check the DGrid manual). DGrid writes the wavefunction data to the ASCII file Mn2CO10_TZP_PBE.adf (the conversion of data from other QM packages yields filenames with corresponding extensions, e. g., “gms” for GAMESS wavefunctions). In some cases, DGrid produces a new wavefunction file, for instance, when domain natural orbitals are utilized; cf. Section 4.4.2. With the wavefunction file Mn2CO10_TZP_PBE.adf at hand, the analysis of the molecule can be performed.

4.2 Grid for electron density The bonding analysis with DGrid is based on the evaluation of field data generated on an equidistant grid. DGrid is capable to calculate around 50 different properties. As an example, let us generate the values of the electron density ρ(r)⃗ on a grid enclosing the ρ(r)⃗ = 0.001 bohr−3 isosurface with the 0.05 bohr spacing between the grid points (termed a 0.05-mesh), using the wavefunction file Mn2CO10_TZP_PBE.adf generated in the previous section. All the information needed for this task is supplied in the socalled control file saved in a file named, for instance, rho.inp:

78 | M. Kohout control file 4.1. ::Mn2CO10 TZP/PBE :------------------------------------wfn_1 = Mn2CO10_TZP_PBE.adf compute :--------------------------------using wfn_1 rho mesh=0.05 rho=0.001 parallel :--------------------------------compute_end

The control file has the following structure. The first readable information must be a title line starting with two colons. Then the data assignments follow, here the file Mn2CO10_TZP_PBE.adf is connected with the descriptor “wfn_1.” Empty lines are not parsed as well as lines starting with single colon (cf, “:—–”). Additionally, each line is usually read format free, i. e., blanks can be introduced as needed. The keywords are read case-insensitive. In the “compute” block (that must be closed by the “compute_end” command) the keyword “using” is obligatory. It states that the file pointed to by the “wfn_1” descriptor is used to compute the electron density (observe the command “rho” given separately in a single line). All necessary grid parameters are determined by DGrid, which interprets the instructions provided in the command line “mesh=0.05 rho=0.001 parallel” of the “compute” block. In this way, the generated grid box will have such extent as to accommodate the 0.001 bohr−3 isodensity surface, with grid points equidistantly spaced by 0.05 bohr. The instruction “parallel” tells DGrid to generate a grid box oriented parallel with the Cartesian axes. To run the program type the following instruction on the command line (the “&” forces the job to run in background): dgrid rho.inp & When the job is finished, the log file rho.inp.DG_LOG gives some information about the run. The density grid is written to the file Mn2CO10_TZP_PBE.adf.rho_r. Observe the automatically generated file-name with “.rho_r” appended, where “rho” stands for the property and “_r” for the position coordinate r ⃗ (in contrast to “_p,” when computed in momentum space). Beside the density values, this file contains also the atomic coordinates and the information about the grid dimensions and number of points. The inspection of the grid-file reveals that the density was computed at 338 × 338 × 465 evenly spaced points. To visualize the grid data, an external visualization tool, as well as suitable interface to read the DGrid files, is needed. The figures shown in this chapter were created with the Avizo package [2]. For instance, the left diagram in Figure 4.1 showing the

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Figure 4.1: Electron density for the Mn2 (CO)10 molecule. Left: isosurfaces with the isovalue 0.5 bohr−3 ; right: molecular graph with red cubes marking the saddle points and spheres for atomic positions (blue oxygen, black carbon, green manganese).

0.5-localization domains of electron density (i. e., the isosurfaces for the 0.5 bohr−3 isovalue) were created using the file Mn2CO10_TZP_PBE.adf.rho_r from the above DGrid run. It can be seen that each atom is enclosed in a separate domain. The domains are called irreducible, because further increase of the isovalue does not split the domains into more parts. The electron density is a scalar field, the topology of which is described by the ⃗ r)⃗ and ∇∇ρ(r)⃗ (the Hessian matrix), respectively. There first and second derivatives ∇ρ( ⃗ rc⃗ ) = 0 which are called are special prominent stationary positions rc⃗ where ∇ρ( critical points. Depending on the principal curvatures (given by diagonalization of the Hessian matrix), the critical point can be either of minimum, maximum, saddle or ring point of the field (for more details, cf. Chapter 3). Utilizing the grid field Mn2CO10_TZP_PBE.adf.rho_r computed with the control file 4.1 (rho.inp), the critical points can be searched with DGrid using the following control file: control file 4.2. ::Mn2CO10 TZP/PBE :------------------------------------field_1 = Mn2CO10_TZP_PBE.adf.rho_r search :--------------------------------all_cps using field_1 crop using field_1 0.001 save cps_1 :--------------------------------search_end icl_graph :--------------------------------using cps_1 saddles :--------------------------------icl_graph_end

80 | M. Kohout The keyword “all_cps” in the “search” block (closed by the command “search_end”) specifies that all critical points, i. e., the maxima, minima, saddles and ring critical points will be searched for in the grid field. The keyword “using” is obligatory, stating in which field (here “field_1”) the search will be performed. The specification of the “field_1” descriptor must precede the “search” block. To avoid the search in regions of very low gradient the domain outside the 0.001 bohr−3 isodensity surface will be excluded by the command “crop using field_1 0.001.” For convenience, there is a second block, delimited by the keywords “icl_graph” and “icl_graph_end,” included in the control file that generates data for visualization of the molecular graph [3] with an external program (“icl” stands for interconnection line [4]). Within the “icl_graph” block, the critical points are utilized that were created in the preceding “search” block (cf. there the command “save cps_1”). The resulting interconnection lines (in case of electron density termed “bond paths”) run from saddle points to the attractors (cf. the keyword “saddles”). Executing the above control file, the “search” block yields data for the critical points found in the grid region. They are written to a file in specific format suitable for further processing by DGrid and connected through the descriptor “cps_1” with the file Mn2CO10_TZP_PBE.adf.rho_r.cps (the name of which is generated automatically, thus need not to be given explicitly). The data from this file are used in the following “icl_graph” block to create the molecular graph. The “icl_graph” block could as well be executed using a separate control file. However, in this case the assignment “cps_1= Mn2CO10_TZP_PBE.adf.rho_r.cps” would be obligatory. The “icl_graph” block delivers the file Mn2CO10_TZP_PBE.adf.rho_r.cps.graph.str that can be visualized as shown in the right diagram of Figure 4.1. The attractors are not shown in the diagram since they coincide with the nuclear positions. Each carbon atom is connected via two saddle points with its neighbors (C–O and C–Mn contacts). There is a saddle point between the Mn atoms. The molecular graph points to possible bonding interactions. However, notice that already the molecular graph for the promolecular density is almost identical to the one for the optimized density [5]. Additionally, the log-file Mn2CO10_TZP_PBE.adf.rho_r.cps.LOG is created. Besides the coordinates and the density values it also contains the density Laplacian and corresponding principal curvatures for the critical points. The log-file indicates that 22 attractors were found at the atomic positions (thus, the curvatures are not given) together with 21 saddle points, which, in case of the electron density, are also termed bond critical points (BCP). No ring points or minima are present in the region. This yields correctly 1 for the Euler characteristic. Below the log-file section with the data for the saddle points is shown:

4 Electron localizability indicator and bonding analysis with DGrid | 81

output 4.1.

/=========================\ | saddles in region= 21 | +-------------------------+ # x y z value Laplacian curvature ellip |V|/G H/rho +-------------------------------------------------------------------------------------------------------------------+ | 1 3.9235 -1.6252 2.5425 0.4616 -0.1503 2.4516 -1.3034 -1.2984 0.00 2.047 -1.799 | | 2 -1.6252 3.9235 -2.5425 0.4616 -0.1503 -1.3034 2.4516 -1.2984 0.00 2.047 -1.799 | | 3 1.6252 -3.9235 -2.5425 0.4616 -0.1503 -1.3034 2.4516 -1.2984 0.00 2.047 -1.799 | | 4 -3.9235 1.6252 2.5425 0.4616 -0.1503 2.4516 -1.3034 -1.2984 0.00 2.047 -1.799 | | 5 3.9235 1.6252 -2.5425 0.4616 -0.1503 2.4516 -1.3034 -1.2984 0.00 2.047 -1.799 | | 6 1.6252 3.9235 2.5425 0.4616 -0.1503 -1.3034 2.4516 -1.2984 0.00 2.047 -1.799 | | 7 -3.9235 -1.6252 -2.5425 0.4616 -0.1503 2.4516 -1.3034 -1.2984 0.00 2.047 -1.799 | | 8 -1.6252 -3.9235 2.5425 0.4616 -0.1503 -1.3034 2.4516 -1.2984 0.00 2.047 -1.799 | | 9 0.0000 0.0000 -6.9598 0.4572 -0.1694 -1.2749 -1.2749 2.3804 0.00 2.055 -1.789 | | 10 0.0000 0.0000 6.9598 0.4572 -0.1694 -1.2749 -1.2749 2.3804 0.00 2.055 -1.789 | | 11 -0.0000 -0.0000 4.5309 0.1467 0.4952 -0.1634 -0.1634 0.8220 0.00 1.337 -0.429 | | 12 -0.0000 -0.0000 -4.5309 0.1467 0.4952 -0.1634 -0.1634 0.8220 0.00 1.337 -0.429 | | 13 -1.6153 -0.6691 -2.6472 0.1287 0.4800 0.7516 -0.1377 -0.1339 0.03 1.281 -0.365 | | 14 -0.6691 -1.6153 2.6472 0.1287 0.4800 -0.1377 0.7516 -0.1339 0.03 1.281 -0.365 | | 15 0.6691 -1.6153 -2.6472 0.1287 0.4800 -0.1377 0.7516 -0.1339 0.03 1.281 -0.365 | | 16 -1.6153 0.6691 2.6472 0.1287 0.4800 0.7516 -0.1377 -0.1339 0.03 1.281 -0.365 | | 17 1.6153 -0.6691 2.6472 0.1287 0.4800 0.7516 -0.1377 -0.1339 0.03 1.281 -0.365 | | 18 -0.6691 1.6153 -2.6472 0.1287 0.4800 -0.1377 0.7516 -0.1339 0.03 1.281 -0.365 | | 19 1.6153 0.6691 -2.6472 0.1287 0.4800 0.7516 -0.1377 -0.1339 0.03 1.281 -0.365 | | 20 0.6691 1.6153 2.6472 0.1287 0.4800 -0.1377 0.7516 -0.1339 0.03 1.281 -0.365 | | 21 -0.0000 -0.0000 -0.0000 0.0250 0.0116 -0.0091 -0.0091 0.0298 0.00 1.594 -0.169 | +-------------------------------------------------------------------------------------------------------------------+

Observe that for each saddle point the principal curvatures exhibit two negative and one positive curvature. The first 8 saddle points correspond to BCPs between the carbon and oxygen atoms of the equatorial CO ligands, while the next two represent the C–O BCP for the axial ligands (with the CO bond distance slightly longer than for the equatorial groups; for more details, [see 6]). The electron density at those BCPs is relatively high and the density Laplacian is negative, i. e., showing charge concentration which, in case of first row atoms, signals covalent bond [3]. The following 10 BCPs, describing the axial and equatorial Mn–C interactions, exhibit a positive density Laplacian, i. e., charge depletion. Since one of the bonding partners is a metal atom (Mn), the charge depletion does not necessarily point to an ionic interaction. Also the last BCP, reflecting the Mn–Mn interaction, shows positive ∇2 ρ value close to zero, thus expressing the pronounced flatness of the electron density between the metals. For classification of the Mn–C and Mn-Mn interactions two other descriptors given in the log-file output 4.1 are used: (i) the ratio |V|/G of the (negative) local potential energy density V(r)⃗ = 1/4∇2 ρ(r)⃗ − 2G(r)⃗ and the positive definite kinetic energy density G(r)⃗ = 1/2∇⃗ ∇⃗ 󸀠 Γ(r,⃗ r 󸀠⃗ )|r=⃗ r󸀠⃗ and (ii) the local energy density ⃗ The ratio |V|/G > 1 indicates stabilization of the interaction, in H(r)⃗ = V(r)⃗ + G(r). which case H(r)⃗ < 0 [7, 8]. As can be seen from the output 4.1, this is the case for all the BCPs for the Mn2 (CO)10 molecule, although with clearly higher values for the C–O interactions (the BCPs number 1–10). To enable the comparison of the energy values H(r)⃗ associated with different bonds, the bond degree parameter H/ρ can be used [9]. With this, the Mn–Mn interaction (BCP number 21) can be classified as a weak covalent bond.

82 | M. Kohout

4.2.1 Comparison between the electron density Laplacian and the one-electron potential The density Laplacian ∇2 ρ(r)⃗ was introduced to the bonding analysis for several reasons. Let us take a wavefunction represented by a single determinant built from orbitals ϕi . Then the evaluation of the kinetic energy density t(r)⃗ in Schrödinger form, i. e., t(r)⃗ = −1/2 ∑i ψ∗i ∇2 ψi , provides the expression: t(r)⃗ =

1 1 ⃗ i (r)| ⃗ 2 − ∇2 ρ(r), ⃗ ∑ |∇ψ 2 i 4

(4.3)

⃗ i (r)| ⃗ 2 . The nonwith the positive definite kinetic energy density G(r)⃗ = 1/2 ∑i |∇ψ classical kinetic energy density part, described by the density Laplacian, can be both positive and negative. Of course, the nonclassical part integrates to zero over the total space. Any suitable space partitioning yielding domains in which ∇2 ρ(r)⃗ integrates to zero would be very appealing for bonding analysis. This leads to QTAIM basins [3]. Additionally, utilizing the virial ratio, connection between the density Laplacian and the local potential energy can be established [7, 8]; see the previous section. Even more, it was found that −∇2 ρ(r)⃗ is capable to describe the atomic shell structure by a sequence of positive (charge concentration) and negative (charge depletion) regions, each pair representing an atomic shell in real space [10]. However, later the evaluation of the shell structure of heavy atoms (starting with the Ca atom) indicated that the valence shell was not always resolved by a separate concentration/depletion pair [11, 12]. This is of course an inconvenient situation when the shell structure is used for the bonding analysis. Instead of the density Laplacian another indicator based on the electron density and its derivatives can be utilized, namely the one-electron potential (OEP): 2

2 1 ∇ √ρ 1 ∇2 ρ 1 ∇ρ = − ( ). 2 √ρ 4 ρ 8 ρ

(4.4)

OEP describes the motion of a single electron after averaging over the motion of the remaining electrons. The negative of OEP can be regarded as the local kinetic energy. Negative OEP marks regions where the electron is classically allowed [13]. In case of free atoms, for each atomic shell a maximum of -OEP is found [12, 14]. This is true even for heavy atoms. Additionally, in contrast to the density Laplacian, atomic shell bounded by minima of -OEP shows reasonable shell occupations [14]. Let us compare the atomic shell structure for the Mn atom using the (high spin) wavefunction Mn_TZP_PBE.adf computed with the ADF program and the following control file oep.inp:

4 Electron localizability indicator and bonding analysis with DGrid | 83

control file 4.3. ::Mn TZP/PBE :------------------------------------wfn_1 = Mn_TZP_PBE.adf compute :--------------------------------using wfn_1 rho OEP

Laplacian

format= grace origin I J K +--------------------------------------------------------------------+ | X | 0.00000000 | 0.00000000 0.00000000 0.00000000 | | Y | 0.00000000 | 0.00000000 0.00000000 0.00000000 | | Z | 0.00000000 | 0.00000000 0.00000000 4.00000000 | +--------------------------------------------------------------------+ | points | 1 1 4001 | +--------------------------------------------------------------------+ :--------------------------------compute_end

Now, two grid fields are computed in a single job, namely the density Laplacian (keyword “rho Laplacian”) and the one-electron potential (keyword “OEP”). Observe that each field must be given on separate line (however, both upper and lower case can be used). The grid is defined by an origin (set to 0,0,0) and 3 vectors I,J,K. In the above control file, only the K-vector is of nonzero length of 4 bohr with 4001 points evaluated. Instead of the default DGrid format the fields are written in the format for the Xmgrace visualization tool (cf. the command “format= grace”). The fields will be saved in the files Mn_TZP_PBE.adf.lap_rho_r and Mn_TZP_PBE.adf.oep_r, respectively. The left diagram of Figure 4.2 shows that the density Laplacian resolves only 3 atomic shells for the Mn atom (there are 3 pairs of positive and negative values, see the alternating red colored blocks; additionally, the red number marks the third shell charge concentration). The -OEP, shown in the same diagram by the black solid line, exhibits 4 maxima marked by black numbers. Here, an atomic shell is assumed to be a region bounded by -OEP minima. Notice that the valence shell (the fourth shell) is completely in the region of negative (classically forbidden) local kinetic energy. However, the region indicates a shell because otherwise the solid line would steadily decay to the ionization potential without exhibiting the minimum found at 1.57 bohr from the nucleus (cf. the black dashed line for the hypothetical situation of missing OEP valence shell). The integration of the electron density over the shells of the Mn atom yields 1.9, 7.9, 11.8 and 3.4 electrons for the shells based on -OEP, which is similar to the occupations of the ELI-D shells (cf. Section 4.5) that amounts to 2.2, 8.1, 12.3 and

84 | M. Kohout

Figure 4.2: Comparison between the density Laplacian and the one-electron potential (OEP). Left: Shell structure for the Mn atom. −∇2 ρ in red color, black solid line for the -OEP, black dashed line simulates missing OEP valence shell situation. The outer shell numbers are given; right: values along the z-axis of Mn2 (CO)10 starting from the Mn–Mn midpoint. The atomic positions are marked by symbols.

2.4 electrons, respectively. For comparison, the 3 atomic shell of Mn based on −∇2 ρ(r)⃗ are occupied by 3.2, 8.6 and 13.2 electrons (the “ideal” shell occupations reads 2, 8, 13 and 2 electrons). We can proceed with the comparison by computing ∇2 ρ(r)⃗ and the OEP for the Mn2 (CO)10 complex along the Mn–Mn axis (the z-axis in case of the wavefunction used here). This can be accomplished with the control file 4.3 using the proper wavefunction file and increasing the size of the K-vector in the grid definition to 10 bohr. As shown in the right diagram of Figure 4.2 the computed line starts at the Mn–Mn midpoint, which is a BCP with density Laplacian slightly above zero (cf. the last data line in output 4.1). ⃗ is similar to the shape given by The contour of the red blocks, displaying ∇2 ρ(r), the black solid line showing the -OEP. However, there are two significant differences. First, the shell occupations differ strongly. For instance, the core shells of the carbon and oxygen atoms are populated in case of ∇2 ρ(r)⃗ by 2.5 and 2.7 electrons, respectively, whereas it amounts to 1.9 for shells given by the -OEP boundaries. However, the main difference is in the region between the Mn atoms. There the -OEP shows a separate valence shell region indicating a bond. In contrast, there is no valence shell of the Mn atoms that could be used for the bonding analysis (one is using the structure of the ∇2 ρ(r)⃗ penultimate shell instead).

4.3 QTAIM basins Utilizing the electron density grid-file, the corresponding (atomic) basins can be determined. A basin is defined as the manifold of all trajectories, given by the field gradient, terminating at its attractor (field maximum, for details cf. Chapter 3). Usually, the density maxima are found only at atomic nuclei (but nonnuclear density attractors can occur [see 15]). In the quantum theory of atoms in molecules, the density basins

4 Electron localizability indicator and bonding analysis with DGrid | 85

are termed QTAIM basins [3]. The control file for the QTAIM-basin determination reads as follows: control file 4.4. ::Mn2CO10 TZP/PBE :------------------------------------field_1 = Mn2CO10_TZP_PBE.adf.rho_r basin :--------------------using field_1 top= 0.5 crop using field_1 0.001 :--------------------basin_end

The name of the grid-file with the electron density is known to DGrid through the assignment “field_1=Mn2CO10_TZP_PBE.adf.rho_r,” In the “basin” block the keyword “using” is obligatory (the number following the descriptors “field” depends on file assignments preceeding the “basin” block). The assignment “top=0.5” instructs DGrid to start the basin determination from regions enclosed by the 0.5-localization domains (otherwise it starts from the attractors found). The impact of the “top” command can be demonstrated by choosing for instance “top=0.2,” i. e., starting from the 0.2-localization domains. As can be seen from the left diagram of Figure 4.3, the isosurfaces now enclose the CO groups. Such domains are called reducible, because an increase of the isovalue will split the domains into parts; cf. Figure 4.1. After the basin search, there would be no separate basins for the C and O atoms. Instead, 10 CO superbasins [16] would be created (here the CO superbasin correspond to basin set [4] for the interconnection value 0.4616, respectively, 0.4572, cf. the BCP values in output 4.1). Reducing the “top” value further to 0.1 bohr−3 (cf. the right diagram of Figure 4.3 with the corresponding density isosurfaces) would result in two superbasins, each one for a separate Mn(CO)5 group.

Figure 4.3: Electron density isosurfaces for the Mn2 (CO)10 molecule. Left: isovalue 0.2 bohr−3 ; right: isovalue 0.1 bohr−3 .

86 | M. Kohout In principle, the oxygen basins of the Mn2 (CO)10 molecule extend to infinity. Of course, the electron density is computed only in limited spatial region given by the grid box. Thus, the oxygen basins are cut by the grid-box faces. This is somewhat inconvenient situation concerning both, the computational demand for basin determination (low ⃗ values) and unique definition of basin volumes. To avoid this, we crop the ρ and ∇ρ basins by a specific density isosurface. Such isosurface is called the molecular envelope [3]. Usually, one of the respective isovalues 10−3 , 10−4 , or 10−5 is used. The command “crop using field_1 0.001,” given on a single line, serves this purpose. It instructs DGrid to crop the basins by the 0.001 density isosurface determined using the data from the grid-file pointed to by the “field_1” descriptor. The units are not given as those are identical with the ones used in the wavefunction file. The resulting basins are written to a file with name derived from the name of the grid-file for which the basins were determined appending the extender “.bsn,” i. e., in our case Mn2CO10_TZP_PBE.adf.rho_r.bsn. The basins are coded in such way that the density value at each grid point is replaced by an integer number indicating to which basin the point belongs to. The QTAIM basins for the Mn2 (CO)10 molecule are shown in the left diagram of Figure 4.4. Due to the cropping with the isodensity surface the basins have defined volume. When discussing the volumes of the outer basins the cropping isovalue should be stated. For the inner basins, the cropping has negligible or even no influence; cf. for instance the QTAIM basins for the Mn atoms shown in the right diagram of Figure 4.4.

Figure 4.4: Mn2 (CO)10 molecule. Left: QTAIM basins cropped by 0.001-isodensity surface (blue oxygen, black carbon, green manganese); right: manganese QTAIM basins.

The log file (named Mn2CO10_TZP_PBE.adf.rho_r.bsn.LOG) is written during the job run. There the progress of the evaluation can be inspected. After successful determination the basin volumes and assignment of the basins to atoms is documented in the log file.

4 Electron localizability indicator and bonding analysis with DGrid | 87

Usually, the real space bonding analysis proceeds with the evaluation of basin population given by the integral of the electron density over the basin volumes. In DGrid, the integration is performed numerically. In the simplest case, i. e., for light atoms and fine grid mesh, the electron density at each basin point is multiplied by the voxel volume (which for our example grid equals 0.053 bohr3 ). The integration is accomplished using the following control file: control file 4.5. ::Mn2CO10 TZP/PBE :------------------------------------field_1 = Mn2CO10_TZP_PBE.adf.rho_r basin_1 = Mn2CO10_TZP_PBE.adf.rho_r.bsn integrate :--------------------using field_1 over basin_1 :--------------------integrate_end

The two commands (each in separate line) in the “integrate” block state that the grid values read from the file assigned to the “field_1” descriptor (the electron density) will be integrated over the basins coded in the file connected to the “basin_1” descriptor. The block is closed by the “integrate_end” keyword. The result of the integration is written to the output file Mn2CO10_TZP_PBE.adf.rho_r.bsn.rho_r_ITG. This is a long name, but one can immediately deduce from it that, based on an ADF calculation (“cf. the file-name ending .adf”), the electron density in position space (“.rho_r”; otherwise “.rho_p” for momentum density) was calculated and the corresponding basins (“.bsn”) determined. Subsequently, the electron density was integrated (“.rho_r_ITG”) over the basins. The total population of all basins amounts to 189.31 electrons, which can be compared to the 190 electrons given in the wavefunction file (bear in mind that the basins are cropped). If heavy atoms participate, then the precision of the core density integral can sometimes be improved by insertion of precomputed core populations (tabulated within DGrid). This is accomplished by the keyword “core_charge” included in the “integrate” block. In the case mentioned above, this leads to an increase of the total population to 189.81 electrons. This procedure is the only possibility for improvement in case of calculation using solid state programs without a DGrid interface to create the wavefunction data. For our example, the wavefunction file Mn2CO10_TZP_PBE.adf is available. Then, the integration precision can be improved by computing additional grid points utilizing a control file with the “refinement” block:

88 | M. Kohout control file 4.6. ::Mn2CO10 TZP/PBE :------------------------------------field_1 = Mn2CO10_TZP_PBE.adf.rho_r refinement :--------------------using field_1 precision= 0.1 :--------------------refinement_end

The procedure tests each grid point of the grid field assigned by the “field_1=” command and if necessary compute additional points until the precision specified by the command “precision=0.1” is reached (usually the precision is significantly below the one given in the block). The precision is given in units to which the field integrates, e. g., in case of the electron density, in electrons. The routine can be heavily time demanding. The log file Mn2CO10_TZP_PBE.adf.rho_r.rfn.LOG shows that around 22 million points were additionally computed for 72840 voxels. The result of the procedure is written to the file Mn2CO10_TZP_PBE.adf.rho_r.rfn where for each processed voxel the integral is given. This file is used in subsequent integration in the “integrate” block of the adapted control file: control file 4.7. ::Mn2CO10 TZP/PBE :------------------------------------rfn_1 = Mn2CO10_TZP_PBE.adf.rho_r.rfn basin_1= Mn2CO10_TZP_PBE.adf.rho_r.bsn integrate :--------------------using rfn_1 over basin_1 :--------------------integrate_end

The only difference to the control file 4.5 is that now the refinement file is assigned to the descriptor “rfn_1” and this descriptor used in the block (instead of the “field_1” descriptor). The result of the integration is written to the output file Mn2CO10_TZP_ PBE.adf.rho_r.bsn.rfn_ITG (see the DGrid manual for each table item in detail):

4 Electron localizability indicator and bonding analysis with DGrid | 89

output 4.2.

/=====================================\ | integrals for 72840 refined cells | +-------------------------------------+ +----------------------------------------------------------------+ | read from file Mn2CO10_TZP_PBE.adf.rho_r.rfn | | ==> | | 22335280 values were computed for inhomogeneity > 0.1000000 | +----------------------------------------------------------------+

BASIN VOLUME INTEGRAL MAXIMUM X Y Z ATOMS DIST +-----------------------------------------------------------------------------------------+ | 1 R 64.360 24.1648 2563.2748+ -0.020 -0.020 -2.800 Mn_2 0.03 | | 2 R 64.362 24.1649 2563.2748+ -0.020 -0.020 2.800 Mn_1 0.03 | | 3 R 69.575 5.1584 93.7169+ -1.320 3.230 -2.550 C_9 0.02 | | 4 R 69.585 5.1580 99.2254+ -3.220 -1.320 -2.550 C_8 0.02 | | 5 R 69.592 5.1590 103.7115+ 1.330 -3.220 -2.550 C_7 0.02 | | 6 R 69.596 5.1585 93.7169+ 3.230 -1.320 2.550 C_1 0.02 | | 7 R 69.602 5.1580 99.2254+ -1.320 -3.220 2.550 C_4 0.02 | | 8 R 69.612 5.1591 103.7115+ -3.220 1.330 2.550 C_3 0.02 | | 9 R 69.618 5.1585 96.9634+ 3.230 1.330 -2.550 C_6 0.02 | | 10 R 69.622 5.1585 96.9634+ 1.330 3.230 2.550 C_2 0.02 | | 11 R 74.742 5.1489 87.2561+ -0.020 -0.020 6.200 C_5 0.03 | | 12 R 74.809 5.1492 87.2561+ -0.020 -0.020 -6.200 C_10 0.03 | | 13 R 130.109 8.9564 250.8449+ 2.180 5.230 2.500 O_2 0.01 | | 14 R 130.123 8.9558 233.0473+ 2.180 -5.220 -2.500 O_7 0.02 | | 15 R 130.123 8.9558 233.0473+ -5.220 2.180 2.500 O_3 0.02 | | 16 R 130.124 8.9564 250.8449+ 5.230 2.180 -2.500 O_6 0.01 | | 17 R 130.132 8.9567 252.8805+ -2.170 -5.220 2.500 O_4 0.01 | | 18 R 130.134 8.9563 289.6270+ 5.230 -2.170 2.500 O_1 0.00 | | 19 R 130.138 8.9563 289.6270+ -2.170 5.230 -2.500 O_9 0.00 | | 20 R 130.140 8.9567 252.8805+ -5.220 -2.170 -2.500 O_8 0.01 | | 21 R 130.540 8.9595 179.3285+ -0.020 -0.020 -8.400 O_10 0.03 | | 22 R 130.540 8.9595 179.3285+ -0.020 -0.020 8.400 O_5 0.03 | +-----------------------------------------------------------------------------------------+ | TOTAL | 2137.178 189.4653 | +-----------------------------------------------------------------------------------------+ | DIFF | -4449.824 | +-----------------------------------------------------------------------------------------+

For each basin the volume (here in bohr3 ), the population, the maximal density value at a grid point (cf. the “+” behind the value) as well as the assignment to an atom is given. The total population within the cropped region amounts to 189.47 electrons. The populations of the QTAIM basins correspond to effective charges (the difference between the nuclear charge and basin population) of +0.84, +0.84 (+0.85), and −0.96 for the Mn, C (axial carbon in brackets), and O atoms, respectively. The relatively high effective QTAIM charges do not necessarily indicate an ionic compound. Bear in mind that the evaluation of the promolecular density grid (i. e., superposition of spherically averaged atomic densities using the same basis set) already yields effective charges of +0.74, +0.48 (0.49), and −0.57 (−0.55) for the Mn, C and O, respectively (axial atoms in brackets) [17]. The value of −4449.824 bohr3 in the “DIFF” row shows how much of the grid-box volume was excluded from the basin search due to the cropping. The cropping is also

90 | M. Kohout the reason why the total population does not reach 190 electrons. The population difference of 0.5 electrons applies mostly to the (outer) oxygen atoms, i. e., roughly 0.05 electrons per oxygen atom is missing. Larger grid box, for instance using the cropping with the 10−5 isodensity surface, would yield almost exactly the total population of the system.

4.4 Delocalization indices The basins partition the space into non-overlapping space filling regions (the cropping with an isosurface is just a convenient analysis technique). The integration of a single particle field over basins yields quantities that are local to the examined basins, like, for instance, the integration of the electron density ρ(r)⃗ yields the corresponding basin populations. It does not describe explicitly some specific connection between the regions. The situation changes when a 2-particle field is involved, like for instance the pair density ρ2 (r1⃗ , r2⃗ ) which, for convenience, can be written as the sum of a quasi-independent pair density ρ(r1⃗ )ρ(r2⃗ ) and the hole-part C(r1⃗ , r2⃗ ) of the pair density [18]: 1 ρ2 (r1⃗ , r2⃗ ) = [ρ(r1⃗ )ρ(r2⃗ ) + C(r1⃗ , r2⃗ )]. 2

(4.5)

Then each coordinate can be integrated over different basins. The integral of the quasi-independent part, which is given by the product of basin populations, does not deliver any explicit information about interactions. It is the integral of the hole-part C(r1⃗ , r2⃗ ), describing the electron sharing between the atomic basins, that is connected to a topological bond order [19–22]. In case of a single-determinantal wavefunction the hole part of the same-spin pair density (the opposite-spin part equals zero) can be written as occ

C(r1⃗ , r2⃗ ) = − ∑ ϕ∗j (r1⃗ )ϕi (r1⃗ ) ϕ∗i (r2⃗ )ϕj (r2⃗ ) i,j

(4.6)

with the sum running over the occupied molecular σ-spin orbitals ϕ. Let us take two basins A and B and integrate the same-spin hole-part over the basins. If both coordinates run over the basin A, then the magnitude of the resulting integral is termed the localization index λ(A) [21]: occ

λ(A) = ∑ SjiA SijA ,

(4.7)

⃗ j (r)d ⃗ r.⃗ SijA = ∫ ϕ∗i (r)ϕ

(4.8)

i,j

with the overlap integrals:

A

4 Electron localizability indicator and bonding analysis with DGrid | 91

If the first coordinate is integrated over the basin A and the second one over B (and vice versa) then the integral magnitude is termed the delocalization index δ(A, B) [21]: occ

occ

i,j

i,j

δ(A, B) = ∑ [SjiA SijB + SjiB SijA ] = 2 ∑ SjiA SijB .

(4.9)

The sum of half the delocalization indices δ(A, B) with all basins B ≠ A is termed the fluctuation σ 2 (A): σ 2 (A) =

1 ∑ δ(A, B). 2 B=A̸

(4.10)

There is an important sum rule concerning the basin population N(A) of the basin A: N(A) = λ(A) + σ 2 (A).

(4.11)

It should be mentioned that the localization λ(A) as well as the fluctuation σ 2 (A) are evaluated as pair-density integrals, thus yielding the number of electron pairs. The sum equals the basin population only formally. Actually, it yields the number of self-pairs (resp., the variance in average basin population, given in units of squared electrons [23]). However, the decision whether to speak about electrons or electron pairs normally does not substantially change the general results of the analysis.

4.4.1 Overlap integrals The basic ingredient to evaluate the delocalization indices (DIs) are the overlap integrals Sij computed for the orbitals from the examined wavefunction. For a singledeterminantal wavefunction built from n orbitals this means n(n + 1)/2 integrals Sij . In case of QTAIM basins the number of integrals can be reduced by evaluating the valence orbitals only. The core orbitals are almost completely inside the atomic basins, thus Sij = δij and no evaluation is needed. The integration of the orbital products over the basin volume; cf. equation (4.8), can be done analytically only for GTOs. Otherwise, numerical methods must be employed, which can be a very time demanding procedure. With DGrid the overlap integrals evaluation for a specific separate basin can be chosen. Then each basin job can be submitted to different processor and the corresponding calculations run in parallel. For instance, the following control file:

92 | M. Kohout control file 4.8. ::Mn2CO10 TZP/PBE :------------------------------------basin_1 = Mn2CO10_TZP_PBE.adf.rho_r.bsn wfn_1 = Mn2CO10_TZP_PBE.adf overlap :--------------------using wfn_1 over basin_1 select_orbitals ----------------: above= -4.0 ----------------: select_orbitals_end select_basins ----------------: 1 ----------------: select_basins_end precision= 0.001 :--------------------overlap_end

states that using the orbitals from the wavefunction file Mn2CO10_TZP_PBE.adf the overlap integrals over the first basin will be computed. The basin is read from the file Mn2CO10_TZP_PBE.adf.rho_r.bsn. The basin number is given in the subblock “select_basins” where more basins, separated by blanks, can be given (remember that strings terminated by a colon, cf. the string “—:”, are skipped). The basin number 1 corresponds to the basin for the Mn atom, see the output 4.2. Only occupied orbitals with energy above −4.0 hartree will be evaluated, as specified in the “select_orbitals” block. The last command “precision=0.001” in the “overlap” block states the precision of the numerical integration (0.001 electrons, which is precise enough for the DI evaluation). The block must be closed by the “overlap_end” keyword. When the job finishes, the overlap integrals are written to the file Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-1. The same procedure can be applied to all basins of interest. The localization/delocalization indices for the chosen basins are determined with the control file:

4 Electron localizability indicator and bonding analysis with DGrid | 93

control file 4.9. ::Mn2CO10 TZP/PBE :------------------------------------sij_1 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-1 sij_2 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-2 sij_3 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-3 sij_4 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-12 sij_5 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-19 sij_6 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-21 DI :--------------------using sij_1 using sij_2 using sij_3 using sij_4 using sij_5 using sij_6 :--------------------DI_end

which states in the “DI” block that 6 overlap files are used for the evaluation. The files contain the overlap matrices for the two Mn basins, as well as for basins of an equatorial (basin number 3 and 19) and the axial (basin number 12 and 21) CO group; cf. the output 4.2. In the output file Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-1.DI (see the output 4.3 below), the localization/delocalization indices can be found: output 4.3.

+------------------------+ | Localization indices | +------------------------+ basin descriptor Q sigma2 LI LIaa LIbb +---------------------------------------------------------------------------+ | 1 Mn_2 14.165 3.230 10.935 5.467 5.467 | | 2 Mn_1 14.165 3.230 10.935 5.467 5.467 | | 3 C_9 3.161 1.619 1.542 0.771 0.771 | | 12 C_10 3.151 1.646 1.506 0.753 0.753 | | 19 O_9 6.954 1.036 5.918 2.959 2.959 | | 21 O_10 6.957 1.027 5.931 2.965 2.965 | +---------------------------------------------------------------------------+ | 48.553 11.787 36.767 18.383 18.383 | +---------------------------------------------------------------------------+ +------------------------------------------------+ | Delocalization indices for total pair-matrix | +------------------------------------------------+ Basin 1 2 3 12 19 21 +--------------------------------------------------------------+ | 1 x 0.1444 0.4813 0.5646 0.0808 0.1053 | | 2 0.1444 x 0.0266 0.0066 0.0099 0.0021 | | 3 0.4813 0.0266 x 0.0480 0.8368 0.0090 | | 12 0.5646 0.0066 0.0480 x 0.0087 0.8259 | | 19 0.0808 0.0099 0.8368 0.0087 x 0.0021 | | 21 0.1053 0.0021 0.0090 0.8259 0.0021 x | +--------------------------------------------------------------+

94 | M. Kohout The localization data are compiled in the first part of the above output. Observe that the basin population “Q” does not include the core electrons (due to the energy window above −4.0 hartree selected in the control file 4.8). Thus, 10 and 2 electrons are missing for the Mn and the light atoms, respectively (compare with the full basin populations in output 4.2). The core electrons can be omitted, because they do not contribute to the electron sharing between QTAIM basins. Twice the values in the column “sigma2,” i. e., the fluctuation σ 2 (A) of equation (4.10), reflects the topological valency of the corresponding QTAIM atom [24]. For the examined complex, it shows the valency of around 2 for the oxygen atoms, around 3.2 for the carbon atoms and even 6.5 for the Mn atoms. The column “LI” tabulates the number of electrons λ(A) = N(A) − σ 2 (A) localized in the atomic basin A. The delocalization indices, which are especially interesting for the bonding analysis, are compiled in the second part of output 4.3. In the first column, the electron sharing between the basin number 1 (which is the atomic basin Mn_2; cf. the table with the localization indices) and the other basins is given. As can be seen, the basin number 1 shares 0.1444 electrons with the basin number 2 (the atomic basin Mn_1). Of course, the basin 2 shares the same with basin 1, cf. the first value in the second column. Thus, the topological bond order between the two manganese basins amount to δ(Mn_1, Mn_2) ≈ 0.29. The electron sharing points to covalency, because of the similarity of the hole-part integral (cf. equation (4.6)), with the exchange integrals (containing the division by |r1⃗ − r2⃗ |). The evaluation of exchange integrals between basins is utilized in the IQA (interacting quantum atoms) approach of Pendás et al. [25–27]. The delocalization index between QTAIM basins can be used to compute a reasonable approximation to the corresponding exchange integral [28]. The topological bond order between the Mn basin (number 1) and the basin (number 3) of the equatorial carbon amounts to δ(Mn_2, C_9) ≈ 0.96, which is slightly less than δ(Mn_2, C_10) ≈ 1.13 for the basin (number 12) of the axial carbon. The opposite is true for the bond order between the atomic basins of the CO groups with δ(C_9, O_9) ≈ 1.67 for the equatorial group versus δ(C_10, O_10) ≈ 1.65 for the axial group. The electron sharing is not limited to neighboring basins. The above table confirms that there is substantial sharing between the Mn atom and the basin (number 21) of the axial oxygen atom as given by the bond order δ(Mn_2, O_10) ≈ 0.21. From this circumstance, the following question arises: what is the overall bond order (given by the electron sharing) between the two Mn(CO)5 fragments of the complex? To answer it one could add up the sharing contributions between all the basins situated on mutually different subunits (the sharing between basins of a chosen fragment contributes to the localization index of the fragment). Instead of this let us determine the two Mn(CO)5 superbasins using the assignment “top=0.1” in the control file 4.4. The resulting basin grid will be automatically saved in the file Mn2CO10_TZP_PBE.adf.rho_r.bsn_001 (to avoid overwriting existing files). For convenience, we rename the basin file to Mn2CO10_TZP_PBE.adf.rho_r.bsn-0.1 The overlap integrals for the two superbasins can be computed with a modified control

4 Electron localizability indicator and bonding analysis with DGrid | 95

file 4.8 (i. e., assigning to “basin_1=” the name of the basin file containing the superbasins). As there are just two superbasins, this procedure yields two files with overlap integrals (with file names ending with “.sij.bsn-1” and “.sij.bsn-2”). Running a control file derived from the control file 4.9 gives the desired result: output 4.4.

+------------------------+ | Localization indices | +------------------------+ basin descriptor Q sigma2 LI LIaa LIbb +---------------------------------------------------------------------------+ | 1 Mn_2 64.733 1.038 63.695 31.847 31.847 | | 2 Mn_1 64.733 1.038 63.695 31.847 31.847 | +---------------------------------------------------------------------------+ | 129.466 2.076 127.390 63.695 63.695 | +---------------------------------------------------------------------------+ +------------------------------------------------+ | Delocalization indices for total pair-matrix | +------------------------------------------------+ Basin 1 2 +--------------------------+ | 1 x 0.7730 | | 2 0.7730 x | +--------------------------+

The population of the Mn(CO)5 fragment does not include the 30 core electrons that were excluded from the overlap calculation. The topological bond order between the Mn(CO)5 fragments is approximately 1.5, which is much larger that the bond order of 0.29 between the Mn basins only. Notice that the fluctuation σ 2 = 1.038 is larger than the sharing 0.773 between the superbasins. This is due to the cropping of the basins. Thus, each fragment shares electrons also with the external cropped regions (bear in mind that around 0.53 electrons are missing in the basin region). This could be amended by using 10−5 density isosurface for cropping.

4.4.2 Domain natural orbitals The delocalization index between two basins A and B is derived from the integral of the hole-part of the same-spin pair density over the basins; see the previous Section 4.4.1. Instead of immediately performing the integration over both coordinates, let us first integrate one coordinate over the basin A, yielding the domain-averaged Fermi hole (DAFH) [29, 30]: occ

⃗ j (r)⃗ g A (r)⃗ = ∑ SijA ϕ∗i (r)ϕ i,j

(4.12)

Further integration of g A (r)⃗ over basin B would lead to the delocalization index δ(A, B). The DAFH is invariant with respect to an unitary rotation of the orbitals. This allows a

96 | M. Kohout simplification of the above expression for g A (r)⃗ by the diagonalization of the overlap integrals: ⃗ 2 g A (r)⃗ = ∑ nAk |ψk (r)| k

(4.13)

The orbitals which diagonalize the overlap integrals are called domain natural orbitals (DNOs). The DNOs represent the wavefunction the same way as the canonical ones. They are orthonormal over the total space and mutually orthogonal in the basin A over which the averaging was performed. The numbers nAk are called the DNO occupations. The sum of the occupations gives the total population of the corresponding basin. There is an interesting feature of the DNOs: usually the DAFH g A (r)⃗ is given by contributions of only few DNOs with significant occupations. Additionally, only squares |ψk |2 are present. Compare this to DAFH represented by canonical orbitals using many ϕ∗i ϕj products. On top of it, the g A (r)⃗ given as sum of orbital squares can easily be decomposed into DNO contributions, thus also the delocalization index can be decomposed in such DNO contributions [30]. In Section 4.4.1, the delocalization indices for the QTAIM basins were computed using the control file 4.9 with overlap integrals saved separately for each chosen basin (files “Mn2CO10_TZP_PBE.adf.rho_rbsn.sij.bsn-1” up to “...sij.bsn-21”). Before the DNOs can be computed the overlap files first must be unified into a single file with the command: dgrid Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.bsn-1 complete This creates the completed file “Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij” utilized in the following control file: control file 4.10. ::Mn2CO10 TZP/PBE :------------------------------------sij_1 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij domain_natural_orbitals :--------------------using sij_1 select_basins_domain 1 select_basins_end :--------------------domain_natural_orbitals_end

The “domain_natural_orbitals” block includes the block “select_basins_domain” (closed by the “select_basins_end” keyword) that specifies for which basin the diagonalization of the overlap matrix will be performed. In the above statement, the overlap matrix for basin number 1 (the Mn_2 atom) was chosen.

4 Electron localizability indicator and bonding analysis with DGrid | 97

Running the above control file yields the wavefunction of the complex in the DNO representation (i. e., using orbitals mutually orthogonal in the basin of the Mn_2 atom) written to the file “Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.dno-b1-Mn_2.” Additionally, a file with the name ending ẃith “.dno-b1-Mn_2.norm,” containing the DNO contributions to the population of each basin (so-called norm parts) will be created. The DNO occupations (the diagonalization eigenvalues; or equivalently, the norm parts for the basin selected in the “select_basins_domain” block) can be inspected either in the wavefunction file or in the log file (of which only the relevant part is shown below): output 4.5.

/===========================\ | Domain natural orbitals | +---------------------------+ +------------------------------------------------------------------------------------+ | reference electron located in basin | | 1 Mn_2 | | populated by 14.164840 electrons | +------------------------------------------------------------------------------------+ | remaining basins | | 1 Mn_2 2 Mn_1 3 C_9 8 C_3 12 C_10 19 O_9 | | 21 O_10 | +------------------------------------------------------------------------------------+

/===============\ | eigenvalues | +---------------+ Orb alpha occ beta occ total occ +------------------------------------------------------------------------------------------+ ... | 65 0.0088546761 | | 25 0.1892413420 0.19 | | 33 0.1893574292 0.19 | | 9 0.2171191886 0.22 | | 5 0.3493192140 0.35 | | 41 0.6473851367 0.65 | | 10 0.8075335274 0.81 | | 43 1.1946034740 1.19 | | 29 1.2834476478 1.28 | | 37 1.2834850600 1.28 | | 22 1.9863516635 1.99 | | 30 1.9863523513 1.99 | | 2 1.9902149573 1.99 | | 1 1.9958796472 2.00 | +------------------------------------------------------------------------------------------+ | 14.16484000000 14.13 | +------------------------------------------------------------------------------------------+ | LI | 10.93480679250 10.95 | +------------------------------------------------------------------------------------------+

It informs over which basin the averaging of the reference electron was performed and what the population of that basin is. All the DNO occupations (eigenvalues) must sum up to this value. Only DNOs with sufficiently high occupation are shown in the above output (DNOs with small occupation do not contribute to the Fermi-hole density, thus neither to the delocalization). Also the 4 DNOs (actually the 3s and 3p atomic orbitals

98 | M. Kohout of the manganese) occupied by 1.99 and 2.0 electrons, respectively, do dot contribute to DI, because those DNOs are completely inside the Mn_2 basin and cannot contribute to the overlap with neighboring basins. The remaining 9 DNOs (with occupations 0.19– 1.28 electrons) contribute to the valency of the Mn_2 atom. The magnitude of the DNO contribution to the bond order can be determined from the norm parts data saved in the file ending with “.dno-b1-Mn_2.norm” utilizing the control file: control file 4.11. ::Mn2CO10 TZP/PBE :------------------------------------norm_1 = Mn2CO10_TZP_PBE.adf.rho_r.bsn.sij.dno-b1-Mn_2.norm DI :--------------------using norm_1 :--------------------DI_end

The resulting file named after the “.norm” file with “.DI” added, contains a table with the percental contribution of the DNOs to the DI: output 4.6.

+------------------------------------------------------------------------------------+ | reference electron located in basin | | 1 Mn_2 | | populated by 14.164840 electrons | +------------------------------------------------------------------------------------+ | remaining basins | | 1 Mn_2 2 Mn_1 3 C_9 8 C_3 12 C_10 19 O_9 | | 21 O_10 | +------------------------------------------------------------------------------------+ +-------------------------------------------------------------+ | DI contributions (%) of both-spin domain natural orbitals | +-------------------------------------------------------------+ DNO\Bsn 1 2 3 8 12 19 21 norm sum +---------------------------------------------------------------------------------+ | 1 : 18.2 : 0.2 | 1.9970 | | 2 : 18.1 : 0.1 0.1 1.2 0.3 | 1.9980 | | 30 : 18.0 : 0.3 0.1 0.1 | 1.9887 | | 22 : 18.0 : 0.9 0.1 0.1 0.2 | 1.9918 | | 37 : 7.5 : 3.0 4.6 2.4 23.0 6.1 38.7 | 1.5993 | | 29 : 7.5 : 3.0 17.9 8.4 23.0 29.6 38.7 | 1.7312 | | 43 : 6.5 : 0.4 18.6 2.9 1.0 33.4 0.1 | 1.4020 | | 10 : 3.0 : 67.6 6.2 53.8 26.4 4.8 9.0 | 1.5614 | | 41 : 1.9 : 0.4 20.9 0.8 0.4 9.3 | 0.9923 | | 5 : 0.6 : 3.6 11.6 4.2 6.9 3.9 1.8 | 0.9564 | | 9 : 0.2 : 16.9 2.1 18.2 16.1 1.3 6.0 | 1.4858 | | 33 : 0.2 : 0.2 5.8 0.4 0.8 2.2 0.2 | 0.5562 | | 25 : 0.2 : 0.2 10.2 1.5 0.8 4.0 0.2 | 0.7991 | ... +---------------------------------------------------------------------------------+ | DI |10.935 0.289 0.963 0.053 1.129 0.162 0.211 | 33.7677 | +---------------------------------------------------------------------------------+

4 Electron localizability indicator and bonding analysis with DGrid | 99

It shows that the DI of 0.289 between the Mn atoms (basin number 1 and 2) is given mainly (to 67.6 %) by one single orbital—the DNO number 10. In contrast, the bond order of 1.129 between the Mn atom and the axial carbon is to large extent represented by the contribution of 3 orbitals—the DNOs number 10, 29 and 37. The DNO number 10 is shown in the left diagram of Figure 4.5. From the shape of the orbital, it is clear that it contributes to both, the Mn–Mn as well as the Mn–C bond. The DNO number 37 is shown in the right diagram of Figure 4.5. It exhibits π symmetry as also does the DNO number 29, which appears rotated by 90 degrees with respect to DNO 37. Both π-like DNOs contribute also to the bond between Mn and an equatorial carbon (cf. the 17.9 % contribution of DNO 29 to the DI for basin number 3).

Figure 4.5: Domain natural orbitals (DNOs) for the Mn QTAIM-basin of the Mn2 (CO)10 molecule (blue oxygen, black carbon, green manganese). Left: the Mn basin (green) and the DNO number 10 (isovalue ±0.07, red/gray); right: the Mn basin (green) and the DNO number 37 (isovalue ±0.07, red/gray). DNO number 29 is of same shape, but rotated by 90 degree.

The inspection of the output 4.4 on page 95 with the delocalization index around 1.55 between the two Mn(CO)5 fragments illustrates that the topological bond order between the fragment is much higher than the sharing contribution of the Mn–Mn basin pair. There must be pronounced sharing contributions between each Mn atom and the CO groups bound to the other manganese as well as between the CO groups themselves. We may ask how many DNOs are needed to represent the bond order (DI) between the Mn(CO)5 fragments. Again, first the two overlap files for the fragment (cf. p. 95) need to be unified into single file using the command: dgrid Mn2CO10_TZP_PBE.adf.rho_r.bsn-0.1.sij.bsn-1 complete thus creating the file Mn2CO10_TZP_PBE.adf.rho_r.bsn-0.1.sij with the full overlap information. Utilizing this overlap file in the control file 4.10 generates the DNOs for the fragment basin number 1. Surprisingly, there is only one single orbital, the DNO number 10 shown in the left diagram of Figure 4.6, with the decisive 64 % contribution to the bond order of 1.55 between the fragments. The next significant contributions of only 4 % each arise from 4 other DNOs. The right diagram of Figure 4.6 present the

100 | M. Kohout

Figure 4.6: Mn2 (CO)10 molecule (blue oxygen, black carbon, green manganese). Left: Bonding domain natural orbital (DNO) number 10 for the Mn(CO)5 superbasin (isovalue ±0.07, red/gray); right: the bonding DNO and the Mn(CO)5 superbasin cropped by 0.001-isodensity surface.

Mn(CO)5 superbasin together with the DNO number 10. The DNO, with high values along the Mn–Mn axis is half immersed in the superbasin. The LCAO expansion (which can be inspected with a DGrid utility) of the DNO 10 shows the largest coefficients for the manganese 3dz 2 and 4pz orbitals, similar to the HOMO.

4.5 Electron localizability indicator In this section, the chemical bonding of the Mn2 (CO)10 complex will be analyzed using the electron localizability indicator in the formulation based on specific electron-pair restriction (ELI-D) [31–33]. The interested reader can find the detailed derivation of ELI-D in the literature. In the following, a brief introduction and some basic ideas will be sketched out. ELI-D is actually a discrete distribution of charges (electron populations) rescaled by a parameter. In a very general sense, it is similar to the distribution of QTAIM electron populations. But whereas the QTAIM populations are computed as density inte⃗ r ⃗ over large basin domains Ωi , ELI-D values are given as density integrals ∫Ω ρ(r)d i grals over extremely small nonoverlapping spatial regions μi called microcells. Additionally, whereas the QTAIM basins are determined by the density gradient field, the microcells are determined by a restriction, generating so-called ω-restricted space partitioning. More specifically, each microcell must contain the same fixed number ωD of electron pairs. If we consider the same-spin αα-pairs, then the α-spin density ⃗ r ⃗ and the symbol ϒαD (Greek letter is integrated over the resulting microcells ∫μ ρα (r)d i Upsilon) is used for ELI-D. Both QTAIM basins and ϒαD microcells generate a nonoverlapping complete partitioning of space (the sum of all domain volumes recovers the total space). Hence, the

4 Electron localizability indicator and bonding analysis with DGrid | 101

sum of all QTAIM populations yields the total number of electrons in system and, similarly, the sum of all microcell charges yields the total number of σ-spin electrons. In case of the QTAIM basins, the domain shape must be determined prior to integration over it. In previous sections, it was shown how to accomplish this task with DGrid. The situation is different for ELI-D while applying, for instance, ωD = 10−10 pairs for the restriction, because at such small dimensions the evaluation over the microcell volume is insensitive to its (compact) shape, i. e., the effects are sharply local. Due to the smallness of the restriction ωD , we can determine the volume Vμ of the micro-cell μ positioned at rμ⃗ from the Taylor expansion around the position rμ⃗ , which for single-determinantal wavefunction simplifies to [34, 35]: Vμ = [

3/8

12ωD ] g(rμ⃗ )

(4.14)

.

The denominator g(rμ⃗ ) is the Fermi-hole curvature, which for single-determinantal ⃗ i |2 . Now it is easy to compute the population ⃗ j −ϕj ∇ϕ wavefunction reads g = ∑σi 0). Panel (c) initial condition (t = 0) for the evolution of an electron in a Morse potential. Panel (d) the time-evolution of the electron wave packet constrained by a Morse potential shows increasingly complicated nodal structure at longer times (t ≫ 0).

A connection can be made with molecular orbitals such as the ones obtained from the Hartree–Fock procedure. These canonical molecular orbitals are stationary solutions of an effective time-independent Schrödinger equation (5.13). These can be compared to the long time electron dynamics, which samples the whole complexity of the potential topology. In a molecule, this potential originating from the Coulomb interactions is inherently multicentered. The stationary solutions eventually find the balance between their intrinsic tendency to spread due to the kinetic energy operator and also to the electron-electron repulsion, and their tendency to remain confined in the attractive potential of the electron-nuclear interactions. Although the analogy between dynamics and stationary solutions is imperfect, it gives an intuitive interpretation to the origin of molecular orbital delocalization. In summary, an electron will tend to spread with time and occupy as much space as possible unless a potential is strong

5 Molecular orbital localization

| 121

enough to confine it to some volume in space. Hence, molecular orbitals will tend to become as delocalized as they can on the molecular scaffold. That is, unless we do something about it.

5.3 Numerical orbital localization techniques As we have seen above, the solution of the Hartree–Fock equation, which defines the set of canonical molecular orbitals, is not unique since it is invariant with respect to unitary transformations; see (5.8). This is a general property of the Slater determinant Ansatz used to minimize the total electronic energy, and the following line of reasoning could also be applied to orbitals stemming from Kohn–Sham density functional theory calculations, for instance. Making use of the invariance of the solution with respect to unitary transformations, it is possible to define transformation techniques that will confer some degree of locality to the set of molecular orbitals. This can be done by either imposing constraints to the shape of the Slater determinant prior to energy minimization, in the so-called a priori techniques, or by simply transforming the canonical molecular orbitals a posteriori. Although the former has many important advantages concerning the transferability of the localized orbitals, the following discussion will focus on a posteriori localization, also known as orbital rotation techniques. The meaning of this rotation will become clear later and, while there is no claim of generality, many different such techniques will be discussed in similar style as the enlightening article by Lehtola and Jónsson [11]. The basic idea of most a posteriori localization techniques is to first define a localization metric in the space spanned by two electrons. This can take the following form: loc

Ω = f (r1⃗ , r2⃗ )

(5.16)

where loc Ω can be any positive semidefinite smooth function f (r1⃗ , r2⃗ ) of two electrons r1⃗ and r2⃗ . Taking the canonical molecular orbitals as an initial basis, it is possible to compute matrix elements for the localization metric in a general form as ⟨ij|loc Ω|kl⟩ = ∫ ∫ ψ∗i (r1⃗ )ψj (r1⃗ )f (r1⃗ , r2⃗ )ψ∗k (r2⃗ )ψl (r2⃗ )dr1⃗ dr2⃗

(5.17)

Summing over selected matrix elements of the localization metric allows to define an objective functional of all molecular orbitals. The higher the value of this objective functional, the more localized the molecular orbitals are. Now it is possible to rotate these orbitals by mixing them together. This can be done by applying a unitary transformation matrix, W, with the same properties as in (5.10), to the set of canonical molecular orbitals, such that a new set of orbitals is obtained ψ̄ i (r)⃗ = ∑ Wij ψj (r)⃗ j

(5.18)

122 | J. C. Tremblay with Wij the elements of the rotation matrix, which can be written as 1 .. (. (0 ( (. W = ∏( ( .. ij ( (0 ( .. . (0

⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

0 .. . cos(θij ) .. . − sin(θij ) .. . 0

⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ ⋅⋅⋅

0 .. . sin(θij ) .. . cos(θij ) .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 .. .) 0) ) .. ) ) .) ) 0) ) .. . 1)

(5.19)

One recognizes readily that the unitary transformation matrix is a product of rotation matrices between pairs of orbitals. The optimal values for all rotation angles θij are determined by minimizing the cost functional J[W] with respect to orbital rotations, i. e., 𝜕J[W] 𝜕W ∗

=0

(5.20)

Numerical minimization of this equation can become quite cumbersome and it is beyond the scope of the present chapter. The interested reader is to the relevant literature for more detail, in particular to the unified framework presented by Lehtola and Jónsson [11]. Many popular molecular orbital localization techniques can be described using the orbital rotation framework presented above. Historically, the method proposed by Foster and Boys [12] has received the most attention due to its conceptual simplicity and computational efficiency. The basic idea here is to constrain the spatial extent of a molecular orbital by ensuring that its spread remains as small as possible, as measured by its standard deviation FB Ω = (r1⃗ − r2⃗ )2 . Treating each orbital independently (note the choice of molecular orbital indices) leads to a cost functional of the form FB

Ωi = ⟨ii|(r1⃗ − r2⃗ )2 |ii⟩ ↔ J[W] = ∑ FB Ωi i

(5.21)

The orbital rotation angles θij in (5.19) are determined simultaneously, which ensures that they are coupled and that all occupied and virtual orbitals are localized in an unbiased manner. By design, the localized molecular orbitals thus obtained minimize the second moment of their distribution. This is often enough for a qualitative description of molecular bonds as well as for understanding localized optical excitations, for example. On the other hand, it is important to recognize that this criterion to define locality is neither perfect nor unique. Since the emphasis is put on the standard deviation, only the spatial extent of the second moment of the orbital distribution is reduced

5 Molecular orbital localization

| 123

by orbital rotations. This concept can be readily extended to higher moments of the distribution, which can be rewritten as (n)

n

⃗ Ωi = ⟨i|(r ⃗ − ⟨i|r|i⟩) |i⟩ ↔ J[W] = ∑ (n) Ωpi i

(5.22)

In the context of local correlation methods, where only transitions from occupied to virtual orbitals that overlap significantly are considered, it was found that more tightly localized orbitals can be obtained by minimizing their fourth moment (i. e., n = 4) [13]. Since the localization metric is here again semipositive definite and essentially the square of the Foster–Boys criterion, it enforces localization of the orbital second moment as well. For an α-helix composed of an alanine oligomer, the canonical orbitals are understandably found to be delocalized over the whole structure because the fragments are of the same nature. It was shown that optimizing the orbitals as in (5.22) using the fourth moment as a criterion creates more compact orbitals, and that these have both a smaller second and fourth moments [13]. Nonetheless, the Foster–Boys scheme and related methods lack some physical motivation, since the chosen localization criterion is purely geometrical. As was shown in the previous section, electrons tend to naturally delocalize over the molecular scaffold. The driving force in this case is not only the kinetic energy but, very importantly, the electron-electron Coulomb repulsion. Based on these premises, Edmiston and Ruedenberg [14] proposed using the orbital self-repulsion energy as a criterion for orbital localization, i. e., ER

Ωi = ⟨ii||r1⃗ − r2⃗ |−1 |ii⟩ ↔ J[W] = ∑ ER Ωi i

(5.23)

This was motivated by the intuition that the resulting orbitals would resemble closely the natural orbitals of the system. In their own words: “We predict that the localized SCF orbitals are those orthonormal basis orbitals in the SCF space which closely approximate the first (N) natural orbitals.” It turned out that this intuition was correct, which represents a strength of this approach to molecular orbital localization: the localized orbitals give a faithful physical description of chemical bonds. Unfortunately, the Edmiston–Ruedenberg localization scheme is computationally more expensive compared with other localization approaches since it requires computing sixdimensional, two-electron integrals in the 1/r12 space for all molecular orbitals. For this reason, it could be preferable to use the localization criterion by von Niessen [15], which is based on the orbital overlap vN

Ωi = ∑⟨ij|δ(r1⃗ − r2⃗ )|ij⟩ ↔ J[W] = ∑ vN Ωi j=i̸

i

(5.24)

The simple form of this choice of metric allows to separate the two-electron integrals as one-electron integrals over products of molecular orbital electron densities, i. e., ⃗ j (r)d ⃗ r⃗ ⟨ij|δ(r1⃗ − r2⃗ )|ij⟩ = ∫ ρi (r)ρ

(5.25)

124 | J. C. Tremblay By doing so, the same physics as the Edmiston–Ruedenberg is recovered at a much reduced computational cost. In hindsight, it appears somewhat strange that the von Niessen localization procedure did not enjoy the same popularity as other localization schemes. Using atomic Mulliken charges as a localization metric, Pipek and Mezey [16] derived a computationally efficient and now widely appreciated localization procedure. It shares some similarities with the procedure introduced 20 years prior by Magnasco and Perico [17, 18], which makes use of the representation of molecular orbitals as linear combinations of atomic orbitals (MO-LCAO), Natoms NA

⃗ = ∑ ∑ CμA i χμA (r ⃗ − RA ) ψi (r)⃗ = ⟨r|i⟩ A=1 μA =1

(5.26)

where χμA (r ⃗ − RA ) is the μA basis function centred on atom A at position RA , and CμA i is the associated coefficient for the ith molecular orbital. The coefficients allow to compute Mulliken projectors on various parts of the molecule as Pi = ∑

∑

A,B μA ,νB ∈Γi

CμA i S̃μA νB CνB i

(5.27)

where S̃μA νB are elements of the atomic orbital overlap matrix inverse. In its original formulation, Γi denotes a subset of atomic orbitals that is chosen to represent the bonds, the inner shell electrons, or the lone pairs. This requires the separation of electrons in different groups prior to molecular orbital localization. The determination of such subspaces can become cumbersome but, by construction, it gives a good separation of core and shell orbitals while allowing for an intuitive interpretation of chemical bonds. This added layer of complexity is possibly the reason why the Magnasco/Perico localization metric remained marginally used. It remains historically important as the first method introducing local Mulliken projectors and exploiting the MO-LCAO structure in the context of molecular orbital localization. Similar concepts are used in the Pipek–Mezey (PM) approach, which defines the localization metric as PM

2

−1

Ωi = (∑(Q(i) A ) ) A

↔ J[W] = ∑ PM Ω−1 i i

(5.28)

Here, Mulliken projectors similar to those in (5.27) but on each atom A are used to define the atomic charges, i. e., Q(i) A = ⟨i|PA |i⟩

(5.29)

Despite the similarities with the Magnasco–Perico approach, a main advantage of the Pipek–Mezey scheme is that no a priori information about the chemical structure of

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the molecule is required, and hence, the tedious partitioning of electrons in bonds, inner shells, lone pairs, etc. is avoided altogether. The physical interpretation of (5.28) is also quite appealing, as the metric implies the subdivision of the molecule to atoms and it yields an approximation to the number of atoms where the orbital is localized. Further, it naturally preserves the separation of σ and π orbitals, and that of core/valence electrons. The major drawback of the PM approach is that Mulliken charges such as the one defined in (5.29) do not have a properly defined basis set limit. They have also been shown to have unphysical behaviour in some cases [19]. The PM localization method could thus in principle inherit this unphysical basis set dependence. Fortunately, there are alternative definitions of atomic charges based on electronic density partitioning ⃗ ⃗ ⃗ Q(i) A = ∫ wA (r)ρ(r)dr

(5.30)

The particular choice of the weighting function wA (r)⃗ on atom A is used to define different estimates for the atomic charge. These are known as Bader, Voronoi, Becke, Hirshfeld or Stockholder, to name but a few. In recent work [20], it was shown that using these different charge estimates lead to very similar sets of localized molecular orbitals, despite having an important influence on the charges themselves. It thus appears that the basis set limit is not so important for the determination of localized molecular orbitals, as one would intuitively believe.

5.4 Conclusion In conclusion, delocalization is a natural tendency for any quantum mechanical wave function not constrained by a potential. Canonical molecular orbitals are therefore generally not localized since they are only required to satisfy an energy minimization criterion in the attractive potential of the nuclei and in the repulsive inter-electronic potential. They are obtained from the solution of the many-electron Schrödinger as a Slater determinant, which is invariant with respect to unitary transformation of the orbitals. This allows defining such a unitary transformation as a rotation of the set of canonical orbitals according to a localization metric, defined in the basis of the molecular orbitals. The choice of this metric is neither unique nor perfect, which explains the large number of different localization procedures. Other recent alternatives [21–23] to established localization techniques have been also neglected to keep the description tractable. In the present chapter, we have deliberately chosen to avoid talking about a priori localization techniques. These impose constraints on the shape of the Slater determinant wave function prior to energy minimization [24]. Generally, the molecular orbitals obtained following such procedure are extremely localized on specific

126 | J. C. Tremblay fragments—atoms, bonds, functional groups—and they can be transferred to other molecules containing the same fragments. This is not the case for localized molecular orbitals obtained from a posteriori unitary rotations of canonical molecular orbitals. The price to pay for this extreme localization is a slightly higher energy for the solution of the constrained Hartree–Fock equations. The interested reader can find more information on extremely localized molecular orbitals or the related absolutely localized MOs elsewhere [25, 26, 24, 27, 28].

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

[17] [18]

Høyvik I-M, Jørgensen P. Characterization and generation of local occupied and virtual Hartree–Fock orbitals. Chem Rev. 2016;116(5):3306–27. Levine IN, Busch DH, Shull H. Quantum Chemistry. vol. 6. Upper Saddle River, NJ: Pearson Prentice Hall; 2009. Lewis GN. The atom and the molecule. J Am Chem Soc. 1916;38(4):762–85. Pauling L. The Nature of the Chemical Bond. 3rd ed. 1960 Ithaca, NY: Cornell University Press; 1939. Shaik SS, Hiberty PC. A Chemist’s Guide to Valence Bond Theory. John Wiley & Sons; 2007. Mulliken RS. Electronic structures of polyatomic molecules and valence. II. General considerations. Phys Rev. 1932;41(1):49. Born M, Oppenheimer R. Zur Quantentheorie der Molekeln. Ann Phys. 1927;20:30. Cohen-Tannoudji C, Diu B, Laloe F. Quantum Mechanics. vol. 1. Paris: Hermann and John Wiley & Sons Inc.; 1977. Heller EJ. Time-dependent approach to semiclassical dynamics. J Chem Phys. 1975;62(4):1544–55. Tannor DJ. Introduction to Quantum Mechanics: A Time-dependent Perspective. University Science Books; 2007. Lehtola S, Jónsson H. Unitary optimization of localized molecular orbitals. J Chem Theory Comput. 2013;9(12):5365–72. Foster JM, Boys SF. Canonical configurational interaction procedure. Rev Mod Phys. 1960;32(2):300. Høyvik I-M, Jansik B, Jørgensen P. Orbital localization using fourth central moment minimization. J Chem Phys. 2012;137(22):224114. Edmiston C, Ruedenberg K. Localized atomic and molecular orbitals. Rev Mod Phys. 1963;35(3):457. Von Niessen W. Density localization of atomic and molecular orbitals. J Chem Phys. 1972;56:4290–7. Pipek J, Mezey PG. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions. J Chem Phys. 1989;90(9):4916–26. Magnasco V, Perico A. Uniform localization of atomic and molecular orbitals. I. J Chem Phys. 1967;47(3):971–81. Magnasco V, Perico A. Uniform localization of atomic and molecular orbitals. II. J Chem Phys. 1968;48(2):800–8.

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[19] Fonseca Guerra C, Handgraaf J-W, Baerends EJ, Bickelhaupt FM. Voronoi deformation density (VDD) charges: Assessment of the Mulliken, Bader, Hirshfeld, Weinhold, and VDD methods for charge analysis. J Comput Chem. 2004;25(2):189–210. [20] Lehtola S, Jónsson H. Pipek–Mezey orbital localization using various partial charge estimates. J Chem Theory Comput. 2014;10(2):642–9. [21] Knizia G. Intrinsic atomic orbitals: An unbiased bridge between quantum theory and chemical concepts. J Chem Theory Comput. 2013;9(11):4834–43. [22] Zimmerman PM, Molina AR, Smereka P. Orbitals with intermediate localization and low coupling: spanning the gap between canonical and localized orbitals. J Chem Phys. 2015;143(1):014106. [23] Heßelmann A. Local molecular orbitals from a projection onto localized centers. J Chem Theory Comput. 2016;12(6):2720–41. [24] Stoll H, Wagenblast G, Preuß H. On the use of local basis sets for localized molecular orbitals. Theor Chim Acta. 1980;57(2):169–78. [25] McWeeny R. The density matrix in many-electron quantum mechanics I. Generalized product functions. Factorization and physical interpretation of the density matrices. Proc R Soc Lond Ser A. 1959;253(1273):242–59. [26] McWeeny R. Some recent advances in density matrix theory. Rev Mod Phys. 1960;32(2):335. [27] Sironi M, Genoni A, Civera M, Pieraccini S, Ghitti M. Extremely localized molecular orbitals: theory and applications. Theor Chem Acc. 2007;117(5–6):685–98. [28] Horn PR, Sundstrom EJ, Baker TA, Head-Gordon M. Unrestricted absolutely localized molecular orbitals for energy decomposition analysis: Theory and applications to intermolecular interactions involving radicals. J Chem Phys. 2013;138(13):134119.

Eric D. Glendening, Clark R. Landis, and Frank Weinhold

6 Natural bond orbital theory: Discovering chemistry with NBO7 6.1 Introduction Both the Tools for Chemical Bonding workshop (Bremen 2019) and this volume address the interest of many contemporary chemistry researchers in employing computational “chemical bonding analysis” methods to complement experimental investigations of chemical structure and reactivity. The Natural Bond Orbital (NBO) concept [1, 2] underlies one such set of methods [3] and their implementation in a computer program (currently NBO 7.0) [4, 5] that provides such analysis of quantum chemical calculations from a variety of electronic structure system (ESS) host programs. The present chapter sketches the concepts, practical “tips” for usage and illustrative applications of NBO-based methods to some representative chemical species and conceptual issues, inviting comparisons with alternative methods described in other chapters. NBO-based description of an ab initio or density functional theoretic (DFT) calculation differs most markedly from other methods in its “natural” (Löwdin-style) [6] focus on inherent (eigen) properties of the first-order density operator (Γ(1) ), rather than the energetic (Hamiltonian or Fock/Kohn–Sham operator [7]) properties featured in “energy decomposition” analyses [8]. Γ(1) rigorously condenses the leading features of the many-electron wavefunction Ψ (1, 2, . . . , N) [including all properties solely dependent on electron density ρ(r)] to the single-particle (orbital-based) sub-domain of the full N-particle Hilbert space. Thus, unlike pure density representations of atoms and molecules, NBO generates a representation that is intrinsically orbital-based and rich in orbital imagery. The density operator Γ(1) is routinely produced by modern quantum chemistry packages and serves as the only essential input to NBO algorithms. NBO-based descriptors of Γ(1) are therefore essentially independent of how Ψ was calculated, including details of basis orbitals (Slater- or Gaussian-type, etc.), accuracy level (single- vs. multiconfigurational, variational vs. perturbative, ab initio vs. DFT, etc.), or assumed functional form (whether MO, VB, CAS, coupled-cluster or whatever) [9]. As the accuracy of Ψ increases, the NBOs and other eigendescriptors of Γ(1) are found to converge smoothly toward those of limiting exact solutions of Schrödinger’s equation. A signature feature of NBO-based methodology is strict maintenance of orthogonality in orbital sets (or higher N-electron wavefunctions) chosen to represent observEric D. Glendening, Department of Chemistry and Physics, Indiana State University, Terre Haute IN 47809, USA, e-mail: [email protected] Clark R. Landis, Frank Weinhold, Department of Chemistry, University of Wisconsin-Madison, Madison WI 53706, USA, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110660074-006

130 | E. D. Glendening et al. able physical properties or systems [10]. Whether orthogonal or nonorthogonal basis functions are chosen to expand variational “trial function” approximations to Ψ is of no conceptual concern, a matter of numerical convenience only. However, when expansion functions are envisioned as representing realistic “states” (eigenfunctions) of physical system operators (necessarily Hermitian!), they run deeply afoul of quantummechanical theorems concerning the mutually orthogonal eigenfunctions of all Hermitian operators [11]. This mathematical conundrum arises most directly in purported “decomposition” of interatomic A⋅⋅⋅B energies in terms of the eigenstates of isolated free atoms, which necessarily overlap as interatomic distance RA⋅⋅⋅B diminishes. All NBO-based “population” measures are therefore based on Natural Atomic Orbitals (NAOs) [12] that minimally distort from free-atom forms as necessary to preserve strict orthogonality at finite interatomic separations. Such strict NAO-based “accounting” underlies the success of Natural Population Analysis (NPA) in largely supplanting traditional Mulliken-type [13] measures of atomic charge or interatomic charge transfer (CT). NPA charge values commonly are similar to those obtained by integration of densities in atomic basins. NBO analysis options are currently supported by a large number of popular ESS host programs [14], including fully interactive ESS/NBO7 interfacing (Gaussian, GAMESS, Orca, Molpro), installed NBO code in the ESS distribution (ADF, Q-Chem, Jaguar, Firefly, TeraChem, PQS), or provision for writing valid NBO archive (.47) files (Psi4, NWChem, deMon2K) that can be analyzed by the stand-alone GenNBO program or NBOPro7@Jmol visualization and data-mining utility [15]. Numerous resources are available to assist novice computational researchers in gaining experience with NBO-based descriptors for calculated systems of interest. These include the tutorials, online program manual and extensive bibliography of the NBO7 website (http://nbo7.chem.wisc.edu/). In addition, the student-level guidebook, Discovering Chemistry with NBOs [16], contains ready-to-run input files and illustrative worked-out examples for many properties of chemical interest. The research-level monograph, Valency and Bonding [17], provides further theoretical background and computational results for a broad spectrum of advanced organic, inorganic and biochemical bonding topics. Additional literature overviews of key NBO methods [18], comparisons with other analysis methods [19], or overall philosophy and “usefulness” of NBO analysis [20] may be consulted as the student advances in proficiency. For present purposes, we largely ignore technical and algorithmic details to primarily focus on the “look and feel” of NBO-based numerical descriptors and orbital imagery that can be obtained for a selection of common chemical species and reaction types, all treated at simple B3LYP/6-311++G** model chemistry level. The selected examples are intended to include some species discussed in other chapters, sufficient to recognize commonalities as well as contrasts in both style and substance of the many available analysis perspectives.

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6.2 Lewis-like analysis of molecular structure NBO methods are commonly associated with valence bond (VB) or Lewis-type descriptions of localized chemical bonding. Indeed, the “prime directive” of variational NBO calculation is to find the optimal pattern of localized 1-center/2-electron (1c/2e) lone pairs and 2c/2e bond pairs, wholly analogous to the familiar electron-dot diagrams of G. N. Lewis. The optimal Natural Lewis Structure (NLS) is the key first descriptor of chemical bonding pattern, showing how the species should be properly drawn (and named!) to most accurately represent the actual e-density specified by input Γ(1) . However, determination of the single best NLS bonding pattern is by no means sufficient to guarantee that a localized Lewis-type bonding description is adequate to represent the input e-density. The NBO algorithm automatically “grades” the NLS in terms of the non-Lewis error-metric ρNL [“rho(NL)” in program output], the number of “remainder” electrons that must be formally assigned to non-Lewis NBOs (not involved in the idealized NLS determinant) in order to exactly describe the input e-density. Alternatively, the accuracy of NLS description can be expressed as the percentage (%ρL ) of computed Lewis density ρL with respect to total N, for consistent comparisons with systems of larger or smaller size. The remarkably high NLS accuracies (often >99.9 % for common laboratory species) are a reassuring result (not predetermined “bias” [21]) of NBO description, offering testimony to the prescience of G. N. Lewis and other bonding pioneers who achieved such insights without access to high-level computational quantum chemistry results that we now take for granted. Figure 6.1 depicts the standard Lewis dot structure representation (upper panel) and NBO-based NLS bonding pattern and accuracy-metric %ρL for a variety of closedshell molecular species: (a) H2 O, (b) HF, (c) H2 CO, (d) H2 SO4 , (e) H3 PO4 , (f) SF4 . In each case, the number of lone pairs (output: “LP”) is represented by a parenthesized pre-superscript [“(n) A”] and bond pairs (output: “BD”) by conventional bond sticks. The panels include two examples [(d), (e)] that are commonly misrepresented in the chemical literature [23], as well as an example [(f)] with evident connections to intermolecular interactions (see below). Despite the somewhat exceptional character of the final three cases, the formal NLS diagrams for the vast majority of common textbook molecules are found to map accurately (with high %ρL ) onto those taught to freshman chemistry students. Similar Lewis-structural mnemonics can be used to depict the optimal NLS bonding pattern for open-shell species. In such cases, the preferred bonding pattern generally differs for electrons of α (“up”; majority) and β (“down”; minority) spin, because the two spin sets experience distinct Coulomb and exchange forces in the openshell environment. Figure 6.2 exhibits such “different Lewis structures for different spins” [24] NLS diagrams for three common open-shell species: (a) molecular dioxygen, (b) nitric oxide, (c) ozone. In each case, the distinct (“competing”) bonding patterns are displayed for α (upper panels) and β (lower panels) spin. Each “BD” bondstick now denotes a 1-electron “half-bond” (like that of H+2 ) and each “LP” a non-

132 | E. D. Glendening et al.

Figure 6.1: Standard Lewis dot diagrams (insets) and NLS Lewis-structural bonding patterns for (a) HF, (b) H2 O, (c) H2 CO, (d) H2 SO4 , (e) H3 PO4 , (f) SF4 , in each case showing bond-sticks for 2c/2e bond pairs and pre-superscripts, in parentheses, for 1c/2e lone pairs.

Figure 6.2: NLS Lewis-structural bonding patterns for (a) 3 O2 , (b) 2 NO, (c) 1 O3 open-shell radical species, showing bond-sticks for half-bonds and pre-superscripts for lone electrons.

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bonded “lone particle” of the radical species. The NLS bonding patterns for 3 O2 and 2 NO radicals differ in the relatively obvious way expected for related closed-shell ions (viz., triple-bonded 1 NO− vs. double-bonded 1 NO+ for 2 NO), whereas the singlet diradical 1 O3 species exhibits the more interesting “resonance-like” bond-shift between α and β spin-structures. One can intuitively expect that the preferred open-shell geometry is an intermediate compromise between the bonding preferences of each spin set (or the associated closed-shell ionic species), a simple “spin-averaging” concept that extends standard freshman-chemistry Lewis-structural reasoning to a broad range of reactive open-shell species. Although we do not consider open-shell species further in this introductory overview, the characteristic features of closed-shell NBO analysis as described below are found to be faithfully echoed throughout the domain of reactive open-shell species. A staple of NBO analysis is the NPA evaluation of “natural charge” (QA ) for each atom (by simply subtracting the summed occupancies of NAOs from the nuclear charge at each atom). Figure 6.3 displays the NPA atomic charges for some common species: (a) HF, (b) H2 O, (c) H2 CO (d) H2 SO4 , (e) SF4 , (f) (H2 O)2 dimer. The displayed QA values differ conspicuously from idealized integer “formal charge” or “oxidation number” assignments, and should never be conceived as idealized “point charges” to estimate dipole moment or other multipole aspects of the spatially distributed e-density. However, numerous investigations [25] show the usefulness of these descriptors for correlations with experimental structural and reactive properties, including charge-transfer (CT) complex formation [as manifested in the altered charges

Figure 6.3: NPA atomic charges for (a) HF, (b) H2 O, (c) H2 CO (d) H2 SO4 , (e) SF4 , (f) (H2 O)2 .

134 | E. D. Glendening et al. of Figure 6.3(b), (f)] [26, 27]. Numerous energy decomposition schemes [8] evaluate the energy change due to charge transfer (ΔECT ) without clear quantification of “atomic charge” or “charge transfer” itself, but all NBO-based descriptors rest firmly on the foundational NPA assignments of these quantities. Table 6.1 displays the Natural Electron Configuration (NEC) for each atom in the species of Figure 6.3, giving the next-higher level of orbital detail (valence NAO occupancies), akin to familiar Bohr-type e-configuration concepts. The NEC values of Table 6.1 are strictly consistent with the NPA charges of Figure 6.3 and allow the electronic charge shifts [e. g., between (b) H2 O and (f) (H2 O)2 ] to be traced to specific angular sub-shells of each atom. Table 6.1: NEC e-configuration assignments for each atom of (a) HF, (b) H2 O, (c) H2 CO, (d) H2 SO4 , (e) SF4 , (f) (H2 O)2 (cf. Figure 6.3), showing parenthesized population of each angular shell of valence NAOs (and most populated extra-valent shell, if appreciable). (a)HF

(b)H2 O

H F

1 2

H O H

1 2 3

(c)H2 CO C O H H

1 2 3 4

1s( 0.45) (d)H2 SO4 [core]2s( 1.91)2p( 5.63)

S

1

O O O H H O 1s( 0.54) (e)SF4 S [core]2s( 1.75)2p( 5.15) F 1s( 0.54) F F F [core]2s( 1.03)2p( 2.65) (f)(H2 O)2 H [core]2s( 1.72)2p( 4.76) O 1s( 0.89) H 1s( 0.89) O H H

2 3 4 5 6 7 1 2 3 4 5 1 2 3 4 5 6

[core]3s( 0.98)3p( 2.29) 3d(0.18) [core]2s( 1.77)2p( 5.06) [core]2s( 1.83)2p( 5.05) [core]2s( 1.77)2p( 5.06) 1s( 0.49) 1s( 0.49) [core]2s( 1.82)2p( 5.00) [core]3s( 1.69)3p( 2.29) [core]2s( 1.95)2p( 5.56) [core]2s( 1.95)2p( 5.56) [core]2s( 1.93)2p( 5.47) [core]2s( 1.93)2p( 5.47) 1s( 0.54) [core]2s( 1.75)2p( 5.19) 1s( 0.51) [core]2s( 1.76)2p( 5.17) 1s( 0.52) 1s( 0.52)

Still further orbital-level detail is provided by the polarization coefficients (cA , cB ) and Natural Hybrid Orbitals (NHOs) hA , hB for each 2-center bond NBO (σAB ) of the NLS, written in conventional form as σAB = cA hA + cB hB

(6.1a)

Each such “Lewis-type” (formally occupied) NBO of the NLS is complemented by the corresponding out-of-phase (non-Lewis-type) antibond NBO (σ ∗ AB ), σ ∗ AB = cB hA − cA hB

(6.1b)

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The normalized polarization coefficients (|cA |2 + |cB |2 = 1) yield the natural ionicity descriptor (iAB ) iAB = |cB |2 − |cA |2

(6.2)

which varies continuously between pure ionic (iAB = −1 or +1) and pure covalent (iAB = 0) bonding limits, analogous to the valence bond (VB) descriptor of “covalent/ionic resonance.” [28] For main-group (MG) atoms, the composition of NHOs can be expressed in familiar Pauling-type “spλ ” notation as linear combinations of s, p NAOs, with fractional p-character = λ/(1 + λ). All such polarization and hybridization details are simultaneously optimized in the NBO variational algorithm that determines the optimal NLS. The resulting shapes and directions of NHOs are found to correspond closely to the valence hybrids of freshman chemistry for the special cases λ = 1, 2, or 3, but more general and subtle features of NHOs also yield numerous research-level extensions [29, 30] that are beyond the scope of present review. Table 6.2 summarizes numerical NBO ionicity and NHO hybridization descriptors for the various σAB bonding NBOs of Figure 6.3 and Table 6.1. Figure 6.4 shows graphical orbital imagery for the full set Lewis-type NBOs (bonding and lone pairs) for (a) HF, (b) H2 O and (c) H2 CO, all shown with preorthogonal (PNBO, PNHO; free atom-type) “visualization orbitals” that allow Mulliken-type estimates of interaction strength from simple orbital overlap conceptions [31–33]. Figure 6.5 displays a corresponding depiction of the two directed (P)NHO hybrids hC , hH that combine to form the σCH (P)NBO of H2 CO [upper-left panel of Figure 6.4(c)], showing the close resemblance to cartoon-like textbook illustrations of covalent bond formation despite the much higher level of chemical information encoded in the NHO shapes and sizes. All such Table 6.2: NBO ionicity (iAB ) and NHO hybridization (hA , hB ) descriptors for bonding NBOs of (a) HF, (b) H2 O, (c) H2 CO (d) H2 SO4 , (e) SF4 , (f) (H2 O)2 (cf. Table 6.1, Figure 6.3). species (a) HF (b) H2 O (c) H2 CO (d) H2 SO4

(e) SF4 (f) (H2 O)2

NBO σH(1)F(2) σH(1)O(2) σC(1)H(3) σC(1)O(2) πC(1)O(2) σS(1)O(2) σS(1)O(3) σS(1)O(7) σO(2)H(5) σS(1)F(3) σS(1)F(4) σO(2)H(1) σO(2)H(3) σO(4)H(5)

iAB +0.554 +0.462 +0.138 +0.307 +0.305 +0.378 +0.297 +0.292 −0.510 −0.553 −0.467 −0.459 −0.501 −0.480

hA

hB

s s

3.95

sp1.97 sp1.98 p sp3.81 sp2.42 sp2.23 sp3.06 ∼p sp7.23 sp3.43 sp2.89 sp3.30

sp sp3.34 s sp1.53 p sp4.16 sp3.27 sp3.12 s sp8.13 sp6.65 s s s

136 | E. D. Glendening et al.

Figure 6.4: Lewis-type NBOs for (a) HF, (b) H2 O, (c) H2 CO, showing characteristic differences between n(σ) -type (on-axis or in-plane; s-rich; low energy) and n(π) -type (off-axis or out-of-plane; p-rich; high energy) lone pairs in all such species.

numerical and visual comparisons indicate the qualitative agreement with freshmanlevel (or Pauling-type) localized bonding concepts that is consistently obtained for a broad range of chosen DFT or ab initio theoretical model chemistries. The lone-pair panels of Figure 6.4 also illustrate the distinction between lowenergy, s-rich “σ-type” (n(σ) ) vs. high-energy, p-rich “π-type” (n(π) ) lone pairs that is a characteristic feature of NBO description (rigorously required in the present highsymmetry species). The n(σ) lone pairs exhibit hybridized spλ mixing and lie along the line (z) or in the plane of bonding interactions with nearby atoms, which break the free-space atomic symmetries. The n(π) lone pairs are “left-overs” of the atomic valence shells that retain the angular character of free-space atoms, with exact equivalence between πx -type vs. πy -type orientations. NBOs are generally found to retain such dynamical-type “local σ/π symmetry” even in species of low overall point-group symmetry, and any attempt to impose “rabbit ears” [34] constraints on their forms

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Figure 6.5: NBO-based graphical depiction of σCH covalent bond formation from bonding hybrids hC , hH in 2D contour (upper) and 3D surface diagrams (lower). (P)NHOs hC , hH are viewed separately in (a), (b) or overlaid (without constructive interference) in (c), for comparison with the final (P)NBO bonding orbital in (d) [cf. Figure 6.4(c), equation (6.1a)].

results in sharp loss of %ρL . One can say more generally that the strict symmetrycompliance and unique forms of NBOs are dictated by their unique occupancies (eigenproperties of Γ(1) ). In this respect, NBO-based “bottom-up” construction is distinguished from other localized molecular orbital (LMO) methods [35–38] that suffer from the inherent indeterminacies (up to arbitrary unitary transformations) [39] in the “top-down” localization of fully degenerate MOs (all occupancies = 2.000). Although the present discussion focuses primarily on main-group (MG) chemistry, it should be emphasized that d-block transition metal (TM) species exhibit closely related Lewis-like localized bonding patterns [40–42], charge distributions and configurational assignments that are fully analogous to those displayed above. The MG/TM parallels begin by replacing the 4-orbital (s + 3p) MG valence shell with the 6-orbital (s + 5d) TM valence shell, thus converting the MG “rule of 8” to corresponding TM “rule of 12” in the d-block. Further details of the many Lewis-like analogies between idealized sdμ bonding hybrids of the d-block and spλ bonding hybrids of the p-block are described elsewhere [43].

6.3 Beyond Lewis structure: Resonance delocalization Characterization of the single best Lewis-structural representation of e-density is only the starting point of NBO analysis. It may seem that successful localized analysis of >99 % of input e-density should be more than adequate for practical chemical pur-

138 | E. D. Glendening et al. poses, but this is far from the truth (as demonstrated below). The NBO toolkit includes a wide variety of methods for further dissecting such “non-Lewis” delocalization effects, which can all be associated with resonance-type corrections to the elementary single-Lewis structure picture. Among the most powerful NBO tools for analyzing non-Lewis corrections to the elementary NLS picture (ΨL ) are the “deletion” techniques of $DEL-energetic analysis [44]. The $DEL options involve orbital energy evaluations that require a 1-electron “effective Hamiltonian” (Fock or Kohn–Sham operator) of MO/DFT type. In effect, $DEL-deletion techniques allow one to “delete” some or all of the non-Lewis (NL) “acceptor” NBOs (formally unoccupied in elementary ΨL description) and recalculate the system properties (reoptimized structure, vibrational frequencies, etc.) as though such NL-type NBOs were absent in nature or never included in the computational basis set. By deleting all such NL-type NBOs ($DEL-deletion “Lewis” type 9), we obtain the single-determinant ΨL (or ρL ) and associated N-electron energy (EL ) as unperturbed starting point for subsequent donor-acceptor perturbative analysis (as discussed below). $DEL-deletion techniques (and variational reoptimizations) can also be applied to individual donor-acceptor interactions between Lewis- (Ωi (L) ) and non-Lewis-type (Ωj (NL) ) NBOs, often allowing one to isolate the specific Ωi (L) → Ωj (NL) “smoking gun” interaction responsible for a particular structural or spectroscopic anomaly [45, 46]. Figure 6.6 illustrates the “chemical significance” of resonance-type corrections to the elementary ΨL description for some simple chemical species. The upper panels of Figure 6.6 show the full B3LYP/6-311++G** optimized structures for (a) SF4 , (b) (H2 O)2 (as considered above; cf. Figure 6.3) and (c) NH2 CHO (formamide), compared in each case with corresponding $DEL-Lewis reoptimized structures with all resonance-

Figure 6.6: Structural bond lengths (Å) of (a) SF4 , (b) (H2 O)2 , (c) H2 NCHO, comparing fully optimized DFT structures (upper panels) with those of corresponding $DEL-Lewis calculations (lower panels) in which all resonance-type (non-Lewis) interactions are absent.

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type NL-corrections deleted. The structural and energetic consequences of neglecting the “small” resonance-type corrections are seen to be profound indeed: Without such corrections, SF4 (initially C2v -symmetric) is seen to split into a long-range F− ⋅⋅⋅SF+3 complex (C3v symmetry) [consistent with Lewis-structural propensity, Figure 6.3(e)]; water-dimer (initially Cs symmetry) is seen to reorient to a weakened long-range complex of expected “dipole-dipole” form (C2v symmetry); and planar formamide (Cs symmetry) is seen to buckle and twist to bizarre “aminoaldehyde” form (C1 symmetry) that could never support α-helix-like polymeric motifs. All such examples emphasize that a “resonance-free world” could not support chemistry and life as we know it. Alternatively, the energetic effect of any particular Ωi (L) → Ωj (NL) (i → j∗ ) donoracceptor interaction can also be estimated by 2nd -order perturbation theory as (L) (2) |F |Ωj (NL) ⟩2 /(εj∗ − εi ) ΔEi→j ∗ = −qi ⟨Ωi

(6.3)

where qi is the occupancy of Ωi (L) (2 for closed-shell; 1 for open-shell) and εj∗ , εi are orbital energies (diagonal expectation value of system F -operator) for acceptor Ωj (NL) and donor Ωi (L) NBOs, respectively. The magnitude of perturbative ΔEi→j∗ (2) estimate (6.3) is generally found to agree well with the corresponding $DEL-deletion estimate [ΔEi→j∗ ($DEL) ] for variational loss of Ωi (L) → Ωj (NL) donor-acceptor stabilization. The tabulated listing of perturbative |ΔEi→j∗ (2) | [“E(2)”] descriptors is usually the first section of NBO output examined by the accomplished NBO user. Figure 6.7 displays the leading NBO donor-acceptor interaction in each species of Figure 6.6: (a) nF(5) → σ ∗ S(1)F(2) in SF4 , (b) nO(4) → σ ∗ O(2)H(3) in (H2 O)2 , and (c) nN(4) → π ∗ C(1)O(3) in NH2 CHO. Each panel includes the parenthesized |ΔEi→j∗ (2) | estimate (kcal/mol) of donor-acceptor stabilization. Although secondary interactions can exercise a modulating influence on structure, one can recognize that the “full” geometry (upper panel of Figure 6.6) is always such as to approximately satisfy the expected “maximum overlap” principle for the dominant orbital interaction shown in

Figure 6.7: Principal NBO Ωi (L) → Ωj (NL) orbital interactions in species of Figure 6.6, showing nF(5) → σ ∗ S(1)F(2) in SF4 (left), nO(4) → σ ∗ O(2)H(3) in (H2 O)2 (center) and nN(4) → π ∗ C(1)O(3) in NH2 CHO (right), with estimated stabilization energy (kcal/mol) in parentheses.

140 | E. D. Glendening et al. Figure 6.7. Of course, one must always offset the purely attractive donor-acceptor interactions against repulsive steric interactions between filled NBOs if attempting to estimate net binding energy. For this purpose, NBO analysis provides complementary STERIC keyword descriptors of donor-donor interactions that are in excellent accord with empirical steric concepts [47, 48], but are not discussed further in this introductory review. Each donor-acceptor interaction of Figure 6.7 can also be associated with a contributing resonance structure, as displayed in Figure 6.8 for the amide nN → π ∗ CO interaction and other common donor-acceptor motifs. In each case, Ωi (L) → Ωj (NL) delocalization corresponds to the stabilizing contribution of a bond-shifted “CT resonance” form, whose weightings and bond-order shifts may be expected to correlate with perturbative |ΔEi→j∗ (2) | or $DEL-variational |ΔEi→j∗ ($DEL) | estimates of resonance stabilization.

Figure 6.8: General correspondence of various NBO donor-acceptor interaction motifs (left) with associated arrow-pushing mnemonic (center) and contributing resonance structure (right).

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The full-fledged extension of NBO concepts to the resonance-theoretic domain is achieved by Natural Resonance Theory (NRT) [49–51]. The basic NRT variational method was recently “rebooted” [52] with the Gram-based convex-solver algorithm [53] that greatly extends the ease, robustness, and generality of NRT applications. As in other NBO-based methods, optimal resonance weights {wR } for each candidate resonance structure and associated e-density ΓR (determined with full NLS-type optimization of the internally generated bonding pattern) are obtained by least-squares matching of the formal NRT e-density operator ΓNRT = ∑ wR ΓR R

(6.4)

with respect to input target e-density Γ(1) . Whereas the original NRT implementation was practically limited to weakly delocalized systems where only one or a few “parent” resonance structures are dominant, new-style NRT extends into the strongly delocalized regime of metallic-like interactions where hundreds or thousands of structures may contribute with near-equal weightings to the composite resonance hybrid. In this limit, only the NRT bond orders {bAB } remain as recognizable chemical descriptors to provide expected correlations with structural, spectroscopic, and reactivity properties of the system [54]. NRT bond orders should be considered to supercede the NAO-based Wiberg bond index values that are sometimes identified as “natural bond orders.” Figure 6.9 exhibits NRT bond orders for the simple SF4 , (H2 O)2 , NH2 CHO species discussed above. By comparison with the |ΔEi→j∗ (2) | descriptors of Figure 6.7, one can see that the bond order shifts from idealized integer (0, 1, 2) values increase in the expected way from the weak intermolecular H-bonding of water dimer (b) to the stronger intramolecular amide resonance (c) and still stronger 3c/4e hyperbonding in sulfur tetrafluoride (a), in accordance with chemical intuition. The more subtle shifts in hydride bOH , bNH , bCH bond orders are also in accord with measureable structural and IR vibrational properties.

Figure 6.9: NRT bond orders for the species of Figure 6.7.

142 | E. D. Glendening et al.

Figure 6.10: NRT bond orders for primary structural (RCC , RCN , RCO ) bonds of GC base pair.

A more challenging NRT application is shown in Figure 6.10 for the guanine-cytosine (GC) base pair of Watson–Crick model DNA. The NRT bond orders for the principal skeletal bonds describe the nuances of GC conjugation, as well as correlated variations of bond lengths, more accurately than do the idealized integer values of the NLS bonding pattern (identical to that shown in every biochemistry textbook). For the thirteen RCN bonds, for example, the NRT bond order-bond length regression is satisfied with reasonably high Pearson correlation coefficient, |χ|2 = 0.85, despite the expected “noisy” statistics of ring-strain distortions. (The corresponding Pearson coefficient is |χ|2 = 0.51 for “textbook” NLS bond orders.) Figure 6.11 displays more detailed NBO/NRT descriptors and orbital imagery for the three GC hydrogen bonds. The miniscule bond orders associated with H-bonding fall virtually into the “noise level” of dominant resonance-type interactions in the full calculation of Figure 6.10. We therefore employed the “local NRT” option [55] to focus on the nine atoms of the three H-bonded triads, as shown in Figure 6.11(a). Such idealized abstraction of the H-bonding motif from the broader conjugative network allows more robust numerical determination of the H-bond descriptors of interest, but may neglect quantitative details of resonance-assisted H-bonding (RAHB) [56] from a more comprehensive NRT model. Figure 6.11(a) shows calculated NRT bond orders for the covalent N–H bond (stick), H-bond (dots), and long-bond [57] (caret) of each 3c/4e triad. Figure 6.11(b) displays corresponding (P)NBO orbital imagery for the leading nN/O → σ ∗ NH interaction in each triad, showing the somewhat canted (and weak∗ ened) n(π) O → σ NH interaction of each O⋅⋅⋅H–N linkage compared to the better aligned (σ) nN → σ ∗ NH interaction of the central N⋅⋅⋅H–N linkage. Table 6.3 summarizes a broader array of NBO and experimental descriptors for the H-bonded triads of GC, including QCT , second-order ΔEnσ∗ (2) and $DEL-deletion

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Figure 6.11: Details of GC H-bonds (cf. Figure 6.10, Table 6.3), showing (a) NRT bond orders for the covalent bond (stick), H-bond (dots), and long-bond ( ̂) of each 3c/4e H-bonding triad; (b) (P)NBO interaction diagrams for nO(24) → σ ∗ N(6)H(10) (upper), nN(4) → σ ∗ N(14)H(25) (middle), nO(3) → σ ∗ N(16)H(26) (lower) donor→acceptor interactions. Table 6.3: NBO, structural, and energetic properties of GC H-bonds, showing correlations with NRT bond orders (summed bB⋅⋅⋅H + bB ̂A ) of Figure 6.11. B⋅⋅⋅H–A bB⋅⋅⋅H + bB a

c

O(24)⋅⋅⋅H(10)-N(6)

O(3)⋅⋅⋅H(16)-N(26)

0.142

0.112

0.071

0.0435

0.0438

0.0233

|ΔEn(B)−σ ∗ (AH) (2) |

19.24

22.35 (14.70(π) , 7.65(σ) )

12.63 (8.68(π) , 3.95(σ) )

|ΔEn(B)−σ ∗ (AH) ($DEL) |

21.28

22.94 (14.95(π) , 8.08(σ) )

13.22 (8.88(π) , 4.28(σ) )

0.0201

0.0266

0.0117

8.77

8.82

5.60

b

d e a

QCT

̂A

N(4)⋅⋅⋅H(25)-N(14)

ΔRAH

ΔEH-bond

difference of σ AH e-occupancies in monomer and dimer; Second-order perturbative estimate of nB → σ ∗ AH stabilization (kcal/mol); c $DEL-deletion estimate of nB → σ ∗ AH stabilization (kcal/mol); d change of RAH bond length from monomer to dimer (Å); e estimated H-bond energy (kcal/mol) from linear regression with QCT [54]. ∗

b

ΔEnσ∗ ($DEL) energies and ΔRNH bond elongation, all known to correlate [54] with overall H-bond strength, ΔEH-bond . From these values, one can judge that the upper two triads in Figure 6.11(a) exhibit considerably stronger H-bonding than the lower O(3)⋅⋅⋅H(26)N(16) triad, with the upper O(24)⋅⋅⋅H(10)N(6) being slightly stronger than

144 | E. D. Glendening et al. the middle N(4)⋅⋅⋅H(25)N(14) interaction [despite the slightly contrary indication of the idealized bond orders in Figure 6.11(a)]. The estimates of net H-bond energy ΔEH-bond (summing to 23.19 kcal/mol for the three triads) are in sensible qualitative agreement with calculated net binding energy (25.71 kcal/mol) for the GC complex. Overall, the tabulated numerical descriptors of H-bonding in GC indicate broad mutual consistency with one another and with the general NBO/NRT picture as found for all known H-bonding species. From these and many similar examples [52, 53], it is evident that NRT descriptors can bridge a broader range of electronic behavior than can the Lewis-centric (NLStype) descriptors of original NBO formulation. The NRT domain automatically encompasses both the localized Lewis-like limit that characterizes many familiar organic species as well as the highly delocalized limit of metallic-like interactions (including exotic “long-bond” phenomena [57]). Analysis methods that depend on specific choice of “reference fragment” or “reference state” [58, 59] inherently obscure such resonance-type features of electronic interactions. Although resonance concepts are sometimes derided as figments of chemical imagination (“unicorns” [60]), NBO/NRT results strongly support the quantum-mechanical conception that resonance-type “superposition” effects underlie all chemical bonding phenomena [61, 62].

6.4 NRT analysis of chemical reactivity A major innovation of NBO7-level NRT implementation is introduction of semilocalized “resonance NBOs” (RNBOs) [63] {Ξk } to the NBO-based family of basis sets. Whereas conventional NBOs {Ωi } and their semi-localized NLMO counterparts are intrinsically tied to a particular NLS bonding pattern, valid only in a limited region of the PES, RNBOs can be computed and applied in continuously variable fashion to any point or path of the potential energy surface (PES), without the discontinuous “jump” from one NLS bonding pattern to another that afflicts NBO-based descriptors and orbital imagery. The Lewis-centric NBOs {Ωi } thereby fail to comply with Pople’s prerequisite [64, 65] for a satisfactory computational model chemistry, namely, that the chosen basis set and wavefunction form be independent of chemical bonding pattern and thereby applicable in democratic fashion to any region or pathway of the PES. (Similar dependencies are found in other analysis methods, as, e. g., in conventional VB formulation of the wavefunction [66, 67], EDA dependence on chosen reference state [58, 59], or QTAIM dependence on a particular pattern of BCPs and bond paths [69].) For didactic purposes (neglecting numerous technical details), RNBOs {Ξk } can be described as the NRT-weighted “average” of NLMOs {Ωk (R) } for each contributing

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resonance structure (R), namely {Ξk } = ∑ wR {Ωk (R) } R

(6.5)

The RNBOs thereby inherit the fully-occupied character of their NLMO constituents but are inherently free to adopt the 3-center (or higher multi-center) character of resonance-type bond shifts as depicted in Figure 6.8. As for other NBO-based localized sets, the interactions of orthonormal RNBOs can be visualized with corresponding pre-orthogonal PRNBOs that are included in PLOT-keyword output. The NBO/NRT applications of previous sections refer to equilibrium chemical species in which resonance-type corrections to a parent NLS representation may (or may not) surpass a conceptual threshold for “chemical significance.” In contrast, chemical reactivity involves competition between two (or more) bonding patterns where resonance-type aspects of the transition-state region are inescapable. In the present section, we therefore focus on NRT- and RNBO-based description of a model exchange-type chemical reaction H− + FH → [H⋅⋅⋅F⋅⋅⋅H− ]‡ → HF + H−

(6.6)

that bears conceptual “inverse electronegativity” relationship to the famous bifluoride (F⋅⋅⋅H⋅⋅⋅F− ) H-bonded species. Figure 6.12 displays the computed IRC (left) and associated bond-order variations (right) in bHF (“left” H; product-like), bFH (“right” H; reactant-like), and bH ̂H (longbond) NRT bond orders in exchange reaction (6.6). The calculated activation energy (ca. 34 kcal/mol) reflects the evident effectiveness of “halogen bond” [70] interaction in reducing the energy penalty (ca. 135 kcal/mol in empirical bond-energy compilations) for breaking the strong HF bond.

Figure 6.12: (a) IRC energy profile (left) and (b) NRT bond orders (right) for H− ⋅⋅⋅FH exchange reaction (6.6).

146 | E. D. Glendening et al. The peaked form of IRC energy profile [Figure 6.12(a)] may suggest that the FH bond “snaps” abruptly from one H to the other near IRC ≈ 0. However, the NRT bond-order plots [Figure 6.12(b)] suggest the more broadly nuanced rise and fall of bHF , bFH orders, augmented by strong bH ̂H long-bonding that becomes the largest single contributor to triad bonding in the transition state region (−0.6 < IRC < 0.6). The essential 3c-resonance character of interactions surrounding the central monovalent member of the triad is fully consistent with general “bond-order conservation” [71] that is observed to hold throughout the IRC profile, bHF + bFH + bH

̂H

≈1

(6.7)

Similar 3c-resonance character is evident in FHF− , I−3 , and numerous other trihalide species that form stable complexes rather than the transition-state species of the HFH− example. Figure 6.13 displays comparisons of NPA/NBO descriptors for [HFH− ]‡ (left) vs. FHF− (right) (employing the reactant-$CHOOSE Lewis structure in the former case). Figure 6.13(a) compares the NPA charge distributions, showing the “inverse electronegativity” relationship between charges in the two species that promotes longbonding in the [HFH− ]‡ case [57]. Figure 6.13(b) similarly compares (P)NBO orbital imagery for the donor-acceptor interactions nH → σ ∗ FH of [HFH− ]‡ vs. nF → σ ∗ HF of FHF− , showing the characteristic difference between s-type nH vs. p-type nF donors and the adverse loss of stabilizing interaction due to reversal of acceptor σ ∗ FH antibond direction in the [HFH− ]‡ case.

Figure 6.13: Comparison of (a) NPA charge distributions and (b) principal NBO n → σ ∗ donoracceptor interaction for transition-state [HFH− ]‡ (left) vs. stable FHF− (right) species.

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Figure 6.14: “Morphing” forms of leading (P)RNBOs of H− ⋅⋅⋅FH exchange reaction (6.6) for (a) longrange limit, (b) intermediate separation and (c) transition-state limit.

Figure 6.14 illustrates the continuous morphing of RNBOs for HFH− from (a) the welllocalized 2c form of the long-range reactant limit (IRC ≈ −5) to (b) semi-localized intermediate separation (IRC ≈ −2) and (c) fully developed 3c character in the transitionstate limit (IRC = 0). The various comparisons presented above illustrate the mutually consistent features of NBO/NRT-based descriptors and orbital imagery that are found to characterize both reactive and equilibrium species. Whether a given triad exhibits an energy well (stability) or barrier (reactivity) can generally be traced to the relative strength of attractive nB -σ ∗ AH (donor-acceptor) vs. repulsive nB -σAH (donor-donor) interactions that can each be quantified and visualized with NBO/NRT descriptors. Further discussion and illustrative NRT/RNBO application to a more complex chemical reaction can be found elsewhere [72].

148 | E. D. Glendening et al.

6.5 Natural energy decomposition analysis of molecular interactions We lastly consider the quantitative interpretation of intermolecular noncovalent forces provided by Natural Energy Decomposition Analysis (NEDA) [73, 74]. NEDA partitions the association energy (ΔE) into electrical interaction (EL), charge transfer (CT) and core repulsion (CORE) components. ΔE = EL + CT + CORE

(6.8)

Electrical interaction is further partitioned into classical electrostatic (ES) and polarization (POL) components. Available for many years for GAMESS/NBO calculations, the method was recently extended to Gaussian-16 (Revision C.01/NBO7). NEDA can be applied to self-consistent field densities (both restricted closed-shell and unrestricted open-shell) evaluated at the Hartree–Fock or DFT levels. The cornerstone of the method is the construction of localized fragment densities from eigenvectors of the Fock/Kohn–Sham operator in the NAO or NBO basis. The resulting CT components of NEDA, which tend to be stronger than CT components of other EDA approaches, are fully consistent with estimates of charge transfer obtained from the perturbative analysis and $DEL-deletions of the NBO method. Table 6.4 shows representative applications of NEDA to water dimer, the GC base pair and the C-C bond of ethane. The ΔE values reported here are evaluated with respect to separated fragments in their unrelaxed geometries of the molecular complex—fragment distortion is not considered. Table 6.4: NEDA descriptors (in kcal/mol) for the hydrogen bonds of (H2 O)2 and the GC base pair, and for the C-C covalent bond of ethane. species b b

(H2 O)2 [Cs ]

(H2 O)2 [C2v ]

GC base pair CH3 –CH3

ΔE

EL

ES

−5.08

−10.84

−3.38

−9.22

−28.00 −109.82

a

POL

CT

CORE

−9.01

−1.83

−9.13

14.90

−6.42

−2.80

−2.45

−53.21

−42.95

−10.26

−60.34

−87.42

−133.30

45.88

−462.89

8.30

85.55

440.49

a

POL values combine the polarization and self-energy (SE, energy penalty to polarize) values reported in NEDA output. b The Cs and C2v geometries of (H2 O)2 are described in the text.

For (H2 O)2 we consider the equilibrium Cs and the “dipole-dipole” C2v geometries of Figure 6.6, although with the O. . . O separation of the latter constrained to that of the Cs dimer. The Cs dimer is bound by 5.1 kcal/mol. NEDA reveals important attractive contributions from electrostatic interaction (−9.0 kcal/mol), polarization (−1.8 kcal/mol)

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and charge transfer (−9.1 kcal/mol), where the last of these is clearly dominated by the single nO → σ ∗ OH interaction (at 8.1 kcal/mol) of Figure 6.7. ES arises from the classical interaction of overlapping relaxed monomer charge distributions (a common component of many EDA methods), and POL is the extra stabilization of the dimer that results from polarization of the charge density, although constrained to maintain monomer orthogonality. The dipole-dipole geometry exhibits a weaker binding energy (by 1.7 kcal/mol) and dramatically reduced CT (by nearly 75 %). Polarization is somewhat stronger in the higher symmetry dipole-dipole geometry, but charge transfer clearly drives the orientation of the monomers to the lower symmetry Cs structure. Finally, we note that NEDA can also be applied to intramolecular “covalent” interactions as shown for the C-C bond of ethane. The reference monomers in this case (as specified by the user) are a pair of separated methyl radicals, one radical with unpaired electron of alpha spin and the other radical with unpaired electron of beta spin. It is of little surprise in this case that CT (−462.9 kcal/mol) dominates the association energy (−109.8 kcal/mol) as the two unpaired electrons strongly delocalize into acceptor hybrids of the neighboring radicals.

6.6 Conclusions The foregoing sections illustrate only a few of the widgets in the NBO7 toolbox. Other program options are available for analysis of steric effects (keyword: STERIC), NMR chemical shielding (NCS) and J-coupling (NJC), dipole moments (DIPOLE), and other physical properties, as well as more detailed localized descriptors of molecular orbital (CMO) or correlated wavefunctions (NPEPA), and so forth. New NBO users may be led in many directions according to personal interests. To conclude this chapter, it may be helpful to address (rather randomly) some “frequently asked” questions that might arise in after-talk Q/A discussions. The following examples supplement those that may be found in the NBO Forum and FAQ links of the NBO7 website [75]. Q1. Can NBO/NRT analysis be applied to solid-state systems? A. Not completely. A partial NPA/NBO implementation [76] is available for the VASP [77] host program. Q2. Is there a linux version of NBOPro7@Jmol? A. Not yet. Q3. Don’t I get the most up-to-date NBO with the most recent Gaussian version? A. No, you get a slightly doctored 1980’s version (“NBO 3.1”) that the NBO developers consider obsolete and unreliable.

150 | E. D. Glendening et al. Q4. Can NEDA be performed with NBOPro@Jmol? A. No. The NBOPro@Jmol utility can only perform noninteractive (GenNBO-level) tasks that don’t require the 2e integrals and other resources of an interactive host program. Q5. What are those “RY” NBOs that fill up the output with zeros? Do they ever matter? A. The RY-type (“Rydberg” = beyond valence-shell) NBOs are necessary to maintain the “completeness” property of all localized NBO sets, but can usually be ignored for chemical purposes. By a rough perturbative rule of thumb [78], orbitals with small occupancy (e. g., 0.0001e) contribute comparably small stabilization energy (in atomic units!: 0.0001 a. u. ≈ 0.06 kcal/mol). Q6. What about the “RDM2” (2nd -order reduced density matrix) option that was supposed to be a feature of NBO7? A. A substantial infrastructure for $WF-keylist input of correlated (multi-configurational) wavefunctions and Γ(2) -based analysis of electron correlation effects is installed in NBO7, but the user interface (keyword I/O syntax) is not yet implemented. When that’s available, announcements will appear on the NBO7 website and licensed NBO7 users will be able (as always) to download the “latest and best” NBO7 revision from the website repository with their original download code. Q7. Why do AIM and NBO charges differ? A. A systematic difference between AIM and NBO charges for atoms of different size was discussed by Perrin [79]. However, comparative studies often indicate surprisingly good correlation between AIM and NBO charges compared to the numerous available alternatives [80, 81]. An extreme AIM/NPA difference (that we believe exposes a vulnerability of NPA in the completely ionized limit) is found in the controversial case of Ca(CO)8 analysis [68]. Q8. Why are there “missing bonds” in NBO results for many hypervalent species? A. NBO depiction of SF4 [Figure 6.3(e)] exemplifies this issue. In symmetric 3c/4e hyperbonding, each “half-bond” can appear in only one of the two (equivalent!) NBO structures that contribute to the resonance hybrid. Q9. Help, I’m overwhelmed by the number of NRT resonance structures. How can I gain any simple picture of what’s going on? A. In a complex molecule with many resonance centers (e. g., of allylic or amide type), the number of significant contributing structures will inevitably grow exponentially with the number of such centers. We recommend focusing on the bond orders around the center of interest, which exhibit correlations with bond lengths and other experimental properties while converging toward well-defined limiting values, even if individual resonance weightings do not. An alternative is to employ local NRT [55], including only atoms near the resonance center of principal interest. Q10. I’m trying to understand vertical trends in π-backbonding for M(CO)3 (M=Ni, Pd, Pt) species, but NBO gives an inconsistent number of M–C bonds in these species (one for M=Ni; none for M=Pd, Pr). How can I get an apples-to-apples comparison?

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A. In such cases where two competing resonance structures have nearly equal weighting, use $CHOOSE-keylist input [82] to insure a consistent NBO bonding pattern for the compared species. Q11. Help, my head is spinning! How can I keep from getting lost in all those NBO-specific acronyms (NPA, NLS, NAO, NEC, NHO, PNHO, $DEL, NRT, RNBO, NLMO, PRNBO, NEDA,…)? A. Unfortunately, NBO 7.0 program usage requires some baseline familiarity with associated acronymic gibberish that afflicts even “introductory” accounts of why and how the program may prove useful to non-specialists. The “N” is practically always “natural” (evoking Löwdin’s seminal development of “natural orbital” concepts in the 1960s), “R” is resonance, and “O” is “orbital,” as often modified in familiar atomic (AO), hybrid (HO), bond (BO) or localized molecular (LMO) variants. The prefix “P” (or “(P)”) distinguishes an idealized “visualization” form of the orbital (preliminary to interatomic orthogonalization) from the actual “computational” object, allowing Mulliken-type maximum-overlap assessments of interaction strength in orbital imagery. The “N” may also be prefixed to other common technical abbreviations—such as “population analysis” (PA), “electron configuration” (EC), “resonance theory” (RT) or “energy decomposition analysis” (EDA)—to compose actual keywords or (with “$” prefix) keylist labels for program input, all connecting to specific entries in the on-line NBO Manual.

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Foster JP, Weinhold F. Natural hybrid orbitals. J Am Chem Soc. 1980;102:7211–8. Reed AE, Curtiss LA, Weinhold F. Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint. Chem Rev. 1988;88:899–926. Weinhold F. Natural bond orbital methods. In: Schleyer PvR, Allinger NL, Clark T, Gasteiger J, Kollman PA, Schaefer HF, Schreiner PR, editors. Encyclopedia of Computational Chemistry. vol. 3. Chichester, UK: John Wiley & Sons; 1998. p. 1792–811. Glendening ED, Badenhoop JK, Reed AE, Carpenter JE, Bohmann JA, Morales CM, Karafiloglou P, Landis CR, Weinhold F. NBO 7.0: Natural Bond Orbital Analysis Programs. Madison WI: Theoretical Chemistry Institute, U. Wisconsin; 2018. Glendening ED, Landis CR, Weinhold F. NBO 7.0: New vistas in localized and delocalized chemical bonding theory. J Comput Chem. 2019;40:2234–41. Löwdin P-O. Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals, and convergence problems in the method of configurational interaction. Phys Rev. 1955;97:1474–89. Parr RG, Yang W. Density-Functional Theory of Atoms and Molecules. London: Oxford U. Press; 1989. Andrés J, Ayres PW, Boto R, Carbó-Dorca R, Ciolowski J, Chermette H, Contreras García J, Cooper D, Frenking G, Gatti C, Heidar-Zadeh F, Joubert L, Martin Pendas A, Matito E, Mayer I,

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Misquitta A, Mo Y, Pilmé J, Popelier P, Rahm M, Ramos-Cordoba E, Salvador P, Schwarz E, Shahbazian S, Silvi B, Solà M, Szalewicz K, Tognetti V, Weinhold F, Zins E-L. Nine questions on energy decomposition analysis. J Comput Chem. 2019;40:2248–83. Foresman JB, Frisch Æ. Exploring Chemistry with Electronic Structure Methods. 3rd ed. Wallingford CT: Gaussian Inc.; 2015. Weinhold F, Carpenter JE. Some remarks on nonorthogonal orbitals in quantum chemistry. J Mol Struct, Theochem. 1988;165:189–202. Levine IN. Quantum Chemistry. 5th ed. Upper Saddle River NJ: Prentice-Hall; 2000. p. 169. Reed AE, Weinstock RB, Weinhold F. Natural population analysis. J Chem Phys. 1985;83:735–46. Mulliken RS. Electronic population analysis on LCAO-MO molecular wave functions. J Chem Phys. 1955;23:1833–40. See the NBO-affiliated programs link of the NBO7 website for current listing and URL contacts. Weinhold F, Phillips D, Glendening ED, Foo ZY, Hanson RM. NBOPro7@Jmol. Madison WI: Theoretical Chemistry Institute, U. Wisconsin; 2018. Weinhold F, Landis CR. Discovering Chemistry with Natural Bond Orbitals. Hoboken NJ: John Wiley; 2012. Weinhold F, Landis CR. Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective. Cambridge UK: Cambridge University Press; 2005. Glendening ED, Landis CR, Weinhold F. Natural bond orbital methods. WIREs Comput Mol Sci. 2012;2:1–42. Weinhold F. Natural bond orbital analysis: a critical overview of its relationship to alternative bonding perspectives. J Comput Chem. 2012;33:2363–79. Weinhold F, Landis CR, Glendening ED. What is NBO analysis and how is it useful? Int Rev Phys Chem. 2016;35:399–440. Zhao L, Schwarz WHE, Frenking G. The Lewis electron-pair bonding model: the physical background, one century later. Nature Rev Chem. 2019;3:35–47. See the online NBO 7.0 Program Manual, Sec. B.4. Suidan L, Badenhoop JK, Glendening ED, Weinhold F. Common textbook and teaching misrepresentations of Lewis structures. J Chem Educ. 1995;72:583–6. Weinhold F, Carpenter JE. The natural bond orbital Lewis structure concept for molecules, radicals, and radical ions. In: Naaman R, Vager Z, editors. Proceedings of the International Workshop on the Structure of Small Molecules and Ions. New York: Plenum; 1988. p. 227–36. Google Scholar for Ref. [12] currently lists about 8000 citations. Reed AE, Weinhold F. Natural bond orbital analysis of near-Hartree-Fock water dimer. J Chem Phys. 1983;78:4066–73. Reed AE, Weinhold F, Curtiss LA, Pochatko DJ. Natural bond orbital analysis of molecular interactions: theoretical studies of binary complexes of HF, H2 O, NH3 , N2 , O2 , F2 , CO, and CO2 with HF, H2 O, and NH3 . J Chem Phys. 1986;84:5687–705. Shaik S, Hiberty PC. A Chemist’s Guide to Valence Bond Theory. Wiley: Hoboken NJ; 2008. Alabugin IV. Stereoelectronic Effects: A Bridge Between Structure and Reactivity. Wiley: Chichester UK; 2016. Alabugin IV, Bresch S, Dos Passos Gomes G. Orbital hybridization: a key electronic factor in control of structure and reactivity. J Phys Org Chem. 2015;28:147–62. Mulliken RS. Quelques aspects de la théorie des orbitales moléculaires. J Chim Phys. 1949;46:497–542. Wolfsberg M, Helmholtz L. The spectra and electronic structure of the tetrahedral ions MnO−4 , CrO4 2− , and ClO−4 . J Chem Phys. 1952;20:837–43.

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[33] Hoffmann R. An extended Hückel theory. I. Hydrocarbons. J Chem Phys. 1963;39:1397–412. [34] Clauss AD, Nelsen SF, Ayoub M, Landis CR, Weinhold F. Rabbit ears hybrids, VSEPR sterics, and other orbital anachronisms. Chem Educ Res Pract. 2014;15:417–34. [35] Edmiston C, Ruedenberg K. Localized atomic and molecular orbitals. Rev Mod Phys. 1963;35:457–65. [36] Foster JM, Boys SF. Canonical configuration interaction procedure. Rev Mod Phys. 1960;32:300–2. [37] Pipek J, Mezey PG. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions. J Chem Phys. 1989;90:4916–26. [38] Knizia G. Intrinsic atomic orbitals: an unbiased bridge between quantum theory and chemical concepts. J Chem Theory Comput. 2013;9:4834–43. [39] Fock V. Näherungsmethode zur Lösung der quantenmechanischen Mehrkörperproblems. Z Phys. 1930;61:126–48. [40] Landis CR, Cleveland T, Firman TK. Structure of W(CH3 )6 . Science. 1996;272:179. [41] Firman TK, Landis CR. Valence bond concepts applied to the molecular mechanics description of molecular shapes. 4. Transition metals with π-bonds. J Am Chem Soc. 2001;123:11728–42. [42] Weinhold F, Landis CR. Natural bond orbitals and extensions of localized bonding concepts. Chem Educ Res Pract Eur. 2001;2:91–104. [43] Ref. [17], Chapter 4. [44] See the online, NBO 7.0 Program Manual, Section B.5. [45] See e. g., Weinhold F. Why do cumulene ketones kink? J Org Chem. 2017;82:12238–45. [46] Nori-Shargh D, Weinhold F. Natural bond orbital theory of pseudo Jahn-Teller effects. J Phys Chem A. 2018;122:4490–8. [47] Badenhoop JK, Weinhold F. Natural bond orbital analysis of Steric Interactions. J Chem Phys. 1997;107:5406–21. [48] Badenhoop JK, Weinhold F. Natural steric analysis: Ab initio van der Waals radii of atoms and ions. J Chem Phys. 1997;107:5422–32. [49] Glendening ED, Weinhold F. Natural resonance theory. I. General formulation. J Comput Chem. 1998;19:593–609. [50] Glendening ED, Weinhold F. Natural resonance theory. II. Natural bond order and valency. J Comput Chem. 1998;19:610–27. [51] Glendening ED, Badenhoop JK, Weinhold F. Natural resonance theory. III. Chemical applications. J Comput Chem. 1998;19:628–46. [52] Glendening ED, Landis CR, Weinhold F. Resonance theory reboot. J Am Chem Soc. 2019;141:4156–66. [53] Glendening ED, Wright SJ, Weinhold F. Efficient optimization of natural resonance theory weightings and bond orders by Gram-based convex programming. J Comput Chem. 2019;40:2028–35. [54] For specific examples of such correlations in the H-bonding domain, see Weinhold F, Klein RA. What is a hydrogen bond? Mutually consistent theoretical and experimental criteria for characterizing H-bonding interactions. Mol Phys. 2012;110:565–79. [55] Ref. [22], p. B-85ff. [56] Gilli G, Bertolasi V, Ferretti V, Gilli P. Resonance-assisted hydrogen bonding. III. Formation of intermolecular hydrogen-bonded chains in crystals of β-diketone enols and its relevance to molecular association. Acta Crystallogr. 1993;B49:564–76. ̂ [57] Landis CR, Weinhold F. 3c/4e σ-type long-bonding: a novel NBO motif toward the metallic delocalization limit. Inorg Chem. 2013;52:5154–66.

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[58] For controversies of this sort, see e. g., Jerabek P, Roesky HW, Bertrand G, Frenking G. J Am Chem Soc. 2014;136:17123–35. [59] Landis CR, Hughes RP, Weinhold F. Bonding analysis of TM(cAAC)2 (TM = Cu, Ag, Au) and the importance of reference state. Organometallics. 2015;34:3442–9. [60] Frenking G, Krapp A. Unicorns in the world of chemical bonding models. J Comput Chem. 2007;28:15–24. [61] Weinhold F. Chemical bonding as a superposition phenomenon. J Chem Educ. 1999;76:1141–6. [62] Weinhold F, Klein RA. What is a hydrogen bond? Resonance covalency in the supramolecular domain. Chem Educ Res Pract. 2014;15:276–85. [63] Glendening ED, Weinhold F. Resonance natural bond orbitals: efficient semilocalized orbitals for computing and visualizing reactive chemical processes. J Chem Theory Comput. 2019;15:916–21. [64] Pople JA. Theoretical Models for Chemistry. In: Smith DW, editor. Proceedings of the Summer Research Conference on Theoretical Chemistry, Energy Structure and Reactivity. New York: John Wiley & Sons; 1973. [65] Hehre WJ, Radom L, Schleyer PvR, Pople JA. Ab Initio Molecular Orbital Theory. New York: Wiley-Interscience; 1986. p. p 31. [66] Cooper DL. Valence Bond Theory. Amsterdam: Elsevier; 2001. [67] Gupta VP. Principles and Applications of Quantum Chemistry. New York: Academic; 2016. 81ff; ref. [25]. [68] Landis CR, Hughes RP, Weinhold F. Comment on ‘Observation of alkaline earth complexes M(CO)8 (M = Ca, Sr. or Ba) that mimic transition metals’. Science. 2019;365:553. [69] For the many available direct comparisons between NBO and QTAIM descriptors, see Weinhold F. Natural bond critical point analysis: quantitative relationships between NBO-based and QTAIM-based topological descriptors of chemical bonding. J Comput Chem. 2012;33:2440–9. [70] For NBO/NRT applications to a variety of such “noncovalent bonding” phenomena, see Jiao Y, Weinhold F. What is the nature of supramolecular bonding? Comprehensive NBO/NRT picture of halogen and pnicogen bonding in RPH2 ⋅⋅⋅IF/FI complexes (R = CH3 , OH, CF3 , CN, NO2 ). Molecules 2019;24:2090. [71] Shahi A, Arunan E. Hydrogen bonding, halogen bonding and lithium bonding: an atoms in molecules and natural bond orbital perspective towards conservation of total bond order, interand intra-molecular bonding. Phys Chem Chem Phys. 2014;16:22935–52. [72] Glendening ED, Weinhold F. Natural resonance theory of chemical reactivity, with application to intramolecular Claisen rearrangement. Tetrahedron. 2018;74:4799–804. [73] Glendening ED, Streitwieser A. Natural energy decomposition analysis: an energy partitioning procedure for molecular interactions with application to weak hydrogen bonding, strong ionic, and moderate donor-acceptor interactions. J Chem Phys. 1994;100:2900–9. [74] Glendening ED. Natural energy decomposition analysis: extension to density functional methods and analysis of cooperative effects in water clusters. J Phys Chem A. 2005;109:11936–40. [75] http://nbo7.chem.wisc.edu. [76] Dunnington BD, Schmidt JR. Generalization of natural bond orbital analysis to periodic systems: applications to solids and surfaces via plane-wave density functional theory. J Chem Theory Comput. 2012;8:1902–11. https://schmidt.chem.wisc.edu/nbosoftware. [77] https://www.vasp.at/. [78] Ref. [16], p. 104. [79] Perrin CL. Atomic size dependence of Bader electron populations: significance for questions of resonance stabilization. J Am Chem Soc. 1991;113:2865–8.

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[80] See, e. g., Wiberg KB, Rablen PR. Comparison of atomic charges derived via different procedures. J Comput Chem. 1992;14:1504–18. [81] Gross KC, Seybold PG, Hadad CM. Comparison of different atomic charge schemes for predicting pKa variations in substituted anilines and phenols. J Comput Chem. 2002;90:445–58. [82] Ref. [22], p. B-20ff.

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Avital Shurki, Benoît Braïda, and Wei Wu

7 Valence bond theory with XMVB 7.1 Basic elements of nonorthogonal valence bond theory The purpose of this section is to introduce to a complete beginner, as briefly and as didactically as possible, the very basic elements of nonorthogonal Valence Bond (VB) theory that are necessary to go through the examples detailed in Section 7.3, and to run one’s first VB calculations on simple molecules. Section 7.2 complements this chapter by shortly presenting the structure of the XMVB program, the algorithms used, and some of the advanced methods and schemes it features. It is strongly advised to refer to the book by Sason S. Shaik and Philippe C. Hiberty [1] for a more complete introduction to VB theory in general, and to the manual of the XMVB program [2–4] for information about the full capabilities of this program and the various options it offers.

7.1.1 Basic ingredients of VB wave functions Valence Bond theory is, with Molecular Orbital (MO) theory, one of the two original branches of wave function theory that emerged in the late 1920s, when the then new quantum theory was first applied to molecules. The starting point of VB is the wave function proposed by Heitler and London for the ground singlet state of the dihydrogen molecule, equation (7.1.1.1), where “a” / “b” correspond to atomic orbitals localized on the left/right hydrogen, respectively. By convention in the following, a bar on the top of the orbital letter indicates that a ß spin function is attached to the corresponding spatial orbital, while the absence of bar indicates an α spin function (the normalization factor is omitted for the sake of simplicity). Note that because inverting two columns in a determinant changes its sign, the two different but mathematically identical expressions shown in equation (7.1.1.1) may usually be found in textbooks Acknowledgement: AS expresses her gratitude to the ISRAEL SCIENCE FOUNDATION (grant No. 1691/17) for funding. Avital Shurki, Institute for Drug Research, School of Pharmacy, Hebrew University of Jerusalem, Jerusalem 91120, Israel, e-mail: [email protected] Benoît Braïda, Laboratoire de Chimie Theorique, Sorbonne Université, CNRS, Paris 75005, France, e-mail: [email protected] Wei Wu, The State Key Laboratory of Physical Chemistry of Solid Surfaces, Fujian Provincial Key Laboratory of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China, e-mail: [email protected] https://doi.org/10.1515/9783110660074-007

158 | A. Shurki et al. and articles. ̄ ΦHL = |ab|̄ + |ba|̄ = |ab|̄ − |ab|

(7.1.1.1)

This wave function can be interpreted as two localized electrons (one on each atom) exchanging their spins. The energy stabilization due to the mixing of the two spin-determinants accounts for the major part of the bond energy in the H2 dimer, and more generally in two-center two-electron covalent bonds, and thus this two determinant state function is also termed as the Covalent VB structure of H2 in usual VB terminology. This initial proposition was soon generalized, and Slater [5] and Pauling [6–8] applied this new approach to a variety of molecules laying the ground of the new Valence Bond theory. A VB wave function generally expresses itself as a combination of VB structures, where a “structure” in the VB terminology is a spin eigenfunction that corresponds to a specific configuration of the electrons, understood as a given distribution of electrons into the VB orbitals, together with a specific spin pairing of these electrons. These VB structures are also called Heitler–London–Slater–Pauling (HLSP) functions. For instance, a full valence VB wave function for the ground state of the dihydrogen molecule expresses itself as a superposition of one covalent (the Heitler–London function) and two equivalent ionic structures: ̄ + C2 |aa|̄ + C3 |bb|̄ ΦVB = C1 (|ab|̄ + |ba|)

(7.1.1.2)

A fundamental ingredient of the so-called “Classical” VB approach, and what constitutes the schism with MO theory, is the use of “strictly” localized orbitals, i. e., orbitals that are atom or fragment centered. Such orbitals are referred to as “HAOs” (for: “Hybrid Atomic Orbitals”) in the XMVB program. As illustrated in Scheme 7.1.1, using strictly localized orbitals is what allows the direct correspondence between the different (covalent and ionic) components of the VB wave function, and the corresponding chemical structures in the Lewis model. In this original formulation, the VB wave function therefore appears as inherently achieving a direct mapping between quantum theory and the Lewis model.

Scheme 7.1.1: Pictorial representation of the Classical VB wave function for the ground state of the dihydrogen molecule, expressed in equation (7.1.1.2). Top: spin-orbitals occupation for each VB determinant. Bottom: corresponding Lewis structures.

Many concepts familiar to chemists, such as resonance/mesomery, hybridization, “arrow-pushing” language for chemical mechanisms, naturally emerge from VB wave

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functions. This explains the immediate initial success of VB theory, and why it is still very appealing to apply VB on chemical problems nowadays. An alternative is to still consider a wave function that is a superposition of VB structures, but to allow the orbitals to partially or fully delocalize onto the whole molecule, following the Coulson–Fischer proposition [9]. In the XMVB program, VB orbitals that are delocalized between two centers only are referred to as “Bond Distorted Orbitals” (BDO), while they are referred to as “Overlap Enhanced Orbitals” (OEOs) when they are free to delocalize onto the whole molecule. This alternative has given rise to the so-called “semilocalized” VB approaches, such as the Generalized Valence Bond (GVB) or Spin-Coupled Valence Bond (SCVB) methods [10, 11]. These methods have the advantage to provide more compact wave functions, because many ionic structures will be implicitly included through the delocalization of the orbitals and, therefore, are usually not explicitly included. However, their interpretative capabilities are also more restricted than in the “Classical” VB approaches, which uses “strictly” localized orbitals (HAOs) and explicit covalent/ionic superposition. For the sake of keeping this chapter short, we will essentially focus on the latter type of VB wave functions and methods in the following. Another common ingredient in practical VB applications is the use of an active/inactive space separation. For instance, to obtain calculated structure weights for Kekule and Dewar Lewis structures of Benzene, it is preferable to consider a VB wave function where only the six π electrons and orbitals are defined as “active” and described at the VB level (i. e., the six π electrons are distributed into localized atomic p orbitals, and spin-coupled together in different ways), while the σ electrons/orbitals are defined as “inactive” and described using delocalized MOs, as illustrated in Scheme 7.1.2. This active/inactive separation allows to keep a compact wave function, and to focus the VB analysis on the electrons of interest.

Scheme 7.1.2: Pictorial illustration of the VB wave function of benzene with the σ system taken as inactive (blue orbitals) and the π system as active (red orbitals and electrons).

A direct consequence is that, when one wants to perform any VB calculation, it is necessary to first define a precise chemical question to address, which in turn will guide the choice of the active space. To illustrate this point, let us consider for instance that the purpose is to study an SN 2 reaction with F− as the attacking nucleophile and Cl− as the leaving group. During this process, the Cl–C bond is broken while a new C–F bond is created. Therefore, two electron pairs should be considered as active, and three active orbitals (those involved in the VB description of the broken Cl–C and new C–F bonds) may be considered. The other electron pairs will not rearrange significantly

160 | A. Shurki et al. during the process, therefore, they could be defined as inactive and described using delocalized MOs. One of the VB structures that will be generated from this choice of active space is shown in Scheme 7.1.3. This key point will be further illustrated through more examples and practical XMVB inputs and outputs in Section 7.3.

Scheme 7.1.3: One of the VB structures involved in the SN 2 reaction of fluorine anion on chloromethane. The three active VB orbitals and four active electrons are in red, while the inactive electrons and orbitals are in blue.

7.1.2 Rumer basis of structures Rumer has proposed an algorithm which, from a given electron configuration and a given spin state, enables to generate a complete nonredundant basis of VB structures. A detailed algebraic presentation of the method could be found elsewhere [12], together with the introduction to other possible spin eigenfunctions. In this subsection, a simple graphical version of Rumer’s algorithm will be introduced through selected basic examples. Let us start with the singlet ground state of the benzene molecule, with the π system taken as active. First, one has to choose a specific electron configuration, i. e., a given distribution of the six active electrons into the six (atomic/fragment) active orbitals. A possible choice, which will lead to the generation of the fully covalent VB structures, is to distribute one electron in each active orbital, as illustrated in Scheme 7.1.4. Then, in a second step, one has to spin-couple the electrons by pairs, generating three singlet pairs that would correspond to one possible covalent structure. There are overall 15 possible ways to spin-couple into singlet pairs six electrons in six orbitals (6e/6o), therefore, there are overall 15 possible covalent structures. However, this set of structures is overcomplete: each of these 15 covalent structures could be expressed as a linear combination of a subset of the others. Therefore, one

Scheme 7.1.4: One electron configuration for the π electrons of benzene with one electron in each active orbitals.

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has to make a selection among all possible covalent structures, to form a complete and nonredundant subset of covalent structures. A systematic method for generating such an independent subset composed of chemically meaningful structures for benzene is the following. First, one has to put the orbitals around an imaginary circle, that does not need to have the shape of the molecule. Second, one has to spin-couple electrons by pairs, in order to generate all possible VB structures not displaying crossing bonds. This “non-bond-crossing rule” is very important, as it allows to keep independent structures (which for cycle like molecules are also chemically meaningful structures) and eliminate redundant ones. In the case of the benzene, as illustrated in Scheme 7.1.5, application of this method leads to a complete set of five nonredundant covalent structures, which could be mapped to the two Kekule and three Dewar structures of benzene. Such a set is called Rumer basis of covalent structures.

Scheme 7.1.5: Application of Rumer’s graphical method to the configuration displayed in Scheme 7.1.4, and for a singlet spin state. Five chemically meaningful structures are selected (the two Kekule and three Dewar structures) to form a complete set of nonredundant covalent VB structures.

Butadiene (Scheme 7.1.6) is a simple case of an open molecule. The procedure remains the same: one first has to order the orbitals into a circle, distribute the electrons, and then spin-couple them in all different ways eliminating structures that display crossing bonds. In this case, a complete set of two covalent structures, corresponding to

Scheme 7.1.6: Application of Rumer’s graphical method to the electronic ground state of butadiene (see text for details).

162 | A. Shurki et al. the usual resonant chemical description, is obtained. Note that without the secondary structure 2, the butadiene would undergo free rotation around its central CC bond at room temperature, the energy stabilization due to the resonance between 1 and 2 being the main cause for the observed rotation barrier. How about ionic structures? First, one has to start with a different electron configuration, now displaying some doubly occupied active orbital(s) and some vacant ones. Then one has to repeat again the same procedure by pairing in different ways the electrons that singly occupy the active orbitals, and to apply the “non-bond-crossing rule” to discard redundant and often nonchemically meaningful structures. Such an example for benzene is shown in Scheme 7.1.7, in the case of a mono-ionic configuration (one doubly occupied and one vacant VB orbital). Let us note that when two electrons occupy the same atomic orbitals, they necessarily bear different spin by virtue of Pauli principle, and will thus be paired together. Therefore, it is important to remember that only the electrons that singly-occupy some of the active orbitals could be paired together in different ways.

Scheme 7.1.7: One possible mono-ionic electron configuration (left), and the Rumer basis of two associated singlet VB structures (right) for the benzene molecule.

Then the procedure shall be repeated for each possible mono-ionic electron configuration to generate a Rumer basis of mono-ionic structures, and then repeated for electron configurations involving two or three ionic pairs, to finally obtain a complete Rumer basis of structures. An important notice regarding the XMVB program. This program features an algorithm that automatically generates a complete set of nonredundant structures (see manual and examples given in Section 7.3). However, the program is not able to “guess”, for a given structure, whether some specific bonds are crossing or not in the real molecule, and thus such an automatically generated set may include nonchemically meaningful structures displaying crossing bonds. The user has to be aware and cautious about that. In such cases, usually a well-chosen permutation of orbitals in the orbital section may help to obtain a basis of structures which are chemically meaningful. The method extends similarly to the case of higher spin states. For instance, covalent structures corresponding to the triplet π states of Benzene could be generated in the same way as illustrated in Scheme 7.1.5, but by spin-pairing only four of the six electrons, leaving two spin-up electrons into two of the active orbitals.

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The method also easily extends in the case where the number of active electrons is different than the number of active orbitals, as shown in Scheme 7.1.8 in the case of cyclopentadienyl anion, for one of the possible electron configurations.

Scheme 7.1.8: One possible electron configuration (left), and the Rumer basis of two associated singlet VB structures (right) for cyclopentadienyl anion.

A little more subtle situation is the case of noneven number of active electrons, such as in radicals. The methodology, in this case, is illustrated in Scheme 7.1.9 for the H3 prototype molecule. The trick is to get back to the situation of an even number of electrons by, in an intermediate stage, adding a fictitious center (“X” in Scheme 7.1.9) bearing an extra active electron. Then the standard graphical method previously detailed can now be applied, which in the case of H3 allows to select two covalent structures out of the three possible ones. In a last stage, the fictitious center is erased and its extra electron removed, leading to the complete set of two covalent structures displayed in the right-hand side of Scheme 7.1.9.

Scheme 7.1.9: Application of Rumer’s graphical method in the H3 ∙ case (see text for details).

7.1.3 The VBSCF and BOVB methods Let us start by briefly summarizing the previous two sections. An active space of electrons is first defined, motivated by a given chemical question. In Classical VB approaches, these active electrons will occupy “strictly” localized nonorthogonal orbitals, while the remaining (inactive) electrons will occupy in pairs more often delocalized molecular orbitals. The VB wave function then can be expressed as an expansion onto a Rumer basis of VB structures: N

Ψ = ∑ CK ΦK k=1

(7.1.3.1)

164 | A. Shurki et al. where N is the total number of nonredundant structures, ΦK a given VB structure with its corresponding coefficient CK . In practical calculations, a basis set is then chosen, typically among the same standard families of gaussian basis sets used for DFT or MO-based methods. Then the VB active orbitals are expanded onto a subset of the whole basis of functions, composed of the basis functions centered onto some specific atom(s) of the molecule. This is the main difference with MO-based wave function or DFT with respect to the orbitals. In the latter case, each MO is in principle expanded onto the whole basis of functions, leading to orbitals delocalized onto the whole molecule. Quite in contrast, in Classical VB approaches the orbitals must remain “strictly” localized on a given atom or fragment, and thus each VB orbital will be restricted to expand on the subset including only the basis functions that are centred onto a specific atom/fragment. The definition of the different fragments / subsets of basis functions is left to the user, because it depends on the initial chemical questions one wants to answer. As a simple illustration, equation (7.1.3.2) displays the expressions of the two atomiccentred VB orbitals a and b used in equation (7.1.1.1) and (7.1.1.2), developed onto the subsets (χμA ) and (χμB ) of the basis functions centred on the hydrogen atoms A and B, respectively: nA

basis { { { a = ∑ cμA χμA { { { { μ=1 { { nBbasis { { { B B { { b = ∑ cμ χμ μ=1 {

(7.1.3.2)

cμA and cμB being the coefficients of the VB orbitals onto the basis (to be optimized),

nAbasis and nBbasis the total number of basis functions centered on atoms A and B, respectively. The procedure to initiate a VB calculation on a new molecule is thus the following. The user should first consider a well-defined chemical question, which will determine the active set of electrons and orbitals, and a specific geometrical fragmentation of the molecule into localized fragments. Then, combined with possible symmetry considerations, this leads to a choice of definition of the different subsets of basis functions. This general approach will be put in practice through examples in Section 7.3. Another technical consequence is that, to keep “strictly” localized VB orbitals, one has to work with nonorthogonal orbitals, which dramatically complicates the algebra of matrix-element evaluation and more generally the formulation of efficient wave function optimization algorithms. The “0 level” of modern ab initio methods belonging to the Classical VB family is the VBSCF method [13, 14, 28]. From a wave function of the type expressed in equation (7.1.3.1), and with VB orbitals (active and inactive) expanded onto chosen basis functions, the VBSCF wave function is then obtained after simultaneous optimization

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of all parameters (structure coefficients, and coefficients of the orbitals onto the basis) in order to minimize the expectation value of the Hamiltonian following the variational principle. Basically, the VBSCF method is just a MCSCF method using localized active and nonorthogonal orbitals. Specific algorithm used in the XMVB program under the context of nonorthogonal orbitals will be briefly discussed in the next section. When a complete basis of (covalent plus ionic) structures is used, the VBSCF wave function includes essentially the same physical content as a CASSCF wave function with a similar active space, i. e., these two functions include only the so-called “static” (nondynamical) correlation, and thus both methods display similar performances as energy differences are concerned (bond dissociation energies, excitation energies, etc.). A pictorial representation of the VBSCF wave function for the F2 molecule, with the F–F bonding pair and corresponding orbitals taken as active, is shown in Scheme 7.1.10 below:

Scheme 7.1.10: Pictorial representation of the VBSCF wave function for the difluorine.

There are many methods available in XMVB that include both static and dynamical correlation effects. The principles of the BOVB method, which has been the most used for applications, and which will be used in examples given in Section 7.3, are didactically presented here. Additional post-VBSCF and hybrids methods available in XMVB are presented in the next section. In the BOVB method [15, 16, 29], the same set of structures as in the VBSCF method (or, more often, a subset of it) is used. Then, similarly as for the VBSCF method, the structure coefficients and all orbital coefficients onto the basis are simultaneously optimized to minimize the total energy. Besides, in the BOVB method, different sets of orbitals are used for different structures, while a common set of orbitals is used for all structures in the VBSCF wave function. Using different sets of orbitals for different structures allows the orbitals to fluctuate in size and shape, so as to fit the instantaneous charges of the atoms on which these orbitals are located, as pictorially illustrated in Scheme 7.1.11 as compared with Scheme 7.1.10.

Scheme 7.1.11: Pictorial representation of the BOVB wave function for the difluorine.

166 | A. Shurki et al. This so-called “breathing orbital effect” brings into the VB wave function the necessary differential dynamical correlation to calculate accurate energy differences, such as bond dissociation energies or Resonance Energies for instance. The BOVB method has been extensively applied on many different systems over the time, and shown to provide consistently good observables. One very appealing feature of the BOVB method, is that it expands on the same minimal structure set that serves the VBSCF wave functions (only orbitals differ), and thus BOVB allows to include dynamical correlation while keeping a very compact wave function. Just to provide one simple example, Table 7.1.1 displays the bond dissociation energy that can be obtained for the difluorine molecule with different levels of theory, illustrating the need for including both static and dynamical correlation to obtain a good estimation of this quantity. Table 7.1.1: Electronic Bond Dissociation Energy (BDE) of F2 , in kcal.mol−1 , obtained from different MO and VB based wave function methods using the Def2-TZVPP basis set (this work). Experimental F–F distance of 1.412Å has been used. Method BDE

RHF

CASSCF

VBSCF

L-BOVB

D-BOVB

SD-BOVB

Estimated exact [17]

−31.3

15.4

12.8

29.4

35.1

38.2

39.0

Different variants of the BOVB method exist. At the lowest level, commonly referred to as L-BOVB, all orbitals, both active and inactive alike, are strictly localized on a given atom or fragment (thus the meaning of the “L” prefix). Because of the simultaneous optimization of several sets of orbitals, the orbital coefficient hypersurface of BOVB wave functions is usually very complicated, featuring many unphysical local minima. Therefore, it is very important to perform a L-BOVB calculation using an already converged VBSCF set of orbitals as a guess. A more accurate level is the D-BOVB level, where the inactive orbitals are fully delocalized over the whole molecule (thus the “D” prefix). In order to avoid instabilities during the optimization or convergence to unphysical minima, and in particular orbital flipping between inactive and active sets of orbitals, it is necessary to adopt a two-step strategy to perform a D-BOVB calculation: (i) first perform a L-BOVB calculation, and then: (ii) perform, using the converged L-BOVB orbitals as a guess, a D-BOVB calculation where the active orbitals are kept frozen while the inactive orbitals are reoptimized and allowed to delocalize. In case of molecules displaying σ/π symmetry, the simpler σ-D-BOVB or π-DBOVB alternatives may be used. In the σ-D-BOVB wave function, all σ orbitals are inactive and delocalized while all π orbitals are localized, whereas it is the opposite for the π-D-BOVB wave function. Because the different subsets of basis functions

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on which the delocalized and localized orbitals expand respectively are completely disjoint, there is no risk of orbital flipping as previously mentioned for the D-BOVB method and, therefore, a direct full optimization (both active and inactive orbitals simultaneously) can be performed starting from a σ-D-VBSCF or π-D-VBSCF orbital guess. Last, a highest SD-BOVB level is also defined, where the “S” prefix means that the active ionic pairs are “split,” i. e., they are described by two singlet coupled electrons into two different orbitals that are localized in the same center or fragment. This splitting of ionic pairs allows to include some extra dynamical correlation for the active electron pairs, usually radial correlation. This level is more delicate to use, and only brings little to moderate improvement to energy differences in most cases. Because each structure will have its own set of orbitals, and all of these sets have to be optimized simultaneously, the cost of the BOVB wave function optimization rapidly increases with the number of structures. Therefore, it is in general necessary to perform a selection of structures, and only keep in the BOVB calculations structures that come out with a nonnegligible weight at the VBSCF level. As a thumb rule, it is recommended to include in BOVB calculations only structures displaying a VBSCF weight larger than 1 %. Another reason for performing this structure selection is that very minor structures will only have a faint to negligible impact on the total BOVB energy and, therefore, their associated orbital coefficients will be source of convergence difficulties during the optimization process. Another very important point regarding the BOVB method is to keep active orbitals strictly localized on a specific atom and fragment, i. e., to use HAO type of orbitals only, and never use BDOs or OEOs (see Subsection 7.1.1 for the definition of these acronyms). Using active orbitals that are delocalized in some way or another together with explicit covalent/ionic superposition will induce severe redundancies in the wave function, because the ionic structures will also be implicitly included in the covalent structures through the delocalization of the active orbitals. At the BOVB level, this typically results in severe convergence issues, and to an unphysical wave function with an abnormally low total energy. A similar issue may arise when very large basis sets are used, particularly when diffuse functions with very small exponents are included into the basis. In this case, a VB orbital that is formally localized onto a specific atom or fragment can in practice become spatially delocalized in real space. In order to avoid this issue, it is recommended to use only moderate-size basis sets, such as typically the 6-311G** or cc-pVTZ basis, and include diffuse functions only when dealing with a real anionic system bearing an overall negative charge.

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7.1.4 Interpretative quantities One of the main interests of using VB theory are its interpretative capabilities. Molecular chemists have based their understanding of the electronic structure of molecules on models featuring localized electrons and electron pairs (Lewis model, VSEPR, etc.). Many interpretative methods have been proposed to connect a calculated quantum wave function or density to the chemists’ localized vision, some among the most popular being presented in this book. In a usual interpretative method, the user always has to proceed in two stages: first choose a quantum chemistry method to compute a wave function or a density, and, in a second stage, apply a given interpretative method to extract a localized vision and interpretative quantities from this pre-calculated wave function or density. By contrast, because they are based on localized orbitals, VB wave functions are directly interpretable in terms of localized electron pairs, and many concepts familiar to chemists naturally come out of it. New VB-related concepts and models have also been developed relatively recently, and formulated using the standard chemists’ language, enriching the VB toolbox. Let us mention in particular the VB diagrams model [18–20], which is a very powerful tool to rationalize in particular trends in reactivity; and the concept of Charge–Shift bonds, which forms a new and unique class of chemical bonds along with the already long-known (polar-)covalent and ionic bonds [21]. The following will only focus on briefly introducing two very common interpretative quantities: the VB structure weights, and calculation of Resonance Energies. A wave function of the type defined in equation (7.1.3.1) could be interpreted as a probabilistic superposition of outcomes described by the different VB structures, each representing a possible localization and spin-pairing of the electrons. The VB structure weights would then correspond to the individual probability of these different possible outcomes for a given state. Because non-orthogonal orbitals are used, the different VB structures overlap with each other, and thus the square of the coefficients CK2 in equation (7.1.3.1) do not directly correspond to the probability of a given outcome. In particular, the CK2 do not sum to 1. Several formulas have been proposed to define proper structures weights, the most popular and commonly used being the Chirgwin–Coulson weight definition [22]. Within this definition, the weight Wk of a given VB structure ΦK is I =K ̸

WK = CK2 + ∑ CK CL MKL 1≤I≤N

(7.1.4.1)

where MKL is the overlap between the structures ΦK and ΦL . The structure weights, which can be directly calculated from an optimized VB wave function, enable to quantify a given chemical description, and directly answer some chemical questions. For instance, it has been shown in the case of 1,3 dipoles that VB weights are an accurate quantitative measure of the diradical character of molecules [23], and that the diradical weight directly correlates with the reactivity of 1,3 dipoles [24].

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Another commonly used quantity for interpretative purposes is the Resonance Energy (RE). In VB theory, RE is generally taken as the energy lowering of the dominant Lewis structure (or bonding mode) when secondary structures are allowed to mix with it. It can thus be interpreted as the energetic stabilization due to the electron fluctuation between different bonding modes. Practically, RE can simply be computed as the difference between the energy of the dominant Lewis structure alone, minus the energy of the full multistructure wave function. Let us illustrate how to compute a RE with the example of formamide. The simplest minimal resonance description of the π system of formamide is a superposition of the two structures displayed in Scheme 7.1.12, a covalent and an ionic structure. A possible question is what is the contribution of the resonance between these two structures to formamide energetic stabilization. This can be easily addressed using VB theory. First, one has to perform a VB calculation (VBSCF and then BOVB, for instance) while including the two structures 1 and 2 in the calculation. One then has to perform a second separate calculation by including only the major VB structure 1. In this second calculation, because 2 is not included, the resonance is basically “switched off” in the VB wave function. These two calculations will provide two different energies, and the RE is then simply obtained as the difference between these two energies. In this particular case, such a simple RE calculation at the BOVB level has enabled to prove that resonance is indeed the main cause for the rotation barrier of formamide [25], a point that was controversial at that time.

Scheme 7.1.12: Illustrating the method for computing the resonance energy in the π state of formamide, using a minimal two-structure VB wave function.

The procedure to compute a RE just illustrated here will lead to the so-called “variational” RE. There is another method for calculating the RE, the “nonvariational” method, which, for the separate structure energies, considers the diagonal terms of the Hamiltonian of the full structure wave function Estruc(i) = E(Φi ) = Hii + Enuc

(7.1.4.2)

where Hii is the diagonal term of the Hamiltonian of the full structure wave function (Ψ(1↔2) in example displayed in Scheme 7.1.12), and Enuc the nuclear energy. This method thus does not need step 2) in Scheme 7.1.12 to be performed, only one VB wave function optimization is enough. However, it will provide RE higher than the

170 | A. Shurki et al. variational method, because the individual structures will not be computed with their optimal set of orbitals, therefore, the “variational” method is the preferred one as a general rule. Practical examples of calculations and interpretations using VB weights and RE, with details of XMVB inputs, will be provided in Section 7.3.

7.2 Xiamen Valence Bond Package Xiamen Valence Bond (XMVB), developed by the Valence Bond Group of Xiamen University, is a quantum chemistry software package written in Fortran 90 for performing electronic structure calculations based on the nonorthogonal valence bond theory. The current released version, which was also used for the practical examples in Section 7.3, is XMVB 3.0. The XMVB package is composed of two programs, XMVB and XMVB-GUI, and several utilities. XMVB is the core program to perform VB calculations. Currently, there are two released editions available for users: module and standalone package. The module edition is a module for the GAMESS-US package [26], while the stand-alone edition performs VB computations as a stand-alone program. XMVBGUI is a graphical user interface for XMVB, which helps users to build XMVB input files, visualize XMVB outputs and provide graphs of VB structures and VB orbitals. Currently, XMVB-GUI supports Windows and Mac OS X platforms.

7.2.1 Structure of the XMVB program The structure of XMVB package is shown in Scheme 7.2.1, in which the procedures in red are only available in the module edition. To run a XMVB calculation, proper input files, with extensions “.xmi” and “.gus” (optional), should be prepared. For the module edition, a GAMESS-US input file, “.inp” should be also provided. After parsing the input files properly, the electronic integrals of basis functions will be computed either by an integral-computing routine, PREINT, in the stand-alone edition or from the GAMESS-US package for the module edition. Cholesky decomposition for electronic repulsion integrals (ERIs), is also implemented in PREINT of the stand-alone edition, which is very helpful for molecules with large inactive electronic space [27]. Nevertheless, the current version of PREINT does not support pseudo potentials yet. Once the electronic integrals are prepared, a requested VB calculation is performed. In the current version of XMVB, VBSCF [13, 14, 28], BOVB [15, 16, 29], VBCI [30, 31] and VBPT2 [32, 33] are available both in the stand-alone edition and in the module edition. The hybrid methods, DFVB, VBEFP and VBPCM, are available only in the module edition due to the fact that the DFT and solvation modules in GAMESS-US are required.

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Scheme 7.2.1: Structure of XMVB package. The procedure in red is available in GAMESS-US module only.

After calculations finish, molecular properties are computed and computational results printed. There are several output files of XMVB, called XMO, ORB, DEN and XDAT files, with extension names “.xmo”, “.orb”, “.den” and “.xdat”, respectively. The XMO file is the main output file that includes all the information of the calculation. The ORB file stores VB orbitals, which can be used later as an initial guess for follow-up calculations. The DEN file stores the density matrix in terms of basis functions and orbitals, respectively, and the natural orbitals and corresponding occupation numbers. The XDAT file records the initial orbital guess, orbital overlaps, charge population, etc. The XDAT and DEN files may be used for visualization.

7.2.2 Orbital optimization The motivation of the development of the XMVB package is to provide chemists with a well-designed and efficient program to perform both “classical” (i. e., based on strictly localized orbitals) and “modern” (i. e., based on semi-localized orbitals) valence bond calculations at the ab initio level. To this end, the XMVB package uses Heitler–London–Slater–Pauling (HLSP) functions as state functions, and VB orbitals in XMVB may be flexibly defined: they may be strictly localized, HAOs, fully delocal-

172 | A. Shurki et al. ized OEOs, or bond-distorted (BDOs), depending on particular applications. As such, the orbital optimization process is much more difficult than in molecular orbital methods, in which orthogonal delocalized orbitals are used. In the XMVB program, orbital optimization is based on the limited memory Broyden–Fletcher–Goldfarb– Shanno (LBFGS) method [34]. Scheme 7.2.2 plots the flow chart of orbital optimization procedure.

Scheme 7.2.2: The flow chart of VBSCF and BOVB orbital optimization (see text for details).

In the current version of XMVB, two computational methods, VBSCF and BOVB, involve orbital optimization procedure. The evaluation of the gradients with respect to orbital coefficients, which are required in the LBFGS algorithm, is performed with various schemes. The recently developed nonorthogonal orbitals based on Reduced Density Matrices (RDM) approach [35] was implemented in the XMVB program. In the RDM-based approach, the energy gradients are expressed in terms of the products of orbital integrals and the 3-body RDMs, which can be efficiently computed by the use of enhanced Wick’s theorem [36]. This is the recommended gradient-based algorithm

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for VBSCF calculation in the XMVB program, and it is called by specifying the keyword “iscf=5” in the input file (see Section 7.3 for examples). Alternatively, a generalized Fock-matrix based algorithm [37] was also implemented in XMVB, in which the energy gradients are explicitly expressed as the products of Hartree–Fock-like matrix elements and transition density matrix elements in terms of basis functions (keyword: iscf=2). As such, Electronic Repulsion Integral (ERI) transformation from basis functions to orbitals is not required. Consequently, this algorithm works efficiently and is the recommended one for BOVB calculations (iscf=5 is not yet implemented for BOVB wave functions). The Hessian-based Newton–Raphson (NR) algorithm is also available for the VBSCF orbital optimization in XMVB (keyword: iscf=6). The Hessian matrix is evaluated by using tensor contractions of electronic integrals and N-body RDMs. The NR algorithm is implemented in XMVB with the help of a recently developed automatic formula/code generator [38]. Newton–Raphson algorithm shares a performance of quadratic convergence, and thus has much better convergence behavior than the gradient-based LBFGS algorithms. However, as usual for NR approaches, it needs a decent initial orbital guess to reach convergence. Note also that this algorithm only supports VBSCF wave function optimization, not BOVB. Whether the gradient-based (iscf=5) or Hessian-based (iscf=6) algorithm should be preferred for VBSCF calculations depends on the situation.

7.2.3 Post-VBSCF methods VBSCF is the basic approach in the ab initio VB theory, which lacks dynamic correlation, like its MO-based analogue, MCSCF. With the optimal orbitals obtained from VBSCF calculation, dynamic correlation can be remedied by utilizing post-VBSCF techniques. Currently, in addition to BOVB two more post-VBSCF methods, valence bond configuration interaction (VBCI) and valence bond second order perturbation theory, VBPT2, are implemented in the XMVB program. Orbitals used in VB calculations are divided into three subsets: inactive, active, and virtual orbitals. In the VBSCF method, the total energy is determined only by the occupied inactive and active orbitals, though virtual orbitals are used for orbital optimization. In the latter two post-VBSCF methods, virtual orbitals are used for building excited HLSP functions, and thus the total energy is virtual orbitals dependent. In XMVB, two different definitions are used for virtual orbitals. One is to define strictly localized virtual orbitals, like occupied HAOs. The other is to employ delocalized virtual ones, even though occupied HAOs are used for VBSCF. The advantage of using localized virtual HAOs is that the excited HLSP functions correspond to classical VB structures unambiguously. To this end, a Schmidt orthogonalization is done for each fragment, and thus these virtual HAOs are orthogonal to the occupied HAOs of the same

174 | A. Shurki et al. fragments, but are nonorthogonal to the others. To enhance the efficiency of the postVBSCF methods, an alternative way to generate virtual orbitals is to employ Schmidt orthogonalization over all the orbitals. The advantage of this definition is that the virtual orbitals are orthogonal to all occupied orbitals, and thus the computational cost for Hamiltonian matrix between excited structures can be reduced dramatically. In XMVB, the first definition of virtual orbitals is used for VBCI, while the VBPT2 method uses the second definition to ensure the efficiency. The VBCI method By utilizing the traditional CI technique, the HLSP functions in VBSCF can be augmented with their corresponding excited HLSP functions, VBSCF ΦCI + ∑ CKi ΦiK K = ΦK

(7.2.3.1)

i

where ΦVBSCF is the HLSP function K in the VBSCF wave function, ΦiK the excited HLSP K function generated from ΦVBSCF , and CKi is its coefficient. To ensure ΦiK corresponds K to the same classical VB structures as ΦVBSCF , the replacement of occupied HAOs only K by those virtual HAOs that belong to the same fragment (subset of basis functions) as the occupied. Thus, the VBCI wave function expresses as CI VBSCF ΨVBCI = ∑ CKCI ΦCI + ∑ ∑ CKi ΦiK K = ∑ CK ΦK K

K

K

i

(7.2.3.2)

where CKi = CKCI CKi . In the VBCI calculation, all coefficients are determined by solving the CI secular equation. The group weights corresponding to classical VB structures are defined as CI WKCI = ∑ WKi i

(7.2.3.3)

where WKCI is a normal weight in VBCI wave functions, and the summation is over all structures corresponding to structure K in VBCI wave function. There are various VBCI approaches implemented in XMVB, according to the truncation levels of excitations. VBCIS involves only single excitations, while VBCISD involves also doubles, and so on. VBCIS simulates orbital relaxation effects through a CI and, therefore, it is physically akin to the L-BOVB method; while VBCISD also includes dynamical correlation for the correlated pairs of electrons. The VBPT2 method In the current version of XMVB, the VBPT2 method is implemented with the reduced density matrix based VB approach [32, 36]. Taking VBSCF wave function as a refer-

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ence, the wave function in the VBPT2 method is written as the sum of the VBSCF wave function and the first-order wave function, Ψ(1) , ΨVBPT2 = ΨVBSCF + Ψ(1)

(7.2.3.4)

In a similar fashion to MO-based perturbation theory, the zero-order Hamiltonian, H , is defined by using the projectors of the VBSCF target state, P, and the first-order interacting space, Q, (0)

H (0) = PFP + QFQ

(7.2.3.5)

where F is a Fock-like operator. The first-order wave function is generated by using the reduced density operator. After careful treatment for redundancy of excitations, the first-order wave function is written as a linear combination of independent internally contracted excited wave functions, Ψ(1) = ∑ C μ Ψ(1) μ μ

(7.2.3.6)

where {Ψ(1) μ } involve singly and doubly excited wave functions, and are generated by internally contracted excited wave functions [38]. Coefficients {C μ } are determined by the working equation, ∑ C ν Wμν = −Vμ

(7.2.3.7)

(0) Wμν = ⟨Ψ(1) − E VBSCF |Ψ(1) μ |H ν ⟩

(7.2.3.8)

ν

Vμ =

VBSCF ⟨Ψ(1) ⟩ μ |H|Ψ

(7.2.3.9)

Thus, the second-order energy is expressed as E (2) = ∑ C μ Vμ μ

(7.2.3.10)

It should be noted that in the current version of VBPT2, the individual structures does not correspond to classical VB structures, as the virtual orbitals involved in excited structures are not strictly localized. Therefore, VBPT2 cannot yet be used to access interpretative quantities such as VB weights and calculation of Resonance Energies.

7.2.4 Hybrid VB methods with GAMESS modules In the module edition of XMVB, several VB methods were implemented by hooking XMVB into GAMESS, including DFVB [39–41], VBPCM [42] and VBEFP [43].

176 | A. Shurki et al. Density Functional Valence Bond (DFVB) DFVB is a hybrid method that combines VBSCF method with density functional theory, where the dynamic correlation effect is added to the VBSCF energy by a density functional. Three approaches are implemented in the current XMVB, dc-DFVB [39], hc-DFVB [40] and λ-DFVB [41]. In the dc-DFVB method, the electronic energy of a molecule is given by adding a correlation functional to the VBSCF energy, E dc−DFVB = E VBSCF + EC [ρVB ]

(7.2.4.1)

where EC [ρVB ] is a correlation functional computed from a KS-DFT calculation with the density of the VBSCF wave function. Clearly, such scheme suffers from the double counting error, as the static and dynamic correlations cannot be rigorously separated. In the hc-DFVB method, the dynamic correlation energy is covered with corrected Hamiltonian matrix elements, hc−DFVB VBSCF corr HKL = HKL + HKL

(7.2.4.2)

By expanding the VB function in terms of Slater determinants, ΦK = ∑ dκK Dκ K

(7.2.4.3)

where Dκ is a Slater determinant, and dκK is its coefficient, taken as 1, −1 or 0, the corr correction terms in equation (7.2.4.2), HKL , can be written in terms of correction matrix corr elements, Hκλ , with respect to determinants Dκ and Dλ , corr corr HKL = ∑ dκK dλL Hκλ κ,λ

(7.2.4.4)

corr In the current version of XMVB, the diagonal elements, Hκκ , are defined as corr Hκκ = EC [ρκ ] + (1 − α)(EX [ρκ ] − Kκ )

(7.2.4.5)

where EC [ρκ ], EX [ρκ ], and Kκ are correlation and exchange functionals, and the exact exchange energy of determinant Dλ , respectively, and the parameter α is determined corr by the functional used in the calculation. The off-diagonal elements, Hκλ , κ ≠ λ, are approximately given by corr Hκλ =(

VBSCF Hκλ

VBSCF + H VBSCF Hκκ λλ

corr corr )(Hκκ + Hλλ )

(7.2.4.6)

VBSCF where Hκλ is the Hamiltonian matrix element of VBSCF. Thus, the electronic energy of hc-DFVB can be given by solving the secular equation with the corrected Hamiltonian matrix computed according to (7.2.4.5) and (7.2.4.6).

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λ-DFVB is a newly developed DFVB method. To overcome the double counting issue, λ-DFVB decomposes the electron–electron interactions with a variable parameter λ. The energy of λ-DFVB is expressed as λ−DFVB E λ−DFVB = ⟨ΨVBSCF |T + Vext + λWee |ΨVBSCF ⟩ + EHXC

(7.2.4.7)

where T, Vext and Wee are kinetic energy, external potential and electron-electron interaction operators, respectively, λ is a coupling parameter that depends on the mulλ −DFVB tireference character of molecule. EHXC in (7.2.4.7) is the complement λ-dependent Hartree-exchange-correlation density functional for electronic density ρ given from VBSCF calculation, defined as λ−DFVB EHXC [ρ] = (1 − λ)(EH [ρ] + EX [ρ]) + EC [ρ] − λ2 (EC [ρ] − EC [ρLD ])

(7.2.4.8)

where EX [ρ] and EC [ρ] are exchange and correlation functionals with density ρ, respectively, and EH is Hartree energy. EC [ρLD ] is the correlation energy determined by the electronic density of the leading determinant (LD), which is a single determinant with the largest coefficient in the VBSCF wave function. In a λ-DFVB calculation, the parameter λ is determined by the free valence of molecule, as λ = K 1/4

(7.2.4.9)

Valence Bond Polarizable Continuum Model (VBPCM) VBPCM is a VB method that incorporates the polarizable continuum model (PCM) into the VB scheme, and thus is able to include solvent effect in VB calculations. In the polarizable continuum model (PCM), the solvent is represented as a homogeneous medium characterized by a dielectric constant; the interactions between solvent and solute molecules are treated as a one-body operator in Hamiltonian. Thus, the SCF equation for VBPCM is written as (H 0 + VR )ΨVBPCM = EΨVBPCM

(7.2.4.10)

where H 0 is the gas phase Hamiltonian, and VR is the interaction potential expressed in terms of one-electron integrals. Thus, VBPCM is implemented in XMVB by building an effective one-electron integrals in the VBSCF routine. The integral equation formalism (IEF) version of the PCM model is used in VBPCM. Valence Bond Effective Fragment Potential (VBEFP) VBEFP is a combined quantum mechanics and molecular mechanics (QM/MM) based VB method in which the QM part is treated by the VB approach, while the solvation

178 | A. Shurki et al. effect is taken into account by the EFP method, which is the polarized force field approach developed by Gordon and coworkers [44–49]. In the VBEFP method, the Hamiltonian of a molecule is written as the sum of the Hamiltonian of the QM part, H 0 , and the effective fragment potential, V EFP , and thus the Schrödinger equation for the VBEFP is written as (H 0 + V EFP )ΨVBEFP = EΨVBEFP

(7.2.4.11)

where V EFP includes three interaction terms, the Coulomb interaction (V elec ), polarization interaction (V pol ), and the exchange repulsion (V rep ), V EFP = V elec + V pol + V rep

(7.2.4.12)

As in the VBPCM method, the contribution of the EFP operator is represented by one-electron integrals in the SCF calculations.

7.2.5 Parallelization XMVB package has been parallelized with OpenMP. In OpenMP, computations are parallelized in different threads in which data is shared. Currently, the parallelized version of XMVB cannot be used in multiple nodes, but it is lightweight and efficient inside a single node. In XMVB, all methods except Hessian based VBSCF (iscf=6) are parallelized. The performance of the speed-up ratio is almost linear to the number of CPU cores when the number of CPU cores used is less than 10.

7.3 Inputs and outputs for XMVB This section focuses on how to perform VB calculations in practice, and to obtain the various properties discussed in the earlier sections using the XMVB program [2–4]. We will illustrate some of the capabilities of the VB including, in particular: (i) energy calculation of the whole system as well as of individual structures, (ii) calculation of resonance energy between different structures, (iii) analysis of the weights of different structures, etc. These capabilities will be exemplified using three different systems. As mentioned earlier there are two versions of the XMVB program package: a module version, which is combined with the GAMESS-US package, and a stand-alone version. The difference is in the way one gets the initial integrals. The remaining of the inputs/outputs are identical. The examples in this tutorial are designed for the stand-alone version. Here again, there are two different ways. The first way involves two separate input files, one which provides the geometry (.inp file) and allows calculation of the integrals using a separate PREINT program, and a second one (.xmi file)

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that includes all the VB related information. The second way involves only one (.xmi) input file that includes all the information. Our examples utilize the first approach (for the other options one can turn to the XMVB manual) [2].

7.3.1 F2 : Calculation of charge shift resonance energy In this section, we will illustrate inputs and partial outputs of L-VBSCF and L-BOVB calculations of F2 and show how one can reach the conclusion that this bond is neither covalent nor ionic but rather a charge shift bond [21]. In all input and output files/excerpts, grey text following the”;” sign as well as tables/schemes in blue boxes stand for comments. Geometry of F2 and integral calculation – input for the PREINT program Here is a valid INP input file for performing integral and RHF/ROHF calculations with the PREINT program:

This file name must end with the “.inp” extension, and the calculation is carried out using the following command: > preint 1st_input.inp The method can be either RHF (singlet states) or ROHF (higher spin states). This file generates various files: x1e.int and x2e.int integral files, INFO file, as well as an output file where one can find the molecular orbitals of the system. For a given geometry, the PREINT program should only be run once, as the integrals will not change. Any successive VB calculations on the same geometry can then follow (e. g., VBSCF followed by BOVB, VBCI, VB of only one structure etc.). In general, the xmvb input file “XMI” is composed of several sections each starting with a $ sign and ending with $END. The $CTRL section, that contains global control options for the VB calculation, is mandatory. Other sections are optional and include for example: $BFI, $STR, $FRAG, $ORB, $GUS and $AIM. The format and content of some of these sections will be detailed in the following examples. Prior to building the input it is highly recommended to draw the molecule you are interested in according to the geometrical orientation given in the INP file. In your

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Scheme 7.3.1: Orientation of F2 in space along with its (schematically represented) orbitals and electron distribution associated with the covalent structure.

drawing, the active and inactive orbitals should be shown together with their electronic occupation. Scheme 7.3.1 shows the F2 molecule in the geometrical orientation specified in the INP file above. This system has overall 18 electrons, where 4 electrons occupy 2 core orbitals (not shown in the figure), 12 electrons occupy 6 valence lone pairs (three on each F) and 2 bonding electrons which will serve as the active electrons in this system. The input (see 1st input below) starts with the $CTRL section that defines a calculation of F2 using two active orbitals and two active electrons, and asks the program to provide all the possible VB structures for this combination. Note that in such cases (when nao=N and str=full is used) the active orbitals will be the last N orbitals in the $ORB section. Similar to MO calculations, VB orbitals are also expanded as linear combinations of basis functions, and the user has to define which basis functions will be used for each VB orbital. Here, the VB orbitals are defined as hybrid atomic orbitals (HAOs) and the hybrids in this case are based on atoms (frgtyp=atom keyword). Namely, the basis functions of each atom form a subset of basis functions (referred to as fragment), and, unless otherwise defined, the order of the fragments follows the order of the atoms in the geometry definition (see Table 7.3.1). There are 18 basis functions in the current example (F2 with 6-31G basis-set). The first and last 9 basis functions are centered on F1 and F2, and thus, define fragments I and II, respectively (highlighted by orange and green in Table 7.3.1). Detailed description of the VB orbitals appears in subsequent sections. Here, the “iscf=5” keyword means that the 5th orbital optimization algorithm will be called (see Section 7.2.2), and the keyword “guess=mo” means that the MO obtained from the previous PREINT calculation will be used as a guess for this VBSCF calculation. The latter option is recommended especially when the fragment type is atom. We note in this respect that another recommended option is to use natural bond orbitals (NBOs) as guess (not shown here). Finally, Boys localization on the final orbitals is requested. This is useful when the VBSCF calculation is followed by BOVB or VBCI as it turns the orbitals of the VBSCF into a better set of guess orbitals.

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L-VBSCF calculation of F2 – 1st input for the XMVB program Here is a valid XMI input file for performing a L-VBSCF calculation on F2 with the XMVB program.

The second section is the $ORB section where the number of orbitals as well as which basis functions each of these orbitals is composed of are defined. As mentioned earlier, there are 18 electrons in this system occupying 2 core orbitals, 6 lone-pairs and 2 bond-

182 | A. Shurki et al. ing orbitals resulting in overall 10 orbitals which are all defined. Table 7.3.1 within INP demonstrates the different VB orbitals along with the subsets of basis functions (fragments) they are composed of, and their corresponding sequential number. For example, the first orbital (which is expected to stand for the core orbital of F1) is composed of the basis functions of F1 (namely fragment I). The first line in the $ORB section lists how many fragments compose each VB orbital. The current calculation is a purely localized calculation (L-VBSCF), hence each VB orbital is centered on either F1 or F2 and is, thus, composed of 1 fragment. This first line could also be written as 10*1 suggesting that there are 10 VB orbitals with each developed on 1 subset of orbitals (fragment). The subsequent lines define the different 10 VB orbitals by describing the identity of the orbital subsets they will be composed of. Often, delocalization of orbitals that possess different symmetry than the active orbitals is allowed. In the current example this would mean allowing orbitals 3, 4, 6 and 7 to expand over basis functions of the two fragments (not shown here). The last section in this input, $GUS, provides the initial guess for the VB-orbitals. In this case, the guess is taken as the MOs displayed in the PREINT output file. The first number in each line corresponds to a VB-orbital (following the ordering in the $ORB section), and the second number corresponds to the MO that will serve as initial guess (following the ordering in the PREINT output file). One must make sure that the properties of the MO and the corresponding VB orbital are similar. For example, the first MO corresponds to one of the core orbitals of F2 and can therefore serve as a guess for the first two VB orbitals (note that in this particular case the 2nd MO could also serve as an initial guess for these two VB-orbitals). Table 7.3.2 lists the different VB orbitals along with their desired character and the corresponding MO assignment each highlighted in different color. Table 7.3.3 lists the RHF/6-31G orbitals of F2 . The actual guess for each VB-orbital is highlighted for clarity by the same color as in Table 7.3.2. Table 7.3.3: RHF/6-31G orbitals of F2—colored according to their assignment as guess for VB orbitals.

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Excerpts from the output of a L-VBSCF calculation of F2 are given. The VB structures that were used in the calculation are dictated by the number of active electrons (2), active orbitals (2) and the spin multiplicity (singlet). Each structure describes all the electrons by the orbitals they occupy. In general, inactive electron pairs appear prior to active electron pairs, which may be lone pairs or covalent bonds. For example, the first structure involves 16 electrons (4 core and 12 inactive valence electrons) occupying 8 doubly occupied orbitals (the first 8 orbitals defined in $ORB). The last two electrons are paired and occupy the 9th and 10th orbitals (which are centered on F1 and F2, respectively), forming a covalent bond (1 in Scheme 7.3.2). We note that in higher spin states, unpaired electrons will appear last in the structure definition (no unpaired electrons in this example). L-VBSCF calculation of F2 – Selected excerpts of the output

The coefficients of the VB structures in the linear combination of the overall wave function (not shown) along with the corresponding weights calculated using different schemes are given in the output. Here, only the Chirgwin–Coulson weights are

184 | A. Shurki et al. shown [22]. It is recommended to examine these weights since a substantial negatively signed value (below –0.01) usually implies that something is wrong with the calculated wave function (misconvergence, ill-defined orbitals, etc.). Based on the present calculation, ∼79 % of the wave function is covalent, and about 21 % is ionic. Additional useful information that can be obtained involves the overlap matrix MKL = ⟨ΦK | ΦL ⟩, where ΦK and ΦL are VB structures, as well as the Hamiltonian matrix HKL = ⟨ΦK | H | ΦL ⟩ which provides the resonance integral for K ≠ L and the nonvariational energy of the K th VB structure for K = L. We note in this respect, that the Hamiltonian matrix is calculated without the contribution of the nuclear repulsion. Thus, in order to compare the energy of a particular VB structure with the total energy of the system one has to add the nuclear repulsion (equation (7.1.4.2)). Scheme 7.3.3 presents the energy of F2 and the lowest energy structure at the equilibrium distance. The latter, in this case, is the covalent structure (F∙ – ∙F) and energies are taken from the current L-VBSCF calculation following equation (7.1.4.2) (referred to as nonvariational) as well as from calculation of only the covalent structure (termed variational—input and output not shown here). The energy of the system at infinite separation (calculated as the sum of two radicals at the ROHF/6-31G) is also given. For convenience, the absolute energies (in Hartree) as well as the relative energies (in kcal/mol) are shown. The results show that the covalent structure at equilibrium is located ∼45 and ∼50 kcal/mol above the system at infinite separation considering the variational and nonvariational values, respectively. This suggests that the energy curve of the covalent structure is repulsive implying that the bond is not a covalent bond, nor an ionic bond, but is rather a “charge shift” bond [21]. That is, the bond energy does not originate mainly from the covalent bond energy (which is repulsive) but rather from the resonance between the ionic and the covalent structures. Equation (7.3.1.1) illustrates how to calculate the charge shift resonance energy, RECS , which is the difference between the energy of the lowest VB structure (calculated variationally), E(ΦI ) and the energy of the whole system, Etot , both at equilibrium distance (Scheme 7.3.3). RECS = E(ΦI ) − Etot

(7.3.1.1)

In order to calculate the charge shift resonance energy (REcs), a better estimate of the overall bond energy is required as the VBSCF energy is too high and insufficient for quantitative assessments. A post-VBSCF calculation is, therefore, often considered. Here, we demonstrate a L-BOVB calculation.

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L-BOVB calculation of F2 – 2nd input file for XMVB program

The BOVB input file involves explicit description of the VB structures for which separate sets of orbitals will be optimized. Normally, it is recommended to include only the VB structures that contribute significantly (wi ≥ 1 %) to the wave function. In this case all three VB structures contribute 10 % or above (based on the Chirgwin–Coulson weights at the VBSCF level) and are thus all included in the BOVB calculation. The iscf=2 algorithm calculating analytical gradients in terms of basis functions with the L-BFGS algorithm is recommended for BOVB calculations (as already mentioned in the previous section, iscf=5 and iscf=6 do not work with BOVB). Core orbitals are not expected to “breath” and change their relative size between the different VB structures and are, for the simplicity of the calculation, kept the same for the different VB structures. In a BOVB calculation, a good guess is necessary. One usually good possibility is to use the orbitals that were obtained at a preceding VBSCF calculation (as is done in this case by, e. g., copying the orbitals into a guess file). > cp vbscf_file.orb bovb_file.gus It is noted that in such cases the differences between the VBSCF and the BOVB input files are minor and include, merely, a change in orbital optimization algorithm

186 | A. Shurki et al. (from iscf=5 to iscf=2) in the $CTR section, the guess=read option (which means that a GUS file should be given), and, optionally, definition of core orbitals. This possibility was used here. Alternatively, one can create a different guess for each structure by, for example, calculating the covalent and each of the ionic structures separately and combining the three sets of orbitals into one set of VB orbitals. In such a case, the bovb keyword cannot be used and definition of all the orbitals should be given in the input. This is of course much more tedious, and usually unnecessary. A well-converged and Boys-localized set of VBSCF orbitals allows a smooth convergence of the BOVB wave function in most cases. L-BOVB calculation of F2 – Selected excerpts of the output

The structures used in the calculation as described in the output are shown pictorially in Scheme 7.3.4. Orbitals 1 and 2 (the atomic core orbitals of the fluorines, not shown in the figure), are shared by all VB structures. Otherwise, the first structure (F∙ – ∙F) involves orbitals 3–10 where orbitals 9 and 10 are active. The second structure (F+ F− ) involves orbitals 11–17 with 17 being the active orbital, and the last structure (F− F+ ) involves orbitals 18–24 with 24 being the active orbital. A common direct result of allowing each structure to have its own set of orbitals optimized for its specific electron distribution is an increase in the overall contribution of VB structures that had a rather small share in the VBSCF wave function (minor structures). Therefore, the contribution of the ionic structures increased to 13 % each at the L-BOVB level compared with 10 % each at the L-VBSCF (based on the Chirgwin–Coulson formula). Another systematic impact is the lowering of the energy. Thus, returning to Scheme 7.3.3, one can see that the L-BOVB dissociation energy is now 14.1 kcal/mol. We note in this respect that

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the basis-set we used in this tutorial was way too small to obtain a decent dissociation energy at the L-BOVB level, lacking in particular necessary polarization functions. A larger basis-set results in much better agreement with experiment, 29.4 kcal/mol at the L-BOVB level and 38.2 kcal/mol at the highest SD-BOVB level, as reported in Table 7.1.1. Nevertheless, already at this simple L-BOVB/6-31G level—the charge shift character of the bond is observed with RECS energy of 59.0 kcal/mol and a repulsive covalent curve suggesting that the bond energy originates from the resonance between the VB structures.

7.3.2 O3 and SO2 : Quantification of the diradical character and its effect on reactivity Ozone and sulfur dioxide belong to a family of 1,3-dipoles. An important property of these molecules is their diradical character, which has been shown to have significant implications on their reactivities [23]. In this section, we will illustrate parts of inputs and outputs of σ-D-VBSCF as well as σ-D-BOVB calculations of ozone, the diradical character of which was considered controversial. We will quantify the diradical character in this system, compare it to sulfur dioxide and explain their different reactivity patterns. To save space the input with the geometry is not given. Instead, the orientation of the molecule in space is given in Scheme 7.3.5 (we used MP2/cc-pVDZ geometry). The ozone has overall 24 electrons, where 20 electrons occupy 10 σ-type orbitals (6 electrons in 3 core orbitals and 14 electrons in σ-type valence orbitals Scheme 7.3.5a) the remaining 4 electrons occupy 3 π-type orbitals (Scheme 7.3.5b). These π-type electrons/orbitals create the active space (and the orbitals are thus localized each on a particular oxygen), all the σ-type electrons/orbitals on the other hand are inactive and differ in symmetry from the active orbitals. These orbitals can therefore be kept delocalized over the whole molecule.

Scheme 7.3.5: Orientation of O3 in space along with (a) its σ frame where each line represents a pair of electrons (either bond or a lone pair) and (b) the px orbitals that compose the π-system and their respective number.

The $CTRL section of the σ-D-VBSCF calculation of O3 defines three active orbitals (their corresponding number appears in Scheme 7.3.5b) and four active electrons. Such a 4e− /3o system involves a set of 6 VB structures: one structure, D, that pairs the electrons on the two terminal oxygens and is responsible for the diradical character of the

188 | A. Shurki et al. molecule, four zwitterionic structures Z1 –Z4 and one multiionic structure in which the central atom has a double positive charge (MI). All 6 structures are considered in this calculation and are both defined in the $STR section and displayed pictorially in Scheme 7.3.6 (for simplicity active electrons are highlighted in purple). Similar to the F2 example, the VB orbitals are defined as hybrid atomic orbitals. The hybrids in this case, however, are based on symmetry (frgtyp=sao) rather than atoms, an approach that further facilitates the calculations, in particular convergence of the BOVB wave function. A $FRAG section is then required, where the different subsets of basis functions are defined, usually called “fragments” in the XMVB program. In this section, each line following the first one defines one specific “fragment” (subset of basis functions). Within the frgtyp=sao option, each subset of basis function is defined based on a combination of two criteria: (i) the desired localization of the different orbitals; and (ii) the symmetry of the molecule, which will determine the type of basis functions that could contribute by symmetry to these specific orbitals. Therefore, each line in the $FRAG section (excluding the first one) provide two types of information that together fully define a specific subset of basis functions: first a list of types of basis functions (such as s, px, dxy, . . . ), then a list or range of numbers corresponding to atoms. All of the basis functions that correspond to the specified types and are centered onto the specified atoms will be grouped as a subset. In the present case, the $FRAG section in the σ-D-VBSCF input of O3 contains four lines after the first one (five in total), and thus defines four different “fragments” (subsets of basis functions). The “σ-DVBSCF” acronym means that the inactive σ orbitals of ozone should be delocalized on the whole molecule (atoms 1–3 which could also be written 1, 2, 3 in the XMVB input). As symmetry is concerned, the C2V molecular symmetry determines that σ orbitals can expand on s, py, pz, dxx, dyy, dzz and dyz types of basis functions. Therefore, combining these two criteria leads to the definition of the first subset of basis functions in the $FRAG section, namely: “spypzdxxdyydzzdyz 1–3“ (the part of that subset which is centered on Oxygen 1 is highlighted in green in Scheme 7.3.7). The active VB orbitals, can only expand on px, dxy and dxz types of basis functions by symmetry, and shall each be localized on only one oxygen. Hence, the next three lines in the $FRAG section define three additional different subsets for the three VB active orbitals (this subset on Oxygen 1 is highlighted orange in Scheme 7.3.7). Last, the first line in the $FRAG section lists how many atoms are involved in each “fragment”: namely, three for the first “fragment” and only one in the subsequent three “fragments.”

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190 | A. Shurki et al. A last note: it is allowed to define overlapping “fragments,” i. e., that some basis functions are common to different subsets. However, it is highly recommended, whenever possible, to use disjoint subsets of basis functions (no basis functions are common to two different subsets). This is the case typically with σ-D-BOVB and π-D-BOVB methods. This is particularly important when valence basis functions are concerned (and is less important for polarization basis functions), when performing a BOVB calculations, as this will very much reduce the risk of convergence issues. Now let us move to the actual definition of orbitals, which is provided in the $ORB section. There are 24 electrons in this system occupying 10 σ-type orbitals (3 core orbitals and 7 lone-pairs and bonding orbitals) and 3 px orbitals. These 13 VB orbitals are composed of one fragment each as described in the first line of the $ORB section. The first 10 VB orbitals are delocalized σ-type composed of the first subset of basis functions (fragment 1) whereas the last three VB orbitals are localized π-type composed of fragments 2–4 (each localized on a different atom). Similar to the previous example of the F2 molecule, the $GUS section in this calculation is based on a MO guess, generated by the PREINT program. The first 10 VBorbitals are σ-type orbitals and, therefore, MOs 1–8, 10 and 11, which are identified as being of σ-type, can be used as a guess. MOs 9 and 12 are of π-type, however orbital 12 has no coefficients on the oxygen atom O1. We have therefore used only MO 9 as guess for the three π-type VB orbitals. Note that these MOs are just guess orbitals that will then be variationally optimized in the VBSCF procedure, therefore, they do not need to be optimal, but need to have approximatively the adequate shape, symmetry and spatial extension for the corresponding VB orbital. For instance, in this particular example we could alternatively use MO 9 as a guess for VB orbital 11 (which is centered on Oxygen 1), and MO 12 as a guess for VB orbitals 12 and 13. This should lead after optimization to exactly the same VBSCF wave function and energy. σ-D-VBSCF calculation of O3 – Selected excerpts of the output

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The total energy as well as the weights of the different structures are shown in the selected excerpts of the output of the σ-D-VBSCF calculation of ozone. The diradical structure (D) has the largest contribution (∼68 %) with the first two zwitterionic structures (Z1 & Z2 ) contributing ∼14.5 % each and the MI structure contributing ∼1 %. The contribution of the remaining structures (Z3 and Z4 ) is negligible. These results suggest that O3 has substantial diradical character. It is further emphasized by looking at the energies of the corresponding VB structures (gray values in Scheme 7.3.8). As mentioned earlier, VBSCF provides good qualitative results, but if one is interested in quantitative results, post-VBSCF calculations are often required. The following input and output excerpts are taken from a BOVB calculation of ozone. Only the $CTRL and the $STR sections of the input are shown as both $FRAG and $ORB sections are identical to those of σ-D-VBSCF input. To facilitate the reading, the changes from the $CTRL section of the σ-D-VBSCF input are highlighted in yellow. Only 4 out of the 6 VB structures are now considered, discarding Z3 and Z4 which were found to contribute less than 1 % in the VBSCF wave function. In a BOVB wave function, different structures have their own specific set of orbitals, except for the core ones usually. Therefore, the first 3 VB orbitals that are core orbitals are kept identical for the different structures (ncore=3), while all others differ in the different structures and are thus assigned different numbers in the output: first structure, D, involves orbitals 1–13 as in the VBSCF calculation, structure Z1 involves orbitals 1–3 (occupied by the 6-core electrons) and 14–23 (where 14–20 are σ-type orbitals, equivalent of 4–10, and 21–23 are π-type orbitals equivalent of 11–13), and so on. σ-D-BOVB calculation of O3 – Selected excerpts of input and output

As expected, the BOVB wave function has a significantly lower total energy than the VBSCF one. In addition, the contribution of the various VB structures changed significantly compared to VBSCF level, showing a rise of the zwitterionic structures weight

192 | A. Shurki et al. (23 % each) and of the MI (6 %), while reducing the contribution of the diradical structure (47 %) to keep balance. Despite these changes, the diradical structure is found to still be the lowest in energy (black values in Scheme 7.3.8) and its contribution is still the dominant one for ozone at this level; however, the two zwitterionic structures combined now weight the same. This result, as demonstrated in Table 7.3.4, is in agreement with results obtained at higher level of calculation [23]. Table 7.3.4: Chirgwin–Coulson weights of the individual VB structures of O3 and SO2 based on the BOVB wave function. VB struct. D Z1 Z2 MI a b

O3 a

O3 b

SO2 b

0.480 0.232 0.232 0.063

0.444 0.238 0.238 0.068

0.159 0.353 0.353 0.136

Values are based on MP2/cc-pVDZ geometries and σ-D-BOVB as demonstrated here. Values are taken from ref. [23] using σ-D-BOVB with cc-pVTZ basis set on MRCI geometries.

Similar calculations on the SO2 system exhibit a different picture. As shown in Table 7.3.4, for SO2 the diradical structure contributes only ∼16 % to the overall wave function. Furthermore, the two zwitterionic structures (the energy of which is the lowest among the different VB structures in this case—data not shown) clearly dominate the character of the sulfur dioxide system. These results nicely correlate with the different reactivity patterns of O3 and SO2 [23]. Ozone with its diradical character is a very reactive species that undergoes much easier H-addition to its terminal atoms, compared with SO2 [50]. Furthermore, ozone easily undergoes cycloaddition reactions with, e. g., ethylene or acetylene, whereas sulfur dioxide with its zwitterionic character is unreactive [51]. In fact, the barriers to cycloaddition reactions of various 1,3-dipoles were shown to have an inverse correlation with the weights of their diradical structures [24].

7.3.3 Benzene: using Overlap Enhanced Orbitals (OEOs) to enable smaller VB set of structures In cases of large systems that involve many active electrons, the number of VB structures may increase dramatically and classical VB then can lose its advantages as an interpretive method. In such cases, one simple alternative is to use a semi-localized approach, such as the Spin-Coupled VB (SCVB) method, especially if energy is the main interest. In this section, this will be illustrated on benzene.

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Inputs for three VB calculations of benzene are demonstrated below, the first two involve excerpts from inputs of classical VB calculations where the first considers only the covalent VB structures and the second considers the full VB set (175 VB structures). The third input is a complete input of a SCVB calculation that uses only two Kekule VB structures. All differences in the $CTRL section between the three different input files are highlighted in yellow. The order of the inputs follows the recommended order of the calculations. Classical σ-D-VBSCF and spin-coupled VB calculations of benzene – Selected excerpts of inputs

Benzene has overall 42 electrons, 36 electrons occupy 18 σ-type inactive orbitals and 6 electrons occupy 6 π-type active orbitals (Scheme 7.3.9). In the VB calculations shown all σ-type orbitals are delocalized over the whole molecule. Thus, the first fragment in the first two calculations involves basis functions that contribute to a σ-type orbital on all 12 atoms in that system. The difference between classical and SCVB calculations is in π-type orbitals which are localized in the former (first and second inputs) and delocalized in the latter (third input) calculations. We note in this respect that in the SCVB method the orbital-type is defined as Overlap Enhanced Orbitals (OEOs) suggesting

Scheme 7.3.9: Benzene along with (a) its various subsets of basis functions (σ and π fragments are in green and orange, respectively) (b) the pz orbitals that compose the π-system (xy is the molecular plane) along with their respective numbering in the calculation, and (c) Rumer’s set of five covalent VB structures.

194 | A. Shurki et al. that the orbitals are not restricted to subsets of basis functions (“fragments”) but can span over all basis functions. Therefore, there is no need for neither $FRAG nor $ORB sections in the input file. Benzene is a 6e/6o system and as such it involves 175 independent VB structures to describe it. As the number of VB structures increases, a good initial guess becomes more important. It is, therefore, recommended to start with a calculation using a small set of major VB structures and use the resulting orbitals as a guess for VB calculation with a complete set. Note that, these same orbitals may also serve as a guess for a SCVB calculation. In the current case of benzene, the covalent set that includes the two Kekule and three Dewar structures (c in Scheme 7.3.9) serves as a small set of major VB structures in the first input. The orbitals obtained in this calculation then serve later as a guess for the two subsequent calculations, namely the calculation which involves the full VB set of 175 structures (second input) and the SCVB calculation (third input). Classical σ-D-VBSCF and spin-coupled VB calculations of benzene – Selected excerpts of output

The results of the first calculation suggest, as expected, that among the covalent structures the two Kekule structures are the most important, exhibiting each a total weight of ∼33 % of the overall wave function. The contribution of the remaining Dewar structures amounts to ∼11 % each. Moving to the second calculation, a dramatic decrease in the contribution of the covalent structures is observed (only ∼11 % for each Kekule structure and ∼3.5 % for each Dewar structure) due to the presence of the many ionic structures. However, careful examination of the weights reveals that the contribution of the various ionic structures does not exceed 2 % per structure and is usually < 0.5 %. For example, Scheme 7.3.10a,b demonstrates six out of the twelve most important ionic structures (each contributes ∼2 %) and four out of the twelve second most important ionic structures (each contributes ∼0.9 %). These are all mono-ionic structures derived from structures K1 and D1 , respectively. The multiple structures make the overall wave

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Scheme 7.3.10: (a) six most important and (b) four second most important monoionic structures in the σ-D-VBSCF wavefunction of benzene and (c) Relative energies of the three different calculations of benzene.

function complex and less intuitive. Still, among all the contributions, that of the covalent structures and in particular the two Kekule structures remain the most significant. Note that it has been shown that the symmetry allowance of mixing of ionic structures with the covalent ones is at the origin of the aromatic vs. antiaromatic behaviors in π delocalized systems [52]. The price paid for the complexity of the wave function is balanced by a significant improvement in energy. The addition of the ionic structures indeed lowers the total energy in this case by 92.7 kcal/mol (Scheme 7.3.10c). Using the spin-coupled approach where both inactive as well as active orbitals are allowed to delocalize is an alternative way to lower the overall energy. Such a calculation involves only the covalent structures (Scheme 7.3.9c) and the resulting wave function is therefore relatively simple. In the third calculation (third input and output) we used only the two Kekule structures (which were found to be the most important) while allowing delocalization of all orbitals. The resulting wave function is thus very compact (50 % contribution of each one of the Kekule structures). The energy, on the other hand, is similar to the energy of the calculation with the full VB set (97.1 kcal/mol lower than the classical VB calculation with the covalent set, Scheme 7.3.10c). Note that a SCVB calculation including only the three Dewar structures of benzene, or the five Kekule+Dewar structures (not shown), would lead to a similar energy as in this two-Kekule-structure calculation. The reason is that delocalized OEOs orbitals implicitly include many ionic structures. A very small set of fundamental structures (either the two Kekule, or the three Dewar) is therefore required leading to an important limitation of using OEOs for interpretation purposes.

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Shaik S, Hiberty PC. Chemist’s Guide to Valence Bond Theory. 2008. p. 1–316. XMVB web site. http://www.xmvb.org. Song LC, Mo YR, Zhang QN, Wu W. J Comput Chem. 2005;26(5):514–21. Chen Z, Ying F, Chen X, Song J, Su P, Song L, Mo Y, Zhang Q, Wu W. Int J Quant Chem. 2015;115(11):731–7. Slater JC. Phys Rev. 1931;37(5):481–9. Pauling L. J Am Chem Soc. 1931;53:3225–37. Pauling L. J Am Chem Soc. 1931;53(4):1367–400. Pauling L. The Nature of the Chemical Bond. 3rd ed. Ithaca, New York: Cornell University Press; 1939. Coulson CA, Fischer I. Philos Mag. 1949;40(303):386–93. Goddard WA, Dunning TH, Hunt WJ, Hay PJ. Acc Chem Res. 1973;6(11):368–76. Cooper DL, Gerratt J, Raimondi M. Chem Rev. 1991;91(5):929–64. Pauncz R. Spin Eigenfunctions. New York: Springer US: Plenum Press; 1979. p. 370. Lenthe JHV, Balint-Kurti GG. Chem Phys Lett. 1980;76(1):138–42. van Lenthe JH, Balint-Kurti GG. J Chem Phys. 1983;78(9):5699–713. Hiberty PC, Humbel S, Byrman CP, Vanlenthe JH. J Chem Phys. 1994;101(7):5969–76. Hiberty PC, Shaik S. Theor Chem Acc. 2002;108(5):255–72. Bytautas L, Ruedenberg K. J Chem Phys. 2005;122(15). Shaik S, Shurki A. Angew Chem, Int Ed Engl. 1999;38(5):586–625. Lai W, Li C, Chen H, Shaik S. Angew Chem, Int Ed Engl. 2012;51(23):5556–78. Usharani D, Lai W, Li C, Chen H, Danovich D, Shaik S. Chem Soc Rev. 2014;43(14):4968–88. Shaik S, Danovich D, Galbraith JM, Braida B, Wu W, Hiberty PC. Angew Chem, Int Ed Engl. 2020;59:984–1001. Chirgwin BH, Coulson CA. Proc R Soc A. 1065;1950(201):196–209. Braida B, Galembeck SE, Hiberty PC. J Chem Theory Comput. 2017;13(7):3228–35. Braida B, Walter C, Engels B, Hiberty PC. J Am Chem Soc. 2010;132(22):7631–7. Lauvergnat D, Hiberty PC. J Am Chem Soc. 1997;119(40):9478–82. Schmidt MW, Baldridge KK, Boatz JA, Elbert ST, Gordon MS, Jensen JH, Koseki S, Matsunaga N, Nguyen KA, Su S, Windus TL, Dupuis M, Montgomery JA. J Comput Chem. 1993;14(11):1347–63. Gong X, Chen Z, Wu W. J Comput Chem. 2016;37(23):2157–62. Verbeek J, Lenthe JHV. J Mol Struct, Theochem. 1991;229:115–37. Hiberty PC, Flament JP, Noizet E. Chem Phys Lett. 1992;189(3):259–65. Song L, Wu W, Zhang Q, Shaik S. J Comput Chem. 2004;25(4):472–8. Wu W, Song L, Cao Z, Zhang Q, Shaik S. J Phys Chem A. 2002;106(11):2721–6. Chen Z, Chen X, Ying F, Gu J, Zhang H, Wu W. J Chem Phys. 2014;141(13):134118. Chen Z, Song J, Shaik S, Hiberty PC, Wu W. J Phys Chem A. 2009;113(43):11560–9. Liu DC, Nocedal J. Math Program. 1989;45(1):503–28. Chen Z, Chen X, Wu W. J Chem Phys. 2013;138(16):164120. Chen Z, Chen X, Wu W. J Chem Phys. 2013;138(16):164119. Song L, Song J, Mo Y, Wu W. J Comput Chem. 2009;30(3):399–406. Chen X, Chen Z, Wu W. J Chem Phys. 19;2014(141):194113. Ying F, Su P, Chen Z, Shaik S, Wu W. J Chem Theory Comput. 2012;8(5):1608–15. Zhou C, Zhang Y, Gong X, Ying F, Su P, Wu W. J Chem Theory Comput. 2017;13(2):627–34. Ying F, Zhou C, Zheng P, Luan J, Su P, Wu W. Front Chem. 2019;7:225. Song L, Wu W, Zhang Q, Shaik S. J Phys Chem A. 2004;108(28):6017–24. Ying F, Chang X, Su P, Wu W. J Phys Chem A. 2012;116(7):1846–53.

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Trevor A. Hamlin, Pascal Vermeeren, Célia Fonseca Guerra, and F. Matthias Bickelhaupt

8 Energy decomposition analysis in the context of quantitative molecular orbital theory 8.1 Introduction

The purpose of this chapter is to provide a compact introduction into the analysis of the chemical bond from the perspective of quantitative, canonical molecular orbital (MO) theory in conjunction with a matching, canonical energy decomposition (EDA) method. We aim to provide an understanding of the chemical bond in terms of a causal relationship between, on one hand, the orbital electronic structure of molecules or molecular fragments and, on the other hand, their physical properties, in particular, their capability to bind, or to repel, each other. To this end, we discuss the core concepts, the formulas, and the way in which they serve to understand, as opposed to only compute and describe structure and stability. For a more elaborate discussion of the technicalities and the physical interpretation of EDA and quantitative (in particular, Kohn–Sham) MO theory, the reader is referred to [1–5]. Our basic philosophy here is that understanding, i. e., being able to answer the question “why is this so?”, arises from elucidating the MO bonding mechanism, whereas the EDA serves to quantify how important each of the features (e. g., electrostatic attraction versus donor-acceptor interactions) in the bonding mechanism is. Thus, we advocate our EDA as a most useful addition and not as a stand-alone method.

Trevor A. Hamlin, Pascal Vermeeren, Department of Theoretical Chemistry, Amsterdam Institute of Molecular and Life Sciences (AIMMS), and Amsterdam Center for Multiscale Modeling (ACMM), Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands, e-mails: [email protected], [email protected] Célia Fonseca Guerra, Department of Theoretical Chemistry, Amsterdam Institute of Molecular and Life Sciences (AIMMS), and Amsterdam Center for Multiscale Modeling (ACMM), Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands; and Leiden Institute of Chemistry, Gorlaeus Laboratories, Universiteit Leiden, Einsteinweg 55, 2333 CC Leiden, The Netherlands, e-mail: [email protected] F. Matthias Bickelhaupt, Department of Theoretical Chemistry, Amsterdam Institute of Molecular and Life Sciences (AIMMS), and Amsterdam Center for Multiscale Modeling (ACMM), Vrije Universiteit Amsterdam, De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands; and Institute of Molecules and Materials, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands, e-mail: [email protected] https://doi.org/10.1515/9783110660074-008

200 | T. A. Hamlin et al.

8.2 Bonding mechanism and canonical molecular orbital theory One of the primary objectives of chemical research is to observe and describe the structure and stability of molecules, either experimentally or computationally, or both. A deeper purpose is to also understand the reason behind the observed or computed structure and stability, and trends therein. To this end, we need models that go beyond the categorization of facts. Our model must be not only quantitatively accurate, which is, of course, a conditio sine qua non, but it is only the basis from which to proceed. The model must in some way reveal the causal relationship between the molecular electronic structure and the molecule’s properties, such as bonding capability, reactivity, spectra, and other properties. It should not just simulate and provide a more or less accurate quantitative prediction, it should first and foremost enable the scientist to explain, qualitatively, why things happen, and argue what should be done in order to modify the behavior of the system into the desired direction. This is where canonical molecular orbital (MO) theory comes into play. Canonical MO theory is a deeply physical model of the electronic structure in which the picture of a molecule as a system composed of nuclei and electrons is made very explicit in the mathematical description (independent-particle model) as well as the associated physical concepts (e. g., orbital interactions). We highlight, at this point, valence bond (VB) theory as a complementary approach that also leads to the type of understanding we are striving for (described in Chapter 7) [6–10]. In wavefunction theory, the independent-particle model leads to Hartree–Fock (HF) theory that features canonical orbitals, the HF orbitals, resulting as eigenfunctions from an effective one-electron Schrödinger equation: F HF ϕl HF = ϵl HF ϕl HF [11–13]. This method, however, constitutes a MO model that is able to cope with many chemical problems only in a semiquantitative manner. It is unfortunately not accurate enough for most of the current problems in chemistry. While wavefunction theory does offer the possibility to improve the accuracy in a systematic and straightforward (although computationally costly) manner by improving the wavefunction (e. g., in a configuration interaction (CI) approach [14, 15]), this regrettably always goes with a breakdown of the mathematical form of the independent particle model, due to the incorporation of significant corrections associated with Coulomb correlation between the electrons’ quantum chemical motion. Nevertheless, there are approaches such as the natural bond orbital (NBO) method that do reintroduce the orbital concept in correlated wavefunctions [16, 17]. Note, however, that NBOs are not eigenfunctions of a one-electron Hamiltonian of a molecule or molecular fragment and, therefore, differ from canonical MOs. In Kohn–Sham density functional theory (KS-DFT) [18, 19], the effect of the Coulomb correlation between the electrons is introduced in a different manner which preserves the formalism of an effective independent-particle model: F KS ϕl KS =

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ϵl KS ϕl KS . An advantage of the Kohn–Sham model is that it has a direct connection with the exact wavefunction and, via the exchange-correlation functional EXC [ρ], the exact energetics. In the original Hohenberg—Kohn–Sham DFT, EXC [ρ] was defined only for ground state densities. However, the domain of KS-DFT can be extended to arbitrary proper densities (integrating to N electrons, positive everywhere) by the constraint search definition of Levy [20]. Within the framework of KS-DFT, the total energy of a molecular system, Etotal , is calculated as Etotal = Ts + VNe + Vee + EXC [ρ]

(8.1)

where, Ts is the kinetic energy of the Kohn–Sham orbitals, VNe is the nuclei-electron attraction, Vee is the electron-electron Coulomb repulsion and EXC [ρ] is the prior discussed exchange-correlation functional which also includes corrections to the kinetic energy. For a more comprehensive discussion of these energy terms as well as of KSDFT, we refer the reader to [1, 18, 19]. To discuss the bond energy, ΔE, associated with the interaction between a molecular complex AB, within the framework of canonical KS-MO theory, two individual fragments, A and B, should be considered. A and B may simply be atoms or larger molecular fragments, but in either case, A and B interact to form molecular complex AB. Selection of molecular fragments, A and B, is critical for obtaining meaningful results and will be discussed in more detail later. When the fragments A and B have the geometry and electronic configuration that they obtain in the complex AB, they can now be considered to be deformed with regards to their optimum geometry. ρA and ρB are the electron densities of the fragments with the corresponding wavefunctions ΨA and ΨB and their respective energies E A and E B . The bond energy, ΔE, can be decomposed into the strain energy term, ΔEstrain , and the interaction energy term, ΔEint , according to equations (8.2) and Figure 8.1: ΔE = E AB − E A − E B = ΔEstrain + ΔEint

(8.2a)

ΔEstrain = E A in AB + E B in AB − E A − E B

(8.2b)

ΔEint = E

(8.2c)

AB

−E

A in AB

−E

B in AB

ΔEint = ΔVelstat + ΔEPauli + ΔEoi

ΔE = ΔEstrain + ΔVelstat + ΔEPauli + ΔEoi

(8.3) (8.4)

Figure 8.1: Illustrative example of the decomposition of the bond energy, ΔE, associated with the interaction between the two fragments A and B forming complex AB, into the strain energy, ΔEstrain , and interaction energy, ΔEint .

202 | T. A. Hamlin et al. The strain energy, ΔEstrain , is the penalty that needs to be paid to deform the individual fragments A and B from their equilibrium structure into the geometry they acquire in the complex AB (equation (8.2b)). It is, therefore, heavily related to the geometrical rigidity of the fragments. The ΔEstrain can be expressed as a geometric and energetic effect, such as the lengthening of a CO bond from the equilibrium geometry of free CO (1.128 Å) to the C–O bond length in a metal carbonyl complex (ca. 1.150 Å) [21–23], but also as a large deformation, for instance, changing the planar CH∙3 radical to the umbrella shape of the CH3 fragment in methane derivatives [24, 25]. The interaction energy, ΔEint , on the other hand, accounts for all the chemical interactions that occur between the deformed fragments A and B (equation (8.2c)). Additionally, we can rationalize and completely understand the ΔEint term by using a canonical energy decomposition analysis (EDA) method (equation (8.3)), which is based on work of Morokuma [26, 27] and the related extended transition state (ETS) method developed by Ziegler and Rauk [28–30]. This EDA method defines ΔEint as a sum of three individual terms, namely, electrostatic interaction (ΔVelstat ), Pauli repulsion (ΔEPauli ), orbital interaction (ΔEoi ), that are all physically meaningful as well as quantitatively accurate within the molecular orbital framework that arises from Kohn–Sham density functional theory. In the following section, the origin of the individual terms of the EDA will be discussed. First, the unperturbed fragment charge distributions of A and B were brought from infinity to the positions they obtain in AB, resulting in a superposition of fragment densities, i. e., ρA + ρB , with the associated change in energy ΔVelstat , which is simply the classical electrostatic interaction between the fragment charge distributions: ΔVelstat = ∑ ∑

αϵA βϵB

Zα Zβ Rαβ

Zβ ρA (r) Zα ρB (r) ρA (r1 ) ρB (r2 ) dr − ∫ ∑ 󵄨󵄨 dr + dr1 dr2 ∬ 󵄨 r12 |Rα − r| R − r 󵄨󵄨󵄨 αϵA βϵB 󵄨󵄨 β

−∫∑

(8.5) We note that the first and last terms in equation (8.5) are repulsive, whereas the second and third are the attractive potential of the nuclei. At a large distance (when ρA and ρB do not overlap), the resulting electrostatic interaction is zero. As fragment A and B approach each other, and ρA and ρB begin to overlap, the contribution of the last, repulsive, term becomes smaller than the other ones. Elementary electrostatics tells us that two interpenetrating charge clouds have a repulsion which is smaller compared to the repulsion between point charges at the centers of charge. Thus, it can easily be seen that for two approaching, neutral, fragments with spherical charge densities the total electrostatic interaction energy between their unmodified charge distributions becomes attractive at the distance range of interest. Next, the Pauli repulsion, ΔEPauli , is calculated by the energy change of going from the product wavefunction, i. e., Hartree wavefunction, ΨH = ΨA ΨB to the intermediate wavefunction Ψ0 . Antisymmetrization of ΨA ΨB by an operator  and renormalization by a constant N ensures that the wavefunction obeys Pauli’s principle: Ψ0 = NÂ{ΨA ΨB }. Pauli repulsion is the raise in energy, from ΨH to Ψ0 that goes with electrons of same spin avoiding each other and experiencing an increase in kinetic

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Figure 8.2: Orbital diagrams of common orbital interactions appearing in the EDA.

energy in the latter wavefunction. More specifically, it is the physical mechanism behind steric effects. In general, Pauli repulsion arises from the two-center four-electron destabilizing interaction between filled orbitals of different fragments (Figure 8.2a). The final step involves the relaxation of Ψ0 , with the corresponding density ρ0 , to the final wavefunction ΨAB and optimized density ρAB of the AB complex, resulting in the orbital interaction energy. By definition, the orbital interaction energy, ΔEoi , is stabilizing as it results from an optimization of the wavefunction. In the KS model, the KS orbitals of the determinantal wavefunction Ψ0 relax, by mixing in virtual orbitals in a simple self-consistent field calculation, to the final KS orbitals that build the exact density and form the KS determinant ΨSCF . If the two interacting fragments are closedshell systems, the orbital interactions will consist of charge transfer or donor-acceptor interactions between occupied orbitals on one fragment and virtual orbitals on the other (Figure 8.2b). At the same time polarization will occur, consisting of occupied– virtual interactions on one fragment due to the presence of the other (Figure 8.2c). Charge transfer and polarization cannot be strictly separated in this EDA. However, detailed orbital analyses involving systematic deletion of virtual orbitals on one of the fragments, in combination with additional electron density analyses, in the framework of atomic charge and/or deformation density analyses, may help to discriminate between charge transfer and polarization [31]. If there are singly occupied orbitals present, often one on each fragment, the orbital interaction will primarily involve the formation of an electron-pair bond by the pairing up of the unpaired electrons in a bonding orbital (Figure 8.2d). Group theory posits that only orbitals of the same symmetry and character under the available symmetry operations can mix and interact. Thus, when possible, it may be convenient to further decompose the total orbital interaction energy, ΔEoi , into individual contributions from each irreducible representation (irrep) Γ of the point group to which AB belongs as originally introduced by Ziegler and Rauk [29]: Γ ΔEoi = ∑ Eoi Γ

(8.6)

Additionally, when the chosen exchange-correlation (XC) functional is supplemented with an explicit dispersion correction, say for example Grimme’s dispersion

204 | T. A. Hamlin et al. correction D4 [32, 33], the ΔEdisp from the correction term is added to ΔEint and the magnitude of this term can also be analyzed using the canonical EDA.

8.3 Application of the method To demonstrate the applicability of the energy decomposition analysis (EDA) method, we study the formation of the NC–CN dimer from two CN∙ fragments [34]. This is a fundamental example of a bonding situation where the interaction mechanism of the π- and σ-orbitals vary considerably (Figure 8.3). First, looking only at the π-orbitals (Figure 8.3a), there is a donor-acceptor interaction between the HOMO 1π orbital of one CN∙ fragment and the LUMO 2π orbital of the other fragment and vice versa. In addition, an electron-pair bond is formed by two singly occupied σ-orbitals (5σ SOMOs) and repulsive interactions arise from lower-lying overlapping doubly occupied orbitals, notably the HOMO 4σ (Figure 8.3b). Note that only one of the two stabilizing interactions (both donor–acceptor and electron-pair) between the CN∙ fragments is shown in Figure 8.3 (green lines). The individual terms of the EDA, into which we have decomposed the interaction energy, act almost always simultaneously and will influence each other. In this example, we will give a description of this mutual influence and the distinction between the energies corresponding to the above-mentioned donor–acceptor and electron-pair bond interactions (Figure 8.3). The simplest interpretation of ΔEoi occurs in the situation of the π-orbitals of the CN dimer, where two closed-shell π-systems are interacting via a donor-acceptor interaction, as depicted in Figure 8.3a. This charge transfer will often coexist with other effects, such as the relief of Pauli repulsion and polarization (occupied/virtual orbital mixing on one fragment due to the presence of the other CN∙ fragment) (Figures 8.2a and 8.2c). One can see the π-bond formation between the two CN∙ fragments as charge transfer interaction from the occupied HOMO 1π of one CN∙ fragment to the virtual HOMO 2π of the other, and vice versa. Meanwhile, the Pauli repulsion, that manifest from the formation of the occupied antibonding combination HOMO 1π–HOMO 1π, is relieved by the mixing of LUMO 2π–LUMO 2π, which, in turn, leads to the occupation of the LUMO 2π and electron depletion from HOMO 1π. Within the current EDA scheme, we cannot decompose these simultaneous processes, but denote them collectively as the ΔEπ term in the decomposition of the orbital interaction energy ΔEoi according to group theory (equation (8.6)). As mentioned earlier, the electron-pair bond between the two CN∙ fragments is formed by two singly occupied orbitals of opposite spin, namely, the 5σ SOMO of both fragments (Figure 8.3b). It is possible, for this example, to continue with the symmetry decomposition of the orbital interactions, and relate the energy associated with the electron-pair bond to the magnitude of ΔE σ . But, the analysis can be carried further

8 Energy decomposition analysis in the context of quantitative molecular orbital theory | 205

Figure 8.3: Schematic orbital interaction diagrams of the (a) π-donor-acceptor interaction and (b) an electron-pair σ-bond plus lower-lying occupied levels between two CN∙ monomers forming a NC– CN dimer, where one of the stabilizing interactions between the two CN∙ fragments are indicated in green and the (Pauli) repulsive interactions in black.

by making an estimate of the energy of electron-pair bond formation versus the donoracceptor interactions that will also be present between the σ-orbitals of the fragments (see [34] for this analysis). Although we will not go into so much detail in the herein discussed example, we mention for completeness this analysis in the last part of this section. In this example, the electron-pair bond of NC–CN, between the 5σ SOMOs of both CN∙ radicals, belongs to the σ symmetrical irreducible representation. Below the 5σ SOMOs, there are fully occupied σ-orbitals, from which the 4σ nitrogen lone pair orbital (HOMO 4σ) is the most important. The intermediate wavefunction Ψ0 is written, in this case, as follows (where A and B are both the CN∙ radical monomers): Ψ0 = N|(occupied orbitals)A (occupied orbitals)B 5σA α(1)5σB β(2)|

(8.7)

Due to the fact that the fragment orbitals in Ψ0 are overlapping, the determinantal wavefunction will not be normalized. This issue is resolved by the added normalization factor. One possible way to evaluate the energy of Ψ0 is to first orthogonalize the

206 | T. A. Hamlin et al. fragment orbitals onto each other. Once this set of orthogonalized orbitals is formed, the corresponding density ρ0 can be written as a sum of the corresponding orbital densities, and the energy can be obtained from the Slater–Condon rules for evaluating matrix elements between determinantal wavefunctions [12]. The HOMO 4σ orbitals of both fragments will overlap with each other, which gives rise to steric repulsion, due to Pauli’s principle. This steric repulsion is easily represented in an elementary MO diagram as a two-center four-electron destabilizing interaction (Figure 8.2a). The SOMO 5σ of both fragments, on the other hand, are orthogonal on account of the spin orthogonality (fragment A has α spin; fragment B has β spin), so the only Pauli repulsion is coming from the orthogonality requirement of the SOMO 5σ of fragment A with the α-spin occupied orbitals of fragment B, and vice versa. In Figure 8.4, a plot of the density difference of the NC–CN molecule with respect to the individual CN∙ fragments (Δρ0 = ρ0 − ρA − ρB ) is presented. Surprisingly, there is an accumulation of electron density (green) at the atoms and the depletion (red) in the overlap region between the two fragments. Especially, the shape of the Δρ0 type of electron depletion in the overlap region is striking. One might indeed expect that there should be a buildup of electron density in the overlap region between the two fragments, which is commonly associated with bond formation [41–43]. However, it is the next step in our analysis of the chemical bond formation that will bring electron

Figure 8.4: Contour plot of the difference Δρ0 between the electron density of NC–CN and the CN∙ fragments (Δρ0 = ρ0 − ρA − ρB ) after orthogonalization. Contour plots contain 20 contours from 3.3e−4 –1.0e−1 au. Red indicates a depletion of electron density and green indicates an accumulation of electron density. Computed at ZORA-BP86/TZ2P [35–40].

8 Energy decomposition analysis in the context of quantitative molecular orbital theory |

207

density back into the overlap region, namely, the relaxation from the wavefunction Ψ0 to the fully converged wavefunction ΨSCF . The change in density from the superposition of the fragment densities to ρ0 , as depicted in Figure 8.4, may be viewed as a manifestation of Pauli’s exclusion principle. This principle, which follows from the antisymmetry requirement of the wavefunction, states that electrons with the same spin are not allowed to be at the same place (vide supra) [44]. Thus, by antisymmetrizing the wavefunction, the probability density in the overlap region is reduced compared to what it would be if the necessary antisymmetry of the wavefunction had not been considered. At last, the wavefunction Ψ0 of equation (8.7) is relaxed to the SCF solution by allowing the interactions with the virtual orbitals. This action results in the orbital interaction energy ΔEoi . This energy term contains, together with the donor-acceptor interactions of the π-system (vide supra), the energy lowering connected to the formation of the electron-pair bond between the fragments, which will, in this case, be part of the orbital energy in σ symmetry. To obtain an estimate of the pair bond energy (i. e., the energy gain by making an electron bond), one might consider introducing an intermediate step by first allowing the pair bond (pb) formation followed by the wavefunction relaxation. The pair bond wavefunction Ψpb can be described as followed [34]: Ψpb = N|(occupied orbitals)A (occupied orbitals)B (5σA + 5σB )2 |

(8.8)

The electrons in the SOMO 5σ orbitals have now been allowed to form an electronpair in the 5σA + 5σB bonding orbital, which results in a stabilizing pair bond energy ΔEpb (Figure 8.5). Note that in the center and left of Figure 8.5, the familiar EDA terms ΔVelstat , ΔEPauli and ΔEoi are depicted and that we are now dealing with a further decomposition of the ΔEoi into ΔEpb and ΔErelax , as shown on the right side of Figure 8.5. We use Ψpb to define the pair bond energy, but an analogous definition would correspond to the valence bond (VB) wavefunction in which Ψ0 is accompanied with the determinantal wavefunction in which the electron spins of the SOMOs 5σ orbitals have been interchanged: |(occupied orbitals) 5σA β(1) 5σB α(2)|. We stress that the resulting wavefunction, with these two VB configurations, gives a slightly lower energy in the case of H2 at equilibrium distance, compared to the KS wavefunction, but it is well known that the interpretation of bond does not differ in the VB and KS-MO cases [13]. The energy stabilization that results upon formation of the electron-pair bond in either the VB or the MO description is triggered by the resonance integral (hopping integral, interaction matrix element) ⟨5σA |heff |5σB ⟩. The MO representation of equation (8.8) fits perfectly in the MO-based presentation we are providing. Note that in the wavefunction Ψpb the 5σA + 5σB orbital will, indeed, overlap with the lower lying σ-symmetric orbitals, for example, the 4σA + 4σB , which, in turn, embodies the Pauli repulsion of the same-spin electrons of the SOMO 5σ with the HOMO 4σ orbitals. Finally, the pair bond wavefunction Ψpb is allowed to relax to the SCF KS solution ΨSCF . The admixture

208 | T. A. Hamlin et al.

Figure 8.5: Diagram of the relation between the various energy changes used in the interaction energy analysis of the formation of NC–CN from two CN∙ fragments, where stabilizing energy terms are shown in green and destabilizing energy terms in red.

of virtual orbitals results in the relaxation energy ΔErelax (see Figure 8.5) and this step contains contributions from both donor-acceptor (charge transfer) and polarization, which partly serve to relieve the steric repulsion that results upon bond formation. The various steps up until this point are elucidated in the σ-orbital interaction diagram of Figure 8.6, where only the HOMO 4σ and SOMO 5σ orbitals are shown. The Ψ0 consists, for a large part, of the two-center four-electron destabilizing interaction between the HOMO 4σ orbitals of both interacting fragments. We have illustrated this effect by a sizable splitting of the stabilizing bonding and destabilizing antibonding orbitals. Additionally, the NC–CN dimer will have Pauli repulsion between the singly occupied orbital (SOMO 5σ) with the lower-lying doubly occupied orbitals of the opposite fragment (HOMO 4σ), due to a destabilizing overlap between these two orbitals both located on a different fragment. This is indicated in the diagram of Figure 8.6 by destabilization of the SOMOs in the Ψ0 wavefunction. The second step in the σ-orbital interaction diagram consists of the formation of Ψpb , by allowing the SOMO 5σ orbitals to pair up in a bonding orbital, which yields the energy lowering ΔEpb in Figure 8.5. We may consider the change from the Ψ0 to Ψpb to occur via the formation of the strongly stabilized 5σA + 5σB orbital (gray lines in Figure 8.6), which again will become destabilized by a repulsive interaction with the occupied 4σA + 4σB orbital. Finally, the intermediate wavefunction Ψpb is relaxed to ΨSCF by means of polarization and charge transfer interactions and manifests itself in a stabilization of the antibonding 4σA −4σB and 5σA − 5σB orbitals.

8 Energy decomposition analysis in the context of quantitative molecular orbital theory | 209

Figure 8.6: Schematic representation of the σ-orbital interaction diagram for the formation of the NC–CN dimer, representing the interaction between the CN∙ HOMO 4σ and SOMO 5σ fragment orbitals.

8.4 Conclusion In this chapter, we have introduced and discussed the canonical energy decomposition analysis (EDA), which represents one of the most powerful methods for analysis of the chemical bond. EDA comes in a variety of “flavors” (e.g., symmetry-adapted perturbation theory (SAPT) [45], interacting quantum atoms (IQA) [46]) and, in general, they provide similar conclusions, yet, the canonical EDA is highly appealing, in the sense that it is based on the Kohn–Sham molecular orbital (KS-MO) model. The KS-MO model is a simple and exact one-electron model that offers physical insight into the chemical bond that is under study, based on explicit calculation of the EDA terms, which include the electrostatic interaction (ΔVelstat ), Pauli repulsion (ΔEPauli ) and orbital interactions (ΔEoi ). The classical electrostatic interaction ΔVelstat emerges from the Kohn–Sham calculations as the Coulomb interaction between exact electronic charge densities and nuclei. The often-overlooked Pauli repulsive orbital interaction (or simply Pauli repulsion) ΔEPauli , which is the physical origin of steric repulsion, is in certain cases a major factor in determining bonding features and conformations. Then, the stabilizing orbital interaction ΔEoi offers insight into the covalent nature of the chemical bond that arises from electron-pair bonding, charge transfer (e. g., HOMO–LUMO interaction), and polarization within the fragments connected by the bond. With the help of group theory, the orbital interaction can be further decomposed into the individual contributions from each irreducible representation. As an example, the EDA method was applied to the formation of the NC–CN dimer to

210 | T. A. Hamlin et al. demonstrate and understand the physical origin of the various energy terms with this decomposition scheme. We take a moment to highlight that the canonical EDA has also been essential for increasing our physical understanding of several organic, inorganic, and biological transformations [47–51]. It is envisaged that the success of the canonical EDA method will continue to provide chemists spanning all specialties with unparalleled insight into the chemical bond and beyond.

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[39] van Lenthe E, Baerends EJ. Optimized Slater-type basis sets for the elements 1–118. J Comput Chem. 2003;24:1142–56. [40] Franchini M, Philipsen PHT, van Lenthe E, Visscher L. Accurate coulomb potentials for periodic and molecular systems through density fitting. J Chem Theory Comput. 2014;10:1994–2004. [41] Albright TA, Burdett JK, Wangbo WH. Orbital Interactions in Chemistry. Hoboken, USA: Wiley; 2013. [42] Clayden J, Greeves N, Warren S, Wothers P. Organic Chemistry. New York, USA: Oxford University Press Inc.; 2001. [43] Housecroft CE. Constable EC. Chemistry: An Introducion to Organic, Inorganic and Physical Chemistry. Harlow, UK: Pearson education ltd; 2010. [44] Fujimoto H, Osamura Y, Minato T. Orbital interaction and chemical bonds. exchange repulsion and rehybridization in chemical reactions. J Am Chem Soc. 1978;100:2954–9. [45] Jeziorski B, Moszynski R, Szalewicz K. Perturbation theory approach to intermolecular potential energy surfaces of van der Waals complexes. Chem Rev. 1994;94:1887–930. [46] Blanco MA, Martín Pendás A, Francisco E. Interacting quantum atoms: a correlated energy decomposition scheme based on the quantum theory of atoms in molecules. J Chem Theory Comput. 2005;1:1096–109. [47] Hamlin TA, Fernández I, Bickelhaupt FM. How dihalogens catalyze Michael addition reactions. Angew Chem. 2019;131:9015–20. Angew Chem Int Ed. 2019;58:8922–6. [48] Vermeeren P, Sun X, Bickelhaupt FM. Arylic C–X bond activation by palladium catalysts: activation strain analyses of reactivity trends. Sci Rep. 2018;8:10729. [49] Vermeeren P, Hamlin TA, Fernández I, Bickelhaupt FM. How Lewis acids catalyze Diels–Alder reactions. Angew Chem. 2020;59:6201–6. Angew Chem Int Ed. 2020;132:6260–5. [50] Sergeieva T, Hamlin TA, Okovytyy S, Breit B, Bickelhaupt FM. Ligand-mediated regioselective rhodium catalyzed benzotriazole-allene coupling: mechanistic exploration and quantum chemical analysis. Chem Eur J. 2020;26:2342–8. [51] van Bochove MA, van Roos G, Fonseca Guerra C, Hamlin TA, Bickelhaupt FM. How Mg2+ ions lower the SN 2@P barrier in enzymatic triphosphate hydrolysis. Chem Commun. 2018;54:3448–51.

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Part III: Bonding descriptors from quantum crystallography

Piero Macchi

9 Introduction to quantum crystallography Quantum information from experimental crystallography

9.1 The scope: What is quantum crystallography? In the recent years, the name quantum crystallography has been widely adopted in science [1], accompanying research work that consists of accurate crystallographic studies within a quantum mechanical framework. This symbiosis should not surprise us, because the two fields have been tightly entangled for a long while, since the first applications of X-rays that were contemporary with the beginning of the quantum era itself. A clue comes from the heterogeneous attendance at what is probably the most famous meeting in the history of science, namely the Solvay Conference on Physics 1927, entitled Electron and Photons, that witnessed the completion of the quantum mechanical theoretical framework.1 In fact, together with the well-known fathers of wave and matrix mechanics (like Bohr, Born, Dirac, Heisenberg, Pauli and Schrödinger) and those who initiated the quantum physics (like Planck, Einstein, Marie Curie), other scientists today reckoned as crystallographers, attended that conference. Among them, Bragg, Brillouin, Compton and Debye. For reasons explained below, one should include in the crystallographic team DeBroglie as well, although he never defined himself a crystallographer.2 What was the expected contribution of Bragg, Compton, Debye and Brillouin at the Solvay conference? It dealt mainly with the development of scattering techniques, regarded as the experimental methods able to provide definitive proofs of quantum mechanical theories and of the atomic model. While the incorrect Bohr model [3] for atoms could explain some spectroscopic evidences, the ongoing development of the so-called second quantum physics needed more sophisticated and in-depth observations. The report of Bragg was not remembered as the most relevant at the Solvay conference and he did not contribute much to the discussion, whereas the report and contribution to the discussion by Compton were probably not fully understood, despite containing a number of intuitions that scientists could appreciate in full only 1 A debate is still ongoing on the interpretation of quantum mechanics. According to Popper [2], the Solvay Conference 1927 was not the place where unification occurred, as generally accepted, but where a schism occurred. 2 Interestingly, on the website of the International Union of Crystallography, De Broglie is included in the list of crystallographers who won the Noble prize, whereas Compton is not. Piero Macchi, Department of Chemistry, Materials and Chemical Engineering, Politecnico di Milano, via Mancinelli 7, 20131 Milano, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110660074-009

216 | P. Macchi years later. Compton anyway contributed very profoundly to the discussion sessions. Even more substantial is the book by Compton [4] X-Rays and Electrons, in which he proposed the first comprehensive quantum theory of X-ray scattering that is at heart of quantum crystallography. While the theoretical scaffoldings of the second quantum physics and of the modern crystallography were quite established already in the late 1920s, the computational3 and experimental parts took much longer to put into practice those intuitions. Thus, the merging into quantum crystallography was much slower. Without doubts, a pioneer was Richard J. Weiss, who attempted to put into practice a famous prediction stated by P. Debye [5]: It seems to me that the experimental study of the scattered radiation, in particular from light atoms, should get more attention, since along this way it should be possible to determine the arrangement of the electrons in the atoms.

In the late 1950s, Weiss tried what is nowadays still challenging, i. e., to determine the electronic configuration of metals by means of X-ray diffraction [6]. Needless to remind, at that time, even the determination of a molecular geometry through X-ray diffraction from single crystal was a challenge, while it is nowadays a routine. Weiss’s experiments were especially difficult,4 because the kinematic theory of X-ray diffraction may not be correct for species like those he investigated. As admitted in his autobiography [7], the results were highly affected by an attenuation of the scattered signal, but models to correct for the primary and secondary extinction [8] were not available at that time. Nevertheless the approach to break down the atomic scattering into orbital contributions and test different electronic population for each shell, was a terrific breakthrough that gave eventually rise to the field of X-ray charge density [9]. One of the challenges proposed by Weiss was the calculation of a wavefunction from X-ray diffraction, combining the elastic Bragg and the inelastic Compton scattering. Again, the idea was using scattered intensities to model the arrangement of electrons in atoms, going beyond the simple determination of the number of electrons in each atomic shell, trying instead to determine precisely the space distribution 3 It is vital to stress here that theory and computations are not at all synonymous, as sometime mistakenly assumed. Computations and observations (through experiments) are the two means to validate or falsify theory within the framework of a natural science. Computations enable to extract quantitative information from the models elaborated from theory, whereas experiments enable to ascertain whether the model is a realistic copy of the realm and, consequently, whether the theory is valid or not. 4 A project of the former commission on “Charge, Spin and Momentum Density” of the International Union of Crystallography, was launched in 2012 with the title “Electron distribution in the metallic bond by QCBED techniques and x-ray diffraction” (where QCBED stands for quantitative convergentbeam electron diffraction).

9 Introduction to quantum crystallography | 217

of the wavefunction (where space could refer to position or momentum space).5 The goal could be reached combining the complementary information available from each technique. Moreover, Weiss proposed an alternative and more feasible roadmap: using X-ray diffraction to correct computed Hartree–Fock wavefunctions. The Hartree–Fock method [10, 11] is a sort of mean field approximation that omits the instantaneous correlation among electrons. The error of such a method is relatively small on the energy of a system, but quite significant for a correct estimation of the energy landscape, hence to determine a correct molecular or crystal geometry, and the accurate mapping of the electron density distribution. Weiss’s intuitions were exploited by many scientists starting from the 1970s, who tried to follow the two above mentioned paths, i. e., using scattering techniques to determine coefficients of a wave function or to improve a wavefunction previously computed using first principles. The direct determination of wavefunction coefficients was certainly easier to approach, but very difficult to achieve. The initial steps are due to Stewart and Coppens. Stewart published a series of studies [12–15] intended to develop the so-called generalized atomic scattering factors (first introduced by Dawson [16]) from the evaluation of Fourier transforms of molecular orbital products. The molecular orbitals came from a wavefunction calculation using the molecular orbital approximation. Coppens et al. [17] introduced a least-square procedure to refine population coefficients of the orbital products from X-ray diffraction intensities. Both these projects were motivated by earlier works showing that features of non-spherical valence electron density distribution could be revealed by accurate X-ray diffraction measurements [18, 19], if the thermal motion of atoms could be sufficiently reduced and possibly accounted by other sources of information, for example, neutron diffraction [20]. Using a different approach, Clinton and coworkers [21–23] attempted to solve an even more complicated problem, which is the calculation of the coefficients of the one electron density matrix. First, they developed a series of equations to enable the determination of density matrices for pure states, and then applied this method using X-rays or γ-rays diffraction data, though necessarily focused on very small systems. The attempts to refine orbital product coefficients of molecular wavefunctions from diffraction data turned out to be too difficult given the small amount of information available from the experimental measurements in spite of the complexity of the problem. The main problem is the presence of two-center orbital products, i. e., elements of the electron density, hence of the scattering object, that depend on orbitals centered on two atoms. These terms arise only for interacting atoms, whereas they could be negligible under the hypothesis of a small overlap of atomic orbital func5 The reader should remember that because in quantum mechanics position and momentum operators do not commute, each electron function can be represented in the space of positions or in the space of momentums.

218 | P. Macchi tions. This induced many authors, especially Coppens and Stewart, to develop simplified models, avoiding the two-atom centered terms and using only one-atom centered terms, the so-called multipolar model (MM). This model preserves some features of the widely adopted independent atom model (IAM), which is the standard model for traditional crystal structure solutions and refinements. The common assumptions of IAM and MM are: a) the Born–Oppenheimer [24] approximation holds (nuclear motion and electron motion are not coupled); b) the one-electron density of a system (molecule, crystal) is a simple sum of atomic terms; c) the thermal motion is approximated with independent atomic oscillators (not necessarily harmonic); d) the atomic electron densities are associated with the atomic thermal parameters and remain constant along the nuclear motion. Instead, the MM differs from IAM because: a) the atomic electron densities are not spherical; b) the atoms are not necessarily neutral; c) the atomic electron density shells can be expanded or contracted. By maintaining an approach conceptually similar to IAM, the MM was successfully received by a broad community, because the model refinement uses similar mathematical algorithms as the traditional crystal structure refinements namely the least squares or the conjugated gradient methods. The main difference, and increased complexity, stands in the much larger number of parameters describing the model (see Chapter 10 for more precise details). Even before the development of the most adopted methods for MM, many aspherical atom centered models have been proposed in the literature. For example, McWeeny [25–28] published a series of papers where he proposed a breakdown of the atomic scattering factors into orbital scattering, including nonspherical atomic states and introducing corrective factors to account for the chemical bonding. Because Slater type orbitals are difficult to handle mathematically, he proposed Gaussian type functions that are also typically adopted in computational quantum chemistry [29]. Later on, Kleinman and Phillips [30] as well as Weiss [31] used hybrid orbital densities or symmetry adapted density functions to model the charge density of carbon atom in diamond (or other elemental solids of the same group, like De Marco and Weiss [32]). Dawson [33, 19] also proposed an aspherical atom model and adopted it for the calculation of the structure factors of diamond. The alternative pathway to the direct refinement of an electron wavefunction or an electron density, was using the experimental observation as a benchmark to adjust a wavefunction calculated from approximated first principles methods. In the appendix of his book on X-ray determination of electron distributions, Weiss [34] suggested that

9 Introduction to quantum crystallography | 219

the Hartree–Fock wavefunction of a system ϕ, which is only an approximation of the exact wavefunction ψ, can be corrected by an orthogonal term χ. Instead of using the ̂, i. e., the electronic energy E of the expectation value of the Hamiltonian operator ℋ system, one can determine the correction factor from the expectation value of another ̂ Within this approximation, the correcoperator, for example, the scattering factor Q. tion factor must be small. The drawback is that this may be difficult to determine. Nevertheless, the term χ can be extremely useful because it ideally includes the electron correlation that is missing in a Hartree–Fock calculation. A few years earlier, a method was proposed by Mukherji and Karplus [35], to calculate molecular wavefunctions constrained to experimental data. In the simplest formalism, the data were observations, which correspond to expectation values of oneelectron operators (hence excluding the electronic energy of the system itself, which depends, in part, on a two electron operator). Mukherji and Karplus [35] used the dipole moment and electric field gradient, taking advantage of available spectroscopic data, but they could have used any other observable. This approach was seamlessly adequate for the suggestion by Weiss, if the observation used as constraints were the X-ray diffracted intensities (which are expectation values of a one-electron operator, see below). The most successful method based on this approach is undoubtedly the socalled X-ray constrained wave function, developed by D. Jayatilaka [36] and Jayatilaka and Grimwood [37]. Given the way in which the approximated ψ is obtained, a better name for the method is X-ray restrained wave function [38], because the bound to experiment is not a hard constraint, but applied through a Lagrange multiplier that can be adjusted in order to increase or decrease the effect of the experimental information. This adjustment is not bound to reproduce a specific observation, and it is normally adopted in order to minimize the average discrepancy with respect to the observed diffracted intensities, within the assumptions of a statistical χ 2 distribution.6 In Chapters 10 and 11, the multipolar model and the X-ray restrained wavefunction approach will be discussed in more detail. Other methods, such as the refinement of density matrices, are beyond the scope of this book and will not be presented. In this chapter, instead, we will review the scattering techniques of quantum crystallographic studies, and we will briefly introduce other measurements that, although more rarely adopted, belong anyway to the arsenal of quantum crystallography.

6 Note that χ here has no connection with the χ term defined above as a perturbative function of the Hartree–Fock wavefunction. The χ 2 statistical distribution is one of the most well-known and adopted distributions in statistics for sums of squared values.

220 | P. Macchi

9.2 Scattering techniques One of the most important interactions between electromagnetic radiation and matter is the scattering, i. e., the emission of radiation by a particle due to the acceleration received from the electromagnetic field exerted on it. The classical, nonrelativistic theory of X-ray scattering is due to Thomson [39]. An isolated charged particle, like the electron with charge −e, when subjected to an electric field E (e. g., that of a photon) emits a radiation in all directions, the amplitude of which depends on the angle φ between the direction of the electron acceleration and direction of the scattered radiation: Eφ = −

Ee2 sin φ rmc2

(9.1)

where m is the electron mass and c is the speed of light.7 The scattering of an ensemble of electrons (e. g., an atom, a molecule, a crystal) was reformulated independently by Debye [5] and Thomson,8 corresponding to the interference of the waves emitted by each electron, taking into account the distance between them. The interpretation of X-ray diffraction by crystals could be explained with the classical Thomson scattering, however, the distribution of electrons in atoms, and their aggregates, does not obey to laws of classical electrodynamics. Moreover, as stated by Compton [4]; Far from explaining the scattering of X-rays on the assumption that radiation spreads in all directions as spherical waves, we seem driven by the recent experiments to consider X-rays as definitely quanta of radiation energy.

Among the “recent experiments,” Compton referred to the change of wavelength of inelastically scattered X-rays, that takes in fact his name (the Compton scattering). A quantum treatment is anyway necessary also for the Bragg scattering, i. e. the elastic scattering of X-rays where the wavelength is of course preserved.

9.2.1 Quantum theory of X-ray scattering The total scattering factor f of a given electronic charge distribution ϱ can be expressed as a sum of individual one-electron operators that combines the shifts among scat7 If the incoming ray is an unpolarized electromagnetic radiation (thus the electron acceleration occurs perpendicularly to the propagation of the incoming ray), then the term sin φ can be replaced by

√(1 + cos2 2ϑ)/2, where 2ϑ is the angle between the scattered and incident direction, an consistent with what used in equation (9.5). 8 In the book, X-ray and electrons, Compton [4] mentioned that he has received a manuscript read by J. J. Thomson in front of the Royal Institution in 1916.

9 Introduction to quantum crystallography | 221

tered waves with respect to the incoming radiation: f (k − k 0 ) = ∫ ψ∗ (r) ∑ ei(k−k 0 )r j ψ(r)dτ j

(9.2)

The operator ∑j ei(k−k 0 )r j comes from the Born approximation, ψ(r) is the wavefunction of the system, k 0 and k the incoming and scattered wave vectors, r j the position of jth electron and dτ the volume element in position space. In other words, this is the quantum mechanical translation of the classical Debye–Thomson model. Because the one-electron charge density is the integration of the wavefunction over all electrons but one, the scattering factor can be correlated to the electron density itself: f (k − k 0 ) = ∫ ϱ(r)ei(k−k 0 )r dτ

(9.3)

This relationship is a so-called Fourier transformation, correlating the position space r and the diffraction space of s = k − k 0 . From equations (2) and (3), we are able to appreciate that: (a) the scattering is inherently a quantum mechanical phenomenon (the electron and the photon are both treated as waves); (b) a connection exists between the observable scattering and the observable one electron density; (c) their interplay involves the wavefunction itself.

9.2.2 X-ray diffraction The best way to observe the scattering is using a “condenser,” for example, a crystal lattice made of infinite repetition of the object (atom or molecule) under study. Because of the periodic homogeneity of crystals, the scattering by the atoms of a crystal accumulates at special positions, defined by the lattice vectors. The Bragg law9 (valid for any radiation) describes the diffraction: 2d sin ϑ = nλ

(9.4)

where λ is the radiation wavelength, n is an integer, d is the distance between two crystal planes normal to the scattering vector s = k − k 0 , and ϑ is the semiangle between the incident beam and the scattered beam. The direction is typically indicated by the Miller indices hkl, a triplet of components of the scattering vector with respect to a lattice (called the reciprocal lattice) defined perpendicular to the planes of the direct 9 Noteworthy, in the original definition, Bragg used the supplementary angle between the incoming radiation and the normal to the crystal plane. This implied the formula being 2d cos ϑ = nλ. In all subsequent papers, however, Bragg changed the geometrical definition and the law took the format normally adopted in crystallography and physics.

222 | P. Macchi crystal lattice [40]. The reciprocal lattice is normalized so that only the nodes of the reciprocal lattice (H), where hkl are integers, comply with the constructive interference among the waves elastically scattered by the atoms in the crystal. The scattered intensity by a vibrating electron density distribution is attenuated compared with a static one. The thermal factor describes this reduction, which depends on the amplitude of the vibration of the atoms (hence on the temperature). As stated above, the assumption in all models is that the atomic electron density term does not change along the coordinate of a vibrational mode. In practice, the effective density “visible” through scattering is the thermally averaged electron density, ϱ(r), i. e., a convolution of the electron charge density ϱ(r) and the probability distribution function of the nuclei P(R) (where R refers to a set of nuclear coordinates, whereas r refers to a point in position space). In the reciprocal space, the total scattering factor due to ϱ(r) is the product of the Fourier transformations of ϱ(r) and P(R). In a crystal, atoms are not isolated but somewhat connected through intramolecular covalent bonds or intermolecular secondary interactions. Nevertheless, an atomistic approximation is valid for both the electron density and the nuclear probability distribution functions. The scattering factor F(H) of a group of atoms, for example, those in the unit cell of a crystal, can be approximated as Nat

F(H) = ∑ fj (H)Tj (H) exp(2iπH.r j ) j=1

(9.5)

where fj (H) and Tj (H) are the atomic electron scattering and thermal factors. In Chapters 10 and 11, different ways to partition the total electron density into atomic form factors are presented. Moreover, in Chapter 11, the partition of the electron density arising from two center orbital products in terms of single centers atomic thermal factors is introduced, which is relevant for the X-ray constrained wavefunction approaches. As evident from the equation (9.5), the form factor is a function the scattering vector H. Because of the reciprocal space definition, |H| = 2 sin ϑ/λ. This implies that the scattering of an atom is a function of the scattering angle or more precisely of the diffraction resolution sin ϑ/λ.10 Both the electronic and the thermal factors decrease as the diffraction resolution increases. In fact, the nature itself of the scattering operator defined in equation (9.2) is such that a larger distance between electrons would produce a smaller f . If all electrons of an atom were steadily located at the nuclear site, then there would be no such dependence. However, this is not possible, again for application of quantum mechanical principles. Two electrons cannot share the same set 10 In scattering techniques, the scattering vector may be defined in different ways, such as |H| = 2 sin ϑ/λ (adopted in this chapter and in general in Bragg X-ray diffraction from crystals) or |q| = 4π sin ϑ/λ (more frequently used in small angle scattering). In all cases, high resolution refers to a large scattering vector amplitude, which implies a small spatial shift between scattered waves.

9 Introduction to quantum crystallography | 223

of quantum numbers (principal n, orbital l, magnetic ml and spin ms ), as implied by the Pauli principle [41]. As a consequence, each atomic orbital cannot host more than two (spin opposite) electrons. Moreover, only the orbital 1s has a finite probability to accumulate electron density at the nuclear site. Anyway, 1s is not strictly confined on the nucleus, but features a spatial distribution in the vicinity, dictated by the solution of the Schrodinger equation. As a result, for all electrons in atomic orbitals, the scattering is not constant but it decreases with the diffraction angle, more rapidly the lager is the spatial extension of a given orbital far from the nucleus. For this reason, core electrons scatter up to very high angle, whereas valence electrons scatter only up to a low resolution (see Figure 9.1).

Figure 9.1: Breakdown of the atomic form factor of carbon into core and valence shell contribution.

The thermal factor is also ϑ dependent for a somewhat similar reason. Tj (H) would be unitary, if the jth nucleus was steady at the position R0 . As this is not possible even at T = 0 K, due to the inherent zero point vibration, the effect of atomic displacements from equilibrium position is that of increasing, on average, the distance between electrons and, therefore, to reduce the atomic scattering (again, in keeping with equation (9.2)). Tj (H) clearly depends on the temperature, but also on the forces experienced by a nucleus in a crystal which affects the amplitude of the atomic displacement. The tighter is the bound between the nucleus and the rest of the crystal, the smaller will be the amplitude of the atomic displacement from the equilibrium position, at a given temperature. As anticipated above, the thermal motion is always

224 | P. Macchi partitioned in atomic terms, assuming that each atom is an independent oscillator. However, there are different ways to treat the atomic motion, assuming for example that the oscillation is always harmonic or otherwise anharmonic. The harmonic approximation implies that P(R) is a normal distribution function, described by a Gaussian function, potentially identical along all directions (isotropic displacement) or not (anisotropic displacement). In reality, the potential around an atom is always anharmonic. If the deviation from harmonic is not too large, a correction is possible as kind of Taylor expansion of the harmonic model (called the Gram–Charlier expansion, Johnson and Levy [42]). This implies corrections of the normal distribution, accounted by the so-called cumulant. In standard structure determination from X-ray diffraction, the harmonic (anisotropic) assumption is typically adopted, whereas in accurate charge density refinement, anharmonic motion may also be considered, albeit very demanding. All issues connected with thermal motion (like inherent convolution with electron density function, reduced scattering, anharmonic contributions, etc.) suggest that a measurement at the lowest possible temperature is preferable. In fact, the smaller is the thermal factor, the easier is the deconvolution, the smaller is the deviation from a harmonic motion, the larger is the scattering from the crystal, especially important at high diffraction angle where anyway the electronic contribution is smaller due to the lack of valence electron contribution. So far, we have discussed about the scattering factor of the thermally smeared electron distribution. A question now arises: what about the actual diffracted intensities measured during an experiment? The so-called kinematic approximation implies that the diffracted beam is independent from any interference of the incident beam or any secondary effect induced by the radiation. In this way, if phenomena like absorption, extinction, refraction and multiple scattering are ignored, the diffracted intensities is simply proportional to the square of the scattering factor F(H). However, those effects, which are almost unavoidable, can be in part corrected maintaining a kinematic like approach, and introducing a dependence of the scattering factor from the radiation wavelength. For example, the absorption of the radiation can be easily corrected using a classical Lambert–Beer equation, where the attenuation of the measurable X-ray intensity depends exponentially on the length of the path of k and k 0 inside the crystal and on the linear absorption coefficient (which is radiation dependent). The atomic absorption is larger for heavier elements and for low energy (long wavelength) radiation. In a simplified model, also the secondary extinction, i. e., the attenuation of the primary beam due to the scattering of more and more planes, can be modelled as a perturbation of the kinematic scattering (similarly to the absorption). Primary extinction and multiple scattering are instead more difficult to correct within the kinematic theory and may require a more sophisticated dynamical theory. From the above discussion, we understand the requisites for a very accurate Xray diffraction experiment, which could enable the extraction of quantum crystallographic information:

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a) low temperature, necessary to reduce the negative effects of large atomic displacements and increase the diffracted intensities; b) high resolution, necessary to increase the number of observations and enable a more sophisticated modelling (see especially Chapter 10); c) precise measurements and accurate corrections of the diffracted intensities,11 because the electron density features that differ from a simple spherical distribution of atomic densities are tiny and, therefore, require very precise and accurate estimations of the structure factors; d) small crystals, to reduce absorption, extinction and multiple scattering; e) short wavelength, to reduce absorption, extinction and multiple scattering and increase the diffraction resolution; f) brilliant X-ray sources, to increase the measured intensity (per unit of time) and enable more precision. A more detailed discussion of the techniques or the instrument is beyond the scope of this book and anyway subject to rapid aging, given the pace at which technological improvements in instrumentation occurs for production or detection of X-rays, and for conditioning of the samples.

9.2.3 Electron diffraction The dual nature of waves and particles is one of the paradigms of quantum mechanics. It is especially thanks to the work of De Broglie [43] that this dualism has eventually become sustainable for the same entities, abandoning the exclusivity and, therefore, closing a longstanding debate that characterized the previous two centuries. De Broglie’s work focused on electrons and in fact, he used the diffraction of electron beams by a crystal in order to prove the wave nature of electrons [43]. In this respect, de Broglie was a special kind of quantum crystallographer who investigated the nature of a radiation using a crystal, whereas crystallographers (and quantum crystallographers as well) normally do the opposite. While both electrons and X-rays can produce images and therefore are suitable for microscopes, the X-ray microscopy is a much more recent achievement of science due to the technical complexity of the focusing. On the contrary, electron radiation has been especially used for microscopy, whereas electron diffraction from crystals has only more recently become widespread and its potential is not fully exploited yet. Like X-rays, electrons are scattered by atoms. However, electrons feel the entire atomic electric potential, generated by both nuclei and electrons. Therefore, their scat11 The reader should be aware of the difference between accuracy and precision. Accuracy means minimizing the systematic errors of a measurement and, therefore, approaching the correct value, whereas precision implies reproducibility of the measurement.

226 | P. Macchi tering is different in nature, although the structure factor has a similar form as previously introduced and the Bragg law is of course equally valid. The electron and X-ray atomic form factors are correlated by the so-called Mott–Bethe formula: fe (u) = ∫ V(r)e(−2πiur) dr =

e (Z − fX (u)) 2πu2

(9.6)

where e the unit electric charge, u = s/2π is the scattering vector, r is the position vector, V(r) the electric potential generated by the protons and the electrons, Z is the atomic number (nuclear charge) and fX (s) the X-ray atomic scattering factor. Notably, the electric potential has replaced the electric charge density in the form factor formula. A notable consequence is that the contribution of electrons decreases more rapidly in reciprocal space than that of nuclei. Moreover, in electron diffraction from crystals, the pseudo-kinematic approximation is not applicable. The much higher electron energy and the much stronger scattering power make it impossible circumventing a properly defined dynamical theory of electron diffraction, thus including the multiple scattering events. The major consequence is that the measured intensity is not proportional to the square of the structure factor amplitude (like for X-rays) but to the structure factor amplitude. Among various techniques, which are not described here, the quantitative converged beam electron diffraction [44] has significantly developed and nowadays it is possible to almost routinely measure bonding-sensitive low-order structure factors with very small uncertainties, as for example recently applied to Al samples [45]. Moreover, determination of molecular structures is becoming more feasible [46]. Because of the equations described above, a multipolar model as well as an Xray restrained wavefunction can be similarly calculated from experimental electron diffraction. The use of a dynamical theory instead of a kinematic theory was emphasized as a way to circumvent some pitfalls of X-ray diffraction techniques [1].

9.2.4 Neutron diffraction Another radiation of interest is that of neutrons. As discussed for electrons, neutrons also produce a diffraction from a periodic lattice in view of their wavelike behaviour. Neutron diffraction however differs from X-ray and electron diffraction quite significantly. Although the Bragg law remains valid and the structure factor maintains an atomistic form, neutrons cannot interact with the electric charges of atoms and therefore they do not map the charge density or the position electron wavefunction. They have a (small) scattering cross section with atomic nuclei and, therefore, they are able to inform us on the equilibrium position and vibrational displacement of atomic nuclei. This information is similar to that obtained with X-rays, but from an independent perspective: the nuclear probability distribution function is the actual observable of a

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neutron experiment, whereas it is deduced via modeling for an X-ray diffraction experiment. At variance from X-ray or electron form factors, the so-called neutron scattering length is independent on the diffraction angle (nuclei can be regarded as point particles),12 whereas the temperature dependent atomic displacement has a similar effect and the refined displacement parameters should coincide with those obtained from X-ray diffraction techniques. For this reason, neutron diffraction has been often employed in early charge density experimental determinations, as a way to circumvent the correlation between thermal smearing and bonding effects in the charge density distribution [20]. The neutron scattering is interesting not only for the possibility to determine independently nuclear positions and displacements, but also, and more importantly within the scope of quantum crystallography, for the magnetic component of the scattering. As a matter of facts, because neutrons are spin active, they interact with the magnetic moments of a system, due to spin or orbital magnetic moments. This component is very small and therefore more difficult to characterize. However, because the magnetic structure may differ (in periodicity) from the atomic structure, it is possible to observe diffraction peaks at positions which are not expected from nuclear scattering only. This enables a better modeling. The magnetic structure is somewhat the equivalent of the atomic structure. The latter consist of positions of electron density maxima, where we ideally locate atomic nuclei. The magnetic structure, instead, consists of magnetic moments centered at positions of the magnetic atoms in the crystal. As such, it does not provide itself precise information on the magnetization density. However, if using polarized neutrons, where the spin direction is selected, it is possible to gain insight in the distribution of spin density of a crystal. This could be modeled like the electron charge density, i. e., with a multipolar expansion of the excess spin density [47, 48], or otherwise included in the restraint equation of an X-ray and neutron restrained wavefunction. Gao et al. [49] have also correlated the electron density and wavefunction with the magnetic anisotropy.

9.3 Nonscattering techniques It would be a mistake to believe that quantum crystallography is associated only with diffraction techniques. There are two main reasons. On the one hand, while the scattering experiment is a fundamental example of the wave/particle hypothesis and crystals offer the magnifying effect of a periodically homogeneous lattice, the quantum theory is also strictly associated with absorption and emission of radiation that inform on 12 The radius of a nucleus is ca. 10−5 Å.

228 | P. Macchi different quantum states. On the other hand, diffraction is not the only technique that makes use of scattering. Therefore, other kinds of experiments may convey quantum mechanical information of a crystal. These techniques fall in the broad classes of spectroscopy and surface imaging (not based on scattering). In both cases, crystals are not as vital as for diffraction techniques; nevertheless, the periodic homogeneity enables an easier interpretation and a simpler modelling. Moreover, here we exploit the crystal as the object of study more than as the vehicle to make the observation (as is, e. g., in X-ray diffraction determination of molecular geometries).

9.3.1 Spectroscopies All the spectroscopies depend on quantum states of matter, perturbed by an electromagnetic field and the response of the system is a function of the difference between the states (in terms of energy, but also of symmetry that determines whether a transition is allowed or not). While some spectroscopies are applied only for a mere chemical characterization (like atomic identification) and, therefore, only marginally relevant for quantum crystallography, others offer instead precious information. Among them, nuclear magnetic resonance (NMR) is receiving ever-growing attention. The possibility to use NMR to determine the atomic structure of crystals is now quite well established [50, 51]: a constant magnetic field polarizes the nuclear magnetic moments and the subsequent application of a radio-frequency induces transitions between the magnetic states of the nuclei. Interactions within a crystal affect the nuclear magnetic moments and the energy levels. They depend on the relative orientation with respect to the applied magnetic field (the “space part” of the interaction), on the magnetic state of the nuclei, and on the orientation of the nuclear magnetic moments with respect to the main magnetic field (the “spin part” part of the interaction). Solid state NMR possesses phase coherence and takes advantage of the continuously modulated orientation of the sample in the magnetic field. Many interactions may be simultaneously active on each nucleus, like the electric current shielding, the indirect nuclear spin coupling mediated by electrons spin coupling, the direct dipolar nuclear spin coupling, the quadrupolar interaction between the nucleus and the electrical field gradient generated by the anisotropic charge distribution, the hyperfine interaction between nuclei and electrons. All these interactions carry information on the thermally smeared electron density, thus somewhat similar to the X-ray scattering. However, modeling all the interactions is difficult, although but it is possible to separate them into smaller pieces and recombine afterwards into a unified picture. So far, crystallographers have made limited use of the spin density information available from a NMR measurement, nevertheless the increasing number and quality of the experiments will likely offer more opportunities.

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Another spectroscopy which is very important for quantum crystallography is the (e, 2e) spectroscopy, which measures the electron momentum. This is possible by knocking one electron out of an atom or molecule by means of another electron of known energy and momentum. The energies and momenta of the knocked-out electron and of the scattered electron are measured, which enables the determination of the momentum of the electron before the collision. When this event is repeated for many electrons of a sample, a statistical distribution of electron momenta is obtained, which depends on the angle of collision and therefore it requires a periodically homogeneous sample like a crystal. Impedance spectroscopy on solids enables the characterization of the electric susceptibilities, meaning the capacity if a charge density to repolarize under application of an electric field. Temperature and frequency dependence of the dipolar processes in a crystal enables to estimate the electric polarizability of atoms and molecules. Recent studies have shown how this can be used in quantum crystallographic studies on materials for electronic applications featuring low dielectric constant [52]. Because the counterpart of the electric dipole is the magnetic dipole, it is also possible to gain information on the coupling of atomic magnetic moments, which is at heart in the rapidly growing field of spintronics. The superconducting quantum interference (SQUID) enables to quantify the magnetic susceptibility. Because this can be correlated with the energy difference between coupled states in a solid (ferromagnetic, ferrimagnetic, antiferromagnetic, or paramagnetic), one is able to retrieve information on the strength of magnetic exchange. This requires a separate knowledge of the atomic structure (e. g., through X-ray diffraction) or of the magnetic structure (by mean of neutron diffraction). A similar information is obtainable from the muon spin spectroscopy because the muon spin is able to probe the local magnetic field generated by atoms in the structure.

9.3.2 Surface microscopies The other class of investigation techniques includes imaging and microscopies. While X-ray or electron microscopy are unable at present to achieve the resolution necessary to visualize the fine details of the electron distribution around atoms, surface techniques are much ahead. In fact, they make use of atomic forces (electrostatic, shortrange repulsion or dispersion) or otherwise they reveal atomic states by tunneling. Both types are inherently linked to quantum mechanics. The scanning tunneling microscopy (STM, Binnig et al. [53]) is based on tunneling of electric current through the interatomic barrier between a tip and the sample, before they actually touch, which is a typical case of experiment that could not be even conceived without the quantum

230 | P. Macchi theory paradigm.13 The atomic force microscopy (AFM, Binnig et al. [54]) is also tightly connected with quantum mechanics, though mainly for the correct interpretation of the results, rather than the experimental hypothesis itself. Both STM and AFM do not strictly require crystalline materials and they are not informing on the inner layers of a sample, but only on the surface. Sometime, the homogeneous crystal surface acts as a sample holder to accommodate a molecule, which is the actual subject of the investigation. AFM is certainly the technique able to achieve the best resolution. Because STM strongly depends on the conductivity of the support, it may not be able to visualize a true image of the sample on the surface due to interference with a conductive support. However, STM is very informative on quantum mechanical states of the system near the Fermi level, i. e., the energy associated with the thermodynamic work necessary to add one electron to the system. Because states close to this level are those of the lowest energy unoccupied orbitals (or bands), the STM image is somewhat correlated with the frontier orbitals of the solid or the molecule lying on its surface. On the contrary, the AFM technique is sensitive to the whole electron distribution, given that long-range attractive forces are either electrostatic or dispersive and short range repulsive forces are mainly caused by the destabilizing energy associated with electrons being too close to each other, in virtue of the Pauli principle. The latter are particularly precise, and they increase in the vicinity of an atomic nucleus, therefore AFM is normally considered better to resolve at atomic level the structure of a molecule or a crystal surface [55]. By using a charged tip, it is also possible to visualize the charge state of atoms on the surface [56], measuring the force of chemical bonds [57] or mapping the spin coupling on a solid surface [58]. These techniques are still in their infancy and reported results are very few so far, however the potential is extremely high.

9.4 Outlook The implications of quantum crystallography for chemical bonding analysis emerges quite clearly from the previous paragraphs and the next two chapters. The electron wavefunction, the corresponding electron density and density matrices, the electron charge and spin distribution, all contain information on chemical bonding, which holds for atoms, molecules, polymers and crystals. When atoms and molecules are embedded in a crystal, the magnifying effect of periodic homogeneity offers the opportunity of depicting the electronic functions with a resolution that 13 This is different from X-ray diffraction. In fact, the diffraction can be anticipated with a classical undulatory theory of radiation and, not surprisingly the first X-ray diffraction experiments were carried out before the wave mechanics of Born and Schrödinger was developed. Quantum mechanics is instead necessary for a correct interpretation of the results.

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can be hardly matched by any other noncrystallographic technique. Progresses in surface microscopy techniques also enable envisaging their application for quantum crystallographic studies. While models for the electron wavefunction or for the electron density distribution are well established and functional, the huge array of observables that become available from new techniques or from improvements of known techniques, implies adaptation of these models. As introduced in this chapter, and discussed more in details in the next chapter, the multipole model features the necessary flexibility to adapt to more detailed information. Even more flexible are wavefunction models, because they couple first principle and experimental sources of information, thus increasing the virtual set of data and, therefore, enabling a higher degree of sophistication.

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[44] Kossel W, Möllenstedt G. DynamischeAnomalievonElek-troneninterferenzen. Ann Phys. 1942;42:287–93. [45] Nakashima PNH, Smith AE, Etheridge J, Muddle BC. The bonding electron density in aluminum. Science. 2011;331:1583–6. [46] Gruene T, Wennmacher JTC, Zaubitzer C, Holstein JJ, Heidler J, Fecteau-Lefebvre A, De Carlo S, Mgller E, Goldie KN, Regeni I, Li T, Santiso-Quinones G, Steinfeld G, Handschin S, van Genderen E, van Bokhoven JA, Clever GH, Pantelic R. Rapid structure determination of microcrystalline molecular compounds using electron diffraction. Angew Chem Int Ed. 2018;57:16313–7. [47] Deutsch M, Claiser N, Pillet S, Chumakov Y, Becker P, Gillet JM, Gillon B, Lecomte C, Souhassou M. Experimental determination of spin-dependent electron density by joint refinement of X-ray and polarized neutron diffraction data. Acta Crystallogr. 2012;A68:675–86. [48] Deutsch M, Gillon B, Claiser N, Gillet JM, Lecomte C, Souhassou M. First spin-resolved electron distributions in crystals from combined polarized neutron and X-ray diffraction experiments. IUCrJ. 2014;1:194–9. [49] Gao C, Genoni A, Gao S, Jiang S, Soncini A, Overgaard J. Observation of the asphericity of 4f-electron density and its relation to the magnetic anisotropy axis in single-molecule magnets. Nature Chem. 2020;12:213–9. [50] Taulelle F. NMR crystallography: crystallochemical formula and space group selection. Solid State Sci. 2004;6:1053–7. [51] Martineau C, Senker J, Taulelle F. NMR crystallography. Ann Rep NMR Spectr. 2014;82:1–57. [52] Scatena R, Guntern Y, Macchi P. Electron Density and Dielectric Properties of Highly Porous MOFs: Binding and Mobility of Guest Molecules in Cu3 (BTC)2 and Zn3 (BTC)2 . J Am Chem Soc. 2019;141:9382–90. [53] Binnig G, Rohrer H, Gerber C, Weibel E. Surface studies by scanning tunneling microscopy. Phys Rev Lett. 1982;49:57–61. [54] Binnig G, Quate CF, Gerber C. Atomic force microscope. Phys Rev Lett. 1986;56:930–4. [55] Gross L, Mohn F, Moll N, Liljeroth P, Meyer G. The chemical structure of a molecule resolved by atomic force microscopy. Science. 2009;310:1110–4. [56] Gross L, Mohn F, Liljeroth R J P, Giessibl FJ, Meyer G. Measuring the charge state of an adatom with noncontact atomic force microscopy. Science. 2009;324:1428–31. [57] Welker J, Giessibl FJ. Revealing the angular symmetry of chemical bonds by atomic force microscopy. Science. 2012;336:444–9. [58] Pielmeier F, Giessibl FJ. Spin resolution and evidence for superexchange on NiO(001) observed by force microscopy. Phys Rev Lett. 2013;110:266101.

Benoît Guillot, Christian Jelsch, and Piero Macchi

10 Multipole modeling with MoPro and XD 10.1 An introduction to multipole modeling The surrounding chapters in this book highlight the pivotal role of the charge distribution in the qualitative and quantitative characterization of chemical bonds, and in the determination of molecular and crystal properties. Hence, the motivation to obtain accurate analytical models of the charge distribution that arises from the electron density (ED). Providing such models is the aim of the multipole modeling, summarized in this section. Next, the XD2016 and MoProSuite programs, two among the most widely used packages in charge density analysis, will be described. These programs utilize the multipolar atom model and allow the determination, representation, and the property analysis of electron densities in crystal structures—be they molecular, polymeric, ionic or elemental, including metals. It is well known that crystal structures can be determined by the means of single crystal X-ray diffraction experiments. As X-rays interact with electrons, diffraction experiments image the ED distribution in the crystal, from which the positions of atomic nuclei in the unit cell can be deduced. The aim of routine X-ray diffraction experiments is indeed often limited to the determination of the three-dimensional structure of the studied compound. In other words, the experimentally measured ED is interpreted only in terms of local maxima, indicating where the nuclei are located in the unit cell. The most common methods to determine the crystal structures resort on a simple model, where the crystal is seen as an assembly of noninteracting atoms and the corresponding electronic distribution is described as a collection of spherically symmetric atomic electron densities, centered on the coordinates of the corresponding nuclei. The atoms may define a molecule or not, but this does not affect the model which is invariant with respect to a prejudicial chemical insight (i. e., the connectivity among atoms). A molecule, whose ED is approximated in this way, is called a “promolecule” within the field of charge-density studies, nowadays termed “quantum crystallography” (see Part III, Chapter 9).

Benoît Guillot, Christian Jelsch, CRM2, Université de Lorraine, CNRS, Faculté des Sciences et Technologies, Laboratoire Cristallographie Résonance Magnétique et Modélisations, BP 70239, 54506 Vandoeuvre-lès-Nancy, Cedex, France, e-mails: [email protected], [email protected] Piero Macchi, Department of Chemistry, Materials, and Chemical Engineering, Politecnico di Milano, via Mancinelli 7, 20131 Milano, Italy, e-mail: [email protected] https://doi.org/10.1515/9783110660074-010

236 | B. Guillot et al.

10.1.1 The shortcomings of spherical atom model In the process of determining a crystal structure, the stage of refinement is especially important and needs to be briefly recalled here. The crystallographic refinement corresponds to the process of adjusting parameters of the structural model against the data provided by the X-ray diffraction experiment. In routine structure determination, the refined parameters are the atomic fractional coordinates and the mean square displacements of nuclei, around their equilibrium positions, due to thermal vibrations. These can either be seen as isotropic, leading to one single parameter per atom, or anisotropic, described by up to six independent values within the framework of a harmonic model. In crystallographic refinements, the approximation of noninteracting atoms is called “Independent Atom Model” (IAM), valid for both molecular and nonmolecular crystals. Indeed, atomic parameters are optimized by minimizing an error function S written as a sum, over all used experimental data, of squared differences ⃗ and computed (Fcalc (H)) ⃗ structure factors amplitudes, or between observed (Fobs (H)) intensities. Such procedure is therefore termed as “least-squares refinement”: 󵄨 ⃗ 󵄨󵄨󵄨 − 󵄨󵄨󵄨Fcalc (H) ⃗ 󵄨󵄨󵄨)2 S = ∑ wH⃗ (󵄨󵄨󵄨Fobs (H) 󵄨 󵄨 󵄨

(10.1)

H⃗

where H⃗ is the scattering vector and wH⃗ a weighting coefficient related to the exper⃗ The calculated strucimental uncertainty associated to the corresponding |Fobs (H)|. ⃗ ture factor Fcalc (H) depends on the model which is adopted for the atomic electron distribution and nuclear vibration. The least squares minimization is inherently nonlinear, but it can be linearized by using as variables the shifts from an initial model of the parameters needed to minimize S. This implies an iterative process that eventually converges to the set of best parameters that minimize the disagreement between ⃗ and |Fcalc (H)|. ⃗ Noteworthy, for a more accurate model, such as the multipole |Fobs (H)| model, the IAM could be a reasonable starting point. For a unit cell containing Nat atoms, the calculated structure factor is written as Nat

⃗ = ∑ fj (H)T ⃗ j (H) ⃗ exp(2iπ H.⃗ rj⃗ ) Fcalc (H) j=1

(10.2)

⃗ is the Debye–Waller factor depending on the thermal where, for the jth atom, Tj (H) ⃗ repredisplacement parameters, rj⃗ is the vector of its fractional coordinates and fj (H) sents its scattering factor. The atomic scattering factor (or form factor) is the Fourier transform of the atomic electron density. In the IAM model, given the spherical approximation of atomic electron densities in real space, the scattering factors are spherically symmetric in the reciprocal space as well. For a given chemical species, they depend only on the norm |H|⃗ = 2sin(θ)/λ of the scattering vector H,⃗ with θ being the Bragg angle and λ the wavelength of the X-ray beam used in the experiment.

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A major drawback of the promolecular ED approximation is that, although acceptable to build a sufficiently correct structural model, it is chemically inconsistent. In fact, the wavefunction that produces such a density does not represent a bound state and the typical features of chemical bonding would be missing. This should not surprise us, because the percent difference between a purely spherical distribution of electrons in atoms and the correct electron density is of the order of magnitude of 1 %. In order to better describe the asphericities, a more sophisticated model is necessary, which relies on extra parameters. The augmented number of parameters of an accurate ED modeling requires significant extension of the diffraction resolution. A consensual resolution limit of sin(θ)/λ = 0.7 Å−1 is often taken as a reference. Above this limit the resolution becomes “subatomic.” If coupled with high precision measurements of the diffracted intensities, deviations from the spherical ED approximation become clearly visible. The main deformations from sphericity affect of course the valence electron density, mainly because of chemical bonding. Importantly, these electrons scatter only at low resolution (i. e., below sin(θ)/λ = 0.7 Å−1 ). Nevertheless, the high-resolution data are important because they allow fixing with precision nuclear positions and atomic displacement parameters as they strongly depend on core electrons that dominate the diffraction at high-resolution. In turn, this enables reducing the correlation between position parameters and the additional parameters necessary to accurately describe the electron density. The quality of the refined model can be assessed by computing Fourier transformation maps of the difference between the observed structure factors amplitudes and their computed equivalents (see equation (10.3)):. ρresid (r)⃗ =

1 󵄨 ⃗ 󵄨󵄨󵄨 − 󵄨󵄨󵄨Fcalc (H) ⃗ 󵄨󵄨󵄨) exp(iφcalc ) exp(−2iπ H.⃗ r)⃗ ∑(󵄨󵄨F (H) 󵄨 󵄨 󵄨 V ⃗ 󵄨 obs

(10.3)

H

⃗ where φcalc represents the phase of the computed complex structure factor Fcalc (H). If using an IAM model, these residual electron density maps reveal accumulations of nonmodeled ED on covalent bonds (where they are found shared by the bonded atoms), or at expected positions of electron lone pairs [1] (Coppens (1967)). An example of residual map can be found in Figure 10.1. High and localized peaks in maps of ρresid (r)⃗ clearly indicate that the spherical atom approximation is insufficient, because the information available from the diffracted intensities is not and cannot be properly exploited with the IAM model. Even worse: the refined positional and thermal atomic parameters are in this case necessarily biased by the non-modeled aspherical distribution of the valence ED. This is especially true for the atomic anisotropic displacement parameters, which represent the only degrees of freedom accessible in the IAM approach to partially account for the anisotropic features of the ED. In this case, the deformation ED and the atomic displacement parameters are said to be convoluted, and the IAM model is therefore wrong in trying to reproduce both the electron density and the atomic displacements.

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Figure 10.1: Fourier residual map in the plane of urea, after IAM refinement with MoPro of the estradiol-urea diffraction data up to sin(θ)/λ = 1.16 Å−1 (Parrish et al., 2006). Contour +/−0.05 e/Å3 . Positive: blue; negative: red; zero: yellow lines.

10.1.2 The multipolar atom model Obviously, the objective of properly modeling the fine electron density details, and at the same time properly de-convoluting atomic displacements parameters and deformation ED features, can only be achieved using a more adequate atomic model than the crude spherical approximation of the IAM approach. This is exactly the purpose of the multipole model, elaborated by, among others, Hansen and Coppens [2], Stewart [3, 4] and Coppens [5]. The multipole model preserves the partition of the electron density in terms of atomic contribution, which characterizes the IAM model. However, the atomic terms are no longer spherical. In fact, the atomic electron density ρatom (r)⃗ result from the sum of three terms, accounting respectively for the core, the spherical and the aspherical deformation of valence electrons: lmax

+l

l=0

m=−l

⃗ + κ3 Pval ρval (κ|r|) ⃗ + ∑ κ󸀠 3 Rl (κ󸀠 |r|) ⃗ ∑ Plm Ylm (θ, φ) ρatom (r)⃗ = ρcore (|r|)

(10.4)

The total molecular electron density ρmol (r)⃗ is then obtained by summing up these atomic contributions describing so-called “multipolar pseudo-atoms”: ρmol (r)⃗ = ∑ ρatom (r)⃗ atoms

(10.5)

Let us now describe in more detail equation (10.4), as it is central in multipole modeling. The first term accounts for the core electron shell of the considered atom.

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As these electrons are usually not involved in interatomic bonds, in the Hansen and ⃗ follows the same spherical Coppens model, they are kept unperturbed so that ρcore (|r|) approximation as atoms in the IAM model with a fixed number of electrons. More pre⃗ is a spherically averaged core ED computed from theoretical wavefunccisely, ρcore (|r|) tions of the ground state electronic configuration of the unperturbed isolated atom. The parameters of the core orbital functions (coefficients and exponents of Slater type functions) come from Roothaan–Hartree–Fock calculations (see, e. g., Clementi and Roetti [6]) or from Dirac–Fock numerical solution of the atomic Schrödinger equation and wave-function fitting (Su and Coppens [7]; Macchi and Coppens [8]). The second term in equation (10.4) describes the spherical part of the valence electron shell. This ED is calculated like the core density from the corresponding valence orbitals of the atomic wavefunction; however, the associated population Pval is allowed to vary in the course of the refinement as well as the associated volume. Indeed, ⃗ term represents here a spherical valence ED, normalized to one electron. the ρval (κ|r|) The Pval parameter allows to represent the charge flow among atoms of the compound, and consequently gives access to an estimation of the experimental atomic charge. Indeed, knowing Nval , the number of valence electrons in the neutral state, the atomic charge q can be defined by simply computing the difference q = Nval − Pval . The spherical valence term depends also on the κ coefficient, which is again an atomic parameter that can be fitted in the course of the refinement. This κ parameter describes the contraction or expansion of the valence electron shell around the atomic nucleus. If Pval increases (e. g., for the most electronegative atoms), the increased electron-electron repulsion produces an expansion of the atomic volume. Be⃗ term, κ appears as a parameter scaling the radial coordinate |r|, ⃗ cause in the ρval (κ|r|) the expansion of the shell implies κ < 1. In fact, when κ < 1, then κ|r|⃗ < |r|⃗ which ⃗ corresponding means that the same valence ED value is found at a larger value of |r|, to an expanded spherical valence shell. The inverse situation occurs when κ > 1: in this case, the valence shell is contracted. In equation (10.4), the κ 3 factor is due to a ⃗ should remain nornormalization factor of the atomic orbital functions, as ρval (κ|r|) malized to one electron. The third term in the Hansen and Coppens model accounts for non-spherical features of the valence electron density deformation. At variance from the other two terms, which are spherical, the third term is written by splitting the radial (function ⃗ and the angular (function of (θ, φ)) dependencies of the aspherical valence of |r|) ED using, respectively, single-ζ Slater-type functions Rl and density-normalized real spherical harmonics Ylm . The outer sum in equation (10.4) runs over a positive integer l, which corresponds to the order of spherical harmonic functions used to model the considered atom. The second summation runs over a second integer m with −l ≤ m ≤ +l. Clearly, this resembles the mathematical representation of the angular part of hydrogenic orbitals in quantum mechanics; however, it is important to understand what the difference is. Orbitals are atomic wave-functions, whereas the multipole formalism models atomic electron densities. The density of an atomic

240 | B. Guillot et al. orbital is the square of the orbital function, which implies squaring also the angular part. In analogy with the atomic orbitals, it is convenient to adopt the spherical coordinates (|r|,⃗ θ, φ). While a global coordinate system (equal for all atoms) would be computationally easier, Hansen and Coppens (1978) suggested instead the possibility to adopt a local coordinate system for each atom [2]. This is defined using orthonormal Cartesian basis vectors chosen to follow the local symmetry around the considered atom or anyway a pseudo-symmetry generated by the chemical environment. The atomic axes are generally defined by using interatomic vectors towards bonded neighbor atoms. The single-ζ Slater functions, representing the radial dependency of the aspherical valence ED, are node less atomic orbitals expressed by the general formulas: ⃗ = Rl (κ󸀠 |r|)

ζ nl +3 n ⃗ l exp(−ζκ󸀠 |r|) ⃗ (κ󸀠 |r|) (nl + 2)!

(10.6)

The parameters nl and ζ are related to coefficients of analogous hydrogenic orbital functions (Clementi and Raimondi (1963) [9]). At variance from core and spherical valence, constructed from Roothaan-type atomic wave-functions (meaning that each orbital is a combination of many Slater-type functions), the aspherical radial part is a single Slater-type orbital transformed into a Slater-type density function (meaning ζ = 2ζorbital ). In addition, the ζ values are taken as averages of energy optimized single-ζ valence orbitals, computed by theoretical methods for the electron shells of isolated atoms. This means that for the valence shell of a second raw atom (e. g. C), only one radial density function is used, with a ζ exponent that is the twice the average between exponents for single-ζ orbitals 2s and 2p. The exponents nl should be in principle determined by the valence orbitals (nl = 2(n − 1)), where n is the principal quantum number of the atomic orbital. However, in order to satisfy Poisson equation, nl ≥ l. This implies that for higher poles (for which l > 2(n − 1)) a direct connection between the radial density function and the valence atomic orbital function is lost. Suitable sets of values are given, for various atomic species, in the library files of specialized multipole modeling programs such as XD2016 or MoProSuite. Being an integer, nl remains fixed during the refinement of a multipole model, whereas the ζ exponents can be optimized using a κ 󸀠 scaling similar to the κ scaling of the spherical valence. A change in κ 󸀠 is like using a larger or smaller exponent of the Slater-type radial density function and, therefore, a contracted or expanded density shell, respectively. Indeed, computing the root of the derivative of ⃗ with respect to |r|⃗ leads to a maximum value located at |r|⃗ = nl /(ζκ󸀠 ). Again, the Rl (κ󸀠 |r|) 󸀠3 κ factor found in the last term of equation (10.4) is required for normalization purposes. The aspherical deformation integrates over all space to zero (assuming P00 = 0) as it describes a reorganization of valence electrons whose total amount is imposed by the Pval value.

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10.1.3 The spherical harmonic functions The real spherical harmonics Ylm (θ, φ) are used as density functions in equation (10.4) to represent the angular dependencies of the valence nonspherical deformation ED. Real spherical harmonics are very important functions found in many fields of mathematics, physics and quantum chemistry. They are related to the real parts of the angular portion of the solutions to Laplace equation Δf (r, θ, φ) = 0 in three dimensions. As Ylm (θ, φ) are solely angular functions, they can be defined on the surface of a sphere of unitary radius, where they are orthogonal. In other words, they form a basis set with closure property, so that they can be linearly combined and complemented by a radial dependency to model any anisotropic three-dimensional shape. This is exactly what is achieved in the Hansen and Coppens multipole formalism. The closure property is very important because it assures that any product of two such functions is a linear combination of spherical harmonics. Because density functions arise from orbital products, and the angular part is equally important as the radial part described above, spherical harmonics are ideal for the atomic multipolar expansion. Core and spherical valence only use one spherical harmonic (the monopole, which is the total symmetric irreducible representation of the spherical symmetry group). On the other hand, combinations of angular real spherical harmonics, modulated by Slater radial functions, are able to model the shape of atomic valence electron densities which are deformed from sphericity upon formation of interatomic bonds. Even if they depend only on angular directions, Ylm functions can be plotted in various ways in three dimensions. For instance, their standard representation consists in distorting a unit sphere, by scaling each point radially by the absolute value of the Ylm function, then coloring it based on its sign. Apart for l = 0 which corresponds to Y00 of purely spherical (monopolar) shape, this method of representation highlights the presence of positive and negative lobes, or “poles,” separated by angular directions for which the function is equal to zero (Figure 10.2). They are orthonormal multipolar functions on which is based the “multipole” modelling. For instance, when l = 1, the three Y1m spherical harmonics are dipolar functions, characterized by one single negative and one single positive pole. The value of lmax found in the external sum of equation (10.4) depends on the nature of the considered atom, as lmax dictates the shapes and the symmetries of the highest order of real spherical harmonics used in the modeling. For instance, hydrogen atoms, involved in a single covalent bond, are often described by a single dipolar function (hence lmax = 1) oriented along the bond. This way, the positive lobe of the single spherical harmonic (for instance, Y11 ) accounts for the accumulation of electrons shared along the covalent bond between the hydrogen atom and its neighbor.

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Figure 10.2: Representations of real spherical harmonics up to the order lmax = 3 using the method given in the text. Positive poles are drawn in green, negative ones in orange. The reference Cartesian basis is represented for each multipole. The associated spherical coordinates system is represented next to the monopolar function (lmax = 0).

10.1.4 Guidelines on the multipolar atoms As rule of thumb, one can consider that lmax = 1 (dipoles) is sufficient for hydrogen atoms, lmax = 3 (octupoles) is adequate for first-row atoms while lmax = 4 (hexadecapoles) should be used for heavier atoms. However, we have to consider which atomic orbitals form the valence shell and what is the atomic stereochemistry. H atoms only have a 1s orbital, but they form one bond and, therefore, feature a preferential direction of polarization of the electron density. For this reason, a dipolar function along this special direction is necessary. The flexibility of the electron density model strongly depends on the treatment of the atomic thermal motion. For instance, using an anisotropic model for hydrogen atoms nuclei enables extending to lmax = 2 (quadrupoles). Alkylic, tetrahedral (sp3 hybridized) C atoms are obviously different from olefinic or aromatic (sp2 hybridized) ones. The former requires more flexible description than the latter, and expansion to hexadecapoles is necessary. Similarly, fluorine is often modeled in the literature using a multipolar expansion up to hexadecapoles in order to better represent its three lone pairs [10]. Transition metal elements have d-type electrons in their valence and their orbital products produce hexadecapolar density functions, which are therefore essential for a proper modeling. For the same reason, f -block elements require lmax = 6.

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Of course, the aspherical deformation term of the Hansen and Coppens model includes parameters that must be adjusted against experimental data in a crystallographic refinement. These parameters are the multipole populations Plm and, as for the spherical valence deformation described above, the expansion/contraction parameter κ󸀠 . The multipole populations, Plm , represent fractions of electrons being “moved” from the region of negative values (a pole) of the corresponding Ylm function toward the region of positive values. This way, the number of valence electrons of the considered atom depends only on Pval , whereas the deformation from sphericity is represented by the set of Plm coefficients. Hence the ED of atoms is deformed when they become involved in chemical bonds, as it is the case in molecules. The P00 population, associated to the Y00 monopolar spherical harmonic is a special case as it ⃗ a spherical component of the valence ED. Often, allows to describe, just as ρval (κ|r|), the P00 valence population is fixed to zero in a multipole refinement, but is deemed necessary mostly to model the diffuse outer s-electron shell of transition metal atoms when Pval is used for the d-electrons. In the crystallographic refinement based on the multipole formalism, as implemented in the XD2016 and the MoProSuite programs, the minimized function S takes exactly the same form than equation (10.1), using structure factors amplitudes (or intensities) depending now on the ED parameters described previously. More precisely, as the atomic scattering factor is the Fourier transform of the atomic ED, the multipolar parameters appear in aspherical scattering factors, Fourier transform of the aspherical electron density defined in equation (10.4). For a given multipolar pseudo-atom j included in the structure factor expression (equation (10.2)), the corresponding aspherical scattering factor fj,mult takes the form: ⃗ = fj,core (|H|) ⃗ + Pval fj,val ( fj,mult (H)

lmax +l |H|⃗ H⃗ ) + ∑ ∑ Plm flm ( 󸀠 ) κ κ l=0 m=−l

(10.7)

⃗ and fj,val ( |H|⃗ ) are the core and valence isotropic form-factors obin which fj,core (|H|) κ ⃗ and ρval (κ|r|) ⃗ electron densities, tained by Fourier transform of the spherical ρcore (|r|)

respectively. Similarly, flm ( κH󸀠 ) is the Fourier transform of the aspherical ED function (third term in equation (10.4)) and depends on the direction of the scattering vector. ⃗ Hence, flm ( κH󸀠 ) is the contribution of the (l,m) multipole function (radial and angular part) to the scattering factor, whose detailed expression and calculation can be found ⃗ ⃗ 󸀠 of the valence form factors in Coppens (1997) [5]. The dependence in |H|/κ or |H|/κ ⃗ in direct space results in an inversely scaled reflects the fact that a scaled ρval (κ|r|) scattering function in reciprocal space. This “inverse scaling” relationship between direct and reciprocal space has an important practical consequence in multipole modeling. Indeed, atomic core electron shells are necessarily more contracted, around the nucleus, than the spherical valence shells. Their scattering factors obey to an inverse relationship: core electrons scatter further in the reciprocal space than valence electrons. Consequently, data related to the valence electron distribution are located in ⃗

244 | B. Guillot et al. the low or the medium resolution ranges of diffracted intensities, while information related to core electrons (and, indirectly, to atomic nuclei) are located also at highresolution. This actually offers a practical way to partially solve the convolution problem mentioned before. Refining the structural parameters using the IAM model and against data located only in the very high-resolution range allows avoiding the bias due to the non-modelled valence deformation ED affecting the lower resolution. Such procedure is termed “high order refinement,” in reference to the larger values of Bragg angles characterizing the high-resolution ranges of diffraction data. To summarize, beyond atomic fractional coordinates and the atomic displacement parameters, the multipole formalism implies extra parameters to model the ED of an atom. There are two κ and κ󸀠 expansion/contraction coefficients, the spherical valence populations Pval and (lmax + 1)2 multipoles populations Plm (including the additional spherical P00 ). Assuming lmax = 4 (hexadecapole level) a total of 37 parameters per atom are necessary, although often P00 is not refined and the contraction/expansion parameters κ and κ󸀠 are cumulatively refined for atoms of the same element in similar chemical environments. This reduces the parameters to 34 per atom, 2 per atom type and one (or more) scale parameter necessary because the measured intensities are on an arbitrary scale.

10.1.5 Extensions of the multipole model For the sake of completeness, it must be noted that several modifications of the original multipole model [2] (Hansen and Coppens (1978)) have been tested and can be found in the literature. All these modifications of the multipole model have in common to further increase the number of atomic parameters needed to represent a given pseudo-atom, but underline its powerful inherent flexibility. For instance, core polarization effects in, mostly inorganic, crystals of very high diffracting power have been modelled by allowing the refinement of an extra core population parameter Pcore associated to a dedicated contraction/expansion coefficient ⃗ term in equation (10.4) by κc [11]. This is achieved simply by replacing the ρcore (|r|) 3 ⃗ in which ρcore is normalized to one electron. The refinement of κ κc Pcore ρcore (κc |r|), core may also be useful when refining against theoretical structure factors which may yield high residual density around the nuclei. The κ core may correct for the mathematical discrepancy between theoretical density issued from Gaussian functions and the modeled density using Slater functions [12]. Another extension of the multipole model is the use of several κ󸀠 parameters each associated to a given level of the multipolar expansion. In other words, in this approach, κ 󸀠 become a function of l and is included in the outer sum of equation (10.4) [13]. Similarly, another noteworthy modification of the multipolar expansion radial functions consists in the duplication of the aspherical valence deformation term of

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equation (10.4), in order to introduce a second ζ Slater exponent [14]. This allows performing double-ζ multipole refinements providing more flexibility to the radial dependency of the aspherical valence term than in the conventional, single-ζ , Hansen and Coppens model. An overwhelming majority of published charge-density studies based on the multipolar formalism reported expansions limited to lmax = 4. However, studying elements with many electronic shells (like heavier elements of the main groups, transition metal atoms, lanthanides or actinides) imposes the use of modified forms of the Hansen and Coppens equation to account for deformation effects occurring in their inner electron shells. An approach which has been successfully tested in such cases consists in attributing one term akin to the full equation (10.4) to each electron shell of the heavy element, and extending its multipole expansion up to lmax = 6 or lmax = 7 to model the ED of their highly aspherical orbitals of large principal quantum numbers [15–17]. This further increases the number of parameters of the model, which can be problematic in a least-square refinement. At first, there is a serious risk of overfitting: increasing the number of degrees of freedom may lead to model experimental noise, or chemically irrelevant features. Second, the least-squares method is sensitive to the “observation over parameter” ratio, which should be maintained over about 10. The inverse of the least-squares normal matrix is the variance-covariance matrix of the least-squares variables. The very last stage of a refinement should include all atomic parameters and an inversion of the full least-squares normal matrix, to ensure a correct final convergence of the refinement and to gain access to uncertainties on the model parameters.

10.1.6 Constraints and restraints This step might be problematic if the number of refined parameters is too large. Fortunately, applying constraints on the structural and ED parameters of the multipolar pseudo-atoms is a way to reduce the size of the least-squares matrix. Alternatively, restraints applied can render the normal matrix definite positive. Constraints give a fixed target to a parameter or to a derived function while restraints allow for a tolerance around the target value. There are two main types of constraints/restraints on the charge density. The first one corresponds to local symmetry. The choice of the local Cartesian axis system associated to a pseudo-atom (Figure 10.2) can be done in such a way that it follows the local pseudo-symmetry of the considered atom’s neighbourhood. In this case, only the multipole populations Plm of real spherical harmonics Ylm satisfying these symmetries will be freely refined. Other Plm values will be constrained to a zero value or restrained to be close to zero within a standard deviation σr . Symmetry constraints, of course, hold also for atoms lying on special positions in the unit cell. The

246 | B. Guillot et al. Ylm that are not invariant under the point symmetry operations of their Wyckoff position, must be discarded and their associated Plm fixed to zero. The second category of constraints/restraints are called chemical equivalences: multipolar pseudo-atoms of the studied compound that are chemically equivalent (same nature, same covalent neighborhood, same hybridization . . . ) are forced to share the very same (constraint) or similar (restraint) ED parameters. Constraints results in a diminished number of refined variables while restraints increase the number of observations. Applying, until the very last stages of the multipolar refinement, such chemical and symmetry constraints/restraints on the ED have been shown to be especially relevant to reduce the risk of overfitting [18, 19].

10.1.7 Assessing the data and model quality A multipole modeling, eventually providing a chemically meaningful and accurate experimental electron density model of a studied compound, is usually not an easy task. At first, it requires experimental data of adequate quality. As already stated, collecting X-ray diffraction data of subatomic resolution is a compulsory requirement, but might not be sufficient. The diffraction experiment must indeed be conducted at cryogenic temperature. Low temperature reduces the thermal smearing of the electron density and for this reason increases the scattered intensities. This makes ED easier to observe and model. At the same time, a stronger scattering implies better precision of the measurement. To yield a well-observable deformation density, the thermal displacement parameters of atoms should be lower than typically 0.015 Å2 . For the same reason, the presence of static disorder in the unit cell will hamper the observation of the deformation density. Anyhow, the overall diffraction data quality should be optimized by targeting close to 100 % completeness, very high redundancy and strong signal over noise ratio of the collected intensities. For crystal containing heavy elements (sulphur, chlorine, bromine, metals. . . ), it is necessary to make proper absorption correction. All resolution shells are equally important for a multipole modelling. As stated earlier, the low and medium resolution ranges carry information related to the scattering of valence electrons while the highest resolution ranges concern core electrons. Moreover, crystals described in highly symmetric space groups, enabling merging of many equivalent reflections are usually more favorable. Centrosymmetric space groups are preferable, if possible, to eliminate the uncertainties of the phase of structure factors. Although the focus is here on the determination of multipolar density models against experimental diffraction data, nothing disallows to perform a multipolar refinement against theoretically computed structure factors. This approach is actually very commonly followed, as it allows comparing experimental and theoretical ED models or their derived properties based on a common formalism [20].

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To conclude, let us enumerate some criteria that can be used to evaluate the quality of a multipolar refinement. Because a multipole model is somewhat an extension of the IAM typically adopted for conventional structure solution, the traditional agreement factors, such as R(F) or wR2(F), and the goodness-of-fit (Gof(F)), hold: ⃗ − |Fcalc (H)|) ⃗ ∑H⃗ (|Fobs (H)| ⃗ ∑ ⃗ (|Fobs (H)|)

(10.8)

⃗ − |Fcalc (H)|) ⃗ 2 ∑H⃗ wH⃗ (|Fobs (H)| ⃗ 2 ∑ ⃗ w ⃗ (|Fobs (H)|)

(10.9)

⃗ − |Fcalc (H)|) ⃗ 2 ∑H⃗ wH⃗ (|Fobs (H)| nobs − nvar

(10.10)

R(F) =

H

wR2(F) = √

Gof(F) = √

H

H

where nobs is the number of experimental data and nvar the number of refined parameters. Alternatively, these discrepancy indices can be written using net integrated in⃗ and |Fcalc (H)|, ⃗ tensities (or squared structure factors amplitudes), instead of |Fobs (H)|

leading to R(I), wR2(I) and Gof(I). Of course, a multipole model should produce a significant improvement compared with IAM and indices should be lower. However, improved R-factors is not a sufficient criterion to ensure the chemical validity of the refined ED model. This is why the model itself and the resulting deformation density must be carefully analysed during the multipolar refinement process. Notably, ED parameters must stay within realistic ranges of values. As a rule of thumb, Plm populations larger, in absolute value, than about 0.5 can be considered suspect. Similarly, κ or κ󸀠 parameters are expected to stay relatively close to unity. κ values are expected to be smaller/larger than unity for electronegative/electropositive atoms, respectively. Notably, κ󸀠 values significantly deviating from unity undoubtedly indicate a problem in the refinement, which might be due to an incorrect definition of the nl or ζ parameters of the radial functions of the corresponding atom. The κ 󸀠 parameters are actually well known to be difficult to refine in a multipole modelling. They are sometimes restrained or even fixed to values taken from the literature or from theoretical computations, especially for hydrogen atoms for which recommended κ’ values are in the [1.2–1.4] range [21]. The multipolar ED model should be visually checked using for instance static de⃗ They are simply obtained by subtracting to formation electron density maps Δρmol (r). equation (10.5) spherical and neutral atomic references (equation (10.11)). This way, such maps highlight the deformation (both spherical and aspherical) of the valence ED and are said “static” as they are obtained directly from the Hansen and Coppens

248 | B. Guillot et al. model without including any effects of atomic displacement parameters: ⃗ Δρmol (r)⃗ = ∑ (ρatom (r)⃗ − ρref (r)) atoms

(10.11)

⃗ + Nval ρval (|r|). ⃗ where ρatom (r)⃗ corresponds to equation (10.4) and ρref (r)⃗ = ρcore (|r|) Examples of static deformation density maps can be seen in Figures 10.3 and 10.4.

Figure 10.3: Example of deformation electron density map in the plane of urea molecule, after charge density refinement of the estradiol/urea crystal [22]. Contours are the same as in Figure 10.1. While the bonding density is well-defined, the electron lone pairs on the oxygen atom are weaker than expected presumably due to the non-centrosymmetric space group and the relatively high thermal motion Ueq > 0.02 Å2 .

Finally, the nonmodeled deformation electron density peaks visible after the IAM refinement should eventually vanish as they are accounted for by the multipole atom model. Hence, the final residual electron density map should be flat, apart from randomly distributed weak peaks related to experimental noise. At the end of the multipolar refinement, it is strongly recommended to use residual density analysis (RDA) tools, such as jnk2RDA [23]. Significant deviations from the ideal distribution of a Gaussian noise might be indicative of an error in the electron density model or of systematic errors in the diffraction data.

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10.2 The MoProSuite software package 10.2.1 Overview of the programs MoProSuite is a versatile least-squares refinement package which implements both the IAM and the Hansen and Coppens multipole models. It is compatible with the structural and the electron density refinement of crystal structures ranging from small compounds to reasonably sized macromolecules. Besides the features related to the leastsquares refinement, MoProSuite also allows the computation and the representation of a wide range of ED-derived properties, such as the electrostatic potential and the topology of the electron density. MoProSuite is made of several modules we shall now describe. The core components of MoProSuite are the MoPro, VMoPro and Import2MoPro programs [24, 25]. Import2MoPro is a utility program for the conversion of common crystal structure file formats (such as CIF, INS, RES, PDB, XD, xyz . . . ) into the one needed by the MoProSuite programs. Import2MoPro can determine suitable atomic local axis systems (needed to orient ED deformation functions of equation (10.4)) following the local pseudo-symmetry of the considered atoms. These optimal atomic axes are written in the MoPro parameter file by Import2MoPro and allow the definition of chemical and symmetry constraints on ED parameters. MoPro is dedicated to the least-squares refinement. It implements the usual chemical equivalences and symmetry constraints used in multipole modeling, but also numerous restraints applying both on structural and on electron density parameters. Available restraints and constraints can be automatically generated by MoPro and written in dedicated text files. Therefore, they can be afterward checked or edited by the user. Restraints in MoPro are of two kinds: target or similarity restraints. Target restraints force parameters or functions of parameters (e. g., interatomic distances) to remain close to target values within a given tolerance. Similarity restraints impose that values of two parameters, or function of parameters, must stay similar within a user-defined tolerance. Restraints are implemented in the least-square refinement by adding terms to the minimized residual function S (equation (10.1)), which becomes 󵄨 ⃗ 󵄨󵄨󵄨 − 󵄨󵄨󵄨Fcalc (H) ⃗ 󵄨󵄨󵄨)2 + ∑ wt (f − ftarget )2 + ∑ ws (g − h)2 S = ∑ wH⃗ (󵄨󵄨󵄨Fobs (H) 󵄨 󵄨 󵄨 H⃗

Nt

Ns

(10.12)

where Nt and Ns are the number of target and similarity restraints of a given type. h and g are parameters or functions of parameters whose values must remain similar. ftarget is the target value of the restrained function f . wt and ws are the weights associated to the target and similarity restraints, respectively. A strong weight will reduce the tolerance of deviations between restrained parameters. The restraints implemented in MoPro increase significantly its versatility, making it compatible with the refinement of protein or nuclei acids structures at atomic or subatomic resolution. In this optic,

250 | B. Guillot et al. MoPro also implements the conjugate-gradient minimization approach which allows when the number of parameters is large to avoid the costly matrix inversion needed in standard least-squares routines. MoPro is interfaced with the ELMAM2 electron density database [26–28]. This library contains transferable multipolar pseudo-atoms describing many common organic chemical groups. These pseudo-atoms have been obtained by averaging multipolar parameters issued from numerous accurate subatomic resolution charge density analyses of small compounds (amino acids, various organic molecules. . . ). Parameters were averaged per “atom types”, i. e., per atoms presenting similar covalent neighborhood, and can be transferred to any molecular structure containing compatible atom types. This approach, rooted in the so-called “transferability principle,” allows fast reconstructions of multipolar ED models of large biological molecules. The resulting transferred model can be exploited directly to compute ED derived properties [29] or be used as a starting model for a constrained multipolar refinement, assuming diffraction data of sufficiently high resolution is available [30, 31]. In addition, MoPro proposes a model combining real and additional virtual spherical atoms as an alternative to the multipole modeling [12, 32]. In this approach, the charge density can be refined using a model based on real spherical atoms and additional dummy charges on the covalent bonds and on electron lone-pair sites. Compared to multipoles, this spherical charge modeling needs fewer parameters to describe the deformation electron density (Figure 10.4). For each atom, only the valence population Pval and the contraction/expansion κ coefficients are refined. Most of the deformation density is modelled and molecular electrostatic properties are very close to those modeled with the multipole model. A database of transferable spherical “real+virtual” atoms issued from theoretical calculations is also available to model structures at lower resolution and bio-macromolecules. VMoPro is the MoProSuite component dedicated to the computation of ED derived properties. It gives access to static electron densities or electrostatic potentials using any contribution of the multipolar charge density (nuclei, core, spherical or deformation valence, etc.). It allows also the computation of Fourier maps, including dynamical electron densities. Any of these properties can be represented with VMoPro in the form of 2D contour plots (in postscript format) or exported as 3D regular grids (e. g., in the Gaussian CUBE format). Topological analyses of 3D scalar fields, within the QTAIM framework (see Part II, Chapter 1) can be performed with VMoPro, using the electrostatic potential, the total ED or its Laplacian (which can also be obtained independently in the form of 2D or 3D maps). Atomic charges can be computed by integrating the charge density over the Bader atomic basins [33]. VMoPro also implements the computation of electrostatic interaction energies using the EP/MM approach [34].

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Figure 10.4: (a) urea molecule with virtual atoms (in green) located on the covalent bonds and on the electron lone pairs sites of the oxygen atom. (b) deformation valence electron density map in the plane of urea modeled using transferred “real+virtual” atoms.

10.2.2 The graphical user interfaces MoProGUI and MoProViewer MoPro runs using a user-created input commands file containing keyword base instructions (e. g., “mopro.inp”) containing keywords-based instructions. A complete description of all available keywords and options can be found in the MoPro documentation. By contrast, VMoPro and IMoPro are interactive programs. They can be used either by prompting instructions directly in the console or giving scripted input files using standard input redirection. Even if convenient in a console-based environment, for instance to execute the programs in batches, that kind of usage can be advantageously replaced by the use of the graphical user interfaces MoProGUI and MoProViewer. MoProGUI is the graphical user interface of MoPro. It allows the user to easily configure and execute a multipolar refinement, and to follow its outcomes, without using the keyword-based approach needed to execute MoPro in command-line mode. Unlike with the IAM model, the multipolar refinement implies the notion of “refinement strategy.” Indeed, the multipole model is based on parameters of different types, either global (e. g., the scale factor), related to the structure (e. g., fractional atomic coordinates) or to the spherical and aspherical deformation electron density (Pval , Plm , κ and κ󸀠 ). Moreover, these parameters present various degrees of correlations between them

252 | B. Guillot et al. (e. g., between Pval and κ, or between Plm and thermal displacement parameters), can be subject of various kind of constraints or restraints and may depend on different resolution ranges of the available diffraction data. Altogether, it means that a multipolar refinement is usually not straightforward, and often implies trial and errors before obtaining an adequate strategy leading to a chemically meaningful electron density model. MoProGUI has been specifically designed to ease the elaboration of multipolar refinement strategies. It appears obvious that the various functionalities of a multipolar refinement program can be sorted in categories: options related to experimental data (e. g., resolution limits, I/σ(I) cut-off. . . ), to the restraints or the constraints, to the refinement (refined parameters, number of refinement cycles . . . ), to data exportation, and so on. This is exploited in MoProGUI, in which the user can graphically build a refinement strategy by combining groups of instructions, named “blocks” in the MoProSuite jargon. The refinement strategy in MoProGUI appears then as a list of successive blocks, which will be executed sequentially by MoPro. As said earlier, each block gathers categories of instructions. The first block of any MoProGUI strategy list is always a “Files” block. The “Files” block allows indeed to specify the (initial) atomic parameter file, the experimental diffraction data file and the constraints / restraints files. In this first “Files” block is also given the location of the library tables, provided with the MoProSuite package, containing default nl and ζ coefficients of the atomic radial functions and the parameters of the orbital functions needed for ρval and ρcore spherical ED (equation (10.4)). One of the most important instruction block in MoProGUI is obviously the “Refinement” block. It is where the user can graphically select (i) a type of parameters to refine, (ii) atoms which will be included in the refinement using inclusion / exclusion logic and (iii) a refinement method and its corresponding options. For instance, a user can, with few mouse clicks, configure a refinement block corresponding to “the refinement of thermal displacement parameters and fractional coordinates of every nonhydrogen atoms using 10 cycles of least-squares matrix inversion method and a damping of parameter shifts of 0.5”. The choice of the data resolution limits used in a given refinement stage is made using a “Resolution” block, whose inclusion in the refinement strategy will affect resolution limits until the next “Resolution” block. Another example of block, as group of instructions, that can be included in a MoProGUI strategy list is the “Preparation” block. This block is intended to automatically prepare various kind of restraints or constraints (either structural or related to the electron density). Once executed, this block in MoProGUI will prompt MoPro to create restraints and constraints files, which can be used in subsequent refinement steps. Hence it has to be executed only once, at early stages of the refinement. Finally, “Output” blocks can be used at any position in the strategy list. Using “Output” blocks, the user can ask, for instance, for the creation of intermediary molecular parameter ⃗ |Fcalc (H)| ⃗ and associated phases (computed files, CIF files or files containing |Fobs (H)|, using the current model parameters) needed for Fourier maps. Instruction blocks can

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be configured in a very flexible way in the MoProGUI interface. Any block can be drag and dropped within the refinement strategy, commented / activated or even included in loops in which groups of blocks will be executed several times, for instance up to convergence of the refinement of a given set of parameters. They can also be renamed and saved for future use, which is especially convenient for “Refinement” blocks. A user can this way create a refinement strategy using his own set of custom preconfigured “Refinement” blocks. A last peculiarity of the MoPro / MoProGUI philosophy is the versioning of molecular parameter files. In MoPro, a parameter file contains basically the crystallographic data (e. g., cell parameters and symmetry operations), the global parameters (scale factor, extinction coefficient . . . ), the list of atoms in the asymmetric unit with their parameters and their local atomic axis systems. The name of a MoPro parameter file contains a version number (e. g., “mycompound.par01”), which will be incremented at each execution of MoPro. At the cost of a larger number of files ending up in the user working directory, this versioning system allows to keep track of what has been done along the refinement and, of course, to restart the refinement from any stage using the adequate parameter file version. Once MoPro is executed through MoProGUI, an “Output panel” appears in the MoProGUI interface. It displays data allowing the user to follow the progression and the outcome of the refinement strategy configured in the Input panel. Notably, the full “mopro.out” file is displayed and updated in real time during the progress of the refinement along with plots showing the evolution of user selected crystallographic agreement factors. MoProViewer is the graphical user interface for VMoPro, and a molecule / crystal structure viewer especially designed for the charge density field [35]. It features specific functionalities related to the multipole modeling such as the representation of atomic local axes systems or chemical equivalences constraints using color-coded symbols. One of important features of MoProViewer, among many others, is its capability to allow the user to configure directly, from the representation of the molecule, the computation with VMoPro of ED-derived properties. For instance, any 2D plot (or 3D relief maps) can be obtained with a few mouse clicks on atoms to define a plane, then MoProViewer runs silently VMoPro, retrieves its results and represent them in a dedicated contour maps viewer. 3D properties can be computed in a similar way, and can be represented in MoProViewer using 3D iso-surfaces, possibly color-mapped by any other 3D property. Similarly, the search for ED critical points with VMoPro can be configured from MoProViewer, and the resulting bond critical points can be displayed with their associated bond paths. Figure 10.6 illustrates some of the representations of ED-derived properties available in MoProViewer. The software has some links to other programs developed by other authors: – The fractal analysis of Meindl and Henn (2008) [23] can be performed on residual (Fobs-Fcalc) maps of XPLOR or Gaussian CUBE format computed by fast Fourier transform.

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The analysis of diffraction data quality with program DRK [36] can be performed on the output reflections file using FCFW option, which replaces σ(Ihkl ) in order to have a unitary goodness-of-fit (gof = 1). The SHADE or SHADE3 servers [37, 38] prepare the values of modelled anisotropic thermal parameters of hydrogen atoms to be inserted in the MoPro constraints file (FIXUIJ).

10.2.3 A practical example: charge density refinement of estradiol/urea with MoProSuite The practical use of the MoProSuite is here briefly described, based on the EstradiolUrea complex (Figure 10.5) published by Parrish et al. in 2006 [22].

Figure 10.5: MoProViewer view of the thermal ellipsoids (50 % probability presence) of the estradiol/urea crystal structure.

A tutorial based on this molecule is actually included in the MoProSuite package. Diffraction data “estradiol.Ihkl” and “estradiol.cif ” files can be found in the “Tutorial_estradiol” examples folder of the MoProSuite installation directory. Upon importation of the CIF file, the program will set automatically the atomic axes systems to orient the multipoles, based on the neighbors. The first refinement step is the adjustment of scale factor. In the next step, the constraints and restraints need to be prepared by the program. Indeed, the stereochemical restraints (or constraints) are necessary for proper treatment of hydrogen atoms. In this case, the following ones can be selected: – X–H bond distances adjusted to standard values from neutron diffraction. – X–H bond distances similarities (d(A-H1) ∼ d(A-H2) ±σd ).

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U iso thermal parameters of hydrogen riding on that of bonded atom (multiplied by 1.2 or 1.5). Equivalent atoms have same/similar charge density parameters κ, κ’, Pval , Plm

Multipoles of some atom obey a local symmetry (mirror, inversion center . . . ). The second refinement step applies on the structural parameters (XYZ and U ij ). After structural refinement, the “experimental deformation density,” or residual ED, can be observed by computing a Fourier residual map (Figure 10.1). The signal can be enhanced by applying a “high order refinement,” i. e., refining XYZ and U ij of non-H atoms against high resolution reflections only (typically sin(θ)/λ > 0.7 Å−1 ). When the Fourier map shows distinct bonding density and electron lone pairs with limited noise, the charge density can then be refined favorably. Several procedures are possible. Charge density parameters may be introduced progressively in the refinement, starting with multipoles Plm ’s. MoPro “refinement” menu allows to do customized refinement where parameters refined are chosen by the user. “automatic refinement” menu proposes automated procedures where all the parameters can be refined iteratively or together. A fully automatic procedure is also available and will most likely work for a structure with good diffraction data and no complications (disorder, anharmonic thermal motion, special positions. . . ). A significant R-factor drop is expected upon multipolar refinement. The refinement can be carried out till convergence. Ideally, all parameters should be refined together in the last stages.

10.2.4 Properties derived from the charge density This paragraph highlights some of the most important properties and results which can be computed from a successful charge density refinement.

Electrostatic potential The electrostatic potential V(r)⃗ (ESP) can be obtained by integration over space of the total charge density ρtot (r)⃗ (i. e. including atom nuclei) divided by the distance: V(r)⃗ = ∫

ρtot (r 󸀠⃗ ) 3 󸀠 d r⃗ |r 󸀠⃗ − r|⃗

(10.13)

The Hansen and Coppens modeling of the electron density enables to compute the ESP generated by a molecule which is far more accurate than that derived from point atom charges placed at the nuclei. Electrostatic potential provides more information than the electron density on the chemical reactivity and the intermolecular interactions. Several types of representations can be selected (Figure 10.6). For instance,

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Figure 10.6: Representations of the molecular ESP using MoProViewer. (a) Generated and shown in the urea plane. Contours: +/−0.05 e/Å, positive in blue, negative in red, zero line in green. (b) A qualitative semitranslucent contour map of static deformation ED is shown in the urea plane. The estradiol molecule is surrounded by a 0.1 e/Å3 total ED isosurface coloured by its ESP, with the color legend ranging from −0.25 to 0.3 e/Å. (c) Hirshfeld surface around the urea molecule. The surface is colored according to ESP values (range −0.6 to 0.3 e/Å). (d) ESP Isosurfaces of the estradiol molecule. Positive contour in grey: +0.2e/Å. Negative contour in red: −0.092e/Å.

the C=O electronegative group of urea forms bifurcated hydrogen bonds with the NH2 group of a neighbor urea molecule (Figure 10.6c); in the molecular dimer, electropositive and electronegative regions are in interaction. The estradiol molecule shows two electronegative lobes at −0.15 e/Å. around the two C–O–H groups. Less electronegative regions at V = −0.09 e/Å appear on both side of the aromatic C6 cycle due to π electrons (Figure 10.6d). The urea molecule generates an electronegative potential around the C=O oxygen atom (Figure 10.6a,c).

Laplacian The Laplacian of the total electron density shown in Figure 10.7 is an alternative way to highlight electron charge local concentrations and depletions. Regions with negative Laplacian correspond to local negative charge concentration. The lone pairs of the C=O oxygen atom of urea appear as two peaks in the Laplacian map.

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Figure 10.7: Laplacian of the total electron density (e/Å5 ). L(ρ) = 𝜕ρ2 /𝜕x 2 + 𝜕ρ2 /𝜕y 2 + 𝜕ρ2 /𝜕z 2 . Contours are quasi-logarithmic ±2, 4, 8 × 10n with n = −1, 0, 1, 2. Positive: blue discontinuous line, Negative: red lines.

Critical points The critical points (CPs) are the region of space where the gradient of a property, here the total electron density ρ, is zero. Minima and maxima correspond to (3,+3) and (3,−3) CPs, respectively, where the Hessian matrix 𝜕2 ρ/𝜕xi xj has 3 positive/negative eigenvalues. Figure 10.8a shows the saddle CPs: the cycle (3,+1) CPs and the bond (3,−1) CPs within the asymmetric unit. After searching with MoPro all symmetry neighbors involved in hydrogen bonds around the estradiol molecule, the (3,−1) CPs were searched with VMoPro and are displayed in Figure 10.8b. After search of the critical points, the properties of the CPs are summarized in GcpVcp.dat file, which can be found in the MoPro working directory. In addition to the electron density ρcp , the Laplacian ∇2 ρcp and the ellipticity, the kinetic and potential energy density values are also retrieved (Table 10.1).

Electrostatic energy To understand the electrostatic forces within crystals, the energy can be computed between neighboring molecules using VMoPro, with or without the help of its graphical interface MoProViewer. In the case of the estradiol-urea complex given here as

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(a)

(b)

Figure 10.8: Critical Points. (a) CPs within the asymmetric unit: bond CPs (in red), cycle CPs (in blue). For the H-bond and H. . . H interaction CPs (in green) the bond path is shown. (b) View of the intermolecular hydrogen bonds around estradiol molecule. The (3,−1) CPs and bond paths are shown in green. Table 10.1: Topological properties at the critical points of the hydrogen bonds. Gcp and V cp are the kinetic and potential energy densities at the CP (kJ.mol−1 .Bohr−3 ). The electron density and Laplacian values are also given. Atom 1

Atom 2

H2O H1O O3 O1 H2NA H4 H12B H4 H17 H14

O3 O2 H1NA H2NB O2 O3 O3 O2 C2 C1

Symmetrya

Gcp

Vcp

Distance (Å)

Density (e/Å3 )

Laplacian (e/Å5 )

43504 34402 54403 55403 44402 45403 43504 34402 43504 43504

120.0 116.8 81.5 48.1 35.6 15.9 12.7 14.3 10.1 10.1

−104.2 −99.3 −71.5 −38.1 −26.1 −10.9 −8.6 −10.2 −7.5 −7.3

1.682 1.704 1.824 2.022 2.120 2.494 2.666 2.673 2.789 2.820

0.2424 0.2315 0.1950 0.1217 0.0890 0.0472 0.0409 0.0490 0.0432 0.0407

4.98 4.93 3.36 2.13 1.66 0.77 0.61 0.67 0.46 0.48

a

Symmetry codes 34402: -X-3 /2 ; -Y-1; Z- 1/2; 43504: X- 1/2; -Y-3 /2 ; -Z 44402: -X- 1/2; -Y-1; Z- 1/2; 45403: -X-1; Y+ 1/2; -Z- 1/2 54403: -X; Y- 1/2; -Z- 1/2; 55403: -X; Y+ 1/2; -Z- 1/2

example, the electrostatic energy is the strongest for a urea. . . urea dimer interacting via a N–H. . . O=C hydrogen bond (Figure 10.8b, at the right bottom). In MoProSuite, the contribution of the electrostatic part to the lattice energy may be approximated by computing the Eelec value between a reference molecule and a surrounding shell of sufficiently large size.

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10.3 The XD2016 software package 10.3.1 Overview of the package Like MoPro, XD2016 [39] is also based on the Hansen and Coppens multipolar formalism [2], while being compatible also with the Stewart formalism [3] as well as with some of the model extensions mentioned in Section 10.1.5, like the double-ζ valence radial function [14] or the core refinement [11]. As mentioned in Section 10.1.4, a reasonable starting point for a multipolar refinement is the structural model obtained with an IAM refinement. For molecular crystals, the gold standard software are SHELX [40] or Olex2 [41], both well tested, reliable and worldwide adopted for many years (in particular SHELX). They both export the output of a refined structure as a crystallographic information file (CIF) or a res file. XD2016 can use both kinds of files to import the basic structural and experimental data, namely: – Unit cell parameters; – Atom types, setting the corresponding atomic form factors; – Atomic fractional coordinates within the asymmetric unit; – Atomic displacement parameters (isotropic or anisotropic, within the harmonic approximation); – Wavelength of the radiation used for the X-ray diffraction experiment; – Scale factor; – Weighting schemes adopted in the refinement (see equations (10.9) and (10.10)) An additional file (extension hkl) contains all the measured structure factors, with their associated uncertainties and potentially the subset number, in case the data have not been previously merged and each reflection is present in several measurements taken in different runs of a data collection. The hkl file may contain also information on the path length of each reflection in the crystal, which could be important for an accurate (anisotropic) correction for secondary extinction. With these data, the appropriate xd files are generated: – A master file (xd.mas) with all fixed parameters (e. g., unit cell, wavelength, atom type) and all instructions to run the different routines of the program package. – An input file (xd.inp) containing all the parameters that are potentially variables of the multipole model (atomic coordinates, atomic displacement, multipole population parameters, scale factors, extinction coefficients). – The reflection file (xd.hkl), containing the same information as the one used by SHELX or Olex, but potentially including also all direction cosines of each reflection in case an extinction correction is applied, for example, with the model by Becker and Coppens [42]. A graphical user interface, WinXD, can read all these files and enable their manipulation, in particular:

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Setting the model for the refinement (IAM model, Multipole Model, IAM or Multipole Model with anharmonic atomic displacement parameters). The anharmonic treatment of the atomic displacement parameters follows the classical treatment summarised by Johnson and Levy [43]. This is an expansion of the harmonic approximation, that requires up to 25 additional parameters per atom and, therefore, cannot be applied to all atoms in a structure, but only those for which it is really necessary. Setting the database for the atomic density functions for the core, spherical valence and deformation valence, as explained in Section 10.1. This means selecting among: – the classical Roothaan–Hartee–Fock atomic wavefunctions of Clementi and Roetti [6] for core and spherical valence and the single-ζ Slater functions from Clementi and Raimondi [9] for the aspherical density; – the relativistic wavefunctions from Su and Coppens [7] and Macchi and Coppens [8] for the core and spherical valence and the single-ζ Slater functions from Clementi and Raimondi [9] for the aspherical density; – the zero-order regular approximation atomic wavefunctions [39] for the core and spherical valence (available for all atoms) and the best single-ζ Slater functions approximating them for the aspherical valence; – a free database of atomic wavefunctions, compiled by the user with the standard XD2016 format. Defining which parameter is a variable of the model and which constraint is applied. By default, an electro-neutrality constraint is activated to guarantee that the total number of electrons in the unit cell remains constant. Additional constraints may involve the κ or κ 󸀠 parameters. In principle, any atom may hold its own set of contraction/expansion parameters, but more conveniently atoms of the same type in a similar chemical environment are grouped to reduce the model instability and the number of parameters. Moreover, different κ 󸀠 may be defined for each multipole level (meaning one for the dipoles, one for the quadrupoles, etc.) but a simple instruction enables defining a single κ󸀠 for all the multipole levels of an atom type, which is the standard.

Moreover, the graphical interface enables running all modules of the program, namely: – The file initialization module XDINI: it reads the structural files from SHELX, Olex2 or a crystallographic information file, and creates the XD2016 files. – The structure factor handling routine XDHKL: it merges the data and calculates intensity statistics. – The least square refinement module XDLSM: it launches the refinement of a model following the specifications. – The Fast Fourier Transformation module XDFFT: it calculates a 3D residual density map and automatically locates the largest residuals (positive or negative).

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The Fourier Transformation module XDFOUR: it calculates a 2D or a 3D residual density map, as well as a deformation density using the difference between the measured structure factors and the IAM calculated structure factors, or a model deformation density, using the difference between the multipole model calculated structure factors and the IAM calculated structure factors. The module for the calculation of the properties based on the refined model XDPROP: it enables running topological analysis of the total electron density or electrostatic potential; computing all electrostatic moments; computing maps of all electron density derived functions (density, gradient, Laplacian, electrostatic potential, electrostatic field, electrostatic field gradient, one electron potential, reduced density gradient, density overlap indicator, single exponential detector, kinetic and potential energy densities, electron localization function, and localized orbital locator). The module for a comprehensive topological analysis in a crystal TOPXD: it calculates all density properties in a periodic 3D framework, whereas XDPROP calculates properties of an isolated fragment of the crystal (e. g., a molecule, a dimer, a small cluster, etc.). The module for graphical representation of the calculated functions, XDGRAPH. The module XDPDF: it calculates the nuclear probability density based on the refined harmonic or anharmonic atomic displacement parameters. The module XDVIB: it calculates atomic displacement parameters from theoretically calculated molecular vibrational frequencies, to set in the model calculated values which do not need further refinement (e. g., for H atoms). The module XDWTAN: it analyses the correctness of the adopted weighting scheme.

The graphical interface is also linked to other routines developed externally by other authors but tightly connected to the multipole refinement: – The routine to calculate residual plots (PIXels stats), following the analysis by Meindl and Henn [23]. This routine is directly inserted in WinXD. – The routine to analyze the residual of intensities (DRK), following the work by Zhurov et al. [36]. This routine is directly inserted in WinXD. – The routine to calculate anisotropic displacement parameters for H atoms, using a rigid body approximation and the parameters of heavier elements in the molecule, following the procedure SHADE introduced by Madsen (2006) and Madsen and Hoser [37, 38]. This routine is linked externally, using the website of the program which provides the calculation. The graphical interface exports the proper files for SHADE or SHADE3 and import the results in XD2016 to continue a refinement. – The software MoleCoolQt [44] is linked externally. This software enables additional graphical representations of calculated functions, as well as setting proper input files for special multipolar refinements with theoretically calculated multipolar coefficients.

262 | B. Guillot et al. Other small routines enable manipulation of XD2016 files, such as operations with functions computed on a grid, update of model from precious refinement strategies, writing tables and crystallographic information files, and creating graphical files in special formats, such as the Persistence of Vision Raytracer. Moreover, all functions can be written in a standard cube file that can be visualized using many software packages available for theoretical chemistry. In Figure 10.9, a scheme of the working procedure of XD2016 is graphically summarized.

Figure 10.9: The flowchart of XD2016 modelling and bonding analysis.

As introduced above, some special multipole refinement may be carried out against synthetic structure factors, calculated by ab initio crystal wavefunctions, or even molecular wavefunctions (embedding the molecule in a virtual unit cell). For these kinds of refinements, the number of variable parameters is much smaller, because

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the structure factors are typically static (i. e., they are not convoluted with the nuclear motion), the atomic coordinates are known and the anomalous scattering of atoms is also neglected in the simulation. This is easily set in the master file of XD2016, which has been in fact often used to produce these models refined against theoretical dataset (see, e. g., [14]). A special feature in this case is the so-called phase constrained refinement, where the phase of each reflection (that are of course known when coming from a simulated dataset) are kept rigidly fixed during the refinement, avoiding potential artefacts for non-centrosymmetric lattices. This may be important when refining a multipole model of a simulated molecular density, calculated with molecular orbital wavefunction and embedded in a unit cell without symmetry (typically with cubic metrics, but simply P1 space group). To run this refinement, it is necessary that the hkl reflection file also contains the calculated phase of the reflection. For the scope of this textbook, the module XDPROP is very important. As mentioned, it calculates the electron density and electric properties of a molecule from the refined multipole model, the interaction energies between two (or more) molecules in the crystal, and the lattice energies. In Chapter 3, the Quantum Theory of Atoms in Molecules (QTAIM; [33]) has been introduced. QTAIM is based mainly on the partition of the one-electron density distribution, which is the quantity that a multipole model is reconstructing. Therefore, a QTAIM chemical bonding analysis can be carried out using a multipole model of the electron density as well as a calculated wavefunction. In fact, the definition of an atomic basin only requires the gradient of the electron density, and many properties at the stationary points of the electron density (the so-called critical points) depend on derivatives of the electron density (e. g., the Laplacian). This perfect correspondence between the wavefunction (theoretical) model and the multipolar (experimental or theoretical) model does not hold true for energy densities, however, because their calculation would require the wavefunction, which is not available from a multipole model (see also discussion in 10.1 concerning the orbital vs multipolar functions). Only approximated quantities of energy densities can be calculated, using some known approximations for the kinetic energy density like the one proposed by Kirzhnits [45]. For this reason, the analysis in terms of electron localization (see Chapter 4) is much less feasible and certainly less accurate because, apart from the experimental error and the approximation of the multipole model, there is an additional approximation to apply for the calculation of the kinetic energy density and all correlated quantities. Therefore, the error propagation may be huge. Even more complicated would be to retrieve from experiment a two-electron (or pair) density, which again requires the knowledge of the wavefunction or of the second-order reduced density matrix” (which is the density matrix corresponding to the pair density).

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10.4 Concluding remarks and outlook The multipolar expansion of atomic electron density has a long historical traditiion; nevertheless, it remains a very useful and reliable method to model the electron density distribution from X-ray diffraction experiments. The major pitfalls concern the data quality and therefore the reliability of the experiments. Nonetheless, being an approximation, there are inherent limitations that may produce severe artefacts. The strength of the multipolar formalism stands in the ease of the interpretation of the resulting model, which maintains an atomistic perspective like the standard structural models for crystallography while being significantly more informative. When the diffraction data quality is not good enough for fitting the ED, the calculation of the model by ED database transfer is nowadays quite feasible and rarely a problem. Such a model is adequate to obtain an accurate structural model of very large molecular systems (up to proteins) for which in fact the measured data may be of poor quality. As discussed in this chapter, the model has also undergone sophistications that enable extracting even more detailed information (e. g., on the polarization of core electrons) that were not forecastable a few years ago. At the same time, approximated theories enable linking the multipole model also with functions that in principle cannot be directly obtained without a wavefunction as, for example, energy densities. It is worth emphasizing that the multipole model is not just better than IAM in modeling the ED. In fact, the additional degrees of freedom in the fit remain chemically sensible because they can be associated with an atomic polarization due to a partial rehybridization or a change of electronic state. In this respect, the multipolar model mimics the linear combination of atomic orbitals, the most popular approximation to solve the Schrodinger equation for a molecule. This constant improvement and modification enable us to expect even more astonishing results in the future, when perhaps very high-resolution diffraction data may become more easily available and of sufficient quality also for organic (small or large) molecules, not only for inorganic systems. Moreover, integration of multipolebased formalism in standard packages for structure refinement, may encourage even more studies to support and extend the routine structural studies, given that the average quality of standard X-ray diffraction equipment available in university laboratories has increased tremendously in the last decade. Based on that, we may conclude that the multipole model will live long and, as all living beings, will continue to transform and evolve.

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Coppens P. Comparative X-ray and neutron diffraction study of bonding effects in s-triazine. Science. 1967;158:1577–9. Hansen NK, Coppens P. Testing aspherical atom refinements on small-molecule data sets. Acta Crystallogr. 1978;A34:909–921. Stewart RF. Electron population analysis with rigid pseudoatoms. Acta Crystallogr. 1976;A32:565–74. Stewart RF. Generalized X-ray scattering factors. J Chem Phys. 1969;51:4569–77. Coppens P. X-ray Charge Densities and Chemical Bonding. New York: Oxford University Press; 1997. Clementi E, Roetti C. Roothaan–Hartree–Fock atomic wavefunctions: basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms, Z ≤ 54. At Data Nucl Data Tables. 1974;14:177–478. Su Z, Coppens P. Nonlinear least-squares fitting of numerical relativistic atomic wave functions by a linear combination of Slater-type functions for atoms with Z = 1–36. Acta Crystallogr. 1998;A54:646–52. Macchi P, Coppens P. Relativistic analytical wave functions and scattering factors for neutral atoms beyond Kr and for all chemically important ions up to I− . Acta Crystallogr. 2001;A57:656–62. Clementi E, Raimondi DL. Atomic screening constants from SCF functions. J Chem Phys. 1963;38:2686–9. Bach A, Lentz D, Luger P. Charge density and topological analysis of pentafluorobenzoic. Acid J Phys Chem A. 2001;105:7405–23. Fischer A, Tiana D, Scherer W, Batke K, Eickerling G, Svendsen H, Bindzus N, Iversen BB. Experimental and theoretical charge density studies at subatomic resolution. J Phys Chem A. 2011;115:13061–71. Ahmed M, Nassour A, Noureen S, Lecomte C, Jelsch C. Experimental and theoretical charge-density analysis of 1, 4-bis (5-hexyl-2-thienyl) butane-1, 4-dione: applications of a virtual-atom model. Acta Crystallogr. 2016;B72:75–86. Tidey JP, Zhurov VV, Gianopoulos CG, Zhurova EA, Pinkerton AA. Experimental charge-density study of the intra- and intermolecular bonding in TKX-50. J Phys Chem A. 2017;121:8962–72. Volkov A, Coppens P. Critical examination of the radial functions in the Hansen–Coppens multipole model through topological analysis of primary and refined theoretical densities. Acta Crystallogr. 2001;A57:395–405. Batke K, Eickerling G. Topology of the electron density of d0 transition metal compounds at subatomic resolution. J Phys Chem A. 2013;117:11566–79. Gianopoulos CG, Zhurov VV, Minasian SG, Batista ER, Jelsch C, Pinkerton AA. Bonding in uranium(V) hexafluoride based on the experimental electron density distribution measured at 20 K. Inorg Chem. 2017;56:1775–8. Gianopoulos CG, Zhurov VV, Pinkerton AA. Charge densities in actinide compounds: strategies for data reduction and model building. IUCrJ. 2019;6:895–908. Zarychta B, Zaleski J, Kyzioł J, Daszkiewicz Z, Jelsch C. Charge-density analysis of 1-nitroindoline: refinement quality using free R factors and restraints. Acta Crystallogr. 2011;B67:250–62. Krause L, Niepötter B, Schürmann CJ, Stalke D, Herbst-Irmer R. Validation of experimental charge-density refinement strategies: when do we overfit? IUCrJ. 2017;4:420–30. Coppens P, Volkov A. The interplay between experiment and theory in charge-density analysis. Acta Crystallogr. 2004;A60:357–64.

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[21] Volkov A, Abramov YA, Coppens P. Density-optimized radial exponents for X-ray charge-density refinement from ab initio crystal calculations. Acta Crystallogr. 2001;A57:272–82. [22] Parrish D, Zhurova EA, Kirschbaum K, Pinkerton AA. Experimental charge density study of estrogens: 17β-estradiol urea. J Phys Chem B. 2006;110:26442–7. [23] Meindl K, Henn J. Foundations of residual-density analysis. Acta Crystallogr. 2008;A64:404–18. [24] Guillot B, Viry L, Guillot R, Lecomte C, Jelsch C. Refinement of proteins at subatomic resolution with MOPRO. J Appl Crystallogr. 2001;34:214–23. [25] Jelsch C, Guillot B, Lagoutte A, Lecomte C. Advances in protein and small-molecule charge-density refinement methods using MoPro. J Appl Crystallogr. 2005;38:38–54. [26] Zarychta B, Pichon-Pesme V, Guillot B, Lecomte C, Jelsch C. On the application of an experimental multipolar pseudo-atom library for accurate refinement of small-molecule and protein crystal structures. Acta Crystallogr. 2007;A63:108–25. [27] Domagała S, Fournier B, Liebschner D, Guillot B, Jelsch C. An improved experimental databank of transferable multipolar atom models–ELMAM2. Construction details and applications. Acta Crystallogr. 2012;A68:337–51. [28] Domagała S, Munshi P, Ahmed M, Guillot B, Jelsch C. Structural analysis and multipole modelling of quercetin monohydrate–a quantitative and comparative study. Acta Crystallogr. 2011;B67:63–78. [29] Fournier B, Bendeif el-E, Guillot B, Podjarny A, Lecomte C, Jelsch C. Charge density and electrostatic interactions of fidarestat, an inhibitor of human aldose reductase. J Am Chem Soc. 2009;131:10929–41. [30] Jelsch C, Teeter MM, Lamzin V, Pichon-Pesme V, Blessing RH, Lecomte C. Accurate protein crystallography at ultra-high resolution: valence electron distribution in crambin. Proc Natl Acad Sci USA. 2000;97:3171–6. [31] Hirano Y, Takeda K, Miki K. Charge-density analysis of an iron-sulfur protein at an ultra-high resolution of 0.48 Å. Nature. 2016;534:281–4. [32] Nassour A, Domagala S, Guillot B, Leduc T, Lecomte C, Jelsch C. A theoretical-electron-density databank using a model of real and virtual spherical atoms. Acta Crystallogr. 2017;73:610–25. [33] Bader RFW. Atoms in Molecules: A Quantum Theory. Oxford: Oxford University Press; 1990. [34] Volkov A, Koritsanszky T, Coppens P. Combination of the exact potential and multipole methods (EP/MM) for evaluation of intermolecular electrostatic interaction energies with pseudoatom representation of molecular electron densities. Chem Phys Lett. 2004;391:170–5. [35] Guillot B. MoProViewer: a molecule viewer for the MoPro charge density analysis program. Acta Crystallogr. 2011;A68:C511–2. [36] Zhurov VV, Zhurova EA, Pinkerton AA. Optimization and evaluation of data quality for charge density studies. J Appl Crystallogr. 2008;41:340–9. [37] Madsen AO. SHADE web server for estimation of hydrogen anisotropic displacement parameters. J Appl Crystallogr. 2006;39:757–8. [38] Madsen AØ, Hoser AA. SHADE3 server: a streamlined approach to estimate H-atom anisotropic displacement parameters using periodic ab initio calculations or experimental information. J Appl Crystallogr. 2014;47:2100–4. [39] Volkov A, Macchi P, Farrugia LJ, Gatti C, Mallinson P, Richter T, Koritsanszky T. XD2016 – A Computer Program Package for Multipole Refinement, Topological Analysis of Charge Densities and Evaluation of Intermolecular Energies from Experimental and Theoretical Structure Factors. User Manual. 2016. [40] Sheldrick GM. Crystal structure refinement with SHELXL. Acta Crystallogr. 2015;C71:3–8. [41] Dolomanov OV, Bourhis LJ, Gildea RJ, Howard JAK, Puschmann H. J Appl Crystallogr. 2009;42:339–41.

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[42] Becker PJ, Coppens P. Extinction within the limit of validity of the Darwin transfer equations. I. General formalism for primary and secondary extinction and their applications to spherical crystals. Acta Crystallogr. 1974;A30:129–47. [43] Johnson CK, Levy HA. Thermal motion analysis using Bragg diffraction data. In: International Tables for X-ray Crystallography. vol. IV. Birmingham: Kynoch Press; 1974. p. 311–36. [44] Hübschle CB, Dittrich B. MoleCoolQt – a molecule viewer for charge-density research. J Appl Crystallogr. 2011;44:238–40. [45] Kirzhnits DA. Quantum corrections to the Thomas–Fermi equation. Sov Phys JETP. 1957;5:64–71.

Alessandro Genoni and Dylan Jayatilaka

11 X-ray constrained wavefunction analysis with Tonto 11.1 Introduction The goal of this chapter is to introduce the reader to the X-ray constrained wavefunction (XCW) technique [1–7] and to the use of this strategy through the quantum crystallographic software Tonto [8]. In particular, in Section 11.2, we will present the fundamentals of the XCW method, starting from the usual concept of wavefunction in quantum mechanics and arriving to the concept of “experimental” wavefunction at the basis of the XCW approach. Finally, in Section 11.3, we will show some examples of basic (XCW) calculations that can be performed through Tonto, which is the program that currently implements most of the existing X-ray constrained wavefunction strategies.

11.2 Initiation to the X-ray constrained wavefunction analysis 11.2.1 The concept of wavefunction and its interpretation Quantum mechanics is one of the fundamental theories of physics. The key idea of this theory is that the state of a system is described by a complex-number valued function: the wavefunction Ψ. Any physical property in which we are interested may be calculated from this wavefunction. For example, within the Born–Oppenheimer (BO) approximation [9] (which allows to separate the motion of electrons and nuclei in a molecule), the electron density for a many-electron system may be defined as [10] N N 󵄨 󵄨2 ρ(r)⃗ = Ne ∫ dσdx⃗2 . . . dx⃗Ne 󵄨󵄨󵄨Ψ(r,⃗ σ, x⃗2 , . . . , x⃗Ne ; {R⃗ i }i=1N , {Zi }i=1N , E,⃗ B)⃗ 󵄨󵄨󵄨

(11.1)

The wavefunction Ψ is a function of the Ne electron coordinates x⃗i = (ri⃗ , σi ). Each electron coordinate comprises the electron position ri⃗ = (xi , yi , zi ) and the electron spin σi , which is a two-dimensional complex quantity with a maximum magnitude of 1/2, Alessandro Genoni, Université de Lorraine & CNRS, Laboratoire de Physique et Chimie Théoriques (LPCT), UMR CNRS 7019, 1 Boulevard Arago, F-57078 Metz, France, e-mail: [email protected] Dylan Jayatilaka, School of Molecular Sciences, University of Western Australia, 35 Stirling Highway, WA 6009 Perth, Australia, e-mail: [email protected] https://doi.org/10.1515/9783110660074-011

270 | A. Genoni and D. Jayatilaka that is, |σi | ≤ 1/2 [11]. Analyzing equation (11.1), we also see that the BO wavefunction Ψ N N parametrically depends on the NN nuclear positions {R⃗ i }i=1N and nuclear charges {Zi }i=1N . It may also parametrically depend on any externally applied electric and magnetic fields (E⃗ and B,⃗ respectively, in equation (11.1)). Actually, Ψ also depends on time, but here we have not written the time coordinate t explicitly because in this chapter we will deal with time-independent processes or processes (such as diffraction) which occur much more slowly that the time dependence in the wavefunction. Therefore, we can ignore it or make approximations so that the time dependence is averaged out. For the sake of simplicity and clarity, we point out that, in the next sections of the chapter, we will not indicate all the variables on which a quantity parametrically depends, unless when they will be really necessary in the context of the discussion. Anyway, looking at equation (11.1), we see that it essentially consists in |Ψ|2 . The integration is performed over all the coordinates except the coordinates of the first electron. It does not matter which of the electron’s coordinates we integrate over. As we will explain later, the coordinates are all identical to each other, so we just choose to integrate over all of them except those of the first electron and then we multiply the result by the total number of electrons to get the total electron density. According to Born, the square of the wavefunction is related to the probability of finding all the particles of the system under exam at their given coordinates [12]. This implies that 󵄨 󵄨2 ∫ dx⃗1 dx⃗2 . . . dx⃗Ne 󵄨󵄨󵄨Ψ(x⃗1 , x⃗2 , . . . , x⃗Ne )󵄨󵄨󵄨 = 1,

(11.2)

namely that the probability of finding all the particles all over the space is equal to 1. From this, it is also easy to show that the electron density exactly integrates to the number of electrons: ⃗ r)⃗ = Ne ∫ drρ(

(11.3)

11.2.2 Calculating the wavefunction In the standard view, the wavefunction is not a measurable quantity. Rather, the time independent wavefunction is obtained by solving the time-independent Schrödinger equation [13, 14]: ̂ = EΨ HΨ

(11.4)

̂ is the energy operator (namely, the Hamiltonian) and E is an allowed energy Here, H of the system. Note that the Hamiltonian operator is a Hermitian operator, namely an operator whose eigenvalues are always real, as in the case expressed by equation (11.4). The Hamiltonian (in atomic units) is a differential operator comprising three-dimensional second derivatives with respect to the electron coordinates, and terms describing the electrostatic interactions between the nuclei and electrons in the

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system; magnetic field effects are weak and usually ignored unless they are specifically of interest: N

N N

N N

e e N ZA 1 1 e e ̂ = − 1 ∑ ∇2 − ∑ ∑ + ∑∑ H i ⃗ ⃗ 2 2 | r − i rj⃗ | i=1 j=i̸ i=1 A=1 |ri⃗ − RA | ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟i=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Kinetic energy Nuclear attraction

(11.5)

Electron repulsion

In addition to satisfying the Schrödinger equation, and being normalized, the wavefunction for electrons is supposed to be identical in particle exchange, that is, Ψ(x⃗1 , . . . , x⃗i , . . . , x⃗j , . . . , x⃗Ne ) = −Ψ(x⃗1 , . . . , x⃗j , . . . , x⃗i , . . . , x⃗Ne )

(11.6)

This is actually nothing else than the Pauli exclusion principle [15] and it ensures that electrons cannot have the same coordinates. The previous equation can be trivially derived by taking the negative square root of the expression 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨󵄨Ψ(x⃗1 , . . . , x⃗i , . . . , x⃗j , . . . , x⃗Ne )󵄨󵄨󵄨 = 󵄨󵄨󵄨Ψ(x⃗1 , . . . , x⃗j , . . . , x⃗i , . . . , x⃗Ne )󵄨󵄨󵄨

(11.7)

Equation (11.7) basically states that if swap two identical electron coordinates, the probability density associated with the wavefunction does not change. The reason why this is true is because electrons are identical, and this is what identical means in quantum mechanics. You may wonder why the positive square root is not taken. This is possible, and a wavefunction for a group of (identical) photons does have this property. They are called bosons [16]. The negative square root holds for electrons. They are fermions [17, 18]. Fermions include not only electrons, but also protons and neutrons. There are only two different types of particles in the universe, and the reason is that there is only a positive or negative square root to the above equation. It is a fact, called the spin-statistics theorem, that particles with half-integral spin are fermions, whereas particles with integral spin are bosons [16–18].

11.2.3 The simplest approximate wavefunctions The Schrödinger equation [13, 14] is very complicated and it is not possible to solve it exactly except for systems with two or three particles. For larger systems, if we want to obtain useful results, we must approximate the exact wavefunction. The oldest approximation, and still one of the most important, is the Hartree product wavefunction Ψ ≈ Φ = ϕ1 (x⃗1 )ϕ2 (x⃗2 ) . . . ϕNe (x⃗Ne )

(11.8)

You can see that the Hartree product wavefunction comprises just a product of spinorbitals, ϕi . Each spinorbital is a function of just a single electron’s (spatial and spin) coordinates. Hence, the spinorbitals are also called one-electron orbitals. They

272 | A. Genoni and D. Jayatilaka can also be called molecular spinorbitals. To simplify the calculations, the (spin)orbitals are usually assumed orthonormal: ⃗ i (x)ϕ ⃗ j (x)⃗ = { ∫ dxϕ

1 0

if i = j if i ≠ j

(11.9)

Hartree used a self-consistent field (SCF) algorithm to work out the form of the orbitals in this product wavefunction [19], almost as soon as Schrödinger derived his equation and before it was officially published in 1926 [13, 14]. Moreover, it is interesting to note that the title of Hartree’s article was “The atomic structure factor in the intensity of reflexion of X-rays by crystals.󸀠󸀠 Furthermore, soon after that, James, Waller and Hartree applied their SCF method to the NaCl crystal to show that zero-point motion of atoms could be detected in the nuclei of that crystal [20]. This shows the close connection in the early years between quantum mechanics, actually what we now call quantum chemistry, and X-ray diffraction. The problem is that the Hartree product wavefunction does not obey the Pauli principle: each of the electrons is different because it is in its own orbital. Luckily, it is easy to satisfy the Pauli principle and make the electrons indistinguishable. All that is required is to sum together Hartree product wavefunctions differing only in the fact that the electron coordinates in each is permuted, and multiplying each permuted term by a positive sign for “even” permutations and by a negative sign for “odd” permutations. The result is the single determinant wavefunction: ̂ 1 (x⃗1 )ϕ2 (x⃗2 ) . . . ϕN (x⃗N )] Ψ ≈ Φ = A[ϕ e e

(11.10)

̂ is the antisymmetrizer that performs the previously mentioned The symbol A permutations-with-sign-changes. The problem is that this will generate Ne ! different product wavefunctions, and the wavefunction will not be normalized anymore. ̂ also includes a normalization factor of (Ne !)−1/2 . Therefore, the antisymmetrizer A Slater noted that this wavefunction could be written in the form of a determinant and, for this reason, it is usually called Slater determinant. To make things clear, consider the example of a three-electron Slater determinant wavefunction. It can be written explicitly as Φ=

1 [ϕ (x⃗ )ϕ (x⃗ )ϕ (x⃗ ) − ϕ1 (x⃗1 )ϕ2 (x⃗3 )ϕ3 (x⃗2 ) − ϕ1 (x⃗2 )ϕ2 (x⃗1 )ϕ3 (x⃗3 ) √3! 1 1 2 2 3 3 + ϕ1 (x⃗2 )ϕ2 (x⃗3 )ϕ3 (x⃗1 ) − ϕ1 (x⃗3 )ϕ2 (x⃗2 )ϕ3 (x⃗1 ) + ϕ1 (x⃗3 )ϕ2 (x⃗1 )ϕ3 (x⃗2 )]

(11.11)

According to Froese Fischer [21], Fock actually published on the antisymmetry requirement before Hartree could [22]. Therefore, if the spinorbitals are determined using the SCF method, (see below) they are called Hartree–Fock spinorbitals. In nature, the most common systems are closed-shell systems. In this case, a further approximation is used for the form of the spinorbitals, namely the spinorbitals

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come in pairs, and they are written as the product of a common spatial part times a spin part which is either “up” or “down.” ⃗ φ(r)α(σ) { { ϕ(x)⃗ = { or { ⃗ { φ(r)β(σ)

(11.12)

This is called the restricted spinorbital approximation. The spatial part φ(r)⃗ is usually called spatial orbital or molecular orbital (MO). For odd-electron systems, we can allow the spatial parts of the α and β spinorbitals to be different, and in this case we would have the unrestricted spinorbital approximation. The spatial orbital parts of the restricted spinorbitals are usually approximated as a linear combination of atomic orbitals (LCAO). Before talking about the linear combination, we discuss these atomic orbitals, which are preferably called basis functions. The basis functions may be any kind of functions, but in most cases they are taken to be atom-centred Gaussian basis functions [23, 24] of this form: χ(r;⃗ n,⃗ A)⃗ = (rx − Ax )nx (ry − Ay )ny (rz − Az )nz exp(−α|r ⃗ − A|⃗ 2 )

(11.13)

In the above equation, the three numbers nx , ny and nz are the Cartesian powers of the Gaussian function, and nx +ny + nz is the total angular momentum of the Gaussian. Thus, if all the powers are zero, we have an s-type Gaussian function, while, if only one of them is one and all the other ones are zero, we have a p-type Gaussian function, following the normal spectroscopic labels for angular momentum, and so on. The vector A⃗ is the centre of the Gaussian and it normally corresponds to the position of one nucleus of the molecule. Different are the reasons why Gaussian basis functions are used: (i) their linear combinations look very much like the exact atomic Hartree–Fock orbitals; (ii) integrals involving Gaussian basis functions may be calculated relatively easily and (iii) nowadays there are many sets of “canned” Gaussian-type basis sets, developed by researchers over the years, that can be used in calculations. Among them, we can mention the following widely used contracted Gaussian basis-sets, such as: – the Slater type orbitals STO-nG basis-sets (with n = 2–6), which are minimum basis-sets; – the popular k-nlmG Pople basis-sets, which are split-valence sets of basis functions, namely basis-sets that split the atomic valence orbitals in different basis functions (generally 2 or 3). Examples are the 3-21G, 6-31G, 6-311G basis-sets and the corresponding ones with polarization and, if necessary, diffuse functions, which are generally indicated with the “*” and “+” symbols, respectively (examples are the 3-21G*, 6-31G**, 6-311++G** basis sets). Note that if there are two “*” symbols in the basis-set specification, the first one indicates a set of polarization functions on atoms from the second period (“heavy atoms”), while the second one indicates a set of polarization functions also on the hydrogen atoms (e. g., 6-31G* is equivalent to 6-31G(d), while 6-31G** is equivalent to 6-31G(d,p)). The same logic is adopted for the “+” symbols;

274 | A. Genoni and D. Jayatilaka –

–

the Ahlrichs type basis-sets, such as the SVP, TZV and QZV sets of basis functions (which are double-zeta, triple-seta and quadrupole-zeta split-valence basis-sets, respectively); the correlation consistent basis-sets, such as the cc-pVDZ, cc-pVTZ and cc-pVQZ sets of basis functions (which are again double-zeta, triple-seta and quadrupolezeta split-valence basis-sets, respectively). They automatically include polarization functions and can be “augmented” by adding diffuse functions, which are indicated through the prefix “aug” (e. g., aug-cc-pVDZ, aug-cc-pVTZ and aug-ccpVQZ basis-sets).

It is also common to use plane-wave basis functions; integrals with these functions are also extremely easy, but they suffer from the problem that a combination of many of them are required to represent atomic and molecular wavefunctions, especially for electrons near the core of an atom. However, for a more complete presentation and discussion on the different types of basis-sets used in quantum chemical calculations we encourage the reader to refer to specialized textbooks on quantum chemistry [25–27]. Whatever you choose for the basis functions, the spatial molecular orbitals are approximated by the basis functions as Nbf

φi (r)⃗ = ∑ Cμi χμ (r)⃗ i=1

(11.14)

Here, the coefficients {Cμi } are the so-called molecular orbital (MO) coefficients, which, for a closed-shell system, comprise a matrix Nbf × (Ne /2) in size, where Nbf is the total number of used basis functions. Not all of these parameters are independent, because the orbitals are generally orthonormal (see equation (11.9)) and Levy and Goldstein have shown that the number of independent parameters in the wavefunction is (Ne /2) × (Nbf − Ne /2) [28]. Anyway, for a given set of basis functions, the MO coefficients are the quantities that must be determined in the calculations; but how to find them?

11.2.4 The Hartree–Fock wavefunction For many types of wavefunction, and for the Hartree–Fock wavefunction in particular, the method that is used to determine the unknown parameters is the variational principle [10]. The variational principle states that the energy of an approximate wavefunction Φ always lies above the energy of the exact wavefunction Ψ, i. e., ̂ x⃗1 , . . . , x⃗N ) ≥ E[Ψ] E[Φ] = ∫ dx⃗1 dx⃗2 . . . dx⃗Ne Φ∗ (x⃗1 , . . . , x⃗Ne )HΦ( e

(11.15)

Here, the energy is a function of a function (in this case, the wavefunction) and so we call it a functional and use square instead of round brackets. Furthermore, the best

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approximate wavefunction Φ (namely, the one that is the closest to the exact wavefunction Ψ in a least-squares sense) is the one that minimises the approximate energy expression E[Φ] above. So, all we have to do is to substitute equation (11.10), into equation (11.15) taking into account the orthonormality condition expressed by equation (11.9). The result is an expression involving the molecular spinorbitals. By minimizing this expression with respect to the spinorbitals, we get the celebrated Hartree– Fock equations [22, 29]: ̂ i = ϵi ϕi Fϕ

(11.16)

where F̂ is the so-called Fock operator, a one-electron operator that can be considered as an effective Hamiltonian taking into account the mean (electric) field due to the other electrons. The values {ϵi } are the eigenvalues associated with the different spinorbitals, the so-called (spin)orbital energies. For a closed-shell system, if we introduce the expansion of the spatial molecular orbitals in terms of the adopted basis-set (see equation (11.14)), the Hartree–Fock equations reduce to the well-known Roothaan–Hall matrix equations [30, 31]: FC=SCE

(11.17)

where F is the matrix associated with the Fock operator F̂ in the adopted basis-set, S is the matrix of the overlaps between the atomic orbitals, E is the diagonal matrix of the orbital energies and C is the matrix of the molecular orbitals coefficients. It is important to note that the matrix F depends on the MO coefficients C. So, the whole procedure starts with an initial set of orbitals (guess molecular orbitals). Then equation (11.17) is solved with this initial guess to obtain better MO coefficients and, consequently, a better Fock matrix F. Obviously, the procedure is iterated until the selfconsistent field (SCF) convergence is achieved, namely, until the mean (electric) field generated by the electrons of the molecule remains invariant.

11.2.5 Kohn’s thoughts on the wavefunction The exact wavefunction is a very complicated function. In his Nobel lecture, Kohn gave arguments for the notion that a wavefunction for systems constituted of more than about a thousand electrons was not a sensible idea [32]. He went on suggesting that we need to develop models based on more simple quantities, such as the electron density that is only a three dimensional quantity. Indeed, Hohenberg and Kohn proved that the electron density for a nondegenerate ground state uniquely determines the external electric potential of the system to within a constant, and hence it uniquely determines the wavefunction [33]. They also showed that the quantum mechanical energy (in the BO approximation) is a unique variational functional of the electron

276 | A. Genoni and D. Jayatilaka density, that is, they proved that E[ρ]̃ ≥ E

(11.18)

This says that there is a functional such that, when a trial electron density ρ̃ is used in it, an energy higher than the exact energy E is always obtained. It thus follows that the wavefunction apparently contains a lot of redundant information. However, the proof of these facts did not initially give any idea of how to find that functional. Luckily, one year later, Kohn and Sham developed a model based on the idea of a single determinant wavefunction that exactly reproduces the exact electron density [34]. This Kohn–Sham model forms the basis of many approximate density functionals that perform very well compared to the Hartree–Fock theory. In fact, the Kohn– Sham wavefunction is nothing else than a single determinant wavefunction that provides the exact electron density, but that minimizes the kinetic energy. The use of orbitals in this method is essential in obtaining a reasonable approximation for the kinetic energy. There is now a systematic way to improve such density functional theory (DFT) functionals, called Jacob’s ladder [35], by fitting parameters in these functionals to quantities obtained from accurate wavefunction calculations. Sometimes experimental results are also used in the fitting process. In any case, DFT is now the go-to-method for fast accurate results on large systems.

11.2.6 Experimental wavefunctions Quantum mechanics is only a model and, according to Kohn, it is not even sensible for large systems. However, the Schrödinger equation is a very good model: for example, it serves as the basis for DFT. This question then arises: would it be possible to determine the wavefunction in another way, perhaps from experimental measurements? The Hohenberg–Kohn theorem states that the wavefunction can be determined from the electron density. On the other hand, the magnitude of the Fourier transform of the electron density can be measured using X-ray diffraction. Depending on the molecular crystal, there can be about 103 –105 pieces of experimental information (namely, experimental data). Thus it would seem that a model wavefunction, such as the Kohn–Sham wavefunction, might be determinable from experiment. Figure 11.1 illustrates the general procedure, which is nothing more than what everyone really thinks a measurement is. However, there is still a problem: in quantum mechanics, it is in principle impossible to measure a wavefunction. On the contrary, the results of the measurements are predicted from the expectation values of Hermitian operators, namely from the expectation values of those operators with real eigenvalues (such as the Hamiltonian operator seen above). Therefore, it is important to state that measuring a wavefunction, by which we mean measuring parameters in a model wavefunction, goes beyond standard quantum mechanics.

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Figure 11.1: Three-stage conception of a measurement, in this case of a model wavefunction: (1) think of a model and an experiment to determine parameters in the model; (2) carry out the experiment; (3) analyze the data and determine the parameters. Figure reproduced with permission of Springer Nature from reference [7] of this chapter.

If we accept that Figure 11.1 is a “correct” depiction of an experiment, then another issue arises, particularly in relation to step (1) of the process. This seems to indicate that a measurement requires theory first in order to be “carried out.” Again, this may go against some notions that consider theories to be built on experimental observations. In this regard, it is worth pointing out that knowing “how Nature works” is an iterative procedure between experiment and observation: theories do not appear miraculously from nowhere and the first step in this iterative procedure is of course experiment, even if what is being done in the experiment is not entirely clear in the first instance. Now, we discuss further remarks concerning the connection of experimental model wavefunctions and different fields of study. First of all, we mention the idea of model or effective Hamiltonians. As the name implies, these are simply Hamiltonians in which there appear parameters, and these parameters are determined in such a way that they fit the results of an energy spectrum. Examples include: (i) the rigidrotor vibrational Hamiltonian, which is used to describe the rovibrational spectra of small molecules; [36] (ii) the spin Hamiltonians, which describe NMR or ESR experiments and (iii) the Hückel Hamiltonians, which describe π electrons. Of course, when diagonalized, all these Hamiltonians give not only energy eigenvalues, but also the corresponding wavefunctions. These wavefunctions may be considered “experimental.” Unlike model wavefunctions, model Hamiltonians are used to describe many

278 | A. Genoni and D. Jayatilaka different states. On the contrary, a model wavefunction is used to describe only a single state, usually the ground state. As a second example of possibility to determine a model wavefunction, we refer to Figure 11.2, which presents the experimentally determined wavefunction for the H atom. The method used in this case is the (e, 2e) coincident electron scattering experiment [37, 38]. This is rather a special case, because that experiment normally determines only the Dyson orbitals, which are the orbitals that “remain” when a single electron of a particular energy is ejected from a system. In this case, for a one-electron system, the Dyson orbital is the wavefunction.

Figure 11.2: The theoretical and (e,2e) experimentally determined momentum-space wavefunction for the hydrogen atom. Reprinted from Physics Letters A, Vol. 86, B. Lohmann & E. Weigold, Direct measurement of the electron momentum probability distribution in atomic hydrogen, Pages No. 139–141, Copyright (1981), with permission from Elsevier (reference [35] of this Chapter).

Scanning tunnelling microscopy (STM) can also provide images very similar to Dyson orbitals, as shown in Figure 11.3 [39]. The authors of that image claimed that they are HOMOs and LUMOs. They are not. Actually they are only scattering images related to the Dyson orbitals [40]. A somewhat better discussion with reference to other experiments claiming to have observed orbitals may be found in a paper published by Schwarz in 2006 [41]. Anyway, it seems clear that the objects seen in the images here are rather similar to the Hartree–Fock orbitals of the same name. Nevertheless, orbitals are not wavefunctions, but they are only a component of a model wavefunction. Finally, we want to mention a work on the determination of wavefunctions or density matrices of a photon, ostensibly for the purposes of optical error correction in quantum optical devices. In this context, the original work goes back to the work of Fano [42], while the group of Raymer has performed pioneering experiments in 1994 [43]. The field has been afterwards reviewed by Paris and Rehacek in 2004 [44],

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Figure 11.3: “Images” of the HOMO and LUMO of pentacene via Scanning tunneling microscopy (STM). Reprinted figure with permission from L. Gross, N. Moll, F. Mohn, A. Curioni, G. Meyer, F. Hanke, M. Persson, Phys. Rev. Lett. 107, 086101, 2011; https://dx.doi.org/10.1103/PhysRevLett.107.086101 (reference [37] of the this Chapter). Copyright (2011) by the American Physical Society.

although those researchers do not draw any connection between their techniques and those developed from scattering theory experiments.

11.2.7 X-ray constrained wavefunctions We mentioned above that the BO wavefunction is a complicated function that is described by many parameters. Some of the parameters in the wavefunction, such as the positions of the nuclei, can be refined in a least squares sense to experimental X-ray diffraction data. This is the idea at the basis of the Hirshfeld atom refinement (HAR) method introduced by Jayatilaka and Dittrich [45] and refined by different researchers in recent years [46–51]. However, the description of this technique is out of the scope of the present chapter. But what about refining (always against experimental X-ray diffraction data) the other parameters in the wavefunction, such as the MO coefficients in a single determinant wavefunction? Levy and Goldstein have pointed out that a unique single determinant wavefunction may be obtained/refined provided that we have more experimental X-ray data than the number of independent parameters in that wavefunction ((Ne /2) × (Nbf − Ne /2)) [28]. This is experimentally achievable. To accomplish this task, one has just to maximise the agreement between the model and the available experimental X-ray diffraction data. In other words, one has to minimize the following χ 2 statistical agreement between experimental and computed structure factor magnitudes

280 | A. Genoni and D. Jayatilaka (Fhexp and Fhpred [Ψ], respectively): χ2 =

(ηFhpred [Ψ] − Fhexp )2 1 ∑ Nr − Np h (σhexp )2

(11.19)

For readers that are not familiar with crystallography, it is worth reminding that the structure factor magnitudes are the square roots of the X-ray diffraction intensities, which are what we can really measure in an X-ray diffraction experiment. It is also important to bear in mind that structure factors are simply Fourier transforms of the unit-cell electron density. In the previous equation, Nr is the number of experimental data, Np is the number of adjustable parameters (as, for example, the external multiplier λ in the XCW approach, see below), h is a vector representing the triad of Miller indices for the reflection under exam, σhexp is the experimental error corresponding to the experimental

structure factor magnitude Fhexp , and η is an overall scale-factor that sets the computed structure factor magnitudes on the same scale of the experimental ones. Furthermore, in equation (11.19) we have explicitly shown that the calculated structure factor magnitudes Fhpred [Ψ] depend on a wavefunction, which itself depends on some parameters, namely the MO coefficients (see subsection 11.2.3). It should also be mentioned that one has to introduce the atomic displacement parameters (i. e., the “thermal ellipsoids” of crystallography) as additional (and fixed) parameters in equation (11.19) in order to produce reasonable structure factor magnitudes. These can be obtained by using other models as, for example, the HAR strategy that we briefly mentioned above. By minimizing equation (11.19) with respect to the parameters (i. e., the MO coefficients), we obtain a model experimental wavefunction. Of course, the more data one has, the better this procedure works. Therefore, the X-ray diffraction experiment seems ideal because it generates a large amount of data, and also because the Hohenberg– Kohn theorem says that the procedure is theoretically well-defined. Unfortunately, Morrison showed that the kinetic energy obtained from such fitted wavefunctions was rather poor [52]. Later, Schwarz and Muller established that this was because the basis function products used in the fitting procedure were not linearly independent [53]. In fact, earlier, Coppens et al. had found that such leastsquares procedures were not very successful [54, 55], probably for the same very reason pointed out by Schwarz and Muller. Nevertheless, there is still some hope that better momentum-space experimental information might solve these problems [56]. However, for the moment, another approach is required and this is where the idea of the X-ray constrained wavefunction [1–7] enters into play. The idea is very simple, although perhaps not so easy to implement. It was initially proposed and carried out by Henderson and Zimmerman [57] using theoretical structure factors in the framework of the Clinton equations [58–61]. Instead of trying to exploit only the experimental information to fit the wavefunction in a least-squares

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way, one instead tries to modify the theoretical wavefunction so that it reproduces the experimental data to some desired level. Thus the wavefunction is constrained to produce a desired value χ02 of the χ 2 agreement statistics by minimising L, which is a combination of the quantum mechanical energy associated with the wavefunction and the least-squares error: L[Ψ] = E[Ψ] + λ(χ 2 [Ψ] − χ02 )

(11.20)

To understand this, let us consider what happens when we use the value λ = 0 and when we assume to work with a single Slater determinant wavefunction. In that case we recover the variational principle and we simply get the Hartree–Fock wavefunction. On the other hand, by using large values of λ, we increase the weight of the least squares term relative to the energy term, and the procedure becomes more like a least squares fit. Hence, we expect that as λ increases, the value of χ 2 decreases. Of course, by using any positive value of λ, the energy of the wavefunction will no longer be a minimum, and so the energy will increase. We cannot have it both ways: either the energy is low and the χ 2 statistical agreement is what it is, or else the χ 2 is lower and the energy is higher. The latter trend is indeed what we observe in any X-ray constrained wavefunction fitting, as shown in Figure 11.4.

11.2.8 How to decide the value of χ 2 at which we should terminate the fit If the σhexp values were reliable, it would not make sense to increase the fit beyond a

value χ 2 = 1, because in that case we would be fitting the data (on average) to one standard deviation. However, if we used very large basis-sets and very large values of λ, we should be able to obtain a perfect fit to the experiment. Nevertheless, in practice, two iconic situations were observed in relation to the termination problem. The first case, which is increasingly common for high-quality X-ray data sets on molecular crystals of simple organic compounds, corresponds to the situation in which the starting χ 2 is less than about four, and the constraint of χ 2 = 1 is easily achieved. In these cases, one observes hardly any change in the constrained wavefunction from its unfitted partner. We have to conclude that in these cases the agreement with quantum mechanics is excellent and we require even better experiments or investigations on systems which are more challenging to model for quantum mechanics (e. g., transition metal complexes). The second (and more problematic case) is the one where, at a certain point, the χ 2 associated with the X-ray constrained wavefunction does not improve significantly, even for large values of λ. The reasons for the difficult convergence are not clear yet. To better understand this problem, (i) one could investigate the Hessian (namely, the matrix of the second derivatives) of the functional L (see equation (11.20)) with respect

282 | A. Genoni and D. Jayatilaka

Figure 11.4: (a) Typical dependence of the χ 2 statistical agreement versus the external multiplier λ; (b) total electronic energy (solid) and negative kinetic energy (dashed) in function of λ. Reproduced with permission of the International Union of Crystallography from D. J. Grimwood, D. Jayatilaka, Acta Crystallogr., Sect. A 2001, 57, 87–100 (reference [3] of this chapter).

to the MO coefficients and establish to what its near-zero eigenvalues correspond or (ii) one could analyze the so-called conflicting data points, namely those reflections that, if fitted, would worsen the agreement of another reflection. The latter solution is probably another way of stating that the Hessian associated with the constrained fitting problem has near-zero eigenvalues, but in that case, one could go back and investigate specific reflections. At the moment, these kinds of studies have not been performed yet. Nevertheless, different empirical criteria to stop the fitting procedure have been introduced over the years, from the idea of halting the calculations at the value of λ for which the SCF cycles stop converging [62], to the empirical rule of stopping the computations when the crystallographic statistical agreement Rw (F) approaches the one of a corresponding multipole model refinement [63]. Later, it was proposed a termination criterion based on the incremental ratio of χ 2 with respect to λ and on the variation of the X-ray constrained energy compared to the unconstrained value [64]. 2

𝜕χ More recently, a rule based on the second derivative 𝜕λ 2 was also introduced, with the suggestion of halting the fittings when a clear inflection point is observed in the curve

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representing the trend of the statistical agreement as a function of λ [65]. Another strategy to deal with such situations is to use extrapolation techniques. In some cases, a good extrapolation might be obtained until χ 2 = 1, but the extrapolation approach has not been investigated in detail yet. Finally, another possibility could consist in 2 using the χfree technique, where one randomly reserves a certain portion of the data (appropriately chosen to be evenly distributed in diffraction angle) and then fits to the remaining ones, monitoring what happens to the agreement statistics of the reserved portion. Fitting is terminated when the agreement statistics of the unused portion increase. However, so far, in this case, it was found that there is no consistency in the value of λ at which an increase occurs. Moreover, to achieve a statistically consistent result, very many trials would be required, and this would become computationally prohibitive. Notwithstanding all these efforts, the choice of the correct λ value in the XCW calculations unfortunately remains an open problem that would probably deserve more and more detailed and dedicated investigations in the near future.

11.2.9 Equations for the X-ray constrained Hartree–Fock wavefunction method Over the years, the XCW approach has been developed also in forms different from the original one, even considering the unrestricted formalism or multideterminant wavefunctions [64–69]. However, in this chapter, for the sake of simplicity, we will only consider the basic one which assumes to work with a single Slater determinant (Hartree–Fock) wavefunction and which consists in finding those molecular orbitals (and consequently the MO coefficients) that minimize the functional described by equation (11.20). It is possible to show that this is equivalent to solving the following modified Hartree–Fock equations: F̂ exp ϕi = ϵi ϕi ,

(11.21)

F̂ exp is a modified Fock operator that can be expressed like this: ̂ + ∑ Λh Im{F pred }Ih,C ̂ ] = F̂ + λv,̂ F̂ exp = F̂ + λ[∑ Λh Re{Fhpred }Ih,R h h

h

(11.22)

where F̂ is the usual Fock operator of quantum chemistry seen in equation (11.16), ̂ and Ih,C ̂ are the real and imaginary parts, respectively, of the structure factor operIh,R ator Iĥ , which is defined as follows: Nm

Iĥ = ∑ e j=1

i2π(Sj r+tj )⋅(B h)

̂ + iIh,C ̂ , = Ih,R

(11.23)

284 | A. Genoni and D. Jayatilaka and Λh is an h-dependent multiplicative factor given by pred exp 2η ηFh − Fh Λh = . Nr − Np (σ exp )2 F pred h h

(11.24)

For the sake of completeness, in the previous equations, B always represents the reciprocal lattice matrix (namely the matrix whose columns are the reciprocal lattice N vectors), while {Sj , tj }j=1m are the Nm symmetry operations of the unit-cell for the crystal

under exam (with Sj as a rotation matrix and tj as a translation vector).

For a closed-shell system, if we introduce the expansion given by equation (11.14) for the spatial orbitals into equation (11.21), we obtain the following matrix equation: F exp C = (F + λv) = S C E

(11.25)

which are also analogous to the Roothaan–Hall equations (see equation (11.17)) and are iteratively solved exactly in the same way. It is worth noting that the matrix v is essentially the matrix of the Kohn–Sham effective potential in DFT. Nevertheless, knowing this potential for one particular molecule does not give the unknown and universal Hohenberg–Kohn energy functional of density functional theory. Before concluding this introduction on the XCW approach, it is worth pointing out that, obtaining a converged X-ray constrained wavefunction, we also directly have the opportunity to extract properties of molecules in the crystal phase (see Subsection 11.3.3), which can be afterwards compared to the corresponding properties in the gas phase. Recent research works support this aspect. For example, X-ray constrained wavefunctions have been successfully exploited to determine molecular polarizabilities, hyperpolarizabilities and refractive indices in crystals [62, 63, 70, 71]. Furthermore, recent investigations have also shown that the XCW technique is able to extract both electron correlation and crystal field effects from X-ray diffraction data [72– 74]. This aspect might have repercussions not only in research domains as theoretical chemistry or quantum crystallography, but also in scientific fields as supramolecular chemistry and crystal engineering.

11.3 Starting to perform XCW calculations with Tonto In this section, we will describe how to put into practice the XCW method that we have theoretically introduced above. To accomplish this task, we will show how to perform some very simple Hartree–Fock and X-ray constrained Hartree–Fock calculations using the quantum crystallographic software Tonto [8], which can be down-

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loaded free of charge from a dedicated webpage on github (https://github.com/dylanjayatilaka/tonto). Through a series of basic examples, we will particularly describe the main keywords used in XCW computations and we will discuss some of the problems/needs that usually arise in XCW analyses. In the first example (Subsection 11.3.1), we will show how to run a basic Hartree– Fock calculation. This example will be instrumental to start practicing with the syntax and the structure of the Tonto input and output files. Afterwards, we will explain how to perform a simple X-ray constrained Hartree–Fock computation (Subsection 11.3.2) and, in particular, we will describe in details the keywords and the data-blocks in the input file that are crucial for this particular type of calculations. In Subsection 11.3.3, we will illustrate how to restart an XCW computation from a previous step and how to extract useful information (e. g., scatter plots) and compute/visualize properties (e. g., deformation and residual densities, electron localization functions [75] (ELFs), electron localizability indicators [76] (ELI-Ds), and other topological quantities) associated with an obtained X-ray constrained wavefunction. Finally, in Subsection 11.3.4, we will describe another interesting functionality for the X-ray constrained wavefunction computations, namely the possibility of performing calculations with surrounding cluster charges and dipoles, which could help in speeding up the convergence toward the desired statistical agreement.

11.3.1 Unconstrained Hartree–Fock calculations on the ammonia molecule As just mentioned above, the goal of the example described in this subsection is to start making the reader confident with the structure of the Tonto input and output files and with its basic keywords. For this reason, we will discuss the input file for a simple unconstrained restricted Hartree–Fock calculation (cc-pVTZ basis-set) on the ammonia molecule. First of all, let us consider some preliminary remarks. It is worth pointing out that the name of the Tonto input files must always be stdin. As it can be seen in Figure 11.5, curly brackets must enclose the content of the whole input file, while any exclamation mark (!) indicates a comment. So, every time an instruction is preceded by an exclamation mark, that instruction becomes inactive. From Figure 11.5, it is also clear that the input file is mainly organized in specific blocks, which are in turn enclosed by curly brackets and which will be analyzed in details below. After opening the left curly bracket for the input file, we need to assign the value to the variable name. This is the title of the job and defines the temporary file names. For example, in this case, we could set “name = NH3_RHF”. Always remember to leave at least one blank space after “=” and before/after any curly bracket. This is required for each input variable and for each type of input file in Tonto. Note that there are many such variables which are needed to enter data into the program, but they always end

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Figure 11.5: Tonto input file for the RHF/cc-pVTZ calculation on the ammonia molecule.

with the “=” sign. A list of all of the variables and keywords available in Tonto may be obtained by simply typing ”?” in the input file; even though there is no documentation, the printed variables and keywords are normally easy to understand. Then we have the output_style_options block (which is used to define the precision of the output). Inside this block, we only need to set the variable real_precision (for example, real_precision = 4). This is followed by the assignment of values to a

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series of typical quantum chemistry variables: “basis_name = cc-pVTZ” (to define the basis-set that we want to use), “charge = 0” (to specify the global charge of the system) and “multiplicity = 1” (to indicate the multiplicity of the system). Afterwards, we have the atoms block, which is used to introduce the information about the atoms that constitute the molecule under investigation. Inside this block, it is necessary to specify the keys and the data blocks, which will be described below. The keys block defines the order of the list of atom data. For example, we can set: keys = { label = { units = angstrom } pos = } This means that, for each atom in the data block (see below), we need to provide, in this order, (i) the label of the atom and (ii) the three atomic coordinates of the atom in Ångstrom (because the “embedded keyword” { units = angstrom } was given before the keyword pos). On the contrary, if we set keys = { { units = angstrom } pos = label= } we would have to give the atomic coordinates (in Ångstrom) followed by the atomic label. The data block defines the data for the atoms constituting the molecule. These data are given according to the order specified in the keys block. In Figure 11.5, after the end of the atoms block, we have the beginning of the block scfdata, which contains all the options to run the SCF calculations. Several variables are associated with this block. Below, we will describe some of them, particularly those that are relevant to perform the restricted Hartree–Fock calculation on the ammonia molecule. In this particular case, we can set: (i) “initial_density = promolecule,” which means that a promolecular density is used to start the SCF procedure (another possibility is “initial_density = restricted,” which means that the calculation restarts from the binary file of a restricted density matrix resulting from a previous calculation, obviously using the same basis-set); (ii) “kind = rhf,” which indicates the type of SCF calculation that we want to perform, in this case restricted Hartree–Fock; (iii) “convergence = 0.00001,” which specifies the convergence threshold on the energy difference; (iv) “diis = { convergence_tolerance = 0.00001 },” which is the diis block (to speed up the convergence through the extrapolation technique direct inversion of the iterative subspace [77, 78] (DIIS)) and contains different keywords (in this case we only specify the keyword convergence_tolerance); (v) “output = YES,” to indicate that we want the information about the SCF cycles printed in the output file; (vi) “output_results = YES,” to indicate that the final information after convergence is also printed in the output file. It is worth noting that, after the block scfdata, we have the simple keyword scf. This keyword is fundamental because it makes Tonto execute the SCF procedure with the settings specified in the scfdata block. In general, commands that do not have the

288 | A. Genoni and D. Jayatilaka “=” sign attached to them make the program execute a task, whereas those with an “=” sign are used to enter data into the program. The last keyword of the input file (before the terminal right curly bracket that closes the input) is delete_scf_archives that allows the user to delete all the files produced by the SCF calculation. Since these files could be useful for following calculations or analyses (see below), this keyword is usually commented by preceding it with an exclamation mark (see again Figure 11.5). Now, it is possible to run the RHF calculation with the parameters specified above in the input file. In particular, to launch a Tonto calculation, it is simply enough to use the following instruction at the terminal line: /auto/tms3/ale/TONTO_NEW/BREMEN_XCW-LECTURE/tonto & Of course, it is necessary to update the path to indicate where the Tonto executable is. If now we open the output file (whose name is always stdout for any Tonto calculation), it is possible to notice that we have the following error message: Error in VEC{BASIS}: read_library_data ... no library basis set This is due to the fact that we did not specify the path for the basis-set. To overcome this problem, we obviously need to reopen the input file stdin and, before the basis_name keyword, give the path where to read the basis-set file by setting the value of the keyword basis_directory. For example, we could set “basis_directory = /auto/tms3/ale/TONTO_NEW/basis_sets” This means that Tonto will look for the file “cc-pVTZ” (which obviously contains the specification of basis-set cc-pVTZ) in the directory “/auto/tms3/ale/TONTO_NEW/basis_sets”. Note that the variable basis_name must correspond exactly to the name of the file in the basis-sets directory. Of course, also in this case, the path must be updated. This time, running again the Tonto executable, the calculation converges, as it can be seen by opening the stdout file and searching for the string “SCF iterations” and “SCF results” (see Figure 11.6). Furthermore, after closing the output file, if we use the instruction “ls –lh” at the terminal line to list the files in the working folder, we have the results reported in Figure 11.7. Other than the input and output files (stdin and stdout, respectively), we have a file (stderr) that contains the information about the nonsuccessful termination of the calculation (in this case it still contains the information about the previous calculation that did not start because the basis-set path was not originally specified). Then we have: – NH3_RHF.density_matrix,restricted, which is the binary file associated with the converged density matrix and can be used to restart the calculations;

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Figure 11.6: Details of the SCF cycle and results for the RHF/cc-pVTZ calculation on the ammonia molecule.

Figure 11.7: Files associated with the RHF/cc-pVTZ calculation on the ammonia molecule.

–

–

NH3_RHF.molecular_orbitals,restricted, which is the binary file associated with the coefficients and of the molecular orbitals and can be also exploited to restart the calculations; NH3_RHF.orbital_energies,restricted, which is the binary file of the orbital energies.

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11.3.2 X-ray constrained Hartree–Fock calculations using X-ray diffraction data for the ammonia crystal The goal of this example is to make the reader confident with the Tonto input file for X-ray constrained Hartree–Fock calculations. In particular, we will describe computations that have been performed by exploiting experimental X-ray diffraction data collected for the ammonia crystal. For the sake of brevity, in this section, we will explain in details only the keywords and the blocks that were not already taken into account in the previous example. In this case, in the stdin file, we set the values of the following variables (already described in Subsection 11.3.1): – name = NH3_XCRHF – inside the output_style_options block, we set real precision= 4 – basis_directory = /auto/tms3/ale/TONTO_NEW/basis_sets – basis_name = cc-pVTZ – charge = 0 – multiplicity = 1 Immediately after the keyword multiplicity, we have the crystal block, which allows to introduce the crystallographic information (see Figure 11.8). In particular, in this block we have: (i) the spacegroup keyword, which is used to specify the space group of the crystal under exam (for the calculation considered in this subsection, we set “spacegroup = { hermann_maugin_symbol = “P 21 3 ” }”); (ii) the block unit_cell used to define unit-cell angles and dimensions through the keywords angles and dimensions with their corresponding units (see Figure 11.8 for the values pertaining to the example considered in this subsection); (iii) and finally the xray_data block, which is used to give the information about the treatment of the X-ray diffraction data that will be used as constraints in the calculation. The possible keywords for the xray_data block are: (i) wavelength, which is used to specify the wavelength of the experiment and which is useful only when we need to introduce secondary extinction corrections (see below). Note that, in our case, wavelength is set equal to 0.55960, which is the shorthand for 0.55960 a. u. In fact, all quantities are generally given in atomic units unless postfixed by a unit modifier, such as “0.55960 ansgtrom”; (ii) thermal_smearing_model to indicate the method for the treatment of thermal smearing applied to basis functions that are not centered on the same nucleus (in our example, it is set equal to hirshfeld, but other options are coppens, stewart and tanaka); (iii) partition_model to specify how the molecular fragment in the crystal has to be partitioned into atoms for the correct calculation of structure factors (in our case, it is set equal to Gaussian, but another option is Mulliken). This keyword is particularly important when, in the X-ray constrained wavefunction calculations, we choose a molecular fragment that is larger than the crystal asymmetric unit to properly take into account the intermolecular interactions (e. g.,

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Figure 11.8: Crystal block in the input file for the XC-RHF/cc-pVTZ calculation that exploits X-ray diffraction data for the ammonia crystal.

hydrogen bonds) that nonnegligibly perturb the electron density distributions of the asymmetric unit; (iv) optimize_extinction to indicate if the treatment of secondary extinctions must be included (TRUE, as in the examined case) or not (FALSE). If included, the Larsen model is exploited and the wavelength of the experiment is used in the calculations (see wavelength keyword above); (v) optimize_scale_factor for the optimization of the scale factor that multiplies the computed structure factor magnitudes (see η in equation (11.19)). Options are TRUE (as for the case under exam) or FALSE; (vi) REDIRECT, which is used to specify the file where Tonto has to look for the structure factor magnitudes that are used as constraints in the calculation. In the example reported in this chapter, the keyword REDIRECT is followed by “nh3.hkl.” This means that the experimental structure factor magnitudes are contained in the file nh3.hkl located in the folder where the calculations are launched. Obviously, a different path can be also specified. For example, we could have “REDIRECT /auto/tms3/ale/TONTO_NEW/nh3.hkl “ The structure of the file nh3.hkl is shown in Figure 11.9.

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Figure 11.9: Structure of the nh3.hkl file that contains the experimental X-ray diffraction data used as constraints in the XC-RHF/cc-pVTZ calculation on the ammonia crystal. For the sake of simplicity, only a subset of experimental data is shown in this figure.

It basically consists in the reflection_data block that is used to specify the X-ray diffraction data used as constraints. It is defined by the keys and data blocks, with the former that defines the order with which the information is introduced in the latter. In this particular case (see always Figure 11.9), we have keys = { h = k = l = f_exp = f_sigma = } This means that the experimental X-ray diffraction data used as constraints in the XCW calculation are provided in the data block as a repetition of Miller indices (h, k and l) followed by the value of the corresponding structure factor magnitude (f_exp) and its experimental error (f_sigma). The file nh3.hkl ends with the keyword REVERT that redirects to the stdin input file. Those familiar with crystal structures will recognize that this format is very similar to the one of the crystallographic information file (CIF). Tonto also offers the possibility to read intensities (and the relative experimental errors) instead of structure factor magnitudes. In that case, the keys block would assume this form: keys = { h = k = l = i_exp = i_sigma = } It is worth noting that the reflections provided in input (in the form of structure factor magnitudes or intensities) must be those that have been preliminarily pruned and merged. In other words, they must be those that are usually deposited with crystallographic structures in .cif and/or .fco/.fcf files. Going back to the input file, below the xray_data block, the atoms block starts (see Figure 11.10). As seen in the example reported in Subsection 11.3.1, the atoms

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Figure 11.10: Atoms block in the input file for the XC-RHF/cc-pVTZ calculation exploiting X-ray diffraction data for the ammonia crystal.

block is fundamental to give the information on the atoms that constitute the molecule under investigation. Also here it is necessary to introduce the keys and the data blocks. However, in this case, the keys block is given by this expression: keys= { label= { axis_system= crystal } pos= { units= angstrom^2 } thermal_tensor= } which means that for each atom of the molecule, we should provide (in this order): (i) its label, (ii) the three atomic coordinates (in fractional coordinates because we have the embedded keyword { axis_system= crystal } before pos=); (iii) the six ADPs (in Å2 because we have the embedded keyword { units= angstrom^2 } before thermal_tensor=). The ADPs must be given in this order: U11 U22 U33 U12 U13 U23 . Afterwards, the data block defines the data for the atoms constituting the molecule. The values in the data block should be provided according to the definition of the keys block just described above. Also for the XCW calculations, after the atoms block we have the scfdata one. As it can be seen in Figure 11.11, in this situation we have much more keywords compared to the case of the unconstrained Hartree–Fock calculation considered in subsection 11.3.1. Below we will discuss in detail only the keywords that were not examined before and those that are generally relevant for the X-ray constrained wavefunction fitting computations. First of all, it is worth noting that the keyword kind is set equal to “xray_rhf” to indicate that we want to perform an X-ray constrained restricted Hartree–Fock calculation. Then we can also see that, other than the simple keyword convergence already discussed in the previous subsection, in this case we have much more keywords to control and/or speed up the convergence of the calculations: (i) the diis block with the associated keywords save_iteration, start_iteration, keep, convergence_tolerance to tune the DIIS extrapolation technique [77, 78]; (ii) the keywords use_damping, damp_factor, damp_finish for the damping technique that helps in avoiding/reducing the instability in the resolution of the (modified) Hartree–Fock equations; (iii) the keywords use_level_shift and level_shift_finish associated with the level-shift

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Figure 11.11: SCFdata block in the input file for the XC-RHF/cc-pVTZ calculation that uses X-ray diffraction data for the ammonia crystal.

approach that also allows to improve the convergence of the Hartree–Fock-based calculations. The values of the “convergence-keywords” appearing in Figure 11.11 are the default ones. Afterwards, we have three other keywords that are fundamental for the XCW computations: (i) initial_lambda, to give the initial value of the external parameter λ (see equation (11.20) and Subsection 11.2.8 for more details on this quantity), (ii) lambda_step, which is the value of the increment of the λ parameter at each step and (iii) lambda_max, which is the final value of the λ parameter. Therefore, using the values for initial_lambda, lambda_step and lambda_max reported in Figure 11.11, Tonto will solve equation (11.21) for all the selected values of λ: from λ = 0 (calculation corresponding to the unconstrained one seen in Subsection 11.3.1) to λ = 0.02 with a step of 0.001 (overall, twenty-one SCF calculations).

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Always in Figure 11.11, it is possible to observe that, also in this case, after the scfdata block, we must use the keyword scf that makes Tonto execute the SCF procedure with the SCF options chosen above. Moreover, the keyword delete_scf_archives is commented again (!delete_scf_archives) to leave all the files produced by the SCF procedure (see below) in the folder where our X-ray constrained restricted Hartree– Fock calculation is launched. To perform the computation with the options specified in the previous paragraphs, we use again the following instruction at the terminal line: /auto/tms3/ale/TONTO_NEW/BREMEN_XCW-LECTURE/tonto & where the path must be obviously updated to indicate where the Tonto executable is. If now we open the output file stdout, and we look for the string “SCF has converged,” we can indeed see that there is an SCF cycle for each value of the external parameter λ and that all of them converged. For the sake if simplicity, in Figure 11.12, we have reported only the part of the Tonto output concerning the last SCF cycle, (namely the one for λ = 0.02) and the final SCF results. Note that, in this case, the “total energy” is −56.2015 a. u., while, for the corresponding unconstrained calculation (see Figure 11.6), the “total energy” was −56.2182 a. u. This confirms what mentioned at

Figure 11.12: SCF cycle and results when λ = 0.02 for the XC-RHF/cc-pVTZ calculation that exploits X-ray diffraction data for the ammonia crystal.

296 | A. Genoni and D. Jayatilaka the end of Subsection 11.2.7 and seen in Figure 11.4, namely that, for any value of λ, the energy is higher than the energy obtained for λ equal to zero. If we use again the instruction “ls –lh” after closing the output file, this time we will obtain many files in the working folder, namely – for each value of λ, (i) a file for the molecular orbitals (e. g., for λ = 0.001, NH3_XCRHF.molecular_orbitals,lambda=0.001000,restricted) and (ii) a file for the orbital energies (e. g., for λ = 0.001, NH3_XCRHF.orbital_energies,lambda= 0.001000,restricted); – the files NH3_XCRHF.density_matrix,restricted, NH3_XCRHF.molecular_orbitals, restricted and NH3_XCRHF.orbital_energies, restricted, which correspond to the maximum value of λ (in our case, 0.0200); – the file NH3_XCRHF.residual_density_map,cell.cube, which is the cube file to plot the residual density and which always refers to the maximum value of λ. – the stdout.fit_analysis file for the comparison between the computed and experimental structure factor magnitudes (e. g., the scatter plot (Fhexp − Fhmodel )/σhexp Vs. Fhexp ). These plots can be visualized using software programs such as Excel, gnuplot or QtGrace (see below).

To quickly extract the value of χ 2 (statistical agreement between calculated and experimental structure factor magnitudes (see equation (11.19)) for each value of λ, we can simply use the following instruction at the terminal line: grep ’chi^2(N_p) ...’ stdout > chi2.txt The χ 2 values are stored in the file chi2.txt and by opening it (see left panel in Figure 11.13), we can see that, excluding the first value (which in this case corresponds to the promolecular electron density used as starting guess for the calculations), χ 2 decreases monotonically as a function of λ. This can be also graphically observed in the right panel of Figure 11.13, where the χ 2 values have been plotted as a function of λ. As seen in Subsection 11.2.8, determining the value of λ at which halting the X-ray constrained wavefunction calculations is still an open problem. However, it is quite well accepted that we should ideally stop the X-ray constrained wave unction calculations when χ 2 is equal to 1 or starts being lower than 1. Analyzing the stdout file and chi2.txt, in our case we should stop at λ = 0.008 (χ 2 = 0.9821). In the next subsection, we will see how to extract results and compute properties only for the X-ray constrained wavefunction obtained at λ = 0.008.

11.3.3 Restarting an X-ray constrained wavefunction calculation and computation of molecular properties In this subsection, we will show how to restart an XCW calculation from a well-defined λ-step and how to extract results associated with a specific X-ray constrained wave-

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Figure 11.13: Left panel: χ 2 values corresponding to different λ values for the XC-RHF/cc-pVTZ calculation exploiting X-ray diffraction data for the ammonia crystal; right panel: plot of the χ 2 values as a function of λ (the red horizontal dotted line represents the desired statistical agreement χ02 = 1).

function. In particular, starting from the example discussed in the previous subsection, we will explain how to perform the XC-RHF computation on the ammonia crystal only at λ = 0.008, for which the χ 2 value is close to 1, and how to obtain properties for the resulting X-ray constrained wavefunction. First of all, to accomplish this task, we have to rename the file NH3_XCRHF.molecular_orbitals,lambda=0.008000,restricted as NH3_XCRHF.molecular_orbitals,restricted. This was one of the files produced by the XCW calculation analyzed in the previous subsection and contains the coefficients of the molecular orbitals corresponding to the X-ray constrained Hartree–Fock wavefunction obtained for λ = 0.008. By renaming the file as indicated above, we simply want to use the molecular orbitals at λ = 0.008 as guess orbitals (see below). As a second step, we obviously need to slightly modify the input file of the example discussed in Subsection 11.3.2. All the modifications have to be done in the scfdata block (see again Figure 11.11) as indicated below: (i) comment (with an exclamation mark) of the instruction related to the initial_density keyword !initial_density = promolecule and addition of the following one: initial_mos = restricted This means that the orbitals in file NH3_XCRHF.molecular_orbitals, restricted will be used to restart the calculation (remember that these are the XC-RHF orbitals obtained at λ = 0.008); (ii) modification of the values for the keywords initial_lambda and lambda_max in this way:

298 | A. Genoni and D. Jayatilaka initial_lambda = 0.0080 lambda_max = 0.0080 After rerunning the XCW computation according to the new modified stdin input file, we obtain an output file stdout, where, unlike the example in Subsection 11.3.2, we have only the calculation at λ = 0.008 and the SCF cycle converges in few iterations. This is obviously due to the fact that, for this new calculation, we used guess orbitals that are already solutions of the X-ray constrained Hartree–Fock equations for λ = 0.008. The final χ 2 value (0.9820, see Figure 11.14) is basically identical to the one obtained in the previous example (0.9821) when λ was equal to 0.008.

Figure 11.14: SCF cycle and results obtained by restarting the XC-RHF/cc-pVTZ calculations on the ammonia crystal for λ = 0.008.

Now, we will see how it is possible to extract some results and compute properties associated with the obtained X-ray constrained wavefunction, namely properties for molecules in crystals. First of all, we can exploit the information contained in the file stdout.fit_analysis, which, as already mentioned above, contains information about the comparison between the calculated and the experimental structure factor magni-

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Figure 11.15: (a) Plots that can be obtained from an XCW calculation through Tonto (this is the heading of file stdout.fit_analysis produced by Tonto after convergence); (b) values for the scatter plot exp pred exp (Fh − Fh )/σh vs Fh corresponding to the XC-RHF/cc-pVTZ calculation on the ammonia crystal for λ = 0.008; for the sake of simplicity, only a part of the values is explicitly show.

tudes (see Figure 11.15a for all the possibilities). We can copy the file stdout.fit_analysis to the auxiliary file auxilary.txt (arbitrary name) and, after opening the latter, we can look for the string “Scatter plot of F_z = (Fexp-Fpred)/F_sigma vs Fexp” (see Figure. 11.15b) and keep only the numerical values related to this kind of scatter plot (we delete also the string). We can afterwards use software programs such as gnuplot, QtGrace or Excel to visualize the scatter plot given by the data contained in auxiliary.txt (see Figure 11.16a). Note that it is possible to proceed in analogous way for the other plots available in the file stdout.fit_analysis (see again Figure 11.15a). Another possibility consists in visualizing the residual density resulting from the fitting of the X-ray diffraction data. This can be done by exploiting the content of the cubefile NH3_XCRHF.residual_density_map,cell.cube through visualization software programs as VMD (see Figure 11.16b). Other cube files corresponding to the obtained wavefunction can be obtained in Tonto, for example, those associated with the deformation density, the electron localization function (ELF) or the electron localizability indicator (ELI-D; see also Chapter 4 of this book). To obtain them, it is necessary to add the block plot_grid after the scf keyword that executes the SCF calculation, as shown in the example reported in Figure 11.17a. Many keywords are associated with this block and, for a comprehensive overview, we suggest the reader to consider all the available testexamples downloadable along with the Tonto software from the Tonto webpage on github. Here, we limit to comment the keywords appearing in Figure 11.17a, which

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exp

Figure 11.16: (a) Scatter plot (Fh

pred

− Fh

exp

)/σh vs Fh

corresponding to the XC-RHF/cc-pVTZ calcula-

tion on the ammonia crystal (λ = 0.008); (b) residual density (in the unit-cell) associated with the XC-RHF/cc-pVTZ calculation on the ammonia crystal for λ = 0.008 (blue isosurface = 0.0125 eÅ−3 , red isosurface = −0.0125 eÅ−3 ).

Figure 11.17: (a) Plot_grid blocks for the generation of ELF and ELI-D cubefiles; (b) ELF (η = 0.9) associated with the XC-RHF/cc-pVTZ calculation on the ammonia crystal (λ = 0.008); (c) ELI-D (isovalue of 1.75) corresponding to the XC-RHF/cc-pVTZ calculation on the ammonia crystal (λ = 0.008).

shows how it is possible to obtain cubefiles to plot the ELF and ELI-D corresponding to the XC-RHF/cc-pVTZ wavefunction for the ammonia crystal when λ is set equal to 0.008 (see Figures 11.17b and 11.17c). As you can see, inside the plot_grid block we have the keywords: (i) kind, which indicates the type of grid that we want (in our cases, kind = ELF or ELI_D, but other possibilities are, among others, electric_potential (for the plot of the molecular electrostatic potential), deformation_density or simply electron_density); (ii) n_points, which gives the total number of points of the grid; (iii) use_bbox_with_shape_axes, which defines a bounding box along the molecular inertial axes, with width twice the projection of the molecule onto those axes; (iii) box_scale_factor, which scales the size of the box by a given amount (a very “tight” box might be scaled by 0.7 or 70 % of the original bounding box size and

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(iv) plot_format, which specifies the format of the cube file (in this case the format of the popular quantum chemistry package Gaussian09). It is important to observe (see again Figure 11.17a) that, after the block plot_grid, we need to insert the keyword plot to make Tonto really produce the desired cube file. For the sake of completeness, it is worth noting that, although the instructions considered above and reported in Figure 11.17a enables to obtain three-dimensional grids, by using proper keywords Tonto also allows the generation of two-dimensional grids for the plots of properties (e. g., residual and deformation densities, ELFs) in a particular plane of the molecule under exam. Interested readers should refer to the available test-examples mentioned above. We also want to point out that Tonto also offers the possibility of obtaining wavefunction files (namely, .wfn files) for subsequent topological analyses with programs such as AIMAll (for traditional Quantum Theory of Atoms in Molecules (QTAIM) analyses [79], see Chapter 3 of this book) or NCIplot (for Non Covalent Interaction (NCI) analyses [80, 81], see Chapter 14 of this book). Again, this option allows to analyze molecular electron densities influenced by the crystal environment. To do that, it is simply enough to add the keyword write_aim2000_wfn_file, always after the keyword scf for the execution of the SCF cycle. In this way, Tonto provides a file job.wfn, where job is the title of the calculation given in input through the keyword name. Finally, to conclude this section, in future versions of the Tonto software, the keyword write_nbo_file will be available to produce an archive file (FILE47 of the NBO7 program, see Chapter 6 of this book), which will allow the Tonto users to perform nearly all analyses of the NBO methods [82] and, in particular, to examine how properties such as atom hybridizations, bond polarizations and delocalizing interactions corresponding to an X-ray constrained wavefunction differ from those of a gas-phase wavefunction.

11.3.4 Cluster charges and dipoles in X-ray constrained wavefunction calculations In this final subsection, we will show another Tonto functionality that can be sometimes used in the XCW analysis: the use of surrounding point charges and dipoles that mimic the crystal environment of the molecule considered in the X-ray constrained wavefunction calculation. The point charges and dipoles are computed from the guess electron density/molecular orbitals (when λ = 0) or from the X-ray constrained wavefunction obtained at the previous step (when λ ≠ 0) and they are placed at symmetrygenerated positions around the molecular unit (not necessarily corresponding to the asymmetric unit of the crystal) for which the wavefunction is calculated. This is usually done within a radius of 8 Å, which means that if any atom of any molecule in the original cluster lies within 8 Å of the center of mass of the central molecular unit, then the charges for the whole molecule (including atoms which might be beyond 8 Å) are

302 | A. Genoni and D. Jayatilaka included. This functionality sometimes enables to speed up the convergence towards the desired statistical agreement (χ02 = 1), although it may lead to some convergence problems, especially with basis-sets comprising diffuse functions. To do that, first of all, it is necessary to add an additional scfdata block that is used to compute (at unconstrained Hartree–Fock level) the charges and the dipoles that need to be placed on the surrounding molecules for the first X-ray constrained wavefunction SCF cycle (see Figure 11.18).

Figure 11.18: Structure the scfdata blocks required in the input files of XC-RHF calculations that use surrounding clusters of charges and dipoles to speed up the convergence towards the desired χ02 statistical agreement.

Then we have to specify the following three keywords inside the second scfdata block (the one for the real X-ray constrained wavefunction calculations, see always Figure 11.18): (i) initial_density, which must be set equal to restricted in order to indicate that the X-ray constrained RHF calculation restarts from the density matrix obtained from the unconstrained Hartree–Fock computation that was performed to determine the clusters of charges and dipoles (note that initial_mos can be used instead of initial_density, if you want to read the unconstrained Hartree–Fock molecular orbitals); (ii) use_SC_cluster_charges, which must be set equal to TRUE to indicate the use of surrounding point charges and dipoles calculated self-consistently

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from the molecules in the central fragment; (iii) cluster_radius, which defines the radius within which the clusters of charges and dipoles are used (we typically have cluster_radius = 8 angstrom). If we use these options for the XCW computation on the ammonia crystal discussed in subsection 11.3.2, it is possible to observe that, for the first λ steps, χ 2 decreases more rapidly when surrounding point charges and dipoles are used, although, in this specific case, convergence (χ 2 ≤ 1) is eventually reached always at λ = 0.008 (see Table 11.1). Table 11.1: Values of χ 2 at different values of λ when surrounding clusters of charges and dipoles are used or not used in the XC-RHF/cc-pVTZ calculation on the ammonia crystal. λ 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

χ 2 without charges & dipoles

χ 2 with charges & dipoles

9.1561 4.5205 2.9148 2.1403 1.6976 1.4162 1.2241 1.0858 0.9821

7.3359 3.8576 2.6097 1.9859 1.6178 1.3775 1.2094 1.0859 0.9915

It is also possible to observe that the discrepancies (in terms of χ 2 ) between the two types of XCW calculations (with and without point charges and dipoles) reduce as we consider larger values of λ. This is due to the fact that, as λ increases, we increasingly include the information contained in the experimental structure factors and, consequently, also the crystal field effects introduced by explicitly considering surrounding clusters of charges and dipoles.

11.3.5 Concluding remarks In the previous subsections, we have illustrated some of the basic functionalities available in Tonto to perform X-ray constrained wavefunction calculations. Of course, much more options and functionalities are available, but, due to the limited space of this chapter, here we could only cover a small part of them. For this reason, we invite again the readers particularly interested in carrying out XCW analyses to visit the above mentioned github webpage (https://github.com/dylan-jayatilaka/tonto), where additional examples and information on how to perform more advanced calculations can be obtained by downloading the full package of the Tonto software.

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Dana Nachtigallová and Pavel Hobza

12 Introduction to noncovalent interactions 12.1 Introduction The beginning of the word of noncovalent interactions can be related to the work of a Dutch theoretical physicist Johannes Diderik van der Waals who in his thesis [1] modified the equation of a hypothetical gas phase PV = nRT,

(12.1)

where V, T and P relate to molar volume, absolute temperature and pressure, n is the number of mole of gas and R is ideal gas constant. Equation 12.1 describes ideal gas in which the molecules are considered as spaceless point articles that do not interact with each other. In his modification, van der Waals generalized the ideal gas law by introducing parameters a and b, which represent corrections to molecular attraction and their nonzero volume, resulting in the van der Waals equation Pa

n2 (V − nb) = nRT. V2

(12.2)

For this work, J. D. van der Waals received the Nobel prize in 1910. Note that for rare gases the a and b parameters are close to zero (e. g., for He a = 0.03 and b = 0.02) but they significantly deviate from zero for polyatomic gases (for C2 H4 a = 4.47 and b = 0.06). The experimental work of Heike Kammerling-Onnes [2], a Dutch physicist, represents the second milestone in the history of the noncovalent interactions. In 1908, he reported on liquefaction of helium proving an existence of attraction forces even between the smallest closed shell atoms. In 1913, he received the Nobel Prize for “his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium.” The origin of these forces was, however, not known for more than two decades until Fritz Wolfgang London, a German physicist, described their origin and named them “dispersion forces” [3]. This discovery can be considered as the third milestone of the world of the noncovalent interactions. This work can be considered as a pioneering work since it represents the first big success of recently invented quantum mechanics. The noncovalent interactions extend the whole science. Just to mention one interesting example, the ability of a gecko to climb smooth vertical surfaces (even flat Dana Nachtigallová, Pavel Hobza, Institute of Organic Chemistry and Biochemistry of the Czech Academy of Sciences, Flemingovo nám. 2, 166 10 Prague, Czech Republic, e-mails: [email protected], [email protected] https://doi.org/10.1515/9783110660074-012

310 | D. Nachtigallová and P. Hobza glass) is enabled by the existence of the noncovalent interactions between keratinous hairs on its feet and the surface [4]. Other examples are less picturesque but more important for nature: – The very existence of a condensed phase is probably the most important manifestation of the noncovalent interactions. Majority of chemical and biological processes occur in solution which significantly affects their course. – The most important solvent—water, the environment where the life started—plays a unique role. Its properties (e. g., boiling and freezing points) are dramatically different from those of similar hydrides due to the existence of the strong and directed noncovalent hydrogen (H)-bonding. – The noncovalent interactions are responsible for the structure and function of bio-macromolecules like DNA or proteins. Double-helical structure of DNA exists due to the concert action of the in-plane noncovalent interstrand interactions between nucleic acid bases, forming base-pairs via the electrostatic and H-bonding, and the out-of-plane noncovalent intra and interstrand dispersion interactions, as well as the solvation effects of bases and sugar-phosphate backbone. Similarly, the structure of proteins is determined by the H-bonding and stacking interactions between amino acids. What is position of the covalent and noncovalent interactions in the nature? The oversimplified answer is that the covalent interactions are responsible for the primary structure of molecules (bond lengths and angles), while the fascinating 3D architecture of bio-macromolecules is due to the noncovalent interactions acting between their building blocks. The real answer is more complicated since the covalent and noncovalent interactions coexist. The structure and properties of an isolated molecule (in the gas phase) are determined mainly by the covalent interactions. When the molecule is placed in a solution, bulk or at the surface its properties are affected by the noncovalent interactions with surroundings molecules frequently playing a key role. This can be shown on the structure of DNA, the molecule, which stores and transfers the genetic information. For this purpose, the double-helical structure which only exists in the polar water environment, is of key importance. Upon reducing polarity of a solvent, for example by adding an alcohol to water, the double-helical structure is deteriorated and collapses. The specific role of the noncovalent interactions in DNA was recognized by James Watson who wrote “On the one hand, they should be strong enough to ensure the preferential binding but on the other hand they should be weak enough to allow disruption of bonding” [5]. In other words, the noncovalent interactions are reversible, while the covalent interactions are nonreversible. Apparently, life takes advantages of reversibility of the noncovalent interactions and their favourable energetics at corresponding temperatures. We could even state: “Not despite the weakness but because of the weakness the noncovalent interactions play a key role in our life.”

12 Introduction to noncovalent interactions | 311

Overlapping of two subsystems with incomplete electronic shells results in an increase of the electron density between them and formation of a covalent bond. Description of products of interaction between species with filled valence shells/molecules is much less unambiguous. To honor the van der Waals’s pioneer work on the noncovalent interactions, these products are frequently called “van der Waals (vdW) molecules” and the relevant forces/interactions are called “vdW forces/interactions.” Since vdW term is in molecular mechanics frequently used for the dispersion energy only, we recommend rather to use the alternative term “noncovalent molecules, forces/interactions.”

12.2 Origin of attractions 12.2.1 Simple molecular orbital picture Let us consider the interaction of two hydrogen and helium atoms, i. e., the simplest atoms with 1s orbital filled with one and two electrons, respectively. Upon the interaction 1s orbitals form bonding σg and antibonding σu∗ molecular orbitals (“g” and “u” refer for gerade and ungerade symmetry with respect to the center of the molecular symmetry), which are stabilized and destabilized, respectively, with respect to the atomic 1s orbitals. In H2 molecule the electrons occupy σg orbital stabilized by energy ΔE1 (Figure 12.1a) and form the covalent bond with energy roughly corresponding to the σg orbital stabilization energy. In the case of the He. . . He complex, both σg and σu∗ molecular orbitals, stabilized by ΔE1 and destabilized by ΔE2 , respectively (Figure 12.1b), are occupied. Resulting complex within this scheme is thus not stable in its ground state. Such picture is, however, oversimplified since it neglects an electron overlap and does not consider other electron configurations (excited configurations). Accounting for these effects leads to the stabilization of doubly occupied σg and σu∗ orbitals (Figure 12.1c). This stabilization energy, ΔE3 , is called the correlation energy and corresponds roughly to the energy of the He. . . He noncovalent bonding. Comparison of the energy of the covalent bond in H2 molecule (∼ 80 kcal/mol) and the noncovalent

Figure 12.1: Interaction of the 1s atomic orbitals located on two atoms; (a) two hydrogens, (b) and (c) two He atoms. In the case (c) the configuration interaction with an excited-state is considered.

312 | D. Nachtigallová and P. Hobza bond in He2 dimer (∼ 0.02 kcal/mol) shows a dramatic difference. It is the strength of the noncovalent interactions which makes them easy to form and also easy to decompose and, thus, gives them their crucial role in many processes.

12.2.2 Electric multipoles The example of helium stabilization raises the question whether the correlation energy is the only contribution to the stabilization also in other noncovalent complexes, e. g., in these formed by polar systems. The answer is not straightforward. The correlation energy, or more precisely, the intermolecular correlation energy contributes to the stabilization of every noncovalent complex. However, in the majority of noncovalent complexes the dominant stabilization originates from the interaction of permanent and induced electrostatic moments, due to a permanent or induced uneven distribution of electrons over the skeleton of atomic nuclei, called electrostatic (coulombic) or induced (polarization) interactions, respectively. The uneven charge distribution creates so called electric multipoles; dipoles, quadrupoles, octupoles, etc. The electrostatic interaction energy (E ES ) results from the interactions between the various multipoles of systems A and B: E ES =

qA qB qA μB μ μ Q Q + 2 + ⋅ ⋅ ⋅ + A3 B + ⋅ ⋅ ⋅ + A5 B + ⋅ ⋅ ⋅ , r r r r

(12.3)

where q, μ and Q are monopole (charge), dipole, and quadrupole moments, respectively. In the expansion (equation (12.3)) the interactions of the higher multipoles sharply decrease with an increasing inter-system distance. Depending on the orientation the electric multipole interactions are either attractive, repulsive, or become zero. The electric multipoles have the ability to induce electric field in its surroundings inducing an electric multipole in the other system and, thus, polarizing the other system. The interactions between the original and the induced dipole, called induced or polarization interactions, are always attractive. In analogy with the electrostatic interaction, the induction interaction energy follows the expression: q2 μ2 (3 cos2 ΘB + 1) 1 + ⋅ ⋅ ⋅] E I =− αA [ 4B + B 2 r r2

q2 μ2 (3 cos2 ΘA + 1) 1 − αB [ 4A + A + ⋅ ⋅ ⋅], 2 r r2

(12.4)

where α, q and μ are the polarizabilities, charge, and dipole moments of systems A and B; Θ is the angle between the axis of one system and the line connecting the gravity centers of the two systems. Both components of the interaction energy can be described in the terms of classical physics. To fulfill the list of electronic contributions to stabilization of the noncovalent systems, the above mentioned dispersion interactions E D for the induced dipole-induced

12 Introduction to noncovalent interactions | 313

dipole (and higher) should be added. Using the perturbation theory, Eisenschitz and London derived the formula for the dispersion energy [3] ED ≈ −

3 IA IB 0 0 −6 α α r = −C6 r −6 , 2 IA + IB A B

(12.5)

where α0 and I are the static dipole polarizability and ionization potential, respectively, which define the dispersion coefficient C6 . If higher multipoles are included, equation (12.5) reads as E D = −C6 r −6 − C8 r −8 − C10 r −10 − ⋅ ⋅ ⋅ ,

(12.6)

where C6 , C8 and C10 contain the dipole-dipole, dipole-quadrupole, and dipoleoctupole and quadrupole-quadrupole interactions. The origin of these interactions can be described by the oscillation of atomic nuclei and electrons which leads to the formation of time-variable multipoles even in nonpolar systems. This depends on the nuclei and electron position. The value averaged over time is zero. At the instant when the multipole has nonzero value, it induces multipole on the neighboring molecules which interacts with time-variable multipole. An alternative view was provided by Feynman [6] who showed that dispersion interactions between two spherical atoms induce a dipole moment oriented with negative ends toward each other and the dispersion energy can be recovered from classical electrostatic calculations [7]. For the interested reader we recommend [8] in which the C6 dispersion coefficients are evaluated based on the electrostatic Hellman–Feynman theorem. To prevent collapsing the molecules, repulsion forces act between the molecules not allowing them to approach each other too close. These forces, called exchangerepulsions (E ER ), start to act when the systems approach one another to a distance at which the overlap of electron clouds is not negligible. The exchange-repulsion as well as dispersion interactions are of a quantum origin. Identical interactions operate also in open-shell systems in the formation of covalent bonds. The repulsive forces operate in a short-range and are proportional to r −12 (or exp(−r)).

12.3 Computational modeling Unlike in the covalent bonds, the changes due to the noncovalent bonding are much more delicate and induce only small changes in electronic structure. Description of such subtle changes requires the use of advanced, highly sensitive experimental and computational methods. Difficulties of the computational approaches come from: (i) Different origin among the particular types of noncovalent interactions, which requires different levels of theory to reach a given accuracy. The H-bond and σ-hole/πhole (see below) interactions formed by the acting of the electrostatic forces and charge transfer are accurately described at almost any quantum-mechanics level. On

314 | D. Nachtigallová and P. Hobza the contrary, the description of the stacking interactions (see below) requires methods which cover a large portion of the correlation energy. Accounting for both types of interactions requires highly accurate methods which are able to handle energy contributions of few kcal/mol. (ii) The limited number of available experimental data connected with difficulties to measure mainly the interaction energies of the noncovalent complexes. Comparison of computational results with experiment gives the most informative test of the reliability of the method. Concerning the properties of the noncovalent complexes, their stability is the most important characterization, which can be measured directly in the gas phase using an adequate spectroscopic method. The number of available stabilization energies is, however, limited. The measured quantity, interaction enthalpy at zero temperature ΔH00 (or dissociation energy D0 ) consists of interaction energy (E int ), deformation energy (E def ), which corresponds to the energy penalty due to the change of geometry of monomer upon complex formation, and the difference of zero point vibrational energies ΔZPVE. Evaluation of a method reliability does not necessary require calculations of observable interaction enthalpy; calculations of E int itself and its comparison with benchmark results can be used to test a method of choice. Note that for accurate calculations of ΔZPVE the noncovalent complexes in general require the use of an anharmonic approach. Two approaches, variation (supramolecular) or perturbative, can be used to calculate E int . In the variation (supramolecular) approach, E int is defined as the difference between the energy of the complex (AB) and the sum of the relevant monomer (A and B) energies E int (AB) = E(AB) − E(A) − E(B).

(12.7)

The use of finite basis set used in E int calculations generates so called basis superposition error (BSSE) resulting from the unbalanced description of A and B in the calculations of the isolated monomers and dimer: In the dimer calculations, each monomer can use the basis set functions located on both monomers. In the monomer calculations, however, the basis set functions of the counterpart are missing, which leads to an artificial stabilization of the complex compared to the dissociated state. Such error is overcome using counterpoise correction (CP) changing the equation for E int (AB) to int ECP (AB) = E(AB) − E(AAB ) − E(BAB ),

(12.8)

where the superscript AB denotes the calculation in the basis set of the whole dimer. The perturbation approach originates from the work of London and coworkers [9, 10], in which the interaction energy is determined as the sum of perturbation contributions to the Hamiltonian of noninteracting A and B. In the terms of relevant operators, the general expression in perturbative framework is ̂ int = H ̂0 + V, ̂ H tot

(12.9)

12 Introduction to noncovalent interactions | 315

̂ int , H ̂0 and V ̂ are the total Hamiltonian, the Hamiltonian of the unperturbed where H tot subsystems and the operator of intermolecular interactions, respectively. London’s method, where the interaction operator is replaced by its multipolar expansions [11] is valid only for large separations of A and B since it diverges for finite separations where overlap and exchange terms are not negligible. As stated by Kutzelnigg, the basic reason is that the unperturbed wavefunction is only antisymmetric with respect to electron permutation within either of two subsystems, while the wavefunction of the whole system is antisymmetric with respect to the exchange of all electrons. One cannot expect perturbation theory to establish such a drastic change of the symmetry of the wavefunction in a smooth way [12]. This difficulty is resolved by including the terms which cover the charge overlap (or penetration effects) in the so-called polarization approach [13]. Such approach, however, still fails to correctly describe van der Waals minima due to the lack of description of electron tunneling between the two subsystems. This problem is overcome in the symmetry-adapted-perturbation theory (SAPT) [14], in which the perturbation expansion and permutation symmetry of the wavefunction are combined. The interaction energy term is described by a sum of first-, second-, and higher-order perturbation contributions. The first-order term includes electrostatic and exchange energies, the second-order term includes dispersion and induction energies, the latter is included also in the third-order term together with charge-transfer energy, which is also part of the higher-order contributions. In the SAPT treatment, the interaction energy is defined in a straightforward manner and all terms have a clear physical meaning. However, the application of the SAPT theory is limited or even unfeasible for larger systems due to necessity to solve the monomer’s Schrödinger equation almost exactly. The accuracy of the E int of noncovalent complexes is strongly influenced by the quality of the basis set which must be sufficiently flexible to avoid discontinuity in the correlated wave-function methods. Dunning [15] developed series of correlationconsistent basis sets designed for these purposes, labeled cc-pVXZ, where X=D (double), T (triple), Q (quadruple), 5 (quintuple) and 6 (sextuple)-ζ (ζ stands for exponent of basis function) which should be further augmented with diffuse functions in augcc-pVXZ [16] basis sets. Requirements on the size of the basis set to accurately describe the noncovalent, mainly dispersion, interactions easily lead to extremely demanding or even unfeasible calculations. This problem can be overcome by the extrapolation of the results obtained with the series of basis sets with increasing size to the complete basis set (CBS) limit [17] or by the use of so called explicitly correlated methods in which the explicit correlation effects are added to a standard method, so called R12 [18] and F12 [19, 20] families giving the results with moderate basis sets, such as, e. g., triple-ζ comparable to those obtained with quintuple-ζ and an uncorrelated method. (iii) The size of the model systems required to rationalize and model chemical processes in noncovalent systems, which are often significantly complex and require

316 | D. Nachtigallová and P. Hobza the use of more or less approximate methods. Their selection, however, requires the knowledge of their limitations with respect to the character of the noncovalent complex to be described. For this reason and due to the problems with the lack of experimental information (see the previous paragraph) good benchmark sets, which contain the results of important characteristics performed with a reliable computational method on a representative set of noncovalent complexes, are of a big advantage. In the following, we will provide only brief overview on more approximate methods based on both vibrational and perturbative approaches. The interested reader is referred to the recent review on this topic [21].

12.3.1 Supermolecular method The major issue of the selection of a suitable method is its capability to describe the correlation energy. In other words, post-HF treatment has to be included via methods based on the configuration interaction (CI) [22], Møller–Plesset perturbation (MP) [23], or coupled-cluster (CC) [24] approaches. CI approaches are not practical to describe the interaction of two subsystems since truncated versions to computationally unfeasible full-CI are not size consistent. Among MP methods, the second order (MP2) approach is frequently used. MP2 accurately describes complexes stabilized by the electrostatic interactions. This method, however, suffers from a well-known problem of the overestimation of the dispersion energy. Attempts to handle this error are based on reducing dispersion contributions more or less empirically. In the improved spin-component-scaled (SCS) approach [25], SCS-MP2 method, different scaling coefficients are applied to the other-spin (singlet) and same-spin (triplet). In the less computational demanding scaled opposite-spin MP2 (SOS-MP2) method [26], only one spin component is evaluated. For review of other approaches and their performance in the modeling of the noncovalent interactions, see [21]. CC-based methods represent the most promising approach to simultaneously cover the noncovalent interactions of both electrostatic and dispersion origins. The accuracy of results can be improved by a systematic increase of the level of excitation operators. Description of the noncovalent interactions requires the use of triple excitation in the coupled-cluster single (S), double (D) and triple (T) excitation methods (CCSDT) [27], which is, however, a very expensive approach. The expenses can be reduced if T are added to CCSD (evaluated using an iterative approach) by means of the perturbation theory (up to fourth order and one term of fifth order) in CCSD(T) method [28]. Using this method in combination with CBS limit (CCSD(T)/CBS) yields the stabilization energies with accuracy ∼1 kcal/mol (chemical accuracy) and reliable description of other properties for all types of the noncovalent interactions. Thus, it is a natural choice to be used for benchmarking. Among the methods used in the supramolecular approach, density functional theory (DFT) methods have their special place due to their computational efficiency.

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These methods provide very good results for selected noncovalent complexes, e. g., Hbonded complexes and other types based on the electrostatic interactions. The main drawback is usually their failure to describe the dispersion interactions. In the past, a great attempt has been made to develop appropriate dispersion corrections. These recent efforts are reviewed in details in [29]. The corrections can be assigned into three groups [29], i. e., based on (i) semiclassical C6 parameter (ii) non-local density functionals and (iii) one-electron effective potential. In this text only the (i) type is discussed (for description of other types of corrections the reader is referred to [29]) in which the corrections for the dispersion energy are added to the DFT electronic energy using various approaches (see [29]), all of them derived from perturbation theory. The advantage of such descriptions is that the dispersion corrections are separated from the DFT energy and can be covered using even a simple pairwise empirical model. The most wide-spread are probably the DFT-D approaches developed by Grimme and coworkers [29]. In particular, the DFT-D3 method [30] in combination with a flexible basis set gives very good agreement with the CCSD(T)/CBS benchmark data for a wide range of systems.

12.3.2 Perturbation method Semiempirical quantum-mechanical methods serve as an alternative way to treat the noncovalent interactions, in particular for more extended systems, due to their significantly smaller computational demands. This is appreciated, e. g., in molecular dynamics simulations involving systems where the noncovalent interactions play an important role. Similar to above described DFT methods also the semiempirical methods suffer from the lack of the dispersion which can be fixed in a similar way as in the DFT methods [31] via, e. g., the self-consistent density-functional tight-binding (SCC-DFTB) [32] and modified neglect of diatomic orbital (MNDO) [33–38] methods. Important improvements in the description of the hydrogen bonds were made by using additional empirical corrections [36, 38]. An important step toward the calculations of medium-sized complexes is offered by an approach which combines the DFT descriptions with the previously introduced SAPT method in the so-called DFT-SAPT method [39–41]. In this framework, the interaction energy is expressed as 1 1 2 2 2 2 E int = EPol + EEx + EInd + EEx−Ind + EDisp + EEx−Disp + δHF,

(12.10)

where superscripts 1 and 2 stand for the first- and second-order perturbation terms, respectively, and subscripts Pol, Ex, Ind, and Disp stand for polarization (electrostatic), exchange, induction and dispersion components, respectively. δHF stands for estimated the higher-order Hartree–Fock (HF) induction and exchange-induction energy

318 | D. Nachtigallová and P. Hobza components. The higher-order energy components are determined as a difference between HF (variation) interaction energy and sum of the first- and second-order energies (except dispersion energy). The DFT-SAPT results are much less sensitive to the choice of the DFT functional compared to supermolecular DFT. In addition, the DFTSAPT converges faster with the basis set than the regular SAPT. However, due to the use of δHF term, determined with the HF (variation) interaction energy, the DFT-SAPT is not BSSE-free, unlike the SAPT approach. Another different approach to determine the electronic structure of complexes is the quantum Monte Carlo (QMC) method (see review on the method and its application in [42]) which solves the stationary Schrödinger equation based on stochastic techniques, i. e., using stochastic processes and sampling wavefunctions in the space of electron positions. Recent advances, based on the fixed-node diffusion Monte Carlo (FN-DMC), made it useful as a new alternative to the wavefunction-based and DFT methods for the treatment of the noncovalent interactions. Important advantages for studies on noncovalent extended complexes are an efficient treatment of dynamic correlations and low-order polynomical scaling which allows for applications to much larger systems compared to the wavefunction-based and DFT methods. QMC method was already applied to systems like, e. g., the thermal-equilibrium clusters and bulk water liquid containing up to 64 water molecules [43], solvation processes using water cluster up (H2 O)20 [40], the interaction energies of large supra-molecular carbonbased host-guest complexes (C60 –C60 H28 ) [44] or host-guest complexes with DNA or proteins. For further applications, see [42].

12.4 Special types of noncovalent interactions 12.4.1 Hydrogen bonding Figure 12.2 shows boiling points of hydrides of the elements of the groups 14 (tetrels), 15 (pnictogens), 16 (chalcogens) and 17 (halogens) of the periodic table. Moving down along the 14th group, that is, from CH4 to SnH4 , the boiling temperature increases almost linearly. For the other series of hydrides, the values for the lightest (NH3 , H2 O, and HF) deviate from linearity, with the most pronounced deviation observed in the 16th group (Figure 12.2). Following the trends of boiling points in this group, i. e., a linear decrease along the group, H2 Te → H2 Se → H2 S, the predicted boiling point of H2 O is ∼ −90 °C, enormously different from the observed value of 100 °C. The only explanation of such large deviation is an existence of rather strong noncovalent interactions in H2 O, partially or completely missing in H2 S, H2 Se, and H2 Te. These interactions were reported on already in 1920[45] but its name, hydrogen bond (H-bond), was probably introduced by Linus Pauling [46] a few years later.

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Figure 12.2: Boiling points of hydrides of atoms of 14, 15, 16 and 17 group of periodic table.

Various examples can illustrate the importance of H-bond, e. g., the structure and unique properties (e. g., boiling and melting points) of water are determined by its H-bonds network; H-bonds play and important role in the structure formation as well as functioning of biomolecules such as DNA and proteins. The specificity of H-bonds comes also from their easily detectable spectroscopic manifestation. Upon formation of the X–H. . . Y hydrogen bond, where Y represents an electron donor (proton acceptor), a small charge is transferred between an electron donor and electron acceptor, in particular from the lone pair (LP) of Y to the σ ∗ (X–H) antibonding orbital. An increase of the electron density in σ ∗ orbital is connected with its weakening and elongation accompanied by a decrease of X–H stretching frequency in IR spectra, so called redshift. For a long time, the red-shift represented a fingerprint of the H-bond and was regarded as a unique spectral manifestation of H-bonding. “No red shift – no stabilization” rule belonged to a dogma in chemistry. Calculations of benzene. . . X complexes (e. g., benzene. . . chloroform, cf. Figure 12.3) show results contradicting this statement; upon complexation the C–H bond in chloroform shortens and the respective stretching frequency shifts to higher energies, i. e., it is blue-shifted. These predictions made from calculations which accounts for anharmonicity have been later confirmed by experiments [47] what prompted the IUPAC organization to introduce a new definition of H-bonding [48]. The short version of definition states: “The hydrogen bond is an attractive interaction between hydrogen atom of a molecule or molecular fragment X–H in which X is more negative than H and an atom or group of atoms of the same or different molecule, in which there is an evidence of bond formation. In general, for the donor, the X–H bond length increases and there is an associated red-shift in the X–H stretching frequency. There are, however, certain hydrogen bonds in which the X–H bond length decreases and a blue-shift in the X–H stretching frequency is observed.”

320 | D. Nachtigallová and P. Hobza

Figure 12.3: A hydrogen bond in benzene. . . chloroform complex. Distance is in Å (C: black, Cl: yellow, H: white).

A different spectral manifestation of Z–X–H. . . Y bond formation can be explained by e. g., electron density transfer from an electron donor to an acceptor. Depending on electronegativities of atoms near the proton donor, this electron density is transferred to σ ∗ (X–H) or σ ∗ (X–Z) orbitals leading to their weakening and elongation. In the latter case, the X–Z bond elongation is accompanied by the X–H bond contraction, which results in the blue shift of the relevant stretching frequency.

12.4.2 Dihydrogen bond Let us consider the two systems X–H and X’–H, where X is more electronegative (e. g., C, N, O, F) than hydrogen. Electrons in X–H covalent bond are not evenly shared resulting in partial negative and positive charges on X and H, respectively. If X’ is more electropositive than H (e. g., Li, Na, B, Si), H carries partial negative charge and the complex X–H. . . H–X’ possesses linear structure with an attractive interaction between two hydrogen atoms, called dihydrogen bond. Such structural pattern has been first observed in metal-containing crystals [49]. The existence of the dihydrogen bond has been later used to rationalize unusually high boiling temperature of H3 BNH3 [50]. The dihydrogen bonds are characterized by H. . . H distances of 1.7–2.2 Å, much shorter than the sum of van der Waals radii of 2.4 Å, and large stabilization energies of 5– 8 kcal/mol[51] comparable to the classical hydrogen bond. Its existence has been observed in the gas phase as well as in the solid phase. The dihydrogen bond plays an important role e. g., in protein chemistry: substituted boron hydrides with negatively charged hydrogens inhibit proteins via protein—ligand interactions. Substituted metallocarboranes have been found to act as potent and specific inhibitors of HIV-1 protease with the strong hydrogen bond between metallocarboranes and protease.

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12.4.3 σ- and π-hole bonding The σ- and π-holes are molecular regions with a positive electrostatic potential [52]. They represent a completely new type of the noncovalent interactions with an important role in physical chemistry, biochemistry, and material sciences. Historically, the halogen bond R1 -X. . . Y-R2 , where X is a halogen and Y is an electron donor was detected and investigated first and we will use this type of the σ-hole bonding for a detail description. Figure 12.4 shows the linear C-Br. . . O halogen bond between bromobenzene and acetone with a short Br. . . O distance.

Figure 12.4: A halogen bond between bromobenzene and acetone with the Br. . . O distance marked (C: black, H: white, O: red, Br: green).

According to the IUPAC[53] the halogen bond “occurs when there is evidence of a net attractive interaction between an electrophilic region associated with a halogen atom in a molecular entity and a nucleophilic region in another, or the same, molecular entity.” This type of interaction is somehow surprising since it assumes an existence of positively charged electrophilic region in electronegative atoms. Explanation comes from the analysis of the molecular electrostatic potential (MEP). In 1992 Brinck et al. [54] showed for the first time that the electrostatic potential around halogen atoms is not isotropic and exhibits a negative region, a belt around the R1 –X covalent bond (X is halogen), and a positive region in the elongation of the bond, lately termed as σ-hole [55]. Other bond types based on the σ-hole bonding are named chalcogen-(group 16), pnicogen-(group 15), tetrel-(group 14), and aerogen-(group 18) bonding [52]. The character of the halogen-bond is, however, specific due to its monovalent character. Thus, halogens possess only one σ-hole, while more σ-holes can be formed in other types of the above mentioned bonds. The halogen bonds tend to be linear contrary to other σ-hole bonds which determines the topology of complexes. Characteristics of σ-hole bonds is determined by two main factors: the nature of X (halogen, chalcogen, pnicogen, tetrel, and aerogen) and its chemical environment. Due to the increasing polarizability and decreasing electronegativity the magnitude of the σ-hole increases going from lighter to heavier atoms within a particular group of the periodic table. Concerning the most electronegative fluorine, the σ-hole does not exist when F binds to C, N, O

322 | D. Nachtigallová and P. Hobza and other elements. However, it does exist in the fluorine molecule (F2 ), in which two σ-holes are localized outside the system and their positions determine the properties of fluorine molecule crystals. The effect of chemical environment can be illustrated on the case of methylbromide (H3 CBr) and trifluoro-methylbromide. (Figure 12.5). The magnitude of the σ-hole in H3 CBr is significantly smaller compared to F3 CBr due to the existence of three electron-withdrawing F atoms in the latter. The strength of the σ-hole bonds, attributed mainly to the electrostatic origin, is surprisingly large, about 5–10 kcal/mol. Recently, we pointed out the importance of the dispersion energy, explained by a close contact between two heavy atoms (smaller than the sum of their van der Waals radii), halogen and electron donor, both with large polarizabilities [52].

Figure 12.5: Molecular electrostatic potential (B97-D3/TZVPP) computed on 0.001 a. u. molecular surfaces of H3 CBr and F3 CBr. The color of the ESP ranges in kcal/mol.

The origin of the π−hole [56, 57] is illustrated in Figure 12.6. The MEP of benzene shows a negative electron density evenly distributed above and below the aromatic framework, in agreement with the chemical intuition of benzene acting as a proton acceptor. Substitution of hydrogens by fluorines or CN groups to form hexafluorobenzene and hexacyanobenzene, respectively, depletes the electron density above and below the benzene ring making these systems electron acceptors, capable of forming highly stable complexes with electron-rich partners (anion or lone pairs). As reported by Wang et al., the stabilization energies of C6 F6 . . . Cu− and C6 F6 . . . Cl− complexes are 198 and 49 kcal/mol, respectively. Very large stabilization energies of anion-π-hole complexes are mainly due to electrostatic interactions, the stabilization due to the dispersion is

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Figure 12.6: Molecular electrostatic potential (B97-D3/TZVPP) computed on 0.001 a. u. molecular surfaces of benzene, hexafluorobenzene and hexacyanobenzene. The color of the ESP ranges in kcal/mol.

less important [57].1 Substitution also changes the negative value of the molecular quadrupole moment to positive which affects the mutual orientation of subsystems in complexes [58], as seen in the comparison of the global minima of C6 X6 . . . C6 X6 (X=H, F, CN) and C6 H6 . . . C6 X6 (X=F, CN) dimers. The charge distributions of C6 H6 and C6 X6 (X=F, CN) quadrupoles can be schematically illustrated as (+−−+) and (−++−), respectively. The repulsive interactions of the quadrupoles of the subsystems in homodimers result in parallel-displaced or T-shaped structures. Due to the sign flip upon substitution with electronegative atoms or groups the quadrupole interactions in stacked C6 H6 . . . C6 X6 (X=F, CN) dimers become attractive stabilizing the dimers. For X=CN, the stabilization energy is 11 kcal/mol (calculated at the CCSD(T)/CBS level) [58].

12.4.4 Stacking interactions This class of the noncovalent interaction was already introduced in the previous paragraph. These interactions result from a large degree of the overlap between the interacting subsystems, e. g., aromatic species, achieved in the stacking arrangement. The 1 Comment on the nomenclature: the mentioned complexes are of anion-π-hole type; the term anion-π, frequently used in the literature, is misleading. Electrostatic energy in the latter complexes is by definition repulsive and the only (weak) attraction can originate from the dispersion energy.

324 | D. Nachtigallová and P. Hobza amount of the complex stabilization is geometrically specific as it depends on the overlap extent. For some species like, e. g., heterocyclic aromatic species, the orientation of subsystem’s dipole moments significantly influences the strength of the stacking interactions. Although the stabilization based on the electrostatic interactions, such as the hydrogen bond, is considered to be stronger, the strength of the stacked interactions is often comparable (see below). The dominant contributions to the stacking interactions come from the dispersion. However, the electrostatic (dipole-dipole and quadrupole-quadrupole) interactions are also known to play an important role in the stabilization of these complexes. For the sake of comparison of the strength of the various types of the noncovalent interactions, we report on the values of the stabilization energies (negative interaction energies) obtained with CCSD(T) and CBS for selected complexes from S22 and S66 benchmark data sets [21] (see Table 12.1). Table 12.1: Stabilization energies (Estab , in kcal/mol)a for selected noncovalent complexes. Hydrogen bonds System c

H2 O dimer (Cs ) NH3 dimer (C2h ) HCOOH dimer (C2h ) Benzene. . . H2 O (C1 ) Adenine. . . Thymine (C1 ) Methyl acetamid dimer (C1 ) a

Estab

Nb

5.02 3.17 18.61 3.28 14.92 8.63

1 1 2 1 2 1

Other noncovalent interactions System

Estab

CH4 dimer (D3d ) C2 H2 dimer stacked (D2d ) Benzene dimer stacked (C2h ) Benzene. . . C2 H2 (C1 ) Adenine. . . Thymine stacked (C1 ) Methyl acetamid. . . C5 H10 (C1 )

0.53 1.51 2.73 1.43 12.23 4.24

Ref. [21], b number of H-bonds, c symmetry of the system.

As discussed above, the H-bond stabilization originates predominantly from the electrostatic contributions, mainly the dipole-dipole term. It is thus not surprising that the H-bond interaction between the subsystems with a large dipole moment, such as water, ammonia and formic acid is strong. Notably, the stabilization of the systems with more hydrogen bonds is significantly larger than that one would expect from additive contributions of the individual H-bonds. Such deviation was found also in the MP2 calculations of water dimer and trimer [59] and can be attributed to a cooperative effect of the hydrogen bonds. The dipole moment of methyl-acetamid is also responsible for the formation of a stable H-bond complex. On the other hand, the first non-zero multipole moment of methane is octupole and the resulting dimer is much less stable. Concerning the stacked ethylene and benzene dimers, the former is less stable due to its smaller size and consequently a smaller polarizability resulting in weaker dispersion interactions. Comparing the relative stabilities of benzene. . . water and benzene. . . ethylene, the larger stability of the former can be explained by a more favorable quadrupole-dipole interaction over quadrupole-quadrupole interaction of

12 Introduction to noncovalent interactions | 325

the latter. An interesting comparison is given for the stability of H-bonded and stacked A. . . T complexes with almost equal strength (Table 12.1). Also the stacked G. . . C complex is surprisingly stable with the stabilization energy of 18 kcal/mol (compare to 28.5 kcal/mol obtained for H-bonded structure with three hydrogen bonds). The stability of these two stacked base pairs complexes is explained by large polarizabilities and multipole moments of the nucleic acid bases.

12.5 Summary and outlook The interaction energy of clusters which consists of tens of atoms can be nowadays calculated with chemical accuracy (1 kcal/mol) using the CCSD(T)/CBS approach. However, this energy term is not an observable and does not include all contributions which determine the structure, properties, and stability of the clusters. All these properties depend on the binding free energy which results from the balance of the enthalpy and entropy terms. Evaluation of the interaction enthalpy at 0 K (ΔH 0 ) requires including the zero-point vibration energy (ZPE) to the interaction energy and additional temperature-dependent corrections to obtain the interaction enthalpy at ambient temperature, the value comparable with the experiment. Calculations of ZPE of clusters of several tens of atoms are feasible at the harmonic level. However, anharmonic effects, which play an important role in the case of the noncovalent interactions, are impractical at present, making the chemical accuracy of the calculations inaccessible. The situation is even worse in the case of the binding free energy which requires additional entropy term. Relative binding free energies are thus much more reliable than their absolute values. Thermodynamic characteristics, including the binding free energies, are mostly evaluated via molecular dynamic simulations. These simulations on bio-macromolecules in a real environment are currently performed only using molecular mechanics (MM) methods. Their applicability is, however, limited since they neglect quantum effects playing an important role for the noncovalent bonding. Further development of linearly scaled semiempirical quantum mechanical methods parametrized for a wide range of elements, including transition metals, is thus of a crucial importance. The method must account for an environment, which is mostly water in the case of biological processes. Modeling of biological processes (in DNA, proteins and larger systems) requires reliable description of the environment (solvent) and relatively long simulation time, in order of several ms. This can be performed via a combined specific (molecular) and non-specific (continuous) solvents model, rather than by specific solvent only. DFT methods possess a unique role in future applications due to their favorable CPU scaling. Its success, however, depends on the reliability of functionals. The quality of the functional plays a crucial role also in the modeling of the noncovalent interactions in materials. The open-shell character of electronic structures of

326 | D. Nachtigallová and P. Hobza relevant molecules, which often contain transition metals, is an additional aspect not present in most organic and biological processes. Theoretical description of processes which include open-shell systems often requires the use of multireference approach and it is, thus, limited to rather small systems. Finally, the real progress in the field of noncovalent interactions and their applications will be only obtained if theory and experiment work hand-in-hand.

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[41] Williams HL, Chabalowski CF. Using Kohn–Sham orbitals in symmetry-adapted perturbation theory to investigate intermolecular interactions. 2001;105(3):646–659. [42] Dubecký M, Mitas L, Jurečka P. Noncovalent interactions by quantum Monte Carlo. Chem Rev. 2016;116(9):5188–215. [43] Alfè D, Bartók AP, Csányi G, Gillan MJ. Communication: energy benchmarking with quantum Monte Carlo for water nano-droplets and bulk liquid water. 2013;138(22):221102. [44] Tkatchenko A, Alfè D, Kim KS. First-principles modeling of noncovalent interactions in supramolecular systems: the role of many-body effects. J Chem Theory Comput. 2012;8(11):4317–22. [45] Latimer WM, Rodebush WH. Polarity and ionization from the standpoint of the Lewis theory of valence. J Am Chem Soc. 1920;42:1419–33. [46] Pauling L. The nature of the chemical bond. II. The one-electron bond and the three-electron bond. J Am Chem Soc. 1931;53(9):3225–37. [47] Hobza P, Havlas Z. Blue-shifting hydrogen bonds. Chem Rev. 2000;100(11):4253–64. [48] Arunan E, Desiraju GR, Klein RA, Sadlej J, Scheiner S, Alkorta I et al. Definition of the hydrogen bond (IUPAC recommendations 2011). Pure Appl Chem. 2011;83:1637–41. [49] Lee JC, Peris E, Crabtree RH, Rheingold AL. An unusual type of H⋅ ⋅ ⋅H interaction: Ir-H⋅ ⋅ ⋅H-O and Ir-H⋅ ⋅ ⋅H-N hydrogen bonding and its involvement in σ-bond metathesis. J Am Chem Soc. 1994;116(24):11014–9. [50] Klooster WT, Koetzle TF, Siegbahn PEM, Richardson TB, Crabtree RH. Study of the N–H⋅ ⋅ ⋅H–B dihydrogen bond including the crystal structure of BH3NH3 by neutron diffraction. J Am Chem Soc. 1999;121(27):6337–43. [51] Fanfrlík J, Lepšík M, Horinek D, Havlas Z, Hobza P. Interaction of carboranes with biomolecules: formation of dihydrogen bonds. Chem Phys Chem. 2006;7(5):1100–5. [52] Kolář MH, Hobza P. Computer modeling of halogen bonds and other σ-hole interactions. Chem Rev. 2016;116:5155–87. [53] Desiraju GR, Ho PS, Kloo L, Legon AC, Marquardt R, Metrangolo P et al. Definition of the halogen bond (IUPAC recommendations 2013). Pure Appl Chem. 2013;85(8):1711–3. [54] Brinck T, Murray JS, Politzer P. Surface electrostatic potentials of halogenated methanes as indicators of directional intermolecular interactions. Int J Quant Chem. 1992;14(21):57–64. [55] Clark T, Hennemann M, Murray JS, Politzer P. Halogen bonding: the σ-hole. J Mol Model. 2007;13(2):291–6. [56] Murray JS, Lane P, Clark T, Riley KE, Politzer P. σ-Holes, π-holes and electrostatically-driven interactions. J Mol Model. 2012;18(2):541–8. [57] Wang H, Wang W, Jin WJ. σ-Hole bond vs π-hole bond: a comparison based on halogen bond. Chem Rev. 2016;116:5072–104. [58] Pluháčková K, Jurečka P, Hobza P. Stabilisation energy of C6H6⋅ ⋅ ⋅C6X6 (X=F, Cl, Br, I, CN) complexes: complete basis set limit calculations at MP2 and CCSD(T) levels. Phys Chem Chem Phys. 2007;9(6):755–60. [59] Vilela AFA, Barreto PRP, Gargano R, Cunha CRM. Ab initio studies of hydrogen-bonded complexes: the H2O dimer, trimer and H2OCO. Chem Phys Lett. 2006;427(1–3):29–34.

Mark A. Spackman, Peter R. Spackman, and Sajesh P. Thomas

13 Beyond Hirshfeld surface analysis: Interaction energies, energy frameworks and lattice energies with CrystalExplorer 13.1 Introduction Understanding the nature of intermolecular interactions in the context of crystal packing is critical to our ability to rationally design new materials with desirable chemical and physical properties, and the importance of this field has fueled the search for new tools to provide such insights. CrystalExplorer provides researchers with graphical and computational tools, within an easy to use interface, for exploring the interactions between molecules in crystals. It was created to give researchers access to a suite of novel tools to extract important chemical information about molecular crystal structures—relatively quickly—on their laptop or desktop computer. Although initially developed to facilitate Hirshfeld surface analysis [1], it has proven to be a valuable platform for the development of other novel methods, such as the ability to identify and map void spaces in crystals [2]. More recent developments have taken advantage of quantum mechanical wavefunctions of molecules, computed by Gaussian [3] or Tonto [4], to obtain reliable estimates of intermolecular interaction energies, and from them energy frameworks and lattice energies. CrystalExplorer has grown tremendously in popularity due to its combination of powerful analysis tools which aid in the creation of informative and compelling visualizations, features that are made possible through a friendly and intuitive graphical user interface. Software development has taken place alongside the timely publication of the methodology and applications of the novel approaches, and for this reason this chapter will not attempt to review and summarize material that is widely available in the crystallographic literature. It will, however, point to key publications where relevant. The most recent version of this software, CrystalExplorer17 (released in May 2017, and referred to in this chapter as CE17) implements the accurate and efficient calculation of intermolecular interaction energies, energy frameworks and lattice energies.

Mark A. Spackman, Peter R. Spackman, University of Western Australia, School of Molecular Sciences (M310), 35 Stirling Highway, Perth WA 6009, Australia, e-mails: [email protected], [email protected] Sajesh P. Thomas, Aarhus University, Department of Chemistry, Langelandsgade 140, 8000 Aarhus C, Denmark, e-mail: [email protected] https://doi.org/10.1515/9783110660074-013

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13.2 Using CrystalExplorer A hardcopy users’ guide or manual does not exist for CE17. Instead, all of the relevant documentation has been made available online [5]. For first-time users, the best place to start is the Quick Start Guide [6], which describes in detail most of the basic functionality, and in particular provides descriptions of all of the icons and toolbars in the main graphics window; detailed descriptions of the more advanced options can be found in the Online User Manual [7]. At its most basic level the functionality in CE17 is much like that of other software for viewing crystal structures but, unlike similar software, it also offers many of the features of programs that calculate and display the results of quantum chemical calculations. This integrated functionality means that CE17 is unusual in being able to display properties such as the electrostatic potential (ESP) of a molecule in the context of its environment in the crystal. Because it was always focused on crystal structure analysis, the only input option is in the form of a crystallographic information (CIF) file. Figure 13.1 gives an example of the CE17 main window, highlighting the toolbars and information dialog boxes surrounding the main graphics window: 1. The main toolbar gives access to the usual rotate, translate and zoom operations, measurement of bond distance, bond angle and torsion angles, as well as com-

Figure 13.1: A typical CrystalExplorer window at start-up, showing the five major toolbars and information dialog regions.

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pletion of molecular fragments, construction of multiple unit cells and clusters of molecules. More advanced functions open dialog boxes for the creation of molecular surfaces, cloning of surfaces, and to initiate energy calculations. Hovering the cursor over individual icons will display a tooltip describing their functionality. 2. The Crystals dialog at top right lists the crystal structures loaded in the current session. The example here is the CSD refcode SCCHRN02; the actual text used to identify a structure here is that provided via the data_ entry in the CIF file. This dialog will list all crystal structures that have been opened in the working session and clicking on individual entries in the list will open that structure in the main graphics window, so the user can switch between structures without exiting CE17. Structures (and all associated calculations and surfaces) can be removed from this list using the keyboard Delete key. 3. This dialog box lists all molecular surfaces computed for the selected structure, in this case just the Hirshfeld surface. Individual surfaces can be viewed or hidden with a tick or cross beneath the Show heading. In the creation of surfaces for a cluster of molecules CE17 does not recompute a surface that already exists; instead it constructs a symmetry-related copy (or copies if the Clone Surface function is used), and each of these copies can be hidden or viewed separately, provided the parent surface is not hidden. 4. The Surface controller at the bottom right of the main window gives details about the surfaces and allows users to change features of the surface, for example, which property is currently mapped on the surface. The Surface controller has three tabs: (i) Options, which allows the user to set all aspects of the surface, including the property mapped on the surface, the color range for the property, and whether the surface is displayed semitransparently; (ii) Info, which displays information such as the surface area and volume, globularity and asphericity (see [8]); and (iii) Property Info, which provides the minimum, maximum and mean values of each property mapped on the surface 5. The toolbar at the bottom allows users finer control over the image in the graphics window through specific rotations about three axes, scaling the image, or viewing along the crystallographic axes. In this way images of different molecules or surfaces, in different crystal structures, can be created in an identical (but arbitrary) scale. Many of the functions accessed via the icons or the CE17 drop down menus can also be accessed through the context menus in the graphics window by right-clicking on the background, or on a specific surface.

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13.3 Hirshfeld surface analysis 13.3.1 The Hirshfeld surface The Hirshfeld surface and what has become known as Hirshfeld surface analysis have been described in considerable detail in several publications [1, 8–11]. The Hirshfeld surface arose from an exploration of possible ways to define a specific region surrounding a molecule in a crystal, for the purpose of integrating the electron distribution ‘belonging’ to that molecule. It extends Fred Hirshfeld’s ‘stockholder partitioning’ scheme [12], which uses a simple scalar 3D function to define atoms in molecules, to define a surface associated with a molecule in a crystal. For any atom a in a molecule, Hirshfeld defined a scalar weight function at any point r as the ratio of the atomic electron density to the electron density of the promolecule: wa (r) = ρat a (r)/

∑

i∈molecule

ρat i (r)

The individual ρat i (r) are spherically averaged electron densities for each neutral atom. In CE17, the electron densities come from tabulations of atomic electron density functions due to either Clementi and Roetti [13] (only for atoms up to Kr) or Koga, Thakkar et al. for He to Xe [14] and Cs to Lr [15]. The CE17 Preferences menu allows the user to choose between these options, but the default is the more extensive set due to Koga, Thakkar and co-workers. In analogy with Hirshfeld’s idea, a weight function can be defined for a molecule A in a crystal as the ratio of the electron density of the promolecule to that of the procrystal: wA (r) =

∑

i∈molecule A

ρat i (r)/

∑

j∈crystal

ρat j (r)

The weight function wA (r) is a continuous scalar 3D function with 0 < wA (r) < 1 everywhere. Isovalues of wA (r) define an infinite number of isosurfaces, but it turns out that the surface defined by wA (r) = 0.5 is rather special. This isosurface—designated the Hirshfeld surface—completely envelops a molecule in a crystal and defines the volume of space where the promolecule electron density exceeds that from all neighboring molecules. It ensures maximum closeness of neighboring molecular volumes, which can at most touch and never overlap due to the nature of the weight function wA (r). The Hirshfeld surface is smooth, and as a result it is not exhaustive (i. e., it does not fill all space), unlike QTAIM partitioning. The Hirshfeld surface for a molecule is generated by selecting the molecule of interest and clicking on the Hirshfeld surface icon on the top toolbar. A common feature

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of the generation of all surfaces in CE17 is the ability to choose the surface resolution, i. e., the approximate grid spacing used to construct the surface. Five options are available, ranging from Very Low (grid spacing of ∼1.5 au) to High (0.20 au) and Very High (0.15 au). Computation times naturally scale with increasing resolution, and the default is High, and this is also the only option for which fingerprint plots can be computed (see below). It is important to recognize that no new wavefunction calculations are involved in constructing a Hirshfeld surface; it is based entirely on the nature of the different (pre-calculated) atomic electron densities in a molecule, its geometry and the geometries and locations of its nearest neighbors in the crystal. In order to make meaningful comparisons between crystal structures, by default CE17 standardizes bond distances to H atoms to average values derived from neutron diffraction experiments [16] along with the value of 1.190 Å for B–H (see [17]). This default is not necessarily desirable for crystal structures derived from neutron diffraction, and it can be disabled via a Preferences→Expert setting.

13.3.2 Properties of the Hirshfeld surface The Hirshfeld surface is a smooth, continuously differentiable, 3D manifold and at any point on the surface it is straightforward to obtain the two principal curvatures of the surface, denoted κ1 and κ2 . In conjunction with the distances from the surface to atomic nuclei inside and outside the surface, these can be used to define several useful properties at every point on the surface. Table 13.1 summarizes these. Table 13.1: Summary of functions of distance and curvature mapped on Hirshfeld surfaces. Function

Symbol and definition

Mapping range

distance from a point on the surface to the nearest nucleus outside the surface

de

red (short) through green to blue (long)

distance from a point on the surface to the nearest nucleus inside the surface

di

red (short) through green to blue (long)

shape index, S, a measure of “which” shape, defined in terms of the principal curvatures

S=

2 π

arctan( κ1 −κ2 )

−1.0 (concave) through 0.0 to +1.0 (convex)

curvedness, C, a measure of “how much” shape, defined in terms of the principal curvatures

C=

2 π

ln √(κ12 + κ22 )/2

−4.0 (flat) through 0.0 to +0.4 (singular)

normalized contact distance, defined in terms of de , di and the van der Waals radii of the atoms

dnorm =

κ +κ 1

di −rivdW rivdW

2

+

de −revdW revdW

red (shorter than sum of vdW radii), through white to blue (longer than sum of vdW radii)

334 | M. A. Spackman et al. All Hirshfeld surfaces have the above properties mapped by default (and hence are able to be readily visualized), as well as the properties None (a monochrome surface whose color can be changed via the Preferences menu) and Fragment Patch, where surface patches adjacent to neighboring molecules are colored separately. Figure 13.2 gives examples of each of these properties mapped on the Hirshfeld surface for saccharin.

Figure 13.2: The Hirshfeld surface for a saccharin molecule (SCCHRN02) mapped with the various properties that are available for all Hirshfeld surfaces.

13.3.3 Fingerprint plots The fingerprint plot was devised to address the challenge of representing in a 2D format (printed page or computer monitor) the mapping of di and de , two different but potentially useful distance measures, on a 3D molecular surface [18]. The resulting— and now commonplace—plot is created by binning (di , de ) pairs in intervals of 0.01 Å and coloring each bin (essentially a pixel) of the resulting 2D histogram as a function of the fraction of surface points in that bin, ranging from blue (few points) through green to red (many points). Fingerprint plots are highly sensitive to the immediate environment of a molecule in a crystal, and they are unique for a given molecule in a particular crystal. Taking advantage of this uniqueness, fingerprint plots have been found to be extremely useful in studies involving comparisons between different polymorphs [19] or structures determined at elevated pressures [20]. In CE17, fingerprint plots can be produced following the calculation of a Hirshfeld surface for a molecule, by clicking on the fingerprint icon in the Surface dialog (4. in Figure 13.1). However, because the appearance of these plots (especially the density of points and resulting

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fractions of surface area) depends on the resolution of the grid points used to compute the surface, a fingerprint plot can only be produced for a Hirshfeld surface that has been computed at the default of High resolution. A particularly striking example highlighting the effectiveness of fingerprint plots in issues of this kind was described by Joel Bernstein in 2011, who referred to the interesting case of 4,4-diphenyl-2,5-cyclohexanedione (refcode HEYHUO), which had been reported to crystallize in four polymorphic forms, one with Z 󸀠 = 4, another with Z 󸀠 = 12, altogether comprising a total of 19 distinct molecular conformations [21]. Fingerprint plots for the molecules in the Z 󸀠 = 12 structure clearly revealed four sets of three closely similar plots (Figure 13.3), indicating only four independent molecules in the asymmetric unit (Z 󸀠 = 4). Furthermore, these four unique fingerprint plots were essentially identical to those for the known Z 󸀠 = 4 structure, indicating the two structures were the same. This relatively simple analysis established that the system was actually trimorphic (not tetramorphic), exhibiting seven (not 19) crystallographically independent molecules. This particular example exploits pattern recognition but makes no use of the detailed information that can be extracted from fingerprint plots; features characteristic of various intermolecular interactions are being ignored. The original publication in-

Figure 13.3: Fingerprint plots for the 12 unique 4,4-diphenyl-2,5-cyclohexanedione molecules in its Z’ = 12 polymorph (HEYHUO). The 12 plots are arranged in three rows of four to reveal that there are only four unique fingerprint plots (and hence molecules) in this polymorph.

336 | M. A. Spackman et al. troducing fingerprint plots [18] provided numerous examples of these plots for structures incorporating hydrogen bonds, halogen bonds, C–H⋅⋅⋅π interactions and π⋅ ⋅ ⋅π stacking, illustrating the features in these plots characteristic of those interactions. More detailed interpretation of fingerprint plots for molecules in the polymorphs of 2-chloro-4-nitrobenzoic acid was included in the 2009 Highlight article in CrystEngComm [1]. An increasingly popular application of fingerprint plots is the ability to filter the plots into surface-mediated contacts between specific atom-type⋅⋅⋅atom-type pairs, as originally reported in 2007 [22] using examples for benzene, naphthalene and anthracene, and paracetamol forms I and II. A simple bar chart comparing percentages of surface area for these various contacts between atom types has been adopted by many researchers as a naïve way of quantifying the relative importance of various interactions such as O⋅⋅⋅H, H⋅⋅⋅H, C⋅⋅⋅C. It is important to bear in mind that these bar charts are not produced by CE17, but they can be readily constructed using the convenient summary of area percentages of atom-type⋅⋅⋅atom-type pairs available through the Information dialog. Figure 13.4 illustrates an example for saccharin, showing the cyclic hydrogen bond 2 (R2 (8) in graph set notation [23, 24]) along with the fingerprint plot filtered by just O⋅⋅⋅H contact pairs. The regions of the Hirshfeld surface (mapped with dnorm ) associ-

Figure 13.4: CrystalExplorer window showing an example of the filtering of the fingerprint plot for a saccharin molecule in SCCHRN02.

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ated with these contact pairs are highlighted on the molecular surface in the graphics window. The total of 52.8 % of the surface identified with O⋅⋅⋅H contacts in the fingerprint plot dialog derives from values of 28.6 % for O(inside)⋅⋅⋅H(outside) and 24.2 % for H(inside)⋅⋅⋅O(outside) given in the summary in the Information dialog. Note also that these totals include all the regions mapped in color on the filtered Hirshfeld surface in the figure, and because of this they do not represent just the cyclic hydrogen bond interaction alone. The other features visible on the lower left of the Hirshfeld surface in the figure (associated with a pair of smaller red spots) arise from a pair of symmetry related C—H⋅⋅⋅O interactions involving the phenyl ring and an oxygen atom of the sulfonamide moiety.

13.4 Procrystal void analysis As discussed above, the promolecule electron density is trivial to compute, as it relies only on pre-calculated spherical atomic electron density functions and the molecular geometry. The promolecule electron density and its isosurfaces can be readily computed and visualized in CE17, and it was shown some time ago that a promolecule isosurface of 0.002 au (∼0.013 e Å−3 ) typically contains more than 98 % of the electron density associated with that molecule [25]. Promolecular surfaces offer a number of advantages over more conventional surface models such as fused sphere (CPK) or smoothed Connolly surfaces, as they avoid the arbitrary choice of van der Waals and probe radii (especially for the latter). Computed surface areas and volumes also closely approximate those from ab initio electron densities [26]. Promolecular surfaces can be used to define the size and shape of a molecule, and to visualize the space belonging to a molecule. Equally, promolecular surfaces can be used to define the region of space not associated with a molecule, or with molecules in a crystal; in other words, voids or channels or cavities in a molecular crystal can be visualized and explored by simply constructing an isosurface of the procrystal electron density. CE17 uses this approach to locate and visualize the void space in crystalline materials, as well as calculate surface areas and volumes of the voids. The method is computationally rapid, and capable of locating and characterizing all ‘empty’ space, and not just the larger cavities and channels, in molecular crystals, organic, metal– organic and inorganic polymers. To produce a void surface the user must select a molecule, then the Hirshfeld surface icon in the top toolbar, choosing the Crystal Voids option. This dialog allows the user to specify the surface resolution and isosurface value; the color of the isosurface can be changed via the Preferences menu. Figure 13.5 shows two screenshots in the calculation of the 0.002 au void surface for saccharin. For this example, the volume (79.3 Å3 or 10.6 % of the unit cell volume) and surface area (284.3 Å2 ) are given in the Surface dialog at bottom right.

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Figure 13.5: Two stages in the generation of 0.002 au procrystal void surfaces for the unit cell of saccharin.

The publication introducing procrystal void analysis gave examples of its application to a variety of crystalline systems, including porous dipeptides, metal–organic frameworks (MOFs) and covalent organic frameworks (COFs) [2]. That work considered several different isovalues of the procrystal electron density, and when compared with actual measurements of % porosity of dipeptides from He pycnometry [27], concluded that a 0.0003 au (corresponding to ∼0.002 e Å−3 ) procrystal isosurface is an excellent choice to probe porosity and empty space in crystals. However, 0.0003 au is also close to the minimum of the procrystal electron density found in most typical molecular crystals at ambient pressure (e. g., the minimum in the procrystal electron density for saccharin is 0.00039 au). For this reason, it is of little use in probing structural changes in molecular crystals with increasing pressure; an isovalue of 0.002 au is a much better choice. This is illustrated in Figure 13.6, which shows 0.002 au isosurfaces for the unit cell of 3-aza-bicyclo(3.3.1)nonane-2,4-dione at pressures from ambient to 59 kbar (5.9 GPa), based on the structures reported in [28]. At ambient pressure, there is a continuous broad void channel running along the c axis, which is reduced considerably at 9 kbar, and disappears completely above 19 kbar. At 59 kbar, there is no longer any void space (as defined by this procrystal isosurface) in the structure, and consequently the molecular geometry becomes increasingly distorted in response to further increases in pressure. This brief analysis complements the original analysis based on Hirshfeld surfaces, fingerprint plots and interaction energies [28]. Procrystal void analysis has been employed in many novel ways, including visualizing lithium [29] and sodium [30] ion diffusion in battery materials, quantifying

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Figure 13.6: 0.002 au procrystal void surfaces for 3-aza-bicyclo(3.3.1)nonane-2,4-dione showing the effect of increasing hydrostatic pressure from 0 to 59 kbar (BOQQUT02 to BOQQUT07).

the effect of the guest molecule on cavity volume in molecular clathrates [31], and rationalizing the anisotropic thermal expansion of an organic-inorganic hybrid Cu(II) complex [32].

13.5 Using molecular wavefunctions As mentioned in the Introduction, powerful new developments in CE17 have taken advantage of quantum mechanical (QM) wavefunctions computed by either Gaussian [3] or Tonto [4]. CrystalExplorer has always made extensive use of the quantum chemistry package Tonto to perform back-end processing, primarily the preprocessing of CIFs and the generation of surfaces, but it can also perform molecular wavefunction calculations and, therefore, provides a free alternative to Gaussian. Although Tonto is included in every CE17 distribution, CE17 seamlessly runs Tonto whenever necessary, and the user never has to leave CrystalExplorer.

340 | M. A. Spackman et al. Naturally, Tonto provides far fewer options for computing molecular wavefunctions than Gaussian. Methods are restricted to Hartree–Fock, MP2, B3LYP, as well as a more general DFT which gives the user choice of exchange (Becke88, Slater, X-alpha or B3LYP) and correlation (LYP, VWN or B3LYP) potentials. Basis sets available via the wavefunction dialog in CE17 are summarized in Table 13.2, and most of these have been sourced from the Basis Set Exchange [33]. Table 13.2: Basis sets available via menu options in CE17. (Additional basis sets can be incorporated from the Basis Set Exchange, or further atoms in some of the basis sets, but this requires significant expertise.) basis set keyword

available for atoms

STO-3G 3-21G 6-31G(d) 6-31G(d,p) 6-311G(d,p) D95V cc-pVDZ cc-pVTZ

H to I H to Cs H to Zn H to Zn H to Kr, I H, Li, B to Ne, Al to Cl H to Kr H to Kr

CrystalExplorer automatically recognizes when the chosen surface or surface property requires a molecular wavefunction and guides the user to perform that calculation. CrystalExplorer then runs the calculation on behalf of the user, freeing them from the burden. If the user has a version of the popular Gaussian package installed on their computer, CrystalExplorer can be configured to use it by setting the path to the Gaussian executable in Preferences→General.

13.5.1 Property surfaces and properties mapped on surfaces CE17 can calculate and visualize Electron Density, Deformation Density, Electrostatic Potential and Orbital surfaces derived from QM wavefunctions. Surface resolution can be chosen as for the Hirshfeld surface, and for each surface the user can specify different isovalues. Isosurfaces of the ESP can be time-consuming and are not always closed surfaces; the default isovalue of 0.05 au is a good choice, but typically only identifies broad regions of negative potential. A better (and less time-consuming) way to convey information about the electrostatic nature of a molecule is to map the potential on a molecular surface, such as a Hirshfeld surface, promolecule or electron density isosurface, where a much greater range of the property can be visualized. Based on a B3LYP/6-31G(d,p) wavefunction,

13 Beyond Hirshfeld surface analysis | 341

Figure 13.7: Examples of various property isosurfaces for saccharin. The quantum mechanical properties are derived from a B3LYP/6-31G(d,p) wavefunction.

Figure 13.7 shows a number of quantum mechanical isosurfaces, and compares that of the ESP with the mapping of the ESP on the Hirshfeld surface. The mapping of molecular ESPs onto Hirshfeld surfaces, in the context of the crystal environment, enhances the discussion of close molecular contacts in the crystal, using the concept of “electrostatic complementarity” between touching Hirshfeld surface patches in adjacent molecules [17]. The examples in that work, combined with additional examples for the molecular crystals formamide and s-triazine, and the cocrystals benzene:hexafluorobenzene, chloroacetic acid:2-amino-5-nitropyridine and 1,4-dicyanobutane:1,2-diiodo-1,1,2,2-tetrafluoroethane [34], demonstrate how mapping a limited range of the ESP on Hirshfeld surfaces provides clear evidence of the almost universal role played by electrostatic complementarity in the way molecules pack in crystals. Molecules arrange themselves in crystals in such a way as to maximize the overlap (or number of close contacts) between electropositive and electronegative regions in adjacent molecules. Generating images of this kind is relatively straightforward, and the following points summarize the steps involved in the creation of a cluster of this kind for the co-crystal of acetylene and formaldehyde (refcode GURNEN, Figure 13.8): – After opening the CIF file and completing molecules, the first step is the creation of the Hirshfeld surface mapped with the ESP for the two molecules (in Figure 13.8, the ESP is based on B3LYP/6-31G(d,p) wavefunctions). – A cluster of arbitrary size and shape can be created using Generate External Fragment in the context menu, followed by Clone Surface to generate copies of surfaces for all additional molecules that were created.

342 | M. A. Spackman et al.

Figure 13.8: A cluster of Hirshfeld surfaces mapped with the ESP (B3LYP/6-31G(d,p) wavefunctions, mapped over the range ±0.025 au) for the acetylene:formaldehyde co-crystal (GURNEN).

–

–

–

Note that Clone Surface acts on the selected surface in the list of parent and symmetry-related surfaces (area 3. in Figure 13.1), so this operation will need to be performed twice—once for the cloned acetylene molecules and once for the cloned formaldehydes. The mapping of properties on all surfaces of one kind (e. g., rescaling the property, or enabling transparency) can be adjusted by selecting the parent surface— either in the graphics window or from the list of surfaces. In Figure 13.8 the ESP is mapped on all surfaces over the range ±0.025 au. This operation can even be used to completely change the mapped property (e. g., from ESP to shape index, or dnorm ). An individual surface can be modified in the same way by selecting just that surface. Additional molecules can be added to the picture via the Show/Hide contact atoms button on the top toolbar, and close contacts or hydrogen bonds can be displayed via the Display menu.

13.5.2 Intermolecular interaction energies Electrostatic complementarity is an important consideration in the packing of molecules in crystals, and images like that in Figure 13.8 can go part way to rationaliz-

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ing observed crystal structures. Close molecular contacts can be discussed in terms of the electrostatic complementarity of touching surface patches in adjacent molecules, and the relative magnitudes of the electrostatic potentials mapped in this manner have been shown [17] to correlate with computed electrostatic energies using Angelo Gavezzotti’s PIXEL approach [35–37]. PIXEL became popular for investigating molecular crystals in terms of pairwise interaction energies, as it provides a computationally tractable way to obtain reliable intermolecular energies constructed from a nonempirical electrostatic energy, and semiempirical polarization, dispersion and repulsion contributions. CrystalExplorer computes pairwise intermolecular interaction energies using an approach inspired by Gavezzotti’s work, but different in many important respects. As full details of this approach are available in the literature [38, 39], only a brief outline is provided here. The basic expression follows that used by Gavezzotti and others, except that each energy term is multiplied by a separate scale factor, Etot = kele Eele + kpol Epol + kdis Edis + krep Erep . The terms in this expression are defined as follows: Eele , the classical electrostatic energy of interaction between unperturbed quantum mechanical charge distributions of the monomers; Epol , the polarization energy, calculated as a sum over nuclei with terms of the kind − 21 αF 2 , where the electric field at each nucleus is due to the quantum mechanical charge distribution of the other monomer and α are average isotropic polarizabilities for atoms and ions; Edis , Grimme’s D2 dispersion correction, summed over all intermolecular atom pairs; Erep , the exchange-repulsion energy calculated between unperturbed quantum mechanical charge distributions of the monomers. Optimum values of the scale factors, kele etc., were determined by calibrating Etot against counterpoise-corrected B3LYP-D2/6-31G(d,p) interaction energies for 1794 molecule/ion pairs extracted from 171 crystal structures, which included metal coordination compounds, organic salts, solvates and open-shell systems, as well as the more usual neutral organic molecules [39]. The mean absolute deviation (MAD) of these model energies from benchmark values is 4.7 kJ mol−1 for HF/3-21G monomer electron densities (denoted CE-HF model energies) and as little as 2.5 kJ mol−1 when B3LYP/6-31G(d,p) monomer electron densities are used (denoted CE-B3LYP model energies). It is quite remarkable that these small MAD values result from optimizing just four scale factors for each of the two energy models, and for pairwise interactions spanning a range of 3.75 MJ mol−1 . CrystalExplorer makes the computation of sets of these pairwise model energies a straightforward process involving a small number of steps: – After opening the CIF file and completing molecules, select the molecule and click on the Calculate Energies icon in the top toolbar. If no cluster of molecules exists CE17 will create a cluster of nearest neighbor molecules, defined as those

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–

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molecules for which any atom is within 3.8 Å of any atom in the selected molecule. (Alternatively, users can create their own cluster of arbitrary size by using the Generate Atoms Within Radius button, followed by Complete Fragments). After creation of the cluster a dialog will open asking the user to select either Accurate CE-B3LYP model energies (based on B3LYP/6-31G(d,p) wavefunctions) or Fast CE-HF model energies (based on HF/3-21G wavefunctions) to be computed. On choosing one of these alternatives, CE17 will calculate the required molecular wavefunction (using Gaussian or Tonto, as requested), and then calculate all model interaction energies for molecular pairs involving the selected molecule (at the center of the cluster) and all surrounding molecules. CE17 recognizes symmetry-related interactions and will only calculate energies for the unique pairs. The calculation of model interaction energies is not limited to neutral species [39]. CE17 recognizes the presence of charged species and asks the user to confirm the charge of each moiety before commencing energy calculations.

On completion of all energy calculations, the second molecules involved in the unique pairwise interactions are identified by different colors in the graphics window, and the energies (and individual energy terms) are obtained by clicking on the Information icon at top right. The Information dialog links the colors of the molecules in the cluster with the number N of these interaction pairs in the window, the center-of-mass distances R (Å) between them and the central molecule, the unscaled energy components, and the total energy Etot . For SCCHRN02 the CE-B3LYP result for the nearest neighbor cluster is shown in Figure 13.9. From this figure we see the strongest pairwise interaction is clearly that for the cyclic R22 (8) hydrogen bond (magenta in the figure), with Etot = −58.3 kJ mol−1 . There are four pairs of molecules for which the interaction with the central molecule is identical (i. e., those with N = 2; one pair is red, the others light blue, dark blue and purple). Several observations are worthwhile regarding these model energies: – As suggested by the values for mean absolute deviations from the DFT benchmark energies, the CE-B3LYP model energies are typically more accurate than those from the CE-HF model. But they also take longer to calculate. As Eele , Epol and Erep are computed from quantum mechanical wavefunctions, computation times scale nonlinearly with the number of basis functions. For saccharin, the B3LYP/6-31G(d,p) wavefunction has 209 basis functions, but the HF/3-21G wavefunction has only 122. For these reasons the CE-HF model can give a relatively rapid estimate of interaction energies, but the CE-B3LYP model is the preferred choice for publication. – Although Etot can be obtained with one (or more) decimal places, it is recommended to report energies to integral values of kJ mol−1 . Given the magnitude of the MAD from benchmark results, this implicitly acknowledges that these model energies cannot be expected to be meaningful at the level of tenths of kJ mol−1 .

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Figure 13.9: CE-B3LYP model energies for interactions between a central saccharin molecule and those in a cluster of nearest neighbours.

–

The steps outlined above apply to the calculation of pairwise energies in crystal structures with only one molecule (or part thereof) in the asymmetric unit. For structures with Z’ > 1, solvates or co-crystals, these model energies can also be readily obtained, but wavefunctions for all unique molecules need to be computed in advance of any energy calculations. The detailed procedure can be found in the online documentation [40].

13.5.3 Energy frameworks The pairwise interaction energies obtained for a nearest neighbor cluster of molecules can be used to construct and visualize an “energy framework” for the molecular crystal. These novel pictures arose in order to better understand crystal packing by combining intermolecular interaction energies with a graphical representation of their magnitude. In this manner intriguing questions, such as why some crystals bend with an applied force while others break, and why one polymorph of a drug exhibits exceptional tabletability compared to others, can be addressed in terms of the anisotropy of the topology of these interaction energies. This approach was described and applied to a variety of organic molecular crystals with known bending, shearing and brittle behavior, to illustrate its use in rationalizing their mechanical behavior at a molec-

346 | M. A. Spackman et al. ular level [41]. Anisotropy of the interaction topology, in the form of strongly bound molecular columns, sheets or slabs, can correlate with mechanical properties such as bending and shearing in crystals. In an energy framework picture, the energies between molecular pairs are represented as cylinders joining the centroids of pairs of molecules, with the cylinder radius proportional to the magnitude of the interaction energy. Separate frameworks can be constructed for Eele (red cylinders), Edis (green) and Etot (blue). Energy frameworks offer a powerful and unique way to visualize the supramolecular architecture of molecular crystal structures. The cylinders that make up the framework represent the relative strengths of molecular packing in different directions. An overall scale factor can be used to expand or contract the framework cylinders, meaning energy framework pictures are only directly comparable if this scale factor is the same for all. To avoid crowded diagrams, weaker interactions below a certain energy threshold can be omitted, and because of this the absence of cylinders along a particular direction in the energy framework does not necessarily imply the complete absence of any stabilizing interaction in that direction. Judicious selection of this threshold is often required. A recent enhancement to energy framework diagrams allowed for the visualization of destabilizing (i. e., positive) interaction energies [39], depicted as yellow cylinders for the Eele and Etot frameworks. Following the calculation of model energies for a nearest neighbor cluster of molecules, energy framework diagrams can be produced with only a couple of steps: – Assuming the model energies for a cluster have been computed as in Figure 13.9, select Reset Crystal from the context menu by a right click on the graphics window background (also from the Display menu). This will return the image in the state at opening of the CIF file. Complete Fragments if necessary. – Energy frameworks are best applied to a larger cluster of molecules, so a sensible starting point is a multiple of unit cell packing diagrams. Select Generate Unit Cells from the top toolbar, choose the number of repeat cells along the a, b and c cell axes, then Complete Fragments. – Energy framework diagrams are produced by an option on the Display menu. This will also open a small dialog box that allows the user to toggle between separate pictures of frameworks for the Coulomb Energy, Dispersion Energy and Total Energy, as well as Total Energy (annotated) which enables the user to identify any pair of adjacent molecules with the appropriate interaction energy. – Clicking on Select Options in the Energy Frameworks dialog allows the user to change the scale for tube size (making the cylinders thicker or thinner as desired) as well as select a threshold energy. Framework cylinders for interaction energy magnitudes less than this value will not appear in the diagram. Figure 13.10 shows separate energy frameworks for SCCHRN02, based on the pairwise interaction energies in Figure 13.9. The Coulomb energy framework clearly highlights

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Figure 13.10: Separate energy framework diagrams for SCCHRN02, based on CE-B3LYP model energies. The same cylinder scale has been used for all diagrams, and energies with magnitude below 10 kJ mol−1 have been omitted. Four unit cells are displayed, with the ab plane horizontal in the figure, and molecules (displayed as tubes with hydrogen atoms omitted) repeat vertically along c.

the dominant role of the electrostatic energy for the cyclic hydrogen bonded dimer, as well as the fact that electrostatics plays a much less important role for all other close interactions. The dispersion energy contributes to all near-neighbor interactions, especially offset π⋅ ⋅ ⋅π stacking. The picture for the total energy suggests that this crystal structure can be described as packing of hydrogen-bonded pairs in approximate layers in the ab plane, with weaker interactions between layers along c. These observations may help to rationalize mechanical properties of the crystal in terms of intermolecular interactions, and can be compared and contrasted with conclusions derived from nanoindentation studies [42].

13.5.4 Lattice energy calculations Although the CE-B3LYP model energies were only calibrated against DFT results for nearest-neighbor pairwise interactions, they have been shown to provide remarkably accurate lattice energies for a wide range of molecular crystals. CE-B3LYP lattice energies for a set of 110 crystals show a mean absolute deviation from benchmark energies (derived from experimental sublimation enthalpies) of only 6.6 kJ mol−1 [43]. This excellent performance is due in part to cancellation of errors, but it lends confidence to the use of CE17 to obtain meaningful lattice energies with modest computational cost. Lattice energies are computed in CE17 by direct summation of total interaction energies over molecules B interacting with a central molecule A until Elat is converged to within a small energy threshold, using a cut-off based on the separation of molecular centers-of-mass, RAB : Elat =

1 ∑ E AB 2 R 0, which means that the electron density is overall convex: locally, there is a depletion of ρ(r). Generally, ∇2 ρ(r) < 0 in the regions associated with covalent bonds because there should be some degree of local charge accumulation. Regions with ∇2 ρ(r) > 0 are not expected to correspond to covalent bonds or charge accumulation whatsoever. In principle, ∇ρ(r)

358 | R. Laplaza et al. can be positive or negative, but must always be small, just as the Laplacian must be small and positive. To understand what the condition in equation (14.6) implies from a chemical perspective, one can picture a simple model. Starting from two far-away local maxima, which can be assimilated to two nuclear positions, ρ(r) will decrease exponentially in all directions until it disappears (at infinity) or meets the tail of the other maximum. If the two atoms are covalently bonded and the two densities add up, one would find some sort of charge accumulation, leading simply to a (3,-1) CP. However, if the two densities were already asymptotically decaying such that ∇ρ(r) ≈ 0, they could smoothly overlap and lead to a region with nearly constant density. with low gradient and small upwards concavity. This is the situation that one would attribute to a NCI in a molecular system. Indeed, the aforementioned situation takes place and can be monitored by plotting s(r) against ρ(r) in an isolated molecule (for instance H2 O in Figure 14.2a) and in aggregates of the same molecule (water dimer and trimer in Figures 14.2b–14.2c). The peak toward s(r) = 0 in Figure 14.2b corresponds to the region of space in which the hydrogen bond takes place. Note that many other regions in which the reduced density gradient will be zero exist, but the density is much higher in those, as is the case of the AIM-CPs that characterize the O–H covalent bonds. Other NCIs will be distinguished by analogous but different peaks in the same region, as it is the case in Figure 14.2c. The leftmost peak accounts for the van der Waals interaction shared by the three molecules, while the peak with higher density, similar to Figure 14.2b, corresponds to the pairwise hydrogen bonds, which is expected to be stronger.

Figure 14.2: 2D plot of s(r) against ρ(r).

The previous example highlights two major considerations with respect to s(r). The reduced density gradient tends to zero in the regions that we attribute to NCIs. Precisely due to that, it must be understood as a nonlocal feature: the region around which s(r) = 0, in which typically s(r) → 0, is equally important for interpretative purposes. This represents a paradigm shift with respect to the pointwise analysis of CPs that is often used for the topological study of ρ(r). This paradigm shift allows overcoming the problems presented in benzene dimer (Figure 14.1b).

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Therefore, isosurfaces of s(r) can be expected to qualitatively enclose NCI regions. Generally, a low isovalue (s = 0.5) is taken to ensure a representative depiction. Which value to take is, in principle, arbitrary. However, s ∈ 0.3–0.5 usually leads to representative pictures. On the other hand, a way to distinguish noncovalent from covalent interactions is needed. As we have seen, ρ(r) has much smaller values in the critical points associated with NCIs, so that classification can often be achieved with a simple density threshold. Hence, it is usually stated NCIs are located in those regions of space where s(r) → 0 and ρ(r) is small (typically s < 0.5 for SCF and s < 0.3 for promolecular calculations).

14.2.3 Topological classification of NCIs In the previous section, we have justified why the isosurfaces of s(r) should be able to identify regions where NCIs take place. However, NCIs are very diverse in strength, nature and origin. As far as strength is concerned, the most straightforward way to assess strength is evaluating the value of ρ(r) in the region under consideration, as it has classically been done in AIM theory with notable success. However, as NCIs are less local, a quantitative approach must imply considering the density in a complete region, and not a single CP. In the same spirit, examining the Laplacian can characterize the regions of space in which both AIM-CPs and non-AIM-CPs of the reduced density gradient take place. In non-AIM-CPs, the positivity of the Laplacian is ensured by the dominating contribution of λ3 (it is usually assumed that λ3 > λ2 > λ1 ), which measures the curvature in the internuclear axis of the closest two atoms. The other two eigenvalues will be measuring the perpendicular curvatures, and thus convey whether the plane under consideration (in AIM-CPs, such plane will be a separatrix between atomic boundaries) is controlled by charge accumulation or depletion. If the second most important eigenvalue, λ2 , is negative, then there is some degree of charge accumulation in the perpendicular plane, and so the interaction can be assumed to be somewhat bonding; if instead λ2 is positive, there is a charge depletion at the separatrix which is characteristic of steric crowdings. Following the previous analysis of λ2 , it follows that if s(r) → 0, λ2 > 0 and ρ(r) ≉ 0, the density left in this saddle point of high curvature is forced there by other interactions, and hence it can be likened to a repulsive interaction or steric clash. On the other hand, analogous regions with λ2 < 0 are concave in only one direction, but convex the perpendicular plane. Therefore, all density accumulated in such regions must be localized in the perpendicular plane, which alleviates the nuclear repulsion of the atoms involved in a very pointwise manner. Consequently, regions with λ2 < 0 and significant density must correspond to strong, stabilizing interactions. If ρ(r) ≈ 0 in the region under study, a very weak influence can be expected locally, and such

360 | R. Laplaza et al. Table 14.1: Classification of interactions according to NCI. ρ approx. (a.u.) >0.015 >0.015 0 λ2 ≃ 0

Lentil Compact Sheets

Strongly attractive Strongly repulsive Very weak

hydrogen bonds steric clash van der Waals

regions are attributed to van der Waals interactions of limited magnitude. We have summarized these results in Table 14.1. To make this easier to understand: if we concentrate on CPs, this means we are associating BCPs with attractive interactions, whereas (3,+1) and (3,+3) points (i. e., rings and cages) are associated with several atoms contributing importantly to a null gradient, leading to a steric clash. Applying these notions, we can now classify the previous diagrams in terms of attractive and repulsive character (governed by the sign of λ2 ) and strength (as measured by the value of ρ(r)). A scale can be constructed in terms of the sign of λ2 ρ(r), which can be used to color the isosurfaces of s(r) (Figure 14.3b). The same scale can be applied to 2D diagrams of s(r) against ρ(r) (Figure 14.3a). Generally, such scale matches our initial assumptions, with steric clashes shown as red isosurfaces, van der Waals interactions as thin, delocalized green regions, and strong attractive interactions (i. e. hydrogen bonds) as localized blue lentils (cf. Figure 14.3b).

Figure 14.3: Standard NCI index representations for an adenine-thymine dimer using a promolecular density. (a) s(r) plotted against sign(λ2 )ρ(r). (b) isosurfaces of s(r) = 0.3 colored by sign(λ2 )ρ(r).

14.2.4 Promolecular calculations The NCI technique coupled with optimized wavefunctions can be easily used to identify noncovalent interactions of small and medium-size systems. For application to

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large biomolecules, for which an SCF calculation is out of question due to the system size, different approximations to the density can be introduced. In NCIPLOT-1.0 [2], promolecular densities were used in order to reduce the computational cost. Within this approximation, the electron density is approximated as the sum of independent and spherical atomic electron distributions, which were tabulated as a database of atomic shell exponentials and directly included in the package. The promolecular approach yields the correct number and shape of peaks, but not their position on the sign(λ2 )ρ axis. Promolecular densities tend to be more diffuse and homogeneous. Hence, the gradient is generally smaller and the density is larger (Figure 14.4 top). However, taking this shift into account, surfaces qualitatively similar to the SCF calculation are obtained. In Figure 14.4 bottom this has been done shifting from s = 0.5 for SCF to s = 0.3 for promolecular. Among different SCF methods, only some differences appear between HF and DFT due to their different effect on electron localization, but very little differences are observed among functional approximations. The reader is referred to [8] for a detailed analysis of the stability of densities depending on the method.

Figure 14.4: NCI index representations for a phenol dimer using promolecular densities or DFTcalculated (B3LYP/def2-SVP) densities. In the 3D representations, s(r) = 0.3 for the promolecular case and s(r) = 0.5 for the DFT case.

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14.3 NCIPLOT 14.3.1 Basic run The latest release of NCIPLOT, NCIPLOT4 [9] is publicly available at: https://github.com/juliacontrerasgarcia/nciplot. There is a manual available in the same repository that covers the details of the current release. The minimal input is composed of two pieces of information: – PIECE 1 (line 1) must contain the number of files to be analyzed – PIECE 2 (the following lines) must contain as many file names as indicated in line 1. The type of calculation to be run (SCF or promolecular) is determined by the extension of the files to be read (which must all be of the same type): – name.xyz (Promolecular approximation). It requires an xyz file and is recommended for big systems – name.wfn or name.wfx wavefunction file in the AIMPAC WFN format (SCF calculation) The number of files is typically 1 for calculations, where all NCIs within the given systems are computed: 1 waterdimer.wfn A greater number of files allows to estimate inter- and intramolecular interactions separately. For example, 2 files are used for analyzing intermolecular interactions between a ligand and a protein. This is done with an extra keyword after the previous lines: 2 ligand.xyz protein.xyz INTERMOLECULAR Additionally, the user can define the cutoffs for ρ and RDG which define the boundaries for the NCIs search. For example, if a rather strong interaction is expected, it is possible to change the default values so that the interaction is not discarded. This is for example the case in transition states. Figures 14.5a–14.5b show the results for the transition state in the reaction LiH+NH3 →LiNH2 +H2 . At the transition state, only the

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Figure 14.5: Changes in the output with the CUTPLOT and the INCREMENTS options.

weak interaction between the Li atom and the rest is revealed by the default values. If the CUTPLOT value is increased, we can also visualize the forming and breaking of the N⋅ ⋅ ⋅H and H⋅ ⋅ ⋅H interactions (Figures 14.5c–14.5d). In this case, the input looks as follows:

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1 linh4.wfn CUTPLOT 0.16 0.5 The value 0.16 has been chosen so as to include the NCI peaks from the 2D plot (Figure 14.5c). This change of default values is also needed when interactions with a hole inside the surface appear, revealing that the default value coincides with a peak, leaving a part of it outside of the study range. A 3D grid is defined around the system of interest. ρ and RDG are then computed on the nodes of this grid, along with λ2 . Depending on the accuracy needed, it is possible to change the size of the step defining the grid. By default, the step is 0.1. Increasing the value allows for a faster calculation, but leads to a coarser representation (Figure 14.5e–14.5f): 1 linh4.wfn CUTPLOT 0.16 0.5 INCREMENTS 0.2 0.2 0.2 Once the input constructed, the code is invoked as follows: nciplot.x < inputfile [ > outputfile] The main results are collected in four output files: – name.dat file collects rho vs. RDG – name-grad.cube file with RDG – name-dens.cube file with sign(λ2 ) × ρ × 100 – name.vmd is a script for visualization of the results in VMD

14.3.2 New features The three-dimensional structure of proteins relies on a delicate balance of NCIs which regulate the folding of the given aminoacidic sequence [10–12]. Accurate evaluation of NCIs in biological systems is fundamental to rationalize the structure and to obtain insight into the function and the dynamics of macromolecules. It is also crucial in many biological processes in which biomolecules bind to ligands such as inhibitors or cofactors[13, 14], even more to other large biological systems like protein-protein interactions [15]. Moreover, protein structures are characterized by flexibility, which allows the exploration of the conformational space under biological conditions, which in turn has been associated to their folding and function [16, 17]. For this reason, the simulation of proteins often takes the form of classical molecular dynamics simulations, where

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the temporal evolution of the three-dimensional structure is explored using a force field. In this way, a trajectory is obtained that is expected to be representative of the dynamical features of the system under the given conditions, which most commonly are the equilibrium ones. NCIPLOT-4.0 [9, 4, 18] has been specially designed to extract more information from these large dynamic systems, and in a more efficient manner. With this aim in mind, three main features have been incorporated: (i) adaptive grids have been introduced, which allow for faster calculations, (ii) new input files can be read, which allow coupling with ELMO libraries for the reconstruction of more precise densities and (iii) integrals of NCI volumes have been implemented to quantify changes in dynamic evolutions.

14.3.2.1 Adaptive grids NCIPLOT-4.0 features the implementation of the adaptive grid approach. Starting from a coarse grid to quickly explore the whole 3D space around the molecule, the user is prompted to define a succession of progressively finer grids to refine NCI results in regions where NCIs are actually detected. In this way, only NCI relevant points are computed accurately, resulting in a ∼ 10-fold speed-up. In order to use this option, the number of adaptative grids and their relative size (finishing with “1,” i. e., the basic INCREMENTS) are defined after the keyword CG2FG (Coarse Grid To Fine Grid). By default, the CG2FG option is set to 1 1 using only onelevel grid, i. e., without acceleration. In practice, we recommend to set CG2FG to 3 4 2 1 or 4 8 4 2 1 with 3- or 4-level grids for acceleration. This leads to an input looking as follows: 1 bigmolecule.xyz CG2FG 4 8 4 2 1

14.3.2.2 Integrals A second feature that is new in NCIPLOT4 is the optional definition of integration ranges, to assess the relative strength of NCIs in different regions (attractive, repulsive and van der Waals) of the system. This tool will be collectively referred to as NonCovalent Interaction Integrals (NCIIs). Starting from a standard 2D NCI plot, which allows the identification of NCIs by pairs of values of s and sign(λ2 )ρ, integration regions can be defined as intervals on the sign(λ2 )ρ axis, which correspond to a window on NCIs of a given strength range.

366 | R. Laplaza et al. Collective integration of ρ in such ranges provides a single number, a NCII, which represents a measure of the given interaction strength window. The use of NCIIs is particularly interesting when dealing with complex systems such as biological macromolecules, e. g., proteins, where the focus is shifted from the study of a specific interaction to that of a specific interaction type [18]. In order to define these ranges, the keyword RANGE is invoked, and the sign(λ2 )ρ intervals subsequently defined: 1 bigsystem.xyz RANGE 3 -0.1 -0.015 -0.015 0.015 0.015 0.1 This input defines 3 ranges of integration: [−0.1, −0.015], [−0.015, 0.015] and [0.015, 0.1] to differenciate hydrogen bonds, van der Waals and steric crowding (following the same ranges as in Table 14.1).

14.3.2.3 wfx and ELMOs To refine the representation of NCIs in large systems, a new approach has been recently incorporated in collaboration with Genoni’s group for exploiting the computational advantages offered by the Extremely Localized Molecular Orbitals (ELMOs) [26]. ELMOs are orbitals strictly localized on small molecular units (e. g., atoms, bonds and functional groups) and, for this reason, easily transferable from one molecule to another, provided that the subunits on which they are localized have the same chemical environment. Due to their intrinsic transferability, ELMOs databases consider all the possible elementary units of the twenty natural amino acids in all their possible protonation states and forms. ELMOs are available for five standard quantum chemistry basis-sets (6-31G, 6-311G, 6-31G(d,p), 6-311G(d,p) and cc-pVDZ) [19–21]. It has been shown that they allow fast reconstructions of quantum mechanically rigorous electron densities and electrostatic potentials of systems ranging from small polypeptides to very large proteins. Therefore, ELMO databases have been coupled with the NCI method, giving rise to the NCI-ELMO approach, which allow instantaneous reconstruction of quantum mechanically rigorous electron distributions of polypeptides and proteins. To use ELMO approximation it is important to activate the nci=.true. keyword in ELMOdb. The ELMOdb package will produce the formatted checkpoint file which can then be transformed to wfn or wfx. Given the formatting of the wfn, wfx is usually recommended because it allows going beyond 999 atoms, and hence large systems

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can be analyzed. Thus, the input reads simply as the previous promolecular cases, but with the ELMO derived wfx file: 1 bigmolecule_from_ELMOs.wfx CG2FG 4 8 4 2 1

14.4 Examples Along this section, elaborated examples of the use of the previous keywords is provided. First, We will compare promolecular and ELMO results in protein systems. Then we will exploit the NCI integrals along MD trajectories in the identification and evolution of non covalent interactions.

14.4.1 NCI-ELMO We will present two examples of this new approach, focusing on two types of NCIs. In both examples, the NCIPLOT input files for the promolecular and ELMO approach are formally identical, in order to properly compare the results of both approximations. As a first example we will consider the hydrogen bond [22]. The system selected is the D192N mutant of Rhamnogalacturonan acetylesterase (PDB: 3C1U) [23]. In this system, a hydrogen bond is present between residues Asp75 and Asp87. As a second example, we will consider the metal-protein interaction in the HIV zinc finger-like domain [24]. The zinc is tetra-coordinated by four residues (three cysteine and one histidine residue in PDB: 2ZNF) [25]. This case is especially interesting since it is fundamental to take into account the diffuse nature of the electron density and oxidation state of the metal center, which cannot be properly represented in a promolecular approach. Since we want to focus on specific interactions in a big system, we have resorted to the keyword RADIUS, which limits the analysis to a user defined region of space defined by the center of the cage and its radius. Moreover, the inclusion of the keyword CUTPLOT introduced previously can help to properly fill the 3D plot. 1 name_file.xyz RADIUS 24.658 17.529 7.708 1.8 CUTPLOT 0.2 0.6 Figure 14.6A shows a strong and attractive hydrogen bonding interaction, corresponding to the blue RDG isosurface depicted in the 3D NCIPLOT. To compare these results with those obtained with the promolecular approach, their 2D NCIPLOT are presented

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Figure 14.6: Strong hydrogen bond in the PDB: 3C1U. (a) RDG isosurface obtained at NCI-ELMO level and showing the interaction under examination; (b) 2D RDG plots obtained at promolecular and ELMO levels for all the considered basis-sets; and (c) Zoomed 2D RDG plots obtained at promolecular-NCI and NCI-ELMO levels for all the considered basis-sets. Reprinted with permission from [26]. Copyright (2019) American Chemical Society.

in Figures 14.6b and 14.6c. Each NCI computation provides an RDG peak occurring at a negative value of the “signed” electron density, which clearly corresponds to the hydrogen bond under examination. We can see that the promolecular peak associated with the hydrogen bond interaction is shifted with respect to the ELMOs results. The RDG peak practically shifts toward less negative values as the size and the quality of the adopted basis-set increase (i. e., 6-31G, 6- 311G, 6-31G(d,p), 6-311G(d,p) and cc-pVDZ). In the second example (Figure 14.7), it is important to recall the construction of the transferable density of the metal center. Zn is a charged metal center. The promolecular density is constructed exploiting spherical densities of neutral atoms while ELMO constructs the density on a model molecule mimicking the chemical environment of the metal in the polypeptide. For the sake of completeness, the DFT computation is included. As expected, Figure 14.7b shows that promolecular charge distribution does not allow to reproduce the reference spikes obtained at the DFT level, while ELMO results show a quantitative agreement. As we can see in the examples previously reported, ELMOdb outperforms the promolecular ones. Of course this comes at a higher but bearable computational cost: for the HIV Zinc finger-like domain (275 atoms and 1098 electrons) the CPU time for

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Figure 14.7: Interactions between the Zn2+ ion and the coordinating residues (Cys3, Cys6, Cys16 and His11) in the HIV Zinc fingerlike domain. a) RDG isosurfaces obtained at NCI-ELMO level and showing the interaction under examination; 2D RDG plots obtained at promolecular-NCI, NCI-B3PW91 and NCI-ELMO levels for basis-sets; b) 6-31G ; c) cc-pVDZ. Reprinted with permission from [26]. Copyright (2019) American Chemical Society.

the promolecular calculation is 1.6”, while for ELMOdb and DFT wavefunctions the timings are 12’40” and 51’36”, respectively, with the cc-pVDZ basis set.

14.4.2 Analysis of MD trajectories We will use NCIIs to analyze selected noncovalent interactions along a classical MD trajectory. First, we will study the interactions responsible for the stabilization of the fibrillar three-dimensional structure of α-synuclein. The misfolding and aggregation of this intrinsically disordered protein is associated to Parkinson’s disease, and the abnormal accumulation of proteins in the brain is more broadly associated to the socalled prion and prion-like diseases, that include a large variety of neurodegenerative conditions [27]. The starting point for the simulations presented in this section is the atomicresolution solid-state NMR structure of an α-synuclein fibril extracted from PDB 2N0A [28]. This structure shows a tightly packed parallel in-register β-sheet core with a greek key motive embedded in flexible entropic briskers (Figure 14.8). A salt bridge between GLU46 and LYS80 concurs in stabilizing this architecture. Each strand

370 | R. Laplaza et al.

Figure 14.8: α-synuclein truncated model (residues THR22 to GLU105) with indication of the regions involved in the NCII analysis. Apolar fragment 1 encompasses residues 45–55, apolar fragment 2 residues 71–78 and apolar fragment 3 residues 82–93.

is composed of 140 residues. In the analysis presented here, we will focus on the structured core of the fibril. Only the central region encompassing residues from THR22 to GLU105 will be maintained, removing the N- and C- terminus regions. This truncated model is shown in Figure 14.8. To present an application example of NCIIs with a temporal resolution, we will analyze noncovalent interactions of the α-synuclein system along classical molecular dynamics trajectories. In particular, we will use NCIIs to compare two different situations: (1) a trajectory obtained as a single 100 ns molecular dynamics simulation, (2) a trajectory obtained as the collection of ten independent 10 ns simulations. The difference between these two situations is the extent of conformational space that the α-synuclein model is allowed to explore: in the case of one 100 ns simulation, the structure is allowed enough time to move further away from the original PDB structure than in the case of ten independent 10 ns simulations. In both cases, the initial structure is PDB 2N0A. The extension of conformational space sampling along a molecular dynamics simulation is particularly critical when dealing with fibril models. For instance, α-synuclein fibrils can reach lengths > 500 nm [29]; such a size is incompatible with atomistic simulations, so that in practice shorter fibril models are used. These short models are not expected to have the same stability as mature fibrils, and for this reason simulations need to be performed with great care not to stray too much from the experimental reference model. NCIIs can be used to follow the collective evolution of the NCI network, to assess how it evolves from the starting PDB structure. The first set of NCI that we will consider is the network of hydrophobic contacts between fragments F2 and F3 (Figure 14.8). The tight packing of hydrophobic residues

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at the core of this fibrils is fundamental for its stabilization. The evolution of the attractive and van der Waals contacts along the dynamics trajectories of α-synuclein are shown in Figure 14.9. Input in this case would look as follows: 2 fragment_f2.xyz fragment_f3.xyz INTERMOLECULAR CUTOFFS 0.2 1.0 INCREMENTS 0.2 0.2 0.2 OUTPUT 1 CG2FG 4 8 4 2 1 RANGE 4 -0.1 -0.015 -0.015 0.015 0.015 0.1 -0.03 -0.1 INCREMENT was set to 0.2 to allow the fast evaluation of a large number of NCIs, whereas OUTPUT 1 means that only the 2D information (the “.dat” file) is collected. Four integration ranges have been included: in addition to the common ones (attractive, van der Waals and repulsive), we also computed the evolution of strong hydrogen bonds (range [−0.03 −0.1]). Comparing the values of the attractive NCIIs in the 10 independent short simulations (Figure 14.9a) and in the single long simulation (Figure 14.9c), we can observe that the contribution of this range of noncovalent interactions is significantly inferior to that of van der Waals NCIs, which are shown in Figures 14.9b and 14.9d, respectively. This is informative on the nature of the noncovalent contacts established between F2 and F3, which is mainly dispersive. This is further confirmed by the low values assumed by the light blue lines of Figures 14.9a and 14.9c: these represent the evolution of an NCII corresponding to a sign(λ2 )ρ range representative of the strongest attractive noncovalent interactions, such as hydrogen bonds. These plots indicate that these contacts are absent between F2 and F3. Looking at van der Waals contacts (Figures 14.9b and 14.9d), a remarkable difference is observed between the trajectory obtained as ten independent simulation and the single long simulation. Indeed, while in the case of the short independent simulations, the van der Waals NCI fluctuates around the initial value of the experimental structure, in the case of the single long simulation we observe a drift, with a smooth decrease of the NCII with simulation time. This reflects a geometrical change that affects the whole structure along the single 100 ns trajectory: the fibril evolves by twisting around the growth axis. This conformational change diminishes the packing of the hydrophobic residues at the fibril core, reducing

372 | R. Laplaza et al.

Figure 14.9: Evolution of the attractive (blue) and van der Waals (green) NCII of F2 and F3 of the greek key motif along ten 10 ns independent simulations and one 100 ns simulation of α-synuclein. Two sets of values are shown in Figures 14.9a and 14.9c. Dark blue corresponds to the integration of sign(λ2 )ρ in the range [−0.100, −0.015], light blue to [−0.100, −0.030]. Values reported in Figures 14.9b and 14.9d refer to the [−0.015, 0.015] range.

both the repulsive and the van der Waals contribution to the NCI network. This phenomenon is not observed in the ten independent 10 ns simulations because this time frame is too short to allow the conformational change to take place. This example shows how NCIIs can be used to track the collective evolution of hydrophobic packing in proteins. Not only can NCIIs be used to follow these collective regions of protein contact, but also to focus on individual residues. To clarify this point, we will follow the evolution of NCIs between THR75 and THR92 (Figure 14.8). The input is formally identical to the previous one, with the only difference of the INCREMENT parameter set to 0.1 Å to obtain a higher resolution representation of NCIs, and now only these residues are included in files:

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2 THR_greek_key1.xyz THR_greek_key2.xyz INTERMOLECULAR CUTOFFS 0.2 1.0 INCREMENTS 0.1 0.1 0.1 OUTPUT 1 CG2FG 4 8 4 2 1 RANGE 4 -0.1 -0.015 -0.015 0.015 0.015 0.1 -0.03 -0.1 THR side chains have a central carbon atom with two substituents, a methyl and an hydroxyl group. In the initial PDB structure, THR75 and THR92 are in contact across the greek-key motif, and are oriented in such a way that the contact takes place through methyl groups, while hydroxyl groups are oriented toward the backbone (Figure 14.10).

Figure 14.10: Orientation of THR75 and THR92 in PDB 2N0A.

If this orientation was to change along the molecular dynamics trajectory, which, we recall, allows a sampling of the protein conformational space, THR75 and THR92 could come in contact through the hydroxyl groups, forming hydrogen bonds. To explore this possibility in our two different simulation setups, we will look at the collective THR75– THR92 contacts across the whole fibril model. NCII plots are shown in Figure 14.11,

374 | R. Laplaza et al. color-coded as in the previous example. Results corresponding to ten short independent simulations (Figures 14.11a and 14.11b) indicate that also for these residues, NCIs do not stray from the initial network of the experimental PDB structure, confirming that the dynamics simulation explores only a narrow portion of the conformational space of the protein. Conversely, results obtained form the single long simulation (Figures 14.11c and 14.11d) show that THR75 and THR92 undergo an important modification of the NCI network. Indeed, we can observe an increase of attractive (blue) contacts, and a concomitant decrease of van der Waals ones. From a structural point of view, this change corresponds to the rotation of THR residues and the formation of hydrogen bonds across the greek-key motif. This increases attractive interactions in general, and allows the formation of contacts in the strongly stabilizing sign(λ2 )ρ range (−0.1 to −0.03, light blue, Figure 14.11c). Since this rotation replaces methyl-methyl interaction with hydrogen bonds, it not surprising to observe a concomitant decrease of van der Waals interactions (Figure 14.11d).

Figure 14.11: Evolution of the attractive (blue) and van der Waals (green) NCII of THR75-THR92 of the greek key motif along ten 10 ns independent simulations of α-synuclein. Two sets of values are shown in Figures 14.11a and 14.11c. Dark blue corresponds to the integration of sign(λ2 )ρ in the range [−0.100, −0.015], light blue to [−0.100, −0.030]. Values reported in Figure 14.11b and 14.11d refer to the [−0.015, +0.015] range.

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Overall, this example shows the ability of NCIIs to follow the temporal evolution of the NCI network of complex biological macromolecules.

Abbreviations Atoms in Molecules – Critical Points Bond Critical Points Cage Critical Point Critical Point Density Functional Approximation Density Functional Theory Extremely Localized Molecular Orbital Kohn–Sham Molecular Dynamics Non Covalent Interaction Integral Nuclear Critical Point Nuclear Magnetic Resonance Protein Data Bank Quantum Theory of Atoms in Molecules Reduced density gradient Ring Critical Point Self-Consistent Field Uniform Electron Gas

AIM-CP BCP CCP CP DFA DFT ELMO KS MD NCII NCP NMR PDB QTAIM RDG RCP SCF UEG

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Appendix This section collects the geometries necessary to interactively go through the examples in Chapter 14. From Figure 14.1a: 6 Water dimer O 1.374799 H 1.041041 H 1.536210 O -1.393299 H -1.898988 H -0.530258

-0.097911 -0.279377 0.856594 0.106380 -0.654297 0.009332

0.074120 -0.817245 0.049892 -0.058089 0.255455 0.383656

0.718506 0.404847 -0.954706 -1.694267 -1.360563 -2.415340 -0.405706 -0.720511 0.955853 1.696846 1.359965 2.414388 -0.719016 1.693854 -0.405253 0.954382 -1.360032 1.360705 -2.414483 2.415504 -0.955575 0.406082 -1.696462 0.720989

-2.256663 -1.424589 -1.168030 -1.803486 -0.084817 0.115215 0.742929 1.582046 0.484693 1.123857 -0.598809 -0.794803 2.257904 1.803914 1.425288 1.168320 0.599236 0.084439 0.795866 -0.116154 -0.484991 -0.743673 -1.124175 -1.583503

From Figure 14.1b: 24 Benzene H C C H C H C H C H C H H H C C C C H H C C H H

dimer -0.376082 -0.993411 -1.180962 -0.709975 -1.965272 -2.107454 -2.560818 -3.168640 -2.375699 -2.839729 -1.592391 -1.442010 0.377973 0.710042 0.994534 1.180927 1.593657 1.964155 1.444276 2.105325 2.375810 2.559471 2.839963 3.166304

https://doi.org/10.1515/9783110660074-015

380 | From Figure 14.2: 3 Water molecule O 1.235960 H 0.508343 H 1.994973

1.013398 0.894197 1.197949

-0.068128 0.554944 0.499053

6 Water dimer O 1.380310 H 0.448708 H 1.861344 O -1.277718 H -0.660562 H -1.400591

0.629547 0.854418 1.440418 0.067519 -0.462458 -0.456623

0.339562 0.498911 0.541837 1.134498 1.657481 0.329874

9 Water trimer O 1.135489 H 0.366948 H 1.909086 O -1.159272 H -1.040338 H -1.035880 O -0.109426 H 0.529340 H -0.554120

1.105567 0.974069 0.881772 0.079097 -0.526924 -0.481670 -1.061922 -0.338922 -0.779195

-0.022582 0.577661 0.509662 1.105315 1.847208 0.305844 -1.165332 -0.969606 -1.974515

30 Adenine-Thymine dimer C 5.529181 0.019673 C 3.910800 0.013821 C 2.020152 -0.041500 C 1.895264 -0.004467 C 3.420984 -0.015172 N 1.281203 -0.033704 N 3.199339 0.020079 N 4.434771 -0.011167 N 5.275676 0.035517

0.572370 -0.920835 0.527584 -1.795649 0.384609 -0.593739 -2.055808 1.314131 -0.771575

From Figure 14.3:

Appendix | 381

N H H H H H C C C C C N N O O H H H H H H

1.417806 5.949726 0.404879 1.990568 1.225631 6.534449 -4.267497 -3.645552 -2.189950 -2.193403 -4.364427 -3.574986 -1.579461 -1.521286 -1.595491 -4.055531 -0.537638 -5.344736 -4.097742 -4.080519 -5.442211

-0.082260 0.059152 -0.048100 -0.040463 -0.000698 0.031830 0.011181 0.016211 0.001741 -0.020671 0.035614 -0.006688 -0.016783 0.005486 -0.034912 -0.009332 -0.027341 0.021211 -0.835981 0.912894 0.044829

1.724292 -1.518529 1.789071 2.547069 -2.646419 0.959035 -0.548500 0.651486 0.661825 -1.811084 1.959017 -1.731214 -0.578743 1.697236 -2.872227 -2.615341 -0.593275 -0.637566 2.554719 2.538179 1.809881

0.736060 1.909807 1.957125 -0.170659 -1.356965 -1.648302 -0.737990 0.447424 0.051661 -2.052450 -2.569258 -0.950191 1.161108 1.957125 0.814227 2.374730

0.392359 -0.082916 0.126550 1.131659 1.583091 1.305602 0.569836 0.113263 1.351331 2.152781 1.657635 0.347568 -0.459684 0.126550 -0.357181 0.754591

From Figure 14.4: 26 Phenyl dimer C -1.884971 O -1.384233 H -0.439003 C -1.122146 C -1.696802 C -3.029631 C -3.787504 C -3.221553 H -0.086217 H -1.095353 H -3.471997 H -4.824516 H -3.796892 O 1.304485 C 1.902977 H 1.903834

382 |

C C C C C H H H H H

1.206481 1.765735 3.010292 3.697605 3.145305 0.242450 1.222725 3.438952 4.663619 3.677286

0.089503 -1.078786 -1.522224 -0.787570 0.382173 0.444097 -1.642956 -2.430385 -1.120371 0.952874

-1.320660 -1.828017 -1.383096 -0.420859 0.097113 -1.657916 -2.573345 -1.781653 -0.067454 0.848883

Index 2nd-order perturbation theory 139 3-center-4-electron bond 23 χ 2 statistical agreement 279, 281, 296, 302 π-D-BOVB 166, 190 π-D-VBSCF 167 π-hole 313, 321, 322 π orbitals 125 σ-bond 115 σ-D-BOVB 166 σ-D-VBSCF 167 σ-hole 313, 321 σ orbitals 125 a posteriori localization 121, 126 a priori localization 121, 125 absolutely localized molecular orbitals 126 active/inactive space separation 159 Adenine:Thymine DNA base pair complex 62 aerogen bond 321 agreement factors 247 Ahlrichs type basis-sets 274 AIM2000 code 64 AIMAll 44, 61, 301 AIMAll code 64–66, 69, 71 AIMPAC 362 AIMPAC code 64 allenes 16 anharmonicity 119, 319, 325 antibond NBO 134 antisymmetrization 202, 207 antisymmetrizer 272 antisymmetry requirement 272 aromaticity 17, 18 arrow-pushing 158 aspherical scattering factor 243 association energy 148 asymmetric unit 290 atomic – charge 56, 59, 68, 124, 125, 239 – charge analysis 203 – coordinates 287, 293 – dipole 56 – displacement parameters 280, 293 – electron-electron repulsion 56 – electron-nucleus attraction 56 – force microscopy 230, 233 – multipole 56

– observable 52, 55, 56, 64, 65, 68 – orbital 115, 273 – population 55, 57, 69 – shells – density Laplacian 82 – ELI-D 83 – OEP 83 – theorem 54 – volume 55 attractor 46, 80 – nonnuclear 48 back donation 12 ball-and-stick representation of the source function percentages 62 basin 48, 56, 57, 60, 65 – atomic 84 – attraction 48 – cropping 86, 89 – ELI-D 105 – integrate block 87 – integration 66, 87 – population 87 – QTAIM 82, 84–90 – virial 55 basin file 86 basin intersections 106 – basin set 106 – bond polarity 107 basis set 65, 273, 275, 287, 314, 315, 317, 318 – basis set superposition error (BSSE) 314, 318 – complete basis set limit (CBS) 315–317, 323, 325 – correlation-consistent basis sets 315 BDO 159, 167, 172 benchmarking 314, 316, 317 – S22, S66 data sets 324 benzene 17, 159, 192 bifurcation of domains 103 binding free energy 325 blue shift 320 bond character 4 bond critical point 14, 49, 50, 52, 61, 62, 64, 68, 80, 354, 357, 360 bond degree 81 Bond Distorted Orbitals 159 bond energy 201

384 | Index

bond formation 115 bond index 5 bond order 141 bond pair 131 bond path 49, 67, 80 bond polarity 107 bond radius 50 bonding – chemical 43, 50, 61 – descriptors 4 – mechanism 200 Born–Oppenheimer approximation 116, 269 bosons 271 BOVB 163 Bragg law 221, 226 breathing orbital effect 166 cage critical point 50, 357 calculated structure factor magnitudes 296, 299 canonical molecular orbitals 113, 118, 121, 126 catalysis 26 – heterogeneous catalysis 26 – nickel-cobalt catalyst 27 cause and effect 60 – in source function analysis 61 CCSD(T)/CBS limit 316, 317, 323–325 CG2FG 365 chalcogen bond 321 charge – concentration 51, 61, 81 – bonded 52 – nonbonded 52 – valence 52 – density 12, 23, 27 – depletion 29, 51, 81 – distributions 202 – transfer 10, 56, 133, 148, 203, 204, 208, 313 Charge–Shift bond 168 chemical accuracy 316, 325 chemical bond 3, 113, 199 chemical bonding 9 chemical interactions 202 chemical structure 3 chemisorption 31 Chirgwin–Coulson weight 168, 183, 192 Cholesky decomposition 170 classical VB approaches 159, 163 classical VB structures 173 Clinton equations 280

closed-shell systems 272, 274, 284 cluster of charges and dipoles 285, 301, 302 cobalt phosphate 27 configuration interaction (CI) method 316 conflicting data points 282 constraints 245 continuity equation 53 contour map 69 contracted Gaussian basis-sets 273 control file 77 – basin block 85 – comments 78 – compute block 78 – DI block 93 – domain_natural_orbitals block 96 – icl_graph block 80 – integrate block 87 – intersections block 106 – isect_analyze block 107 – job run 78 – mesh definition 78 – refinement block 87 – search block 79 – select_basins block 92 – select_basins_domain block 96 – select_orbital block 92 – title line 78 coordination site 31 coordination sites of d-metals 28 core polarization 10, 244 core repulsion 148 core/valence electrons 125 correlation consistent basis-sets 274 correlation energy 311, 312, 314, 316 correlation of electronic motion 101 Coulomb interactions 116, 312 Coulson–Fischer proposition 159 counterpoise correction 314 coupled-cluster (CC) method 316, 317, 323, 324 covalency 94 covalent bonding 3, 115, 310 CPU 325 CRITIC2 code 64 critical point 45, 46, 48, 49, 52, 62, 65, 79–81, 257, 356, 357 – (3,+1) 357 – (3,+3) 357 – (3,−1) 357 – (3,−3) 357

Index |

– attractors 80 – degenerate 52 – nondegenerate 46 – principal curvature 81 – rank 46 – search block 79 – signature 46 crystal field effects 284 crystal structure 5 CRYSTAL17 code 64 crystallographic information file (CIF) 292 current density 53, 54 CUTPLOT 363, 367 cyclic alkyl(amino) carbenes (cAACs) 13 D-BOVB 166 DAFH 95 – expressed with DNOs 96 damping technique 293 dative bonding 17 dc-DFVB 176 Debye–Thomson model 221 deformation density 29, 248, 285, 299, 301 deformation density analysis 203 dehydration 27 dehydrogenation 27 deletion technique 138 delocalization 114, 118, 138 delocalization index 57, 59, 69, 90–100 – DI block 93 – DNO contributions 98, 99 – topological bond order 90 – topological valency 94 – using DAFH 95 – using norm parts 98 Density Functional Approximations (DFA) 356 density functional theory (DFT) 29, 200, 276, 284, 316, 317, 325, 355 – DFT-D method 317 – DFT-SAPT method 317 – self-consistent density-functional tight-binding (SCC-DFTB) method 317 density Functional Valence Bond 176 density functionals 276 density Laplacian 82–84 – compared to OEP 84 density matrix 288 – 1-RDM 75 – diagonal part of 75, 76

385

– first order 58 – N-matrix 75 – second order 56 density of states 28 Deoxyribonucleic acid (DNA) 310, 318, 319, 325 – double-helical structure 310 descriptor – basin_1 87 – cps_1 80 – field_1 80 – rfn_1 88 – wfn_1 78 determinantal wavefunction 205 determinants 118 Dewar structures 194 Dewar–Chatt–Duncanson model 17 DFVB 170, 175 diagonal relationship 20 diffraction resolution 222, 225 dihydrogen bond 320 DIIS 287, 293 dioxygen 131 dispersion forces 309 – dispersion coefficients 313 – dispersion energy 311, 313, 315, 316, 318, 322, 343, 347 – dispersion interactions 203, 310, 312, 313, 317, 324 – induced dipole-induced dipole interactions 313 dissociation energy 314 dmpkf 77 DNO – occupation 97 – unified 96 domain – bifurcation 103 – DAFH 95 – DNO 96 – ELI-D 102 – f-localization 79 – irreducible 79, 103 – lone-pair 103 – natural orbitals 95–100 – reducible 85, 103 domain_natural_orbitals block 96 donor-acceptor interaction 203, 204, 207, 208 donor-acceptor stabilization 139 dynamical system 45

386 | Index

Dyson orbitals 278 Edmiston–Ruedenberg 124 effective Hamiltonian 275, 277 Ehrenfest force 55, 56, 66 eigenvalue 46, 50 eight-electron rule 21 electric field 312 electric multipole 312 – dipole 312 – octupole 312, 324 – quadrupole 312, 323 electrical interaction 148 electron – acceptor 319, 322 – accumulation 206 – coordinates 269–271 – correlation 116 – correlation effects 284 – delocalization 56 – density 9, 23, 44, 48, 50, 51, 53, 55, 59–64, 67, 69, 75, 206, 235, 269, 270, 275, 276 – charge concentration 81 – cusp 48 – field 47 – grid generation 77 – Laplacian 51, 52, 54, 59, 64, 68, 81 – promolecular 80, 89 – density database 250 – depletion 206 – diffraction 225 – donor 319–322 – localizability indicator (ELI) 44, 51, 100–109, 285, 299, 300 – atomic shells 104, 105 – basin intersections 105–108 – basins 104, 105 – basins, cropping 104 – bonds 105 – charge decomposition 108 – charge decomposition, DNOs 109 – domain bifurcation 103 – ELID_core 104, 105 – Fermi-hole curvature 101 – grid 102 – lone-pair 105 – orbital contributions 109 – pair-volume function 101 – singlet/triplet pairs 101

– topology 102 – localization 56, 57 – localization function (ELF) 44, 285, 299–301 – positions 269 – spin 269 electron-electron Coulomb repulsion 120, 123, 201 electron-nuclei attraction 201 electron-pair bond 203, 204, 207 electronegativity 14, 20, 320, 321 electronic density partitioning 125 electronic ground state 113 electronic Schrödinger equation 117 electronic structure system 129 electrostatic complementarity 341, 343 electrostatic energy 258, 312, 315, 343, 347 electrostatic interaction 202, 207, 312, 313, 316, 317, 322, 324 electrostatic moment 312, 313, 323–325 electrostatic potential 321 – molecular electrostatic potential (MEP) 321 electrostatics 148 ellipticity 12, 23, 50 ELMAM2 250 energy decomposition analysis 199, 202, 204, 209 energy expectation value 118 energy framework 345, 346 energy minimization 121 enhanced Wick’s theorem 172 enthalpy 314, 325 entropy 325 equidistant grid 77 exchange energy 315, 356 – exchange-induction 318 – exchange-repulsion 313 exchange integral 94 exchange-correlation – density 59 – energy 59, 70 – functional 201 – hole 57 exchange-repulsion energy 343 expectation value 55, 61 experimental electron density 76 experimental structure factor magnitudes 291, 296, 299 experimental wavefunction 269, 277, 280 experimental X-ray diffraction data 292

Index | 387

explicitly correlated methods 315 extremely localized molecular orbitals 126, 365, 366, 368 Fermi-hole curvature 101 fermions 271 field 45, 46 – gradient 45, 46, 48, 65 – Laplacian 51 – lines 45 – scalar 51, 64, 65 – vector 45 final wavefunction 203 fingerprint plot 333–336 first-order density operator 129 fluctuation 91, 94 – sum rule 91 Fock operator 275 formal charge 21, 133 Foster–Boys 123 Fourier transform 276, 280 fourth moment 123 fractal analysis 253 fragment population 95 functional 274, 276 GAMESS 44 GAUSSIAN 44 Gaussian basis functions 273 Gaussian function 273 Gaussian wave packet 119 Gaussian-type basis sets 273 generalized atomic scattering factors 217 Generalized Valence Bond 159 geometrical deformation 202 goodness-of-fit 247 grid definition – parallel 78 grid file – density Laplacian 82 – OEP 82 – rho 78 – visualization 78 Group 14 11, 17 group theory 203 GVB 159 H2 molecule 311 half-integral spin 271

halogen bond 321 Hamiltonian operator 270, 276, 314 Hansen & Coppens model 239 HAO 158, 167, 171 HAR 279, 280 Hartree product 117, 271, 272 Hartree wavefunction 202 Hartree–Fock calculation 284 Hartree–Fock equations 275 Hartree–Fock (HF) method 43, 57, 67, 117, 316–318 Hartree–Fock orbitals 278 Hartree–Fock spinorbitals 272 Hartree–Fock theory 200 Hartree–Fock wavefunction 274, 281 hc-DFVB 176 Heitler–London function 158 Heitler–London–Slater–Pauling functions 158 helium dimer 311, 312 Hellman–Feynman force 56, 66 Hellman–Feynman theorem 313 Hermitian – condition 54 – operator 54, 270, 276 Hessian matrix 45, 46, 50, 281, 282 hexasilabenzene 17 high resolution diffraction 225 Hirshfeld atom refinement 279 Hirshfeld surface 329, 331–334, 336, 337, 340, 341, 350 Hirshfeld surface analysis 329, 332 HLSP 158, 171 Hohenberg & Kohn theorem 275, 276, 280 HOMO 278 homometallic cooperativity 31 hybrid atomic orbitals 115, 158 hybridization 135, 158 hydrogen atom 114, 242 hydrogen bond energy 144 hydrogen bond (H-bond) 141, 310, 313, 317–319, 324 imides 21 Import2MoPro 249 INCREMENT 371, 372 Independent Atom Model 10, 218, 236 independent-electron molecular orbitals 117 independent-particle model 200

388 | Index

induction interaction 312, 315 – exchange-induction 318 influence function 60 integral spin 271 integrate – integrate block 87 – refinement block 87 intensities 292 interacting quantum atoms (IQA) 58, 59, 66, 70, 209 interaction – closed-shell 52 – shared-shell 52 interaction energy 59, 71, 201, 202 interatomic surface 48, 66, 70 interference effects 119 intermediate wavefunction 202, 205 intermolecular interaction energies 342 intermolecular interactions 290 internuclear repulsion 116 intersections block 106 ionic structures 162 ionization potential (IP) 313 irreducible representation 203, 205 isect_analyze block 107 – bond polarity 107 isosurface 69 IUPAC 319, 321 Jacobian matrix 45 Jacob’s ladder 276 Kekule VB structures 193 keyword – all_cps 80 – core_charge 87 – crop 80, 86 – ELID_core 104 – format 83 – mesh 78 – OEP 83 – parallel 78 – precision 88 – rho Laplacian 83 – saddles 80 – save_cps_1 80 – top 85 – using 78, 80, 85 kinematic approximation 224, 226

kinetic energy 55, 56, 58, 201 – atomic 54, 56, 66 – density 54 – operator 54, 116, 119, 120 Kohn–Sham density functional theory 200 Kohn–Sham effective potential 284 Kohn–Sham (KS) 276, 355 Kohn–Sham molecular orbital model 199, 209 L-BOVB 166, 179 L-VBSCF 179, 182 λ-DFVB 176 Laplacian 12, 23, 256 lattice energy 347 LBFGS 172, 173 LCAO 273 least squares fit 281 least-squares refinement 236 level-shift 293 Lewis model 158 Lewis structure 3, 15, 115, 131 Lewis-type description 131 limited memory Broyden–Fletcher–Goldfarb–Shanno method 172 linear combination of atomic orbitals 273 linear space 119 local NRT 142 local source 60 localization index 57, 69, 71, 90 localization metric 121 localization technique 121 localized chemical bonding 131 localized correlation methods 113 localized molecular orbital 113, 137 log file 78 – basins 86 – critical points 80 – DNOs 97 – integration 87 – refinement 88 lone electron pair 15 lone pair 131 low oxidation state 11 LUMO 278 Magnasco–Perico 124 magnetization 26 many body-wave function 113

Index | 389

many-body electronic Schrödinger equation 117 many-electron wave function 125 matrix elements 121 mean field 275 mesomery 158 micro-cell 100 Miller indices 280, 292 MO coefficients 274, 279, 280, 283 model Hamiltonian 277 model wavefunction 276, 278 modern valence bond theory 115 modified Fock operator 283 modified Hartree–Fock equations 283 modified neglect of diatomic orbital (MNDO) method 317 MOLCAS 44 molecular graph 80 molecular mechanics (MM) 311, 325 molecular orbital 43 molecular orbital bonding mechanism 199 molecular orbital delocalization 120 molecular orbital diagram 206 molecular orbital localization 122 molecular orbital theory see MO, 199, 200 molecular orbitals 4, 116, 273, 289, 296 molecular orbitals as linear combinations of atomic orbitals (MO-LCAO) 124 molecular scaffold 121, 123 molecular spinorbitals 272 molecular structure 116 Møller–Plesset (MP) perturbation method – MP2 316, 324 – scaled opposite-spin (SOS-MP2) 316 – spin-component-scaled (SCS-MP2) 316 Monte Carlo (MC) method – fixed-node diffusion Monte Carlo (FN-DMC) method 318 – Quantum Monte Carlo (QMC) method 318 MoPro 249 MoProGUI 251 MoProSuite 240, 249 MoProViewer 251 Morse potential 119 Morse relationship 47 Mott–Bethe formula 226 Mulliken charges 125 Mulliken projectors 124 multipolar expansion 241 multipolar model 10, 218, 219, 226, 235, 238

multipolar pseudo-atoms 238 multipolar refinement strategies 252 multipole model refinement 282 MULTIWFN code 64 N-heterocyclic carbene 11 Natural Atomic Orbitals 130 natural bond orbital (NBO) 129, 200 natural bond orbital (NBO) code 64 natural charge 133 Natural Electron Configuration 134 Natural Energy Decomposition 148 Natural Hybrid Orbital 134 natural ionicity 135 Natural Lewis Structure 131 natural orbitals 113, 123 Natural Resonance Theory 141, 144 NBO methods 301 NBO7 129, 130, 149 NCI 301, 353 NCII 366 NCIPLOT 301, 362 neutron diffraction 226 Newton–Raphson algorithm 173 Non Covalent Interaction analysis 301 non-AIM-CPs 357 non-Lewis-type 134 non-redundant covalent structures 161 non-redundant structures 164 noncovalent interaction index (NCI) 44 noncovalent interactions 310 nonnuclear attractor 84 norm parts 97 – DI 98 NPA charge 130, 146 NR 173 nuclear charges 270 Nuclear Critical Points (NCP) 357 nuclear magnetic resonance 228 nuclear positions 270 nuclear-electron attraction 116 numerical integration 65, 68 objective functional 121 occupied orbitals 122 OEO 159, 167, 172 OEP 82 ω-restricted space partitioning 100 one-electron potential 82

390 | Index

open-shell species 131 orbital 43, 44, 76 – electronic structure 199 – energies 275, 289, 296 – functions 239 – Gauss type 76 – hybridization 115 – interaction 202, 203, 207 – numerical 76 – overlap 123 – relaxation 207 – rotation 122 – self-repulsion energy 123 – Slater type 76 – space 4 ORCA 44 organometallic chemistry 11 orthogonality 129 orthonormality condition 275 Overlap Enhanced Orbitals 159, 192 overlap integrals 91 – core orbitals 91 – parallel run 91 overlap region 206 oxidation number 133 oxidation state 10, 11, 17 ozone 131, 187 p-type Gaussian function 273 pair bond formation 207, 208 pair bond wavefunction 207 pair density 56, 58, 76, 90 – hole-part 90 – integral 90 – same-spin 100 – singlet/triplet pairs 101 pair orbitals 113 pair-volume function 101 paired electrons 114 partial spin density reconstructions and effect on the wavefunction method 62 particle exchange 271 Pauli principle 117, 118, 202, 206, 207, 223, 230, 271, 272 Pauli repulsion 202, 204, 206–208 perturbation approach 313, 314, 317 perturbative analysis 138 Pipek–Mezey 124 plane-wave basis functions 274

pnicogen bond 321 polar bond 14, 21 polarizability 321, 324 – dipole polarizability 313 polarization 148, 203, 208 polarization coefficient 134 polarization energy 343 polarization interactions 312 polyimido sulfur 21 Pople basis-sets 273 population analysis 5 positive semidefinite 121 post-VBSCF 173 potential 119 potential energy density 81 PREINT 170 probability 75 procrystal 332, 337, 338 procrystal void analysis 337, 338 PROMOLDEN code 64 promolecular 361, 367 promolecular density 80, 89, 287 – basin charges 89 promolecule 332, 337, 340 propellane 20 properties in the crystal phase 284 proteins 318–320, 325 proton transfer reaction 145 QM packages 76 QTAIM 45, 47, 48, 50, 57, 58, 61, 64, 65, 301, 354, 356 – basins 55, 56, 60, 70 – subsystem 53–55 QTAIM software 64 quantum chemical topology (QCT) 44, 46, 60 quantum chemistry 272 quantum crystallography 215, 231 quantum effects 325 quantum mechanics 3, 43, 53, 55, 269, 272, 276 quantum numbers 114 quantum theory of atoms in molecules see QTAIM Quantum Theory of Atoms in Molecules (QTAIM) 10, 301 quantum-mechanical observable 3 RADIUS 367 RANGE 366

Index | 391

ratio – virial 82 RDM 172 reactivity 144 real space 4 real spherical harmonics 241 “real+virtual” atoms 250 reciprocal lattice matrix 284 red shift 319 reduced density 44 Reduced Density Gradient (RDG) 356 Reduced Density Matrices 172 refinement 10 – block 87 – file 88 – integration 88 – isect_analyze block 107 relief map 48, 49, 69 repulsive interactions 310, 313, 314, 323 residual density 285, 296, 301 resonance 21, 137, 146, 158 resonance energy 168, 184 resonance integral 207 resonance NBOs 144 resonance weight 141 resonance-assisted hydrogen bonding 142 restarting an XCW calculation 296 restraints 245 restricted density matrix 287 restricted Hartree–Fock calculation 287, 288 restricted spinorbital approximation 273 Ring Critical Points (RCP) 49, 357 Roothaan–Hall equations 275, 284 rotation matrices 122 Rumer basis 160 s-type Gaussian function 273 scale-factor 280, 291 scanning tunnelling microscopy 229, 278 scatter plot 299 scattering factor 236 SCF algorithm 272 SCF calculation 287, 288 SCF cycle 295, 298 SCF method 272 SCF procedure 287, 295 Schrödinger equation 200, 270, 271, 276 SCVB 159, 192 SD-BOVB 167

second moment 122 secondary extinction corrections 290, 291 select_basins block 92 select_basins_domain block 96 select_orbital block 92 Self Consistent Field (SCF) 272, 275, 361 self-energy 58, 59 separatrix 46, 48, 65 SHADE 254 silicon chemistry 11, 17 silicon(0) compounds 14 silicon(II) compounds 11 silylene 11 silylone 13 similarity restraints 249 single determinant 43, 69 single determinant wavefunction 272, 276, 279 single reference 113 single Slater determinant wavefunction 281, 283 single-ζ Slater functions 240 single-crystal X-ray diffraction 5 single-molecule magnets 25 singlet-triplet gap 13 Slater determinant 76, 118, 125, 272 Slater type orbitals 273 Slater-Condon rules 206 small molecule activation 11 software 5 software package 44, 64 source function – analysis 60 – analysis of binuclear transition metal complexes 62 – and σ/π holes 62 – and CC (carbon-carbon) bonds 62 – and chemical interactions 62, 64 – and chemistry 60 – and DFT 61 – and DNA base pair complexes 62 – and hydrogen bonds 64 – and metal-metal bonds 62 – and model independency 61 – and spin density delocalization 62 – and the choice of reference points 62 – and use in HPLC separation design 62 – atomic 60 – ball and stick model representation 61 – descriptor 59

392 | Index

– electrostatic potential (reconstruction) 61, 62 – from X-diffraction intensity data 61 – (partial) density reconstruction in the DNA base pair complexes 62 – percentage contributions 61 – reconstructed partial densities 62 – reconstructed partial densities vs standard partial densities 62 – reconstructed partial densities (definition) 62 – reconstructed partial spin electron densities 62 – spin (partial) density reconstruction in azido Cu(II) molecular complexes 62 source function electron spin density – reconstruction 61 space group 290 spatial delocalization 119 spatial molecular orbitals 275 spatial orientation 115 spherical harmonic 56 spin polarization 356 Spin-Coupled Valence Bond 159 spin-statistics theorem 271 spinorbitals 271, 272, 275 spreading 119 stacking interactions 323 standard deviation 122 static deformation electron density maps 247 stationary solutions 120 statistical (in)dependence 57 steric crowding 359, 366 steric repulsion 206, 208 strain energy 201, 202 stress tensor 55 structure factor 10, 236, 280 structure factor magnitudes 279, 291 structure factor operator 283 structure–property relationship 26 sublimation enthalpy 349 sulfur-nitrogen chemistry 20 sulfuric acid 24 superbasin 85, 94 surface term 53, 54 sustainability 27 symmetry operations 284 symmetry-adapted-perturbation theory (SAPT) 209, 315 synthesis – synthetic strategy 9

TAPE21 76, 77 – dmpkf 77 target restraints 249 tetrel bond 321 thermal ellipsoids 280 thermal smearing 290 thermodynamic characteristics 325 time-dependent Schrödinger equation 119 time-independent Schrödinger equation 116 Tonto 269, 284, 285, 287, 288, 291, 292, 294, 295, 299, 301, 303 Tonto input file 285, 290, 297 Tonto output file 285, 295, 298 topological analysis 50, 65, 301 – electron density 44 topological bond order 90, 94 topological valency 94 topology 12, 44, 47, 49, 53, 64, 65 – of the Laplacian 51 TOPOND code 64 two-center four-electron interaction 203, 206, 208 uncertainty principle 119 unconstrained Hartree–Fock calculations 285 unconstrained restricted Hartree–Fock calculation 285 Uniform Electron Gas (UEG) 356 unitary transformation 118, 121, 122, 125 unrestricted spinorbital approximation 273 valence atomic overlap 114 valence bond configuration interaction 173 valence bond polarizable continuum model 177 valence bond second order perturbation theory 173 valence bond theory 114, 157, 200, 207 valence bond wavefunction 207 valence electron density 237 valence expansion 21, 23, 25 valence shell charge concentration 12, 29 van der Waals (vdW) equation 309 van der Waals (vdW) interactions 311 van der Waals (vdW) molecule 311 van der Waals (vdW) radius 320, 322 variance-covariance matrix 245 variation (supramolecular) approach 314 variational principle 274, 275, 281 VB diagrams model 168

Index | 393

VB structure weights 168 VB structures 158 VB wave function 158 VBCI 173, 174 VBCIS 174 VBCISD 174 VBEFP 170, 175, 177 VBPCM 170, 175, 177 VBPT2 173, 174 VBSCF 163, 164 virial 54 – atomic 55 – basin 55 – surface 55 – theorem 55, 58 virial ratio 81, 82 virtual orbitals 122, 123 VMD 364 VMoPro 249 VSEPR 52 Watson–Crick base pair 142 wave packet 119 wavefunction 43, 53, 54, 63, 269, 270, 275, 276, 278, 279 – file 66 wavefunction file 76, 77, 301 – conversion ADF 77 wavefunction interaction 207 wavefunction relaxation 207 wavefunction theory 200 wfx 366

X-ray charge density 216 X-ray constrained Hartree–Fock calculation 284, 285, 290 X-ray constrained Hartree–Fock wavefunction method 283 X-ray constrained restricted Hartree–Fock calculation 293, 295 X-ray constrained wavefunction 269, 279–281, 283, 284, 285, 297 X-ray constrained wavefunction calculation 269, 283, 285, 293, 293, 296, 301, 303 X-ray diffraction 9, 48, 60, 216, 217, 220–222, 224, 226–230, 232, 233, 272, 276 X-ray diffraction data 279, 284, 290 X-ray diffraction experiment 280 X-ray diffraction intensities 280 X-ray restrained wave function 219 XCW analysis 285, 303 XCW termination criterion 282 XCW termination problem 281, 296 XD-2006 code 64 XD2016 240, 259 Xiamen Valence Bond 170 XMVB 170 XMVB-GUI 170 ylide 23 zero point vibration 223 zero-flux – condition 65 – surface 46, 68 zero-point vibration energy (ZPE) 325

Erratum to: Chapter 2 Chemical concepts of bonding and current research problems, or: Why should we bother to engage in chemical bonding analysis? By Dietmar Stalke published in: Simon Grabowsky, Complementary Bonding Analysis, 978-3-11-066006-7

Erratum The original version of Chapter 2 “Chemical concepts of bonding and current research problems, or: Why should we bother to engage in chemical bonding analysis?” has been revised by the author. The subject index has been revised accordingly.

The updated original chapter is available at DOI: https://doi.org/10.1515/9783110660074-002 https://doi.org/10.1515/9783110660074-016