Intermolecular Forces and Clusters I (Structure and Bonding, Volume 115)

115 Structure and Bonding Series Editor: D. M. P. Mingos

Intermolecular Forces and Clusters I Volume Editor: D. J. Wales

Springer

Berlin Heidelberg New York

The series Structure and Bonding publishes critical reviews on topics of research concerned with chemical structure and bonding. The scope of the series spans the entire Periodic Table. It focuses attention on new and developing areas of modern structural and theoretical chemistry such as nanostructures, molecular electronics, designed molecular solids, surfaces, metal clusters and supramolecular structures. Physical and spectroscopic techniques used to determine, examine and model structures fall within the purview of Structure and Bonding to the extent that the focus is on the scientific results obtained and not on specialist information concerning the techniques themselves. Issues associated with the development of bonding models and generalizations that illuminate the reactivity pathways and rates of chemical processes are also relevant. As a rule, contributions are specially commissioned. The editors and publishers will, however, always be pleased to receive suggestions and supplementary information. Papers are accepted for Structure and Bonding in English. In references Structure and Bonding is abbreviated Struct Bond and is cited as a journal.

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ISSN 0081-5993 (Print) ISSN 1616-8550 (Online) ISBN-13 978-3-540-28194-8 DOI 10.1007/b101390 Springer-Verlag Berlin Heidelberg 2005 Printed in Germany

Series Editor Prof. D. Michael P. Mingos Principal St. Edmund Hall Oxford OX1 4AR, UK [email protected]

Volume Editor D. J. Wales Chemistry Department University of Cambridge Lensfield Road Cambridge CB2 1EW, UK [email protected]

Editorial Board Prof. Peter Day

Prof. Herbert W. Roesky

Director and Fullerian Professor of Chemistry The Royal Institution of Great Britain 21 Albermarle Street London W1X 4BS, UK [email protected]

Institut for Anorganic Chemistry University of Göttingen Tammannstr. 4 37077 Göttingen, Germany [email protected]

Prof. Jean-Pierre Sauvage Prof. Thomas J. Meyer Department of Chemistry Campus Box 3290 Venable and Kenan Laboratories The University of North Carolina and Chapel Hill Chapel Hill, NC 27599-3290, USA [email protected]

Faculté de Chimie Laboratoires de Chimie Organo-Minérale Université Louis Pasteur 4, rue Blaise Pascal 67070 Strasbourg Cedex, France [email protected]

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Anthony Stone

Intermolecular Forces and Clusters: Contributions in Honour of Anthony Stone David J. Wales Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, UK [email protected]

It is a great pleasure to introduce this collection of papers in honour of Anthony Stone. The title of the collection reflects Anthony’s ground-breaking contributions to our understanding of intermolecular forces [1], and how this work has been exploited to describe the structure and dynamics of molecular clusters. I personally feel indebted to Anthony for all the insight he has given me over my career, starting with undergraduate lectures that set the highest standard for clarity and accuracy. I know that the contributors who were fortunate enough to be guided by Anthony for either PhD or post-doctoral projects share these feelings, and several of them are represented by chapters in this collection. The two volumes of Structure and Bonding in this series are completed by papers from other researchers who have been influenced or helped by Anthony in their endeavours. In organising this collection I found that similar unsolicited testaments to Anthony appeared from many of the authors. For example, his depth of understanding, patience in discussions, and quiet good humour were frequently mentioned. His service to the community, in developing and maintaining programs such as ORIENT, and in providing an outstanding textbook [1], has undoubtedly contributed to his far-reaching influence on the field of intermolecular forces. Some of the contributions address the calculation of intermolecular forces at a fundamental level, while the majority are concerned with applications, ranging from water clusters, through surfaces, to crystal structures. Szalewicz, Patkowski and Jeziorski provide a timely review of how perturbation theory can be used to address intermolecular forces in a systematic way. In particular, they describe a new version of symmetry-adapted perturbation theory, which is based on a density functional theory description of the monomers. The interpretation of bonding patterns for both intra- and intermolecular interactions is addressed in Popelier’s review, which focuses on quantum chemical topology. He suggests a novel perspective for treating several of the most important contributions to intermolecular forces, and explains how these ideas are related to quantum delocalization. Water clusters continue to be a particularly active area of research for both theory and experiment. The chapters authored by Xantheas and by Christie and Jordan both address the structure and bonding in water clusters using

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ab initio electronic structure theory. Christie and Jordan show how an nbody decomposition of the binding energy can be used to perform MP2-level calculations for clusters with up to fifty water molecules. Xantheas also considers an n-body decomposition, and analyses different approaches to the systematic development of empirical intermolecular potentials based upon ab initio data. Millot’s review extends some of these ideas to molecular dynamics simulations using both empirical and quantum mechanical treatments of the electronic structure. He compares results obtained for hydrated halide ions within these different frameworks, and also considers dynamical simulations of silicon clusters. Ewing also explores the theme of hydration in his chapter, this time in terms of adsorption of water on the surface of a sodium chloride crystal. Here we see how intermolecular forces determine the efficiency with which salt is dispensed from a shaker, and learn that the actual mechanism by which salt dissolves in water is still poorly understood. Tsuzuki’s review concerns intermolecular forces that involve the microscopic surfaces of aromatic rings. These interactions are important for molecular recognition in biological systems, and play a major role in determining the crystal structure when aromatic rings are involved. Crystal structure prediction itself is addressed directly by Price and Price. They review the critical role of the intermolecular potential in providing useful predictions to discriminate between polymorphs, and highlight recent progress in this field. Here the distributed multipole analysis and distributed polarizabilies, pioneered by Anthony Stone [1, 4–10], play a crucial role in the development of accurate potentials. The application of such ideas to the structure of condensed matter provides a logical extension of earlier work for van der Waals complexes [11, 12]. Anthony Stone has also made influential contributions to fullerene research and to the bonding in inorganic clusters. The chapter by Soncini, Fowler and Jenneskens considers ring currents and aromaticity in fullerenes, using selection rules based on angular momentum theory to determine the aromatic and anti-aromatic terms. This approach ties in neatly with Anthony Stone’s tensor surface harmonic (TSH) theory, which enables us to understand and predict electron counts of main group and transition metal clusters [13–16]. It also prompts me to conclude this overview with some of my own reminiscences. The TSH theory was largely worked out before I began my own PhD with Anthony in 1985. However, I remember that in my first term of research Anthony received a suggestion from Christopher Longuet-Higgins (his former PhD supervisor [17]) that it might be helpful to apply a symmetry-based approach to the π system of C60 . This idea proved to be very fruitful, and we quickly produced a descent-in-symmetry picture for C60 , based on the wavefunctions for a free particle-on-a-sphere. At the time the suggestion that C60 might be particularly stable as a truncated icosahedron [18] was unproven. Furthermore, the descent-in-symmetry

Intermolecular Forces and Clusters: Introduction

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Fig. 1 Top and side views of the pathway connecting the lowest two fullerene isomers of C60 calculated with a plane-wave density functional approach using the LDA and BLYP functionals [2]. The middle panel is the transition state and the end panels are the two minima in each case. The asymmetric path was calculated using the LDA functional with a cutoff of 30 rydberg and the symmetric path corresponds to the BLYP functional and a cutoff of 40 rydberg [2]. Reproduced with permission from D. J. Wales, Energy Landscapes, Cambridge University Press, Cambridge (2003)

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approach suggests that other reasonably spherical arrangements of carbon atoms should have similar π delocalisation energies. We therefore continued our study by considering a few alternative C60 fullerenes, and as an afterthought we decided to think about how such structures might interconvert [19]. The main result was the process illustrated in Fig. 1, which has since come to be known as the ‘pyracylene’ or ‘Stone-Wales’ (SW) rearrangement [20, 21]. The most recent calculations conducted to characterise the SW process have identified two possible transition states, both corresponding to concerted, single step mechanisms with similar barriers [2, 22] (Fig. 1). Furthermore, the same generic mechanism connects a whole hierarchy of C60 fullerenes [23, 24]. The energy of these structures increases systematically with the number of rearrangements, and when combined with the large barrier height produces the ‘willow tree’ disconnectivity graph shown in Fig. 2 [2, 25]. The urge to study rearrangements and global potential energy surfaces was therefore imprinted upon me at an impressionable age, and has influenced virtually all my research ever since. I am therefore led to reflect almost every

Fig. 2 Disconnectivity graphs for minima and transition states in the five lowest StoneWales stacks [3] of C60 calculated using a plane-wave implementation of density functional theory [2]. The graphs on the left and right correspond to the LDA and BLYP functionals, respectively. The vertical energy scales are the same and the energy zero has been shifted to buckminsterfullerene in both cases. The structures of six minima are indicated, including buckminsterfullerene and the next-lowest structure with C2v symmetry. Reproduced with permission from D. J. Wales, Energy Landscapes, Cambridge University Press, Cambridge (2003)

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day, on how lucky I was to have Anthony Stone as a mentor on this journey. Although it was not easy to persuade him that his achievements should be highlighted, I hope he will be happy with the following tributes from some of his friends. June 28, 2005

David J. Wales

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

Stone AJ (1996) The Theory of Intermolecular Forces. Clarendon Press, Oxford Kumeda Y, Wales DJ (2003) Chem Phys Lett 374:125 Austin SJ, Fowler PW, Manolopoulos DE, Zerbetto F (1995) Chem. Phys. Lett. 235:146 Stone AJ (1981) Chem Phys Lett 83:233 Stone AJ (1985) Mol Phys 56:1065 Stone AJ, Alderton M (1985) Mol Phys 56:1047 Price SL, Stone AJ (1987) J Chem Phys 86:2859 Stone AJ (1989) Chem Phys Lett 155:102 Stone AJ (1989) Chem Phys Lett 155:111 Williams GJ, Stone AJ (2003) J Chem Phys 119:4620 Buckingham AD, Fowler PW (1985) Can J Chem 63:2018 Buckingham AD, Fowler PW, Stone AJ (1986) Int Rev Phys Chem 5:107 Stone AJ (1980) Mol Phys 41:1339 Stone AJ, Alderton MJ (1982) Inorg Chem 21:2297 Stone AJ (1984) Polyhedron 3:1299 Mingos DMP, Wales DJ (1990) Introduction to Cluster Chemistry. Prentice-Hall, Englewood Cliffs Mills I, Salem L, Stone A (2005) Mol Phys 103:141 Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE (1985) Nature 318:162 Stone AJ, Wales DJ (1986) Chem Phys Lett 128:501 Fowler PW, Manolopoulos DE (1992) Carbon 30:1235 Fowler PW, Manolopoulos DE, Ryan RP (1992) J Chem Soc, Chem Comm 408 Bettinger HF, Yakobson BI, Scuseria GE (2003) J Amer Chem Soc 125:5572 Liu X, Klein DJ, Seitz WA, Schmalz TG (1991) J Comp Chem 12:1265 Manolopoulos DE, May JC, Down SE (1991) Chem Phys Lett 181:105 Wales DJ (2003) Energy Landscapes. Cambridge University Press, Cambridge

Contents

Quantum Chemical Topology: on Bonds and Potentials P. L. A. Popelier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Angular Momentum and Spectral Decomposition of Ring Currents: Aromaticity and the Annulene Model A. Soncini · P. W. Fowler · L. W. Jenneskens . . . . . . . . . . . . . . . .

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Modelling Intermolecular Forces for Organic Crystal Structure Prediction S. L. Price · L. S. Price . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Molecular Dynamics Simulations and Intermolecular Forces C. Millot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Interactions with Aromatic Rings S. Tsuzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Author Index Volumes 101–115 . . . . . . . . . . . . . . . . . . . . . . 195 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Struc Bond (2005) 115: 1–56 DOI 10.1007/b135617 © Springer-Verlag Berlin Heidelberg 2005 Published online: 19 July 2005

Quantum Chemical Topology: on Bonds and Potentials Paul L. A. Popelier School of Chemistry, University of Manchester, Sackville Site, Manchester M60 1QD, UK [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Quantum Chemical Topology . . . . . . . . . . . . . . . . . . . . . . . . .

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3 3.1 3.2

The Chemical Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Computable Bond? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 6 8

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Attractive or Repulsive Interactions? . . . . . . . . The Birth of the Bond Path . . . . . . . . . . . . . . The Bond Path in Action . . . . . . . . . . . . . . . The Debate Starts . . . . . . . . . . . . . . . . . . . Input from High-Resolution X-ray Crystallography The Debate Continues . . . . . . . . . . . . . . . . . Non-Covalent Interactions in Biomolecules . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . .

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5 5.1 5.2

Reflections and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . The Sudden (Dis)Appearance of Critical Points . . . . . . . . . . . . . . . . Energy as a Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27

6 6.1 6.2 6.3 6.4 6.5

Towards a QCT Force Field . . . . . . . . . . . . . . . . Convergence . . . . . . . . . . . . . . . . . . . . . . . . The Electrostatic Potential . . . . . . . . . . . . . . . . The Electrostatic Interaction: Structure and Dynamics Coulomb Energy . . . . . . . . . . . . . . . . . . . . . . Auxiliary Insight . . . . . . . . . . . . . . . . . . . . . .

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Future Perspectives on QCT Potentials . . . . . . . . . . . . . . . . . . . . The Need for Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Short Range Interactions and Overlap . . . . . . . . . . . . . . . . . . . . .

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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Proof of the Falsehood of Cassam-Chenaï and Jayatilaka’s Counterexample

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Covalency and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . .

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Abstract In this essay-like review two different aspects of the “Quantum Chemical Topology” (QCT) approach are critically discussed at great length. One is the ability to compute bonding patterns in single molecules, complexes or crystals based on the topology of the electron density, via the concept of the bond path. The other aspect is the development of an intermolecular potential based on multipole moments of topological atoms. Although bond paths have been embraced by a wide community and successfully applied to reveal interesting insights, there is an ongoing debate about their interpretation. The ongoing work on QCT potentials leads to a consistent force field for proteins and liquid water. Radical ideas on the modelling of polarisation, the short-range regime, overlap, and the nature of quantum delocalisation are presented. Keywords Atoms in molecules · Chemical bonding · Force fields · Intermolecular potential · Multipole moments · Quantum chemical topology · Electrostatic potential · Coulomb interaction · Reduced density matrix · Polarisation · Quantum mechanics · Convergence

1 Introduction It is with great pleasure that I am contributing to a volume in honour of Anthony Stone, a scientist whose scholarship and sense of rigour I have always much appreciated. One of my first memories of Anthony, as a newly arrived postdoctoral fellow at Cambridge, was standing in his office and noticing a captivating card on the wall, which read ‘Think!’. In response to my curiosity about this card Anthony explained that it reflected the philosophy of his PhD supervisor, Longuet-Higgins, who insisted that researchers spent more time thinking. In the current climate of information deluge and mechanical research metrics [1] I have often been tempted to put a similar sign up in my own office! I have tried to write my contribution in the spirit of Anthony’s card by weaving some thoughts, several of which may appear controversial or unpolished, in what would otherwise be a mere review. Indeed, I shall present some new data and ideas and critically discuss (most) recent and older papers published in the area of QCT. In this sense my contribution shares features with an essay and complements two “uncritical” but extensive reports [2, 3]. Taken together, these reports exhaustively surveyed QCT papers from June 1998 until June 2001, as well as selected papers from the Bader group from the 1960s up to 2001. I have also taken this opportunity to air some rather philosophical comments, which I hope will add insight to ongoing debates or inspire future work. In this Sect. 1 focus on two topics: the assignment of chemical bonds within clusters and the electrostatic interaction, a paramount component of intermolecular forces. The latter topic involves the well-known multipolar expansion [4, 5], performed in the spherical tensor formalism [6], which An-

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thony successfully promoted via his well-known Distributed Multipole Analysis (DMA) [7, 8]. However, consistent with the title and remit of this chapter, I will define atomic multipole moments according to QCT. It is worth mentioning here that the exploration of potential energy surfaces and molecular simulations are still dominated by point charge potentials. Nevertheless, force field designers and the wider community of users seem to recognise the advantages of introducing multipoles [9–17]. When really accurate electrostatic potentials are required, as in the challenging area of polymorphism modelling [18], then one must resort to multipole moments. Finally, consistent with the scope of the Series of “Structure and Bonding”, I will discuss possible future research directions.

2 Quantum Chemical Topology The name Quantum Chemical Topology, refers to a research program originating in the theory of “Atoms in Molecules” (AIM), developed by Bader and co-workers. Over the last decade or so this research program has been carried out by about a dozen groups worldwide, while up to a hundred groups have benefited from its applications. The theory of AIM has been reviewed elsewhere [19–22] or even didactically presented [23]. This is why the allotted space is better allocated to explaining what exactly the name Quantum Chemical Topology covers and why it is appropriate. I stress that the discussion below is beyond mere semantics. Instead, it seeks to clarify QCT’s initial agenda, in the shape of AIM, and the possible new directions it may take in the future. The actual paper [24] that marks the birth of AIM proposes a partitioning of molecular electron densities, generating special portions of real space that have a unique and well-defined kinetic energy. These subspaces were dubbed virial fragments and were identified with atoms, as they appear inside a molecule; hence the name Atoms in Molecules. Of course, in that context this name adequately covers what AIM intended to achieve at that point. Subsequent papers published by the Bader group increasingly deepened the physical foundation of the virial fragments, ultimately arriving at Schwinger’s quantum action principle [25]. This principle constituted a powerful starting point to reformulate the theory of AIM as subspace quantum mechanics, in parallel with the more familiar “total space” quantum mechanics. Schwinger introduced a single dynamical principle from which one can not only define observables but also obtain their equations of motion and the commutation relations. The Bader group generalised this principle to atomic subspaces, thereby naturally recovering the expectation values of atomic properties on the same footing as the expectation values for molecular (i.e. “total system”

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or “full space”) properties. Moreover, proceeding along these parallel lines one is in an excellent position to derive [26] the so-called atomic theorems, such as the atomic force, virial, continuity, torque and power theorem. Although this advanced reformulation of AIM is often promoted as a crowning achievement it should not overshadow the central idea of the “topology”. What really sets the approach underpinning AIM apart from other theories (e.g. Molecular Orbital or Valence Bond theory) and other techniques (e.g. population analyses, partitioning schemes) are the concepts and tools of topology. Strictly speaking, the name topology refers to a vast and mature branch of mathematics. Only a very small part of the mathematical branch called topology is used in AIM, so small that some authors prefer to use the name topography. This name was also used by Bader and co-workers when they first defined [27] the bond path. In any event, it would be very exciting to explore to which extent a broader application of topology can come to fruition in AIM. The sub-branch of dynamical systems is perhaps better designated as the source of mathematical insight that guided AIM’s development. AIM keywords such as critical points, gradient vector field, homeomorphism, bifurcation catastrophe all originate in the area of dynamical systems. Soon after the establishment of topological atoms as proper quantum mechanical subspaces, the Bader group investigated [28] the topology of the Laplacian of the electron density. This investigation applied the concepts of dynamical systems to this new scalar function, which is more complicated than the electron density itself. Although this work on the Laplacian’s topology, as it is loosely referred to, is an integral part of AIM, the name Atoms in Molecules makes less sense in this context. The Laplacian’s topology proposes one solution to the continuing challenge of electron pair localisation, as put forward in the Lewis model. It turned out that there was a remarkable but not perfectly faithful mapping between the Laplacian’s critical points and Lewis’ electron pairs. It is clear that such results do not really add to the definition of an atom in a molecule. The name Atoms in Molecules would only make sense in this context if one regards the Laplacian’s topology as an aid to understand how a free atom changes as it becomes part of a molecule. If one regards AIM in the original context as an electron density partitioning scheme to define rigorous atoms inside molecules, then the Laplacian’s topology adds nothing. However, this is not the only reason why the name AIM, or QTAIM (“Quantum Theory of Atoms in Molecules”) as it is sometimes referred to in the literature, does not stand up to scrutiny. Already in the nineties, AIM studies [29–36] started to appear on crystalline solids, such as diamond, silicon, germanium, metallic alloys, spinels, alkali halides and ionic oxides. Since these materials are non-molecular the topological analysis according to AIM yields ions in crystals rather than atoms in molecules. Furthermore, one may wish to isolate molecules in van der Waals complexes via the topology of the complex’s electron density. Admittedly, such “cut-out” molecules consist of atoms

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(in an additive manner) but the main point is that the end-product of the partitioning is not an atom but a whole molecule. There are more reasons why a name such as AIM becomes unsustainable over the long term. Over the last decade the group of Silvi in Paris has exhaustively studied the topology of ELF [37–44], which stands for electron localisation function [45]. Although this impressive work coherently addresses a multitude of chemical problems it does not, as admitted by Silvi himself, enjoy the quantum mechanical depth of the theory of AIM. In the literature one often finds ELF juxtaposed to AIM, as if they were two types of independent and equivalent types of analyses. The “ELF methodology”, since its inception [37] in 1994, was inspired by AIM and essentially applies dynamical systems to the scalar function ELF rather than the electron density or its Laplacian. In this sense it is appropriate to make both AIM and ELF part of a wider theory called Quantum Chemical Topology. This name acts as an umbrella under which topological investigations of other scalar functions can be housed, such as the electrostatic potential [46, 47], Electron Localisability Indicator (ELI) [48], Localised Orbital Locator (LOL) [49], the virial field [50], the magnetically induced current distribution [51], the total energy (catchment regions) [52] and the intracule density [53]. Future research on other scalar functions, possibly of dimension higher than three, could also be incorporated under QCT. In fact, this acronym leaves room for any developments that embrace the power of dynamical systems in order to partition information and thus extract chemical insight from quantum mechanics. Even if chemical insight is not being pursued, QCT’s strength in defining transferable fragments, especially in the electron density, can be used to design a force field. Work on a polypeptide force field, based on topological atoms taken from small accurate electron densities, has been initiated in our group. If successful, the QCT approach will for the first time make predictions on the structure and ideally also the dynamics of proteins. One may suggest that, here also, a name such as Atoms in Molecules would somehow disguise the core and remit of this piece of research. Finally I would like to rationalise and hopefully justify the name QCT, well aware that this acronym clashes with another older one. QCT also stands for Quasi Classical Trajectory, a totally unrelated concept from the field of Reaction Dynamics. The name Quantum Topology already appeared in 1979, in the title of a paper by Bader, Anderson and Duke [54]. Without intending to be pedantic, one could argue that Quantum Topology wrongly refers to modern theories of space-time and “elementary particles”, where for example string theory, quantum geometry and Calabi–Yau [55] spaces feature. The word “Chemical” is definitely warranted to avoid this confusion. The more compact expression “Chemical Topology” introduces confusion with a much older field, originating in the work of the 19th century mathematician Cayley. Here graph theory, random trees, isomer enumeration and the Wiener index, for example, would feature. None bear any relationship with QCT.

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3 The Chemical Bond 3.1 A Brief History The behaviour of matter at broadly ambient conditions, that is, avoiding plasmas or conditions of the early Universe, is extremely rich. This is why chemistry evolved into a separate science, focusing on a bewildering number of observations about the appearance and transformation of matter at various levels of organisation. Although the whole of chemistry is embedded in physics, it has developed its own “language”. In its earlier development physics focused on a less complex observational arena than chemistry, which is why it connected quicker with mathematics. This is probably also the reason why physics was faster than chemistry in formulating a set of laws and principles that could explain and predict all observed phenomena. Linking the language of chemistry, i.e. its full armoury of concepts, such electronegativity, aromaticity, reactivity, acidity, covalency, the anomeric effect, functional groups, hardness and softness, and indeed the chemical bond itself, to the underlying principles of physics, i.e. quantum mechanics, is an ambitious research program that continues to challenge. In tackling this challenge one should not fall into the trap of naïve reductionism but resort to the elegant idea of emergence and view chemistry as a science emerging from physics. To use an analogy: if physics unravelled the rule of chess, chemistry worked out the different strategies and styles of playing chess. Using this analogy it becomes clear that it is not easy to characterise an aggressive playing style, for example, in terms of the rules governing the moves of the bishop and the queen, for instance. Bearing these thoughts in mind I will now discuss chemical bonding, what QCT offers in this context and which issues remain. There is no need to repeat Sutcliffe’s illuminating account [56] on the development of the idea of a chemical bond, but some statements from his conclusion are worth highlighting in the light of the discussion below, involving QCT. Sutcliffe believes that most chemists would agree with Coulson “in recognising the bond as a figment of our imagination [but that it is] worthwhile to adopt pragmatic schemes for getting molecular structure out of wave functions whenever possible.” Frankland [57] was apparently the first to feel the urge to imagine the chemical bond. He wrote that “by the term bond, I intend merely to give a more concrete expression to what has received various names from chemists, such as an atomicity, an atomic power, and an equivalence. A monad is represented as an element having one bond, a dyad as an element having two bonds, etc.” The concept of the bond thus predates modern formal quantum mechanics by at least six decades. Stronger even, it

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predates by about four decades the notion of atoms and molecules, which was not universally accepted until the 1910s. The notion of a bond goes back to the observation that the atoms of a given element have a characteristic combining power, that is, each atom can form a certain number of bonds called its valence [58]. It is remarkable that such an intuitive concept as the chemical bond has survived for more than a century in spite of the tremendous developments in molecular spectroscopy, quantum chemistry and computation. Indeed, the bond ended up as this magic stripe in the Lewis structure diagrams, deeply couched in the chemist’s way of communicating. This state of affairs should be contrasted with the extensive testimony of Ruedenberg [59] entitled “The Physical Nature of the Chemical Bond”, which starts with the sentence “as yet, the physical nature of the chemical bond is little understood in many essential details, and the reason for this must be seen in the mathematical difficulties that are encountered in solving molecular quantum-mechanical problems.” No solace can be found in the search for the or even a definition of the chemical bond by consulting a recent chemistry textbook, whether general or more specialised. The nearest one can come to finding any definition at all is in Pauling’s well-known book [60]. He states that “...there is a chemical bond between two atoms or groups of atoms in the case that the forces acting between them are such as to lead to an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent molecular species.” It appears to me that the circularity of this “definition” renders it virtually useless. The vague words “sufficient” and “convenient” seem to indicate the existence of the chemist’s preconceived idea of what constitutes a bond. If so, he or she need not consult this definition in the first place. Secondly, the meaning of the phrase “forces acting between the atoms” must be interpreted very carefully. When atoms form an “independent molecular species” at equilibrium, the forces on these atoms, or more precisely on the nuclei, must manifestly vanish. The forces Pauling mentioned seem to refer to the process of forming (“lead to”) the molecule from its constituent atoms, in which case they would of course not vanish. However, as discussed below, this particular formulation could cause serious confusion if a bond is wrongly associated with attracting forces. Pauling is not explicitly saying whether the forces are zero or not, but on first reading of his definition, many would be left with a picture of atoms coming together to form a molecule, driven by the right (“are such as to”) forces, that is attracting ones. In his 1951 Tilden Lecture [61] entitled the “Contributions of Wave Mechanics to Chemistry” Coulson took the electronic charge density, or in short the electron density, as the origin of binding. He stated that “we might say that the description of a bond is essentially the description of the pattern of the charge cloud... Indeed in the very last resort, we cannot entirely separate the charge cloud for one bond from that for another bond.” The electron density is also the starting point of standard QCT to formulate its definition of

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the chemical bond, albeit along lines very different to Coulson’s thinking. Although the QCT definition has spurned an ongoing controversy, which we will discuss in detail below, it has two advantages over Pauling’s definition. First it is much more explicit, less ambiguous and not circular in nature. Secondly it renders computable the decision where bonds appear in a molecule, provided one has a wave function of adequate accuracy. Admittedly, the last condition can lead to ambiguity, practically speaking. However, modern computers are able to generate electron densities of, in principle, arbitrarily high accuracy. Moreover, they can routinely yield electron densities that are “topologically stable” from a given accuracy onward. This means that further enhancing the accuracy of the electron density beyond the given one, will not change the pattern of critical points found at this given accuracy. All these reasons most likely explain the large numbers of papers [2, 3] that used QCT to detect and study unusual bonds. 3.2 A Computable Bond? In the next Section I introduce a sequence of papers that unfolds a controversy surrounding QCT’s approach towards chemical bonding, but first I briefly mention an alternative approach, which is also related to the electron density. In 1951 Berlin used [62], in what is now textbook material, the Hellmann-Feynman (electrostatic) theorem to shed new light on chemical bonding in diatomics. He characterised a precise and computable region of space as binding if electronic charge located in it tends to draw the two nuclei together. Conversely, electron density accumulated in antibinding regions pulls the nuclei apart. In 1978 Koga et al. generalised this approach to polyatomic molecules [63]. In their example on the water molecule they showed how electron density reorganisation at non-equilibrium geometries occurs in such a way as to facilitate restoring the molecule to its equilibrium geometry. However, in 1991 a little known paper by Silberbach [64] appeared (with a three-year time lapse between the received and accepted date). In this work, which so far attracted only 10 citations, Silberbach carefully reinvestigated Berlin’s approach and found it to be erroneous. He points out that others, including Koga, had encountered problems with the Berlin picture, but he was the first to prove that the notion of binding and antibinding regions could not be maintained. Finally, it should be pointed out that the previous and the following discussion assumes that molecules exist in the way that Löwdin defined them in his 1986 talk at a conference in honour of Daudel. Löwdin defined a molecule in quantum mechanical terms as follows: “A system of electrons and atomic nuclei is said to form a molecule if the Coulombic Hamiltonian – with centre of mass motion removed – has a discrete ground state energy.” However, scrutinising this definition in the light of developments made since 1986,

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Sutcliffe concluded [65] in 2002 that “a definition of a molecule in quantum mechanics that would be acceptable to chemists continues to elude us in spite of the helpful beginning made by Löwdin.” The validity of this conclusion will not be examined here; there is already enough to consider if one wishes to discuss chemical bonding in a molecule. I continue to assume the validity of the Born–Oppenheimer approximation, because diagonal non-Born–Oppenheimer corrections are negligible for most molecules [66], but more important for molecules containing hydrogen atoms. Interestingly, the concept of chemical bonding, which at the Born–Oppenheimer level is an electronic phenomenon, is in the non-Born–Oppenheimer approach described as an effect derived from collective dynamical behaviour of both electrons and nuclei [67].

4 Attractive or Repulsive Interactions? 4.1 The Birth of the Bond Path Five years after the introduction [24] of virial fragments the Bader group published a paper [27] defining bond paths after closer inspection of the “topography of the molecular charge distribution.” The paper defines stationary points (now called critical points) as points where the gradient of the electron density vanishes. Then it presents two types of saddle points, which it called internuclear and ring saddle points, where the former would now be called bond critical points. The bond path is then stated to be “the two gradient paths that originate at the same (internuclear) saddle point and terminate at each of the two nuclei.” The authors regard the unit that emerges as the fundamental carrier of chemical information as atom-like and not bond-like. Furthermore they assert that the only property of the observable charge distribution that is bond-like in nature is the bond path. Intriguingly, about seven years earlier Martenson and Sperber suggested [68] that one define “bond lines” as tracks of maximum charge density. The authors expressed uncertainty as to which density function should be used. Their study on cyclopropane, for which they used a single bonded molecular orbital and a valence density, failed to deliver a chemically satisfactory picture. They concluded that any definition of bond line would be arbitrary, without ever trying the total molecular electronic charge distribution! However, the Bader group did use the latter density in their study of cyclopropane, which revealed bond paths that were outwardly displaced from the respective internuclear axes. The emerging picture was intuitively appealing and appeared to capture the ring strain, known to be present in cyclopropane. This observation

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is somehow compatible with work [69] carried out by Coulson and Moffit, who obtained a value of 22◦ for the angle formed between the direction of the maximum in a hybrid orbital on carbon and the carbon-carbon internuclear axis. Bader and co-workers finish off their seminal paper by pointing out that “if two fragments interact strongly within a molecular system, then the electron density will exhibit a saddle point in the region between the nuclei, and the nuclei will be linked by a bond path.” They then go on to say that “this link will exist whether the strong interaction is a bonded one or a repulsive one, as in He2 , for example.” Thus, if nuclei are bonded they show a bond path, but if there is a bond path it does not necessarily mean there is a bond. In their own words, “thus the existence of a bond path is a necessary (but not sufficient) condition for two atomic fragments to be bonded to one another.” They added a note in proof stating that the sign of the Laplacian of the electron density at the saddle point could be used to determine if an interaction is attractive or repulsive. Further work on the characterisation of atomic interactions [70] eventually led to the view that one needed to include the forces on the nuclei to be able to decide if a bond path indicates the existence of a bond. The term atomic interaction line was coined to clarify the situation and avoid possible confusion. The atomic interaction line consists of two gradient paths originating at the same saddle point and terminating at two different nuclei. If the forces disclose that the molecular state is bound then the atomic interaction line becomes a bond path [21]. By invoking this force criterion, which is external to the topological features of the electron density, one recuperates the bond path as a necessary and sufficient condition for bonding. This means, in principle, that one can unambiguously predict where the bonds are located in a molecule. On a positive note this result is a true achievement because, to my knowledge, it is the first and so far only proposal of a contradiction-free and computable criterion for chemical bonding. However, this is not the end of the story. This definition of the chemical bond has weathered several attacks, with subsequent rebuttals, but ultimately still calls for future research, in my opinion. The least one may feel uncomfortable about is that this QCT definition of the chemical bond remains silent on molecules that are not in an equilibrium geometry. There is clearly a need to be able to assign bonds in a non-equilibrium situation. For example, at finite temperature molecules vibrate and consequently they spend more time away from equilibrium than at it, loosely speaking. Secondly, the interpretation of bonding along a reaction path forces one to consider more non-equilibrium geometries than equilibrium ones.

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4.2 The Bond Path in Action Let us pause here and briefly mention a few successes the bond path enjoyed over the last decade and a half. The following collection of examples is by no means exhaustive but simply serves to create a context for the “repulsive interaction controversy”, which I will unfold below. The description of bonding in boranes [54, 71] turned out to be a straightforward application of QCT. This, in spite of boranes being notorious for their difficult rationalisation in terms of the two-centre electron-pair bond model, an issue that has attracted much discussion [72–74]. Organolithium compounds have unusual geometries that challenge traditional ideas of bonding [75] but again, they can be rigorously characterised by the topological analysis [76] QCT offers. The character of the carbon-lithium bond was a matter of debate until it was settled as “predominately ionic” [75, 77, 78]. This conclusion is consistent with the QCT analysis. Organolithiums are classified according to the coordination number of the lithium. In QCT this number is simply defined as the number of bond paths terminating at the lithium in question. As such, the coordination number becomes computable and the classification ensues in a clear-cut manner. As a final point, Ritchie and Bachrach never observed [76] Li–Li bond paths in any of their 23 compounds. This is confirmed by some of our own unpublished work on highly symmetrical (optimised) configurations of the model compounds (LiNH2 )2 , Li2 CH3 NH2 and (LiCH3 )2 at MP2/6-311++G∗∗ level. In these respective systems an extraordinarily long NN, CN and CC bond was A, 3.36 ˚ A and 3.61 ˚ A, respectively. These compounds found, measuring 3.10 ˚ can be thought of as hydrazine, methylamine or ethane, deformed by a duet of ionic Li atoms. Lithium invariably donates almost an entire electron to the molecule it perturbs, thereby creating highly charged amines and/or carbanA) in π-[TCNE]2– ions. Exceptionally long CC bonds (≥ 2.9 ˚ 2 were the subject of a recent and extensive account [79], which presented them as two-electron four-centre cation-mediated π ∗ –π ∗ bonding interactions. In other unpublished work we have looked [80] at a few compounds containing B–Li bonds because their as of yet elusive synthesis is at the forefront of inorganic synthetic chemistry. The dimers in Fig. 1a serve as model systems for the actual target molecule shown in Fig. 1b. Figure 2 shows the B3LYP/6-31+G∗∗ optimised geometry of dimer 1. Each lithium nucleus is connected via a bond path (not shown) to the bond critical point located half way between the two boron nuclei. This situation is reminiscent of a so-called conflict structure. Thus even in these circumstances two lithium nuclei are not directly connected by an uninterrupted bond path. Typically a non-nuclear attractor would be found between two lithium nuclei, as in the Li2 molecule for example. Lastly, the atomic charges of dimer 1 are worth listing. Because of symmetry the following list is complete: 0.13 for Li;

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Fig. 1 (a) Schematic structures of three model dimers of the monomeric unit LiBN2 C2 H4 . (b) Synthetic target molecule, showing the full anticipated stabilizing coordination of the B–Li bond

Fig. 2 Optimized geometry of dimer 1. The dotted line is drawn just to guide the eye

1.45 for B; – 1.78 for N; 0.50 for C; 0.43 for a H bonded to N; and 0.05 for a H bonded to C (the total sum yields a spurious excess charge of – 0.04e, marking the integration error). An electropositive element such as Li is now seen to give away only about 0.1e in the presence of boron atoms, instead of the usual 0.9e in the presence of heavier elements. In 1988, Carroll and Bader published their seminal paper [81] on classical hydrogen bonds, which characterised topologically a large set of Base...HF complexes. This work inspired a topological study [82] on less conventional C–H...O hydrogen bond complexes, which were controversial in the 1960s. In that study, clearly formulated criteria for hydrogen bond detection were proposed, meant to be independent from and additional to hydrogen bond features encountered in NMR, crystallography or IR spectroscopy. The successful application of these criteria in characterising [83] the novel dihydrogen bond (coined [84] in 1995) instigated a host of papers on dihydrogen bonds. Note that the proposed hydrogen bond criteria are systematically violated (except for the inevitable presence of a bond critical point) by the agostic bond [85]. The fact that the agostic bond is “orthogonal” to the hydrogen bond in such a clear fashion emphasises the discriminatory power of these hydrogen bond criteria. The boundaries of topological bond analysis were pushed further in an article [86] on van der Waals dimers and trimers, none of which contain a hy-

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drogen bond. Thirty-six configurations of 11 van der Waals complexes were studied, containing Ar, CO2 , C2 H2 , OCS and SO2 . Figure 3 shows the cyclic global minimum of the ethyne trimer as an example of such analysis. Intuitively, bonds are regarded as signatures of stability, expressed as pairwise interactions. Therefore, the appearance of C...H bonds between the ethyne molecules should not come as a surprise. This figure clearly illustrates that the topological analysis extends beyond the recovery of covalent “Lewis bonds”, and sheds new light on the stability of perhaps rather exotic molecules. We note that the electron density at the “closed-shell” bond critical points (i.e. located along each of the three curved C...H bond paths) is virtually identical to the sum of the respective values for the isolated monomers at the corresponding point. This observation can be generalised to other van der Waals complexes. Bone and Bader pointed out that the difference between these two quantities is an order of magnitude less than the difference found with hydrogen-bonded systems. Interestingly, the magnitudes of the electron density at bond critical points in a representative covalent bond, a hydrogen bond and a “van der Waals bond” approximately follow the ratio of 100 : 10 : 1 respectively. The latter are described as “features of the electron density that are true counterparts of hydrogen bonds, but between pairs of heavy atoms.” The authors conclude that the QCT analysis has provided a description of their van der Waals structures that goes beyond sheer geometry. Indeed, the key atom-atom interactions cannot always be predicted from mere proxim-

Fig. 3 The molecular graph of the cyclic global minimum of the ethyne trimer, superimposed on a plot of isodensity contours. The bond critical points are marked by squares and the ring critical point by a triangle. (from J Phys Chem, 100, 10892 (1996), with permission from the American Chemical Society)

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ity or routine intuition. A dramatic example of this finding is illustrated in Fig. 4, which shows a curious “O – O double bond” in a non-equilibrium configuration of (CO2 )2 . Considering the totality of the configurations studied, however, Bone and Bader conclude that their topology is no different from that encountered in “regular” molecules, in the sense that there are bond and ring critical points and catastrophe and conflict structures. The crystallographers Destro and Merati presented [87] another example of bond critical points appearing in unexpected places and surprisingly not appearing in expected places. In their low-temperature X-ray study of syn1, 6 : 8, 13-biscarbonyl [14] annulene (BCA) they found a bond critical point between the two bridging carbons and none between either pair of carbons (inside BCA’s 14-membered ring) that are geometrically closer than the bridging carbons are from each other. The authors suggested that this unanticipated CC bond aided in guiding BCA’s thermal decomposition, agreeably confirming the observed major fragmentation products. An interesting question is to what extent the topology of the electron density coincides with that of the total energy, if at all. In particular, we ask ourselves if a conflict structure (electron density) occurs at the same nuclear geometry as a transition state (energy). In 1981 the Bader group [88] reported two examples of reactions where this is indeed the case, allowing for a small window of discrepancy. One example is the thermal isomerisation of HCN to CNH. Hydrogen’s migration from being bonded to carbon to

Fig. 4 Atomic Interaction Lines in a non-equilibrium configuration of the CO2 dimer, superimposed on a plot of isodensity contours. The bond critical points are marked by squares and the ring critical point by a triangle. (from J Phys Chem, 100, 10892 (1996), with permission from the American Chemical Society)

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being bonded to nitrogen is monitored via an angle θ. This angle is formed between two vectors, both with their origin at the centre of mass of the CN moiety. One vector connects this centre of mass with the C nucleus, the other vector connects the centre of mass with the H nucleus. While linear HCN corresponds to θ = 0◦ and linear CNH to θ = 180◦ , the transition state was predicted to lie at θ = 73.7◦ [89]. The conflict structure was found [88] to lie in the interval 72.1◦ ≤ θ ≤ 72.4◦ . As a second example, the isomerisation of CH3 CN shows a transition at θ = 79.2◦ , while the conflict structure occurred for a θ between 84◦ and 85◦ . The paper [88] that reports these two examples also presents an almost heuristic justification for this virtually perfect coincidence, invoking the Hellmann-Feynman theorem. However, Mezey claimed in a paper [90] appearing two years later, that the topology of the total energy (“catchment regions”) and that of the electron density (“molecular graphs”) are non-comparable for most chemical systems. He stated that “whereas the chemical structure concept based on catchment regions distinguishes between all chemical systems separated by energy barriers, the structure concept of molecular graphs does not distinguish between rotamers, cis-trans isomers and optical isomers.” The perhaps counterintuitive consequence of the catchment region approach is that conformers are considered to be different structures although they can be interconverted into each other without breaking or creating bonds. Returning to the work [86] on van der Waals complexes, the mismatch between the topology of the potential energy surface and the electron density is amply illustrated. For example, conflict structures (i.e. electron density) may correspond to both minimum energy conformations and transition states (i.e. energy). 4.3 The Debate Starts Then, in 1990 and 1991, a research group reported a number of unusual bonds in their customarily scattered and repetitive way. During this flurry of papers, chronologically listed as [91–97] according to “Date Received”, the authors changed their mind about whether the bond paths they observed were indicative of bonds rather than “steric interactions”. In the first paper [91], unusual bonds were hit upon in C(NH2 )3 C(CN)3 , a hexasubstituted “push-pull” ethane. The optimised geometry (equilibrium configuration) of A). More an isomer with C3v symmetry displayed a very long CC bond (3.25 ˚ surprisingly, the authors noticed three long and very curved bond paths, each connecting two N nuclei, one in a cyano group and one in an amino group. In the second paper [92] two types of “weak bonds” were distinguished, based on the following five examples: benzene-tetracyanoethylene (TCNE), C(NH2 )3 C(CN)3 , Ne@C60 , C(NO2 )3 – and kekulene. The weak bonds in the first three compounds have low ellipticities and have their major axis (u2 ) (almost) parallel to the corresponding ring surfaces. They were thought of

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as stemming from “delocalised electron interactions”, and include very long CC bonds. On the other hand, the weak bonds in the fourth and fifth compounds were deemed to result from steric interactions and include OO and HH “contacts”. They have low ellipticities and their major axes are perpendicular to the ring surfaces. All topological features were independent of the (to modern standards small) basis sets. The authors state that they “refrain here from concluding if these examples demonstrate limitations of the definition of bonds proposed by Bader or deceptiveness of “chemical intuition”.” This opinion is echoed in the third paper [93], where it is repeated that C(NO2 )3 – “provides a new challenge to our understanding of chemical bonding”. In the fourth paper [94] kekulene was revisited, now in the context of “superaromaticity”. Kekulene, a highly symmetric polyaromatic consisting of a hexagon of twelve fused benzene rings, is also referred to as superbenzene due to the resemblance of its six-sided ring to the benzene molecule. The remarkable (highly curved) bond paths linking the six inner hydrogens are again presented but the authors are “unable to attach any physical significance to these bonds”. They assume that the overcrowding of the inner hydrogen atoms is destabilising the molecule, but that it is overcome by additional conjugation around the molecular ring over and above the normal conjugation of the individual benzenes. The fifth paper is a more elaborate study [95] on push-pull hexasubstituted ethanes, where the curious bond paths are now also seen to connect N nuclei with O nuclei. In the sixth paper [96], focusing on “hydrogen-hydrogen” non-bonding interactions, the authors, now more confident, stated that “the term bond path should be reserved for the interaction lines describing ordinary strong bonds.” To gain deeper insight into the nature of the H – H interaction lines they varied the central torsion angle ϕ (C2 C1 C1 C2 ) in biphenyl. They studied its effect on the local topology of the interaction between the hydrogens bonded to C2 and C2 , respectively. The hydrogen-hydrogen distance acts as a controlling variable determining if the H – H interaction line is present. This line is observed in the planar transition state (ϕ = 0◦ ), but not in the local minimum when ϕ = 45◦ . In the seventh and final paper [97] of the flurry, only three extra molecules (ortho-substituted biphenyls) are added to the set already studied. Based on the topology of conformers of 2,6-difluorobiphenyl, 2, 2 -difluorobiphenyl and 2, 2 , 6, 6 -tetrafluorobiphenyl, the authors claim a rigorous definition of sterically crowded molecules, superior to that obtained from van der Waals radii. Three years later the same group produced one more paper [98] on the subject of steric overcrowding, based on new calculations on perhalogenated cyclohexanes, dodecahedranes and [60]fulleranes. Remarkably, the condition that the bond critical point have a low ellipticity and its major axis perpendicular to the ring surface is not invoked to distinguish, according to their own theory, the steric repulsion interaction lines from weak bonds. For example, the six interaction lines extending between the 1,3-diaxial halogen atoms in C6 Cl12 and C6 Br12 are simply taken to be steric interactions. It is

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emphasised that such steric interactions are not mere consequences of proximity, because for example, the gauche 1,2-equatorial-axial FF distance is shorter than the 1,3-diaxial one. In general, substantial lengthening of CC bonds was observed upon increased halogenation. The authors conclude that “the overcrowding is partially relieved by an initial lengthening of all bonds, followed in highly congested molecules by a preferential elongation of the CC bonds.” In his sole author paper [99] Bader totally rejected the notion of steric repulsion expressed as interaction lines, calling the phrase “interaction lines between non-bonded atoms” an oxymoron. He argued that “the question is not how the final geometry is attained in some mental process involving passage over a repulsive barrier, a situation that is in fact common to most chemical changes, but rather how the mechanics determines the final distribution of charge.” The crux of Bader’s rebuttal is that the virial field and the electron density are homeomorphic [50]. In more colourful terms this means that “every bond path is mirrored by a virial path, a line linking the same neighbouring nuclei, along which the potential energy density is maximally negative, i.e. maximally stabilising, with respect to any neighbouring line.” [99] The virial field is a three-dimensional scalar function equal to the trace of the stress tensor. It can also be written as – 2G(r) + (
2 /4m)∇ 2 (r) where G(r) is a kinetic energy density and the second term the Laplacian. The kinetic energy density is not homeomorphic to the electron density. It is remarkable that when G(r) is combined with the complex topology of the Laplacian (which is not at all homeomorphic to the electron density) one obtains the topology of the virial field, which is homeomorphic to the electron density. However, the latter homeomorphism is not mathematically proven but only observed for a finite (and fairly small) number of molecules, including second period hydrides (with extra structural work on water), Li2 , cyclopropane, bicyclobutane, [1,1,1]propellane, tetrahedrane, B2 H6 , arachno-B4 H10 , closo-C2B3 H5 , and more recently [100]phenanthrene, chrysene, dibenz[a,j]anthracene, planar biphenyl, tetra-tert-butyltetrahedrane, tetra-tert-butylcyclobutadiene and tetra-tertbutylindacene. The second issue is that the homeomorphism is not perfect because of a few exceptions. The two fields behave differently for Li2 and also in B2 H6 , where the virial graph shows a path linking the two borons, which is absent in the electron density. Also, in the work on the structure diagram for water [50], the virial field yielded the same sequence of structures and the same structure diagram as the electron density. Nevertheless, the unstable bifurcation structures occurred at different geometries for the two fields. Since the homeomorphism is not perfect, and since the virial field is judged [99] to have the ultimate authority to decide on bonds, one may wonder why the electron density is still consulted. Should one then not reexamine the whole question of bonding solely from the point of view of the virial field?

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The Bader group elaborated the bonding picture supported by the virial graph in a recent study [100] that explicitly presented hydrogen-hydrogen bonding as a stabilising interaction in molecules and crystals. This type of bonding accounts for the existence of solid hydrogen, and as it falls in the class of “van der Waals” interactions, no different in kind, for example, from the intermolecular Cl – Cl bonding in solid chlorine [101]. The authors point out that these H – H interactions are ubiquitous and that they should not be confused with dihydrogen bonds (see above). However, H – H bonds may be found to merge with dihydrogen bonds as the disparity in the charges on the two participating hydrogens increases by at least tenfold, from a few thousandths to a few hundredths. In the same study Matta et al. showed that each H – H interaction makes a stabilisation contribution of up to 40 kJ mol–1 to the energy of the molecule, and categorically deny the validity of the “nonbonded steric repulsions”. They warn that they have not attempted to define a H – H bond energy, since one would then inevitably be faced with the “usual ambiguities”. Instead the authors refer to a stabilisation energy E(H – H), which ambiguously refers to the contribution to the lowering of the energy of the entire molecule associated with the hydrogens involved in H – H bonding. According to their approach there is no steric repulsion between the ortho-hydrogens in biphenyl; rather the resultant H – H bonding contributes a stabilising contribution to the molecule’s energy. The authors never mention nor discuss the orientation with respect to the ring surface of the major axis at the HH bond critical point. In their work on F – F interactions Alkorta and Elguero [102] added an interesting new angle to the debate on the meaning of atomic interaction lines by establishing a link with NMR. Scalar coupling constants between nuclei are associated with the existence of bonds between atoms involved. Based on this link they explored 12 systems with F – F coupling constants across in order to ascertain the presence of bonds. With one arguable exception all cases where strong F – F coupling constants were observed experimentally, a bond critical point was detected. However, in captivating work [103] on self-discrimination the same authors interpret their F...F and H...H interactions rather casually as destabilising. 4.4 Input from High-Resolution X-ray Crystallography One area where QCT has become a new paradigm is high-resolution crystallography [104]. An experimental study on Mn2 (CO)10 [105] invoked topological analysis to characterise metal-metal and metal-ligand bonds. This paper proposed a general chart classifying covalent, polar-covalent, dative, metallic, ionic and van der Waals bonds in terms of semi-quantitative conditions of various energy densities evaluated at the bond critical point in question. Increasingly, high-resolution crystallographers bring in ab initio

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calculations, sometimes published separately from their experimental work, in order to enhance insight in the systems they are interested in. The recent report [106] on 1-phenyl-o-carboranes, which exhibits C – H...H – C bonds, is a typical example. According to the researchers, disentangling the factors (steric repulsion, electronic and stereoelectronic effects) that influence the only C – C distance in the carborane, will ultimately help in the design of non-linear optic materials. Lyssenko et al. [107] combined an ab initio study with their X-ray experiment on the transannular interaction in [2.2]Paracyclophane. They were surprised to find no bond critical points at all between this molecule’s two benzene rings (which are connected by two ethylene moieties). They confidently discussed how this finding adds to their understanding of this non-linear optic material. In another crystallographic paper Ritchie et al. [108] applied QCT to the dinitramide ion, (NO2 )N(NO2 )– , where a remarkable O...O interaction between the “inner” oxygens featured. Although aware of the “steric repulsion” controversy the authors did not engage in a debate about it. Similarly, DuPré [109] was content interpreting the bonding in his molecular proton cage (1,6-diazabicyclo[4.4.4]tetradecane) in the canonical way, again aware of the “confusion about the meaning of bond path and bond critical point”. However, one researcher who attacked [110] the canonical interpretation of bond critical paths and bond paths in ionic crystals from a crystallographer’s point of view was Abramov in 1997. His starting point was a study [101] by Tsirelson, who in collaboration with the Bader group, described the molecular chlorine crystal as consisting of primary interactions, i.e. intramolecular interactions, and of secondary interactions, that is, between molecules. The latter are associated with bond paths, and can also occur between ions of similar charge. Abramov focused on ionic crystals, such as LiF and NaF, arguing that the predominately ionic interaction taking place in such crystals makes a F– – F– bonding pair interaction improbable. He then regarded a stable LiF crystal as consisting of primary Li – F interactions, stabilised by almost complete charge transfer from Li to F, and of secondary Li – Li and F – F interactions that are destabilising. To support his views, Abramov introduces the ionic procrystal, which is a hypothetical crystal constructed from the free spherical ions placed at the same sites as atoms in the real crystal. He concluded that a bond critical point arose as a result of a secondary interaction, while the chemical bond was absent. The author added that additional examples can be found in perovskites ABX3 but that for “the final answer on the question under consideration further detailed quantum-mechanical studies of the periodic systems seem to be needed.” In his 1998 paper [99] that established bond paths as universal indicators of bonded interactions Bader refuted Abramov’s stance. He rejected the electrostatic model (spherical ions in contact) that Abramov used because it was incapable of accounting for electrostatic contributions to the potential arising from the distortion of the spherical ion densities. It is the Ehrenfest force (which is equal to minus the

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divergence of the stress tensor) and not just the electrostatic force that governs the motion of electrons and the electron density. In more general terms Bader also contended that “if repulsive forces were present, the crystal would be unstable and would either atomise or distort to an equilibrium geometry of lower energy.” Fully aware of the controversy Luaña et al. found [111] “clear evidence in favour of the real occurrence of O – O secondary bonds in many crystals.” They warn that the properties of O – O bonds display a clear continuity from their occurrence as strong bonds to the weak secondary interactions appearing in many oxides. It is instructive to terminate my passing excursion into the realm of crystallography by highlighting a more traditional study [112] that aimed at the distinction between the weak hydrogen bond and the van der Waals interaction. Steiner and Desiraju performed a “sociochemical” analysis of geometrical information stored in the Cambridge Crystallographic Database and zoomed in on C – H...H – C contacts. They claimed that a fundamental difference between hydrogen bonds and van der Waals contacts is that the former are inherently directional, while the latter are isotropic. Based on nearly 4000 examples they showed that the mean C – H...H angle of C – H...H – C contacts between methyl groups was 128.6(3)◦ . The angular distribution was almost ideally isotropic in the range of 120◦ to 180◦ , which confirmed the picture expected for non-directional van der Waals interactions. Commenting on C – H...O interactions the authors were adamant that representing the typical C – H...O/N hydrogen bond as nothing more than a classical van der Waals interaction is false. They emphasised that “a C – H...O hydrogen bond does not become a van der Waals contact just because the H...O distance crosses an arbitrary threshold.” Another interesting angle to the issue of bonding originated in the recent developments in very high pressure techniques [113]. For example, the triple bond in N2 breaks upon pressure increase [114], and Li unexpectedly dimerises at very high pressure [115]. Since little is known about the evolution of the electron density’s topology for molecules far from equilibrium geometry, Costales et al. undertook a piece of research [116] that led to some interesting conclusions. They found that a given pair of atoms followed a universal sequence of bonding regimes entirely controlled by the interatomic distance. The closed-shell interaction is typical for large internuclear separations, and shared interactions appear upon approach. Non-nuclear maxima can occur in some cases, until the inner shells of the atoms start to interact. The main message from this work is that the type of bond that best describes a given compound is actually a consequence of the internuclear equilibrium distance. This important observation again pointed to the inseparable relationship between energy and electron density, and reminds us that a given topology of the electron density exists by the grace of the governing potential energy surface.

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4.5 The Debate Continues In 2004 the QCT bonding controversy was re-opened by a computational paper [117] on the inclusion complex of He in adamantane. With their case study the authors specifically address the question of the “postulate of the sufficient and necessary condition for bonding”. The necessary condition is equivalent to stating that if there is a bond then there will be a bond path. The contentious issue is the reverse statement, i.e. the sufficient condition, namely that if there is a bond path then there is a bond. The topological analysis showed that the He atom is connected to four tertiary carbon atoms in the cage by atomic interaction lines. The authors do not seem to be aware of the distinction between an atomic interaction line and a bond path (see above) but one can infer from their work that they are actually discussing bond paths, because their complex is an energy minimum. They found that the dissociation energy of the complex is negative (– 645 kJ/mol). In other words, the energy of the separated He and adamantane was lower than that of the complex. They concluded that the HeC mean bond energy was also negative (a quarter of – 645 or – 161 kJ/mol) and that therefore the interaction was antibonding. The latter term is reminiscent of MO theory although their paper does not explicitly mention it, let alone apply it throughout. Although Haaland et al. concede that the overwhelming majority of atomic interaction lines correspond to bonding or stabilising interactions, their article showed that “these lines may indicate destabilising or antibonding interactions, the interpretation must in each case be judged on its merits.” I believe this is an unsatisfactory state-of-affairs because it would remove QCT’s ability to make (independent) predictions or distinctions. If and when the bonding controversy is finally settled one expects from an unambiguous topological interpretation that, at least, it provides a clear and widely accepted criterion distinguishing between a repulsive interaction or a weak bond. Ironically, a step in that direction was made in the paper [92] that started the controversy in the first place! As discussed above, it suggested that the direction of the major axis (perpendicular/parallel) and the ellipticity at the bond critical point (low/high) is such a criterion. Haaland et al. do not refer to this paper, and incongruously, they do not seem to be aware of the fact that this paper recognises bonding between the Ne and the carbon cage in Ne@C60 , a system with great similarity to He in adamantane. In this sense the argument has come full circle and there is little doubt that the paper by Haaland et al. will soon encounter a rebuttal paper. In any event, confident applications [118] of the QCT paradigm, with titles such as “Where to draw the Line in defining a Molecular Structure” continue to appear. At the 2003 fall meeting of the American Chemical Society Lichtenberger presented a paper on the bonding in the adduct resulting from the reaction of HSiCl3 with Cp(CO)2 Mn, complaining that valence bond

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models are inadequate to account for the electronic structure of organometallic molecules. Bader et al. demonstrated how QCT can be used to complement MO models to obtain an increased understanding to Lichtenberger’s problem. 4.6 Non-Covalent Interactions in Biomolecules Some years ago we looked at citrinin, an extensively studied fungal metabolite, whose experimental electron density was determined by a very careful high-resolution crystallographic study [119] at 19 K. Although we were mainly interested in quantitatively comparing topological features of computational and experimental electron densities, we came across an unexpected atomic interaction. Two tautomeric forms [120] of citrinin (pquinonemethide (citrinin a) and o-quinone (citrinin b)) are depicted in Fig. 5, both of which were fully optimised at the B3LYP/6-311+G(2d,p)//B3LYP/6311+G(2d,p) level, with proper characterisation of Hessian eigenvalues to confirm the geometry as a true minimum. Citrinin A turned out to be 0.6 kJ/mol more stable than citrinin B. We discovered an unusual H – H interaction between the methyl group located at position 3 in Fig. 5 and one of the hydrogens attached to the ring. The corresponding bond critical point is marked in Fig. 6 as “BCP”. The two connected hydrogens (again marked) are 2.03 ˚ A apart. In an attempt to decide if this is a bond or a steric interaction we inspected a number of topological features, including a “characteristic ratio” mentioned in [96]. The latter quantity does not put one in a position to clearly distinguish a bond from a steric interaction. The ellipticity feebly points towards a bond in our case but the strongest discriminator is the observation that the major axis is essentially parallel to the ring plane (the angle between the vector connecting the bond critical point and corresponding critical point, and the second eigenvector u2 is about 40 ). This criterion would classify the detected H – H interaction as a bond. More work is needed to decide whether

Fig. 5 Schematic diagram of two tautomers (a) and (b) of citrinin

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Fig. 6 Conformation of citrinin A displaying an unexpected H – H interaction

it is a dihydrogen bond or a H – H bond. In a similar conformation of citrinin b, optimised at the HF/6-31G∗ level, the net charge q on the methyl hydrogen is + 0.007 and that on the other hydrogen is + 0.017, which is not indicative of either type of bonding. In spite of the need to obtain a more complete and consistent picture, such H – H bonds are not an isolated phenomenon. For example, they arise in the biomolecules retinal, tuftsin [121] and between the two methyls (3 and 4 in Fig. 5) in citrinin a, again optimised at the HF/6-31G∗ level. The idea that methyl forms a hydrogen bond is familiar to those studying weak complexes. For example, Komasa et al. performed fourth-order Møller–Plesset and CCSD(T) calculations [122] on an ethane...HCN complex. Lacking a topological analysis the authors describe a hydrogen bond between one of ethane’s carbon atoms and the hydrocyanide hydrogen. Jensen et al. [123] used the topological analysis to specifically study the hydrogen-bonding ability of a methyl group in quantum mechanical geometry-optimised structures of selected molecules and ions. They suggested that the C – H...X angle should be larger than 100◦ for hydrogen-bond formation to occur. Even C – H...C hydrogen bonds were postulated [124] to exist with remarkable stability where the proton acceptor carbon is situated within an ylide. Returning to citrinin there is one final experiment to report. Rotating the methyl group (position 3 in Fig. 5) around the CC bond that attaches it to the ring system leads to a change in the topology of the electron density. The torsion angle τ, set to 0◦ at the optimised geometry shown in Fig. 6, was the only internal coordinate allowed to vary; no optimisation was performed for a given non-zero value of the torsion angle, i.e. the molecular skeleton was not permitted to relax. When τ = 30◦ the H – H bond critical

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point disappeared, only to re-appear when τ = 60◦ but now between the two methyl groups. Again, the major axis lay in the ring plane. If H – H bonds widely occur in biomolecules one may speculate what their full meaning is, in terms of the localisation of molecular stability. Should the view of apolar side chains merely causing “steric steering” in protein folding be revised, and be replaced by one in which hydrocarbon fragments display as of yet unappreciated relationships of attraction? QCT could also be instrumental in explaining frequently unexpected fragmentation patterns in peptide mass spectrometry. 4.7 Summary In summary, QCT has enjoyed wide application in terms of detection and characterisation of bonds. It has helped researchers to better understand chemical bonding in the systems they study. Surely, this reflects the fact that all has not been said with regards to Pauling’s often-quoted definition of a bond; the community does seem to eagerly seize a computable decision on the existence of a bond. I believe that QCT is unique in delivering this option but, unfortunately for users, this approach repeatedly faced scrutiny and criticism from various corners. There is evidently a history of oscillating results without a clear “Pauling Point”! The latter expression, coined at a meeting in 1957 attended by Pauling and Mulliken, refers to a local minimum in the “error-versus-effort curve”. This is a fluctuating curve, with ebbs and flows, which characterises scientific progress. One may have the impression that a theory is correct until the next experiment (“too far”) disproves it. The art is knowing where to stop. Reviewing the literature on the steric repulsion controversy revealed some “cross purpose” communication. Certain views, criteria or examples of systems seem to have been forgotten as time progressed. I believe there is a need for a comprehensive “final” study that, perhaps starting from a review more complete than the current one, involves many more systems than hitherto studied and more angles of attack. It will be fiendishly difficult to find funding for such a project, maybe because it only aims at putting one of the oldest cornerstones of Chemistry on a firm practical and theoretical footing. The only benefit to society would be that both lecturers and students (as well as teachers and pupils) feel better about the material being taught. In the next section, I lodge a few remarks, perhaps of a rather philosophical nature, that may help in outlining new (computational) experiments to solve the problem of bonding. Let me apologise beforehand for the utterly raw and untested nature of these ideas.

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5 Reflections and Perspectives 5.1 The Sudden (Dis)Appearance of Critical Points A concern often raised in connection with bond critical points marking the existence of a bond is the observation that a small variation in the nuclear geometry can make it disappear. Actually, this concern is not well founded because of the assertion that bond paths only materialise for geometries associated with local energy minima. So, strictly speaking, a small change in the nuclear geometry, away from its energy minimum, results in the bond path becoming just an atomic interaction line. In that situation, no statement about bonding can be made. In spite of this serious caveat, I still wish to comment on the sudden nature of the appearance/disappearance of critical points. Of course, there are many examples in daily life, where discrete functions (yes/no) have discontinuous character in time. For example, one is married or not, found guilty in court or not, or dead or alive. Each transition from one state to another could in principle be characterised by an arbitrarily short time span (although this task becomes increasingly more difficult the shorter this span). Any chemist will accept that a given bond, present at the outset of a reaction, is broken at the end of the reaction. The issue is not of yes/no character itself, but rather how narrow the transition regime is allowed to be. Are there are other phenomena in Nature that share the typical discontinuity that the topology displays? An obvious example that springs to mind is a phase transition. Pressure or temperature act as control variables in the same way that nuclear coordinates control the topology of the electron density and hence the existence of a bond critical point. In a phase transition a tiny change in temperature (or pressure) leads to a new state of matter, characterised by very different measurable physical properties (e.g. heat capacity, density). Actually, the transition is only infinitely sharp for an infinitely large sample size, a point to which I return later. A crucial question is: for a single molecule, is there a measurable physical property that can be tightly associated with the topology of the electron density, such that when it changes abruptly, the physical property changes abruptly as well? If such a molecular property existed then one could independently corroborate bonds, via the topology, by measuring this property. This idea makes sense for crystalline materials (i.e. as opposed to a single molecule), an area where Eberhart utters his excitement about the topology being a guide, beyond mere geometry [125], able to explain why materials break [126]. I suggest that one regard the shattered and the original intact piece of matter as two different “phases” of the same material. The sudden disappearance of

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bond critical points upon sufficiently large distortion of the material would then cause the shattering. Even if there is no hard evidence (yet) for this theory, it illustrates what we are searching for in a single molecule. Our search for a discontinuous physical property of a single molecule is discouraged, or at least diverted, by the development of the concept of continuous symmetry measures [127]. Some time ago, Zabrodsky et al. advocated that it was more natural to analyse symmetry properties in terms of a continuous scale rather than in terms of “yes and no”. In order to justify their view they invoked examples such as symmetry distortions due to vibrations, changes in the “allowedness” of electronic transitions due to deviations from an ideal symmetry and so forth. In a similar vein and around the same time Buda et al. [128] defined a “degree of chirality” as a continuous function that is zero if, and only if, the object is achiral. Kanis et al. [129] were motivated to use continuous symmetry measures in their study of the hyperpolarisability of non-linear optic materials in order to provide correlations between structure and physical or chemical properties. Their continuous symmetry metric provided a quantitative characterisation of the extent to which the inversion centre is present for a particular geometry. It then allowed the comparison of computed hyperpolarisabilities with the extent to which the inversion centre remains. Can one change the way the topology decides if a bond exists by generalising the way it operates? A simple suggestion is to measure the onset of the formation of a bond critical point. Any critical point can be detected in a 2D contour plot by a succession of nested loops encircling the critical point. One can think of geometrical measures that gauge the flatness of the electron density and, given the right local curvature, indicate a zone in which the bond critical point could materialise. Similarly, when a ring critical point is present, one could measure the distance between it and a given bond critical point. The smaller this distance, the closer the bond critical point is to annihilation. This well-known quantity of stability acts as a continuous measure of a bond being broken. Instead of working with a yes/no decision, based on the sudden disappearance of a bond critical point, one has access to a parameter that announces the likelihood that this happens. A nuclear skeleton that vibrates due to thermal motion generates a multitude of electron densities, one for each static geometry, and hence a multitude of topologies. Depending on small changes in the nuclear positions, the bond critical point could “flicker” in and out of existence. Ignoring the issue of the non-vanishing forces on the nuclei (which is quite forceful!), should one conclude that the corresponding bond is constantly broken and reformed? Or would it be more convenient to resort to a continuous measure? Perhaps one could average the electron densities according to thermal smearing with Boltzmann weights and then apply the topology. Another alternative is making the topology operate on a sequence of fluctuating electron densities, which in the limit become the real electron density.

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In the Born interpretation of the electron density, an electron is a point that is either exactly here or exactly there, but we cannot predict where it will occur other than with a certain probability. Each time the electron precipitates somewhere in space (i.e. its position is measured) we can mark this spot with a dot. Suppose that we have access to this probability via a realistic but still approximate (molecular) electron density for a one-electron system. We divide space up in tiny boxes. The probability that an electron is found in one of these boxes is given by its volume times the electron density. Now we use a random generator to populate, with dots or “hits”, the tiny boxes in accordance with the known electron density. A box with many hits is located in a region of high electron density. It is only in the limit of an infinite number of hits that the original electron density will emerge. Any electron density obtained by a finite number of hits will be topologically rougher than the original one. This means that there will be many spurious local minima, maxima and saddle points that eventually all coalesce and annihilate into the final topology corresponding to the smooth and crisp original electron density. Whereas a finite sample of hits creates fluctuations around the original electron density, these fluctuations disappear in the limit of an infinite number of hits. This means that a clear and unique bond critical point will emerge as the random generator continues to populate boxes. The same is true for an interatomic surface. It is tempting to draw an analogy with phase transitions, since phase boundaries become sharper as the number of particles in the system increases. The particles are then the equivalent to the snapshots of electron dots. The purpose of this excursion is to emphasise that the computed electron density we usually work with is an idealised object. In high-resolution X-ray crystallography we actually measure structure factors, which after a substantial amount of processing, lead to data that we fit a model electron density to. A topological analysis can only be performed on such a fitted model density, not on the “real object”. Is the problem surrounding the sudden character of topological change maybe artificial in that it refers to a model electron density only? Of course one could argue back that the predictions made by the Schrödinger equation agree extremely well with measured spectra; hence the corresponding electron densities must be true and real as well. 5.2 Energy as a Guide Hopefully the following analogy is not too far-fetched or flawed in formulating the essence of the problem of the chemical bond. Imagine a group of people attending a standing reception. A person stands for an atom and the group of conversing people for a molecule. We are concerned about the driving force that holds the group together. If each person stood by themselves, without conversation, the reception would soon be over or not even

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take place. Groups form naturally and remain stable for a long period of time because, overall, each member of the group feels happier being in it than being alone. An enjoyable conversation stands for a chemical bond. When two people have a pleasant one-to-one conversation they will stay in each other’s proximity and form a small stable group. We assume that when the conversation is over, they part, which is equivalent to the break-up of a diatomic molecule. Conversely, a repulsive interaction would stand for an unpleasant conversation, perhaps even an argument, in short, a reason to break up. Now imagine three people forming a group. If one person talks to the two others at the same time, one would have an analogue of the molecule water. The fact that the two others do not talk to each other is analogous to the absence of a bond between the two hydrogens. Even if they start talking to each other and an argument develops between, they could still stay together as a group of three, each individual staying in his or her physical position. The group does not break up because overall the three individuals still feel better inside the group than on their own. One could imagine this situation with a charity donor and two possible recipients, or two sisters who admire the same gentleman. When observing a larger number of people, forming a stable group, the task at hand is to disentangle the overall togetherness into pairwise interactions, whether embodied as pleasant or unpleasant conversations, or the lack of them. Returning to chemistry I now put the spotlight on energy as opposed to the electron density. From the point of view of energy, defining a bond reduces to the following question: how can one “factorise” a multi-dimensional energy surface E(r1 , r2 , ..., rn ) to a set of one-dimensional energy curves, where {ri } are a set of internal coordinates? For a diatomic molecule, matters are simple: if E(r1 ) shows a (finite) minimum, then the system is bound. This is expressed by the atomic interaction line that connects them. This line becomes a bond path at the minimum of the energy profile. An important question is whether this connection between a one-dimensional energy curve and a bond path (or even an atomic interaction line) can be extended to a polyatomic. For a nonlinear molecule there are 3N – 6 internal coordinates, given N nuclei. The total number of bond paths in a general polyatomic is less than 3N – 6. For example, methane has four bond paths but needs 3 × 5 – 6 = 9 internal coordinates for its potential energy surface to be completely described. In summary, one lacks degrees of freedom if bond paths determine the factorisation of the energy landscape; if internuclear distances establish energy factorisation, then there are too many degrees of freedom. The following simple observation is crucial: suppose an energy surface is expressed in terms of a set of internal coordinates. Concentrate on an equilibrium geometry, which corresponds to a point on the energy surface with zero gradient and a Hessian yielding only positive eigenvalues. We then know that the whole molecule is stable and bound. This means that we necessarily recover a minimum along each of the eigenvectors associated with the positive

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eigenvalues. Thus all the one-dimensional energy curves (i.e. energy versus displacement) show a minimum. If we associate an atomic interaction line (or a bond path) with such an internal degree of freedom, then we recover a true bond, both in the mathematical and physical sense. The analysis of molecular spectra should be mentioned here. Within the harmonic approximation one defines normal modes as an independent synchronous motion of atoms or groups of atoms that may be excited without leading to the excitation of any other normal mode. The normal mode can be regarded as the outcome of unambiguous factorisation of a high-dimensional molecular energy surface into one-dimensional profiles. However, the problem is that each mode always involves all the atoms, in general. This means that the aspiration of being able to associate a one-dimensional energy profile with a two-centre bond (i.e. linking two nuclei) evaporates. Obviously, certain normal modes are localised in one area of a molecule, sometimes even in a bond; this effect is what turns IR spectroscopy into a structure determination tool, in that one can associate certain frequencies with particular bonds. Heavy atoms generally move less than light atoms in normal modes. To illustrate the elusive energy-bond association we take carbon dioxide, a linear molecular that has four vibrational modes. As expected, the antisymmetric-stretch normal mode involves a collective motion of all three nuclei. When combined with the symmetric normal mode one can define a stretching mode with one CO bond and another with the other CO bond. This permissible description has the disadvantage that when one CO bond vibration is excited, the motion of the C atom sets the other CO bond in motion. Hence energy flows backwards and forwards between the two (dependent) modes. Interesting research has focused on the central question to what extent the molecular vibrations are localised in individual bonds. Some time ago Henry reviewed [130] the use of local modes in the description of highly vibrationally excited molecules, and their ability to account for overtone spectra. If one continually pumps energy into the vibrational motions of a molecule, one will generate dissociation, which is a markedly anharmonic phenomenon. Dissociation along the totally symmetric CH stretching normal mode in benzene would require the simultaneous breaking of all six bonds. This is a prohibitively high-energy process, and the rupture of one or two CH bonds is more likely. It should be recognised that any realistic potential energy surface must develop valleys in the directions of the bond coordinates as the energy increases towards dissociation. As an example, local behaviour was clearly established [131] at the second excited vibrational level of water. Returning to the potential energy surface of water, resulting from straightforward ab initio calculations, one can conceive the following computational experiment. Imagine geometries of water restricted under C2v symmetry. This reduces the degrees of freedom from 3 × 3 – 6 = 3 to only 2, without loss of generality, because we are only interested in reducing multidimensional energy landscapes into independent (or very weakly coupled)

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one-dimensional ones. Working with two dimensions is sufficient to investigate the principal feasibility. Figure 7 shows water in the XY plane. Since the C2v symmetry must be preserved the free motion of the hydrogen nucleus on the right is mirrored by the hydrogen on the left. Imagine that a sufficiently realistic 2D energy surface is computed for an arbitrary displacement
of the hydrogen nucleus of the right. The energy can be fitted via E(x, y) = amn fm (x)fn (y), with a convenient set of basis functions { fi }. The mn

question is whether one can find more compact factorisations with a superior choice of coordinates. Can one recover the view that there is a strong energetic relationship between O and H and a much weaker one between H and H? Figure 7 shows how one can move Hright without changing the rOH coordinate, i.e. the distance between oxygen and hydrogen. It is also possible to move Hright along the line parallel to the y-axis, a motion that does not change the rHH coordinate. Note that Hright can be repositioned such that only the OH distance changes, or only the HH distance. The purpose of this excursion is to find out which internal coordinates have the most influence on the total molecular energy in terms of energy rises. This question is equivalent to the one above: a bond is a special interaction between a pair of atoms being part of a molecule. What gives us the right to single out such special pairs, is that they represent the total stability of the molecule very well. Indeed, if we change one bond length (i.e internal coordinate associated with a bond), then the total energy goes up dramatically. So-called non-bonded interactions, which can also be expressed via internuclear distances, do not influence the total energy that much. Although progress along these lines of thought may deal with the essence of a bond in terms of energy considerations there is an issue left. This type of analysis leads to a continuous picture. In other words, what is the threshold to call something a bond or not? Arbitrary cut-offs should of course be avoided. The topology of the energy surface could help, but that leads us back

Fig. 7 Schematic representation of water under C2v symmetry. The y-axis is the twofold rotation axis C2 . The hydrogen nucleus on the right can move freely (described by coordinates x and y) but the hydrogen nucleus on the left must mirror this motion in order to preserve the symmetry

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to Mezey’s catchment regions. This interesting concept, couched in perhaps excessively formal language, is not implemented in any computer package, to my knowledge. Especially when topological basins in spaces of dimensionality higher than three are required, exciting software development may be on the horizon. A local minimum in the multidimensional potential energy surface is then an attractor (or maximum) in a “stability function”, which is conveniently defined as minus the total energy. The basin, dominated by the attractor, corresponds to a collection of nuclear geometries that, when relaxed or “geometry optimised”, collapse into the geometry of the local minimum. There must be separatrices (“interatomic surfaces”) in the high-dimensional energy surface, that mark special geometries that can collapse to two competing local minima. The topological complexity of higher-dimensional spaces can soon become overwhelming, since there are N – 1 types of saddle points in an N-dimensional space, for example. I refer to Sect. 2 of Chap. 4 in an introductory book [23] in an attempt to avoid the confusion that lingers in the debate on bonds versus repulsive interactions. I will not reproduce the discussion of that section but simply reiterate the terminology that was introduced in order to clarify the debate’s semantics. Let us assume access to a well-defined and one-dimensional energy curve for one bond, independent or very weakly coupled with that of other bonds. If the second derivative of the energy with respect to a change in bond length is positive then the interaction is attractive. If the second derivative is negative then the interaction is repulsive. This is a description of an equilibrium situation, where this equilibrium could be metastable, i.e. not restoring. The terms “attractive” and “repulsive” do not say anything about how the equilibrium was reached. Neither do they say what the constituent terms are that govern it, taking for granted that one can break down the energy in meaningful contributions. We only know that the repulsive profile is unstable and that the right one is stable or restoring. In both cases there is an atomic interaction line between the nuclei and in the attractive case this line becomes a bond path because the forces on the nuclei vanish. All that can be said about the repulsive case is that it can never arise between two atoms that are part of a molecule at a local energy minimum. However, this situation is possible in a transition state. Should we then call it an unstable bond or is this a genuine case of a repulsive interaction? The second pair of terms that is useful in the debate uniquely refers to the first derivative of the energy profile. I propose the terms attracting and repelling to describe the case where the first derivative is positive and negative, respectively. If the first derivative is zero then neither term applies. These terms are clearly “dynamic” in the sense that they describe an action or a direction of change if an external constraint is removed. Thus for a bond we cannot use the terms attracting or repelling because it refers only to the (dynamic) process that led to the formation of the bond. This is an important distinction because in discussions, and in the literature, features of first and second derivatives seem to be mixed up. Using

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this language, it is desirable to extend the computational experiment on water (see above) to other molecules and complexes that featured in the controversy unfolded in Sect. 4 of this article. It is quite possible that a chemist judges two nuclei connected via an unusual atomic interaction line to be part of two “clashing” groups, but ultimately the mathematics of the energy surface says that these nuclei “want to be located there along any internal degree of freedom”. Any imaginable “initial resistance” of these nuclei as they approach each other (during the optimisation process) is eventually compensated by other rearrangements of the nuclear skeleton such that a zero force minimum energy conformation is obtained “pleasing all nuclei”. This situation of balance and compromise is compactly expressed in terms of a bonding scheme. One should be aware that there are other internal degrees of freedom, such as the valence and torsion angles, which are not expressed in the topology of the electron density. Imagine that in the factorisation of the energy hypersurface one hits a well-isolated energy subspace involving three nuclei, rather than two. Ring critical points could be associated with such irreducible units. A virulent paper [132] by Cassam-Chenaï and Jayatilaka exposed difficulties with AIM’s interpretation of excited vibrational states. Unaware of the successful AIM analysis [133] of the density distribution of excited Rydberg states of methanal and ammonia, these authors pointed out that the vibrationally averaged electron density in the v = 1 state of N2 shows four atoms instead of two. Their paper criticised other aspects of AIM, including the assertion that the Schwinger subsystem variational principle cannot be applied to zero flux domains. Their argument is based on a single counterexample, which I prove to be false in Appendix A. Due to these and other criticisms their paper provoked an unparalleled response from various corners [134–137], leading to a sustained combative reply from the authors [138].

6 Towards a QCT Force Field 6.1 Convergence In this Section I review our work on force field design, which is based on the transferability of atomic multipole moments. We adopt Stone’s spherical tensor formalism and the idea of a multi-centre multipole expansion, one site for each nucleus. However, the moments are obtained via integration over an atomic basin of the appropriate tensor times the electron density, rather than via the DMA route. Starting with the electrostatic potential I move on

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to electrostatic interaction, molecular simulation, atom types, and end with polarisability. Given that this volume is dedicated to Anthony Stone, perhaps a good starting point is his excellent, frequently consulted and already reprinted monograph entitled “The Theory of Intermolecular Forces” [4]. In Sect. 7.4 of the 1996 Edition two alternative methods to calculate distributed multipole moments, Voronoi polyhedra and Bader’s Atoms in Molecules treatment, are briefly discussed. It is concluded that “neither are suitable for application to intermolecular forces because they have poor convergence properties.” This conclusion was based on early work [8] by Stone and Alderton, focusing on the nitrogen atom in N2 , where the atomic boundary is a trivial plane. Interestingly, in the same year (1985), Cooper and Stutchbury published [139] successful geometrical predictions of seven HF-containing van der Waals complexes, using AIM multipole moments. Also, in 1996, on the back of a paper [140] on integration of topological atoms, a minimum in the N2 dimer potential energy surface was correctly reproduced with AIM multipole moments, and a comparison with DMA showed a discrepancy of only A in centre of mass to centre of mass 0.02 kJ/mol in total energy and 0.002 ˚ separation. Due to these contradictory findings we decided to investigate the convergence issue systematically, starting with the atomic electrostatic potential. 6.2 The Electrostatic Potential The electrostatic potential was chosen as a starting point because it is a special case of electrostatic interaction, where one of the interacting partners is just a proton. For the first time the exact electrostatic potential generated by an AIM atomic fragment was computed [141]. The premise that the multipole expansion associated with bounded fragments in real space, such as QCT atoms, has poor convergence proves to be wrong. One does not need an excessively large number of QCT multipole moments to reproduce the exact ab initio molecular electrostatic potential. The atomic population (or rank-zero multipole moment) is just one term of the expansion of a physically observable quantity, namely the electrostatic potential. Hence the QCT charges cannot be judged on their reproduction of the electrostatic potential. Instead, they must be seen in the context of a multipole expansion of the atomic electrostatic potential. An advantage of the finite (i.e. sharply bounded) nature of topological atoms is that they have a finite convergence sphere. This means that the convergence condition associated with the multipolar expansion can be exactly obeyed, which is impossible with atoms that extend to infinity. This point is illustrated in Fig. 8. Finally we computed the exact atomic electrostatic potential and its value obtained via multipole expansion for molecules including molecular nitro-

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Fig. 8 Since topological atoms are finite it is possible to monitor their formal convergence. The atomic electron density is totally contained within the blue volume. Since |r | < |r| outside the convergence sphere the potential (V(r)) converges exactly in this region (with permission from the American Chemical Society)

Fig. 9 The deviations in the exact atomic electrostatic potential and the one obtained from the multipole moments up to the octupole (l = 3) for the Cα atoms in alanine. The part of the picture in front of the plotting plane is deleted in order to show the interior of the object. The largest deviations occur near the cusp-like edges of the atom and the region of closest proximity. Color code (in kJ/mol): white < 0.1 < grey < 0.2 < blue < 0.3 < green < 0.4 < yellow < 0.5. (with permission from the American Chemical Society)

gen, water, ammonia, imidazole, alanine and valine. Figure 9 shows the favourable convergence behaviour of the electrostatic potential of the central carbon in the free amino acid alanine, for a modest number of multipole moments. We then tried to understand the cause of this excellent convergence behaviour. How can it be compatible with the admittedly highly non-spherical shape of the topological atoms? The answer [142] lies in the exponentially decaying electron density. The convergence behaviour of the electron density inside an atomic basin is due to its decay rather than to the atom’s shape. Indeed when the atom is filled with a uniform density the convergence worsens,

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often by more than an order of magnitude. We confirmed that finite atomic shapes have undesirable convergence properties but made clear that this phenomenon is in practice not relevant due to the profile of the actual electron density inside. The electrostatic potential can be used as an appropriate and convenient indicator of how transferable an atom or functional group is between two molecules [143]. The potential generated on a grid by the terminal aldehyde group of the biomolecule retinal was compared with the corresponding aldehyde group in smaller molecules derived from retinal. The terminal amino group in the free amino acid lysine was treated in a similar fashion. Each molecule was geometry-optimised by an ab initio calculation at the B3LYP/6311G+(2d,p)//HF/6-31G(d) level. The amino group in lysine turned out to be very little influenced by any part of the molecule further than two C atoms away. However, the aldehyde group in retinal was influenced by molecular fragments six C atoms away. This dramatic disparity was ascribed to the difference in saturation in the carbon chains; retinal contains a conjugated hydrocarbon chain but lysine an aliphatic one. A disadvantage of the traditional multipole expansion is that it introduces a divergence sphere within which the expansion diverges. Because of their finite size, topological atoms yield a small divergence sphere. However, the introduction of an alternative continuous multipole expansion [144] reduced the divergence region even further. The new method allows the electrostatic potential to be evaluated accurately at short range, which is illustrated for a pair of simple molecules. 6.3 The Electrostatic Interaction: Structure and Dynamics Going beyond the potential, we performed [145] a careful test of the convergence of the electrostatic interaction, in the context of the Buckingham– Fowler model, using improved algorithms. Particular attention was paid to the convergence of both the energy and the geometry of a set of van der Waals complexes, with respect to the rank L of the multipole expansion. We contrasted the interaction energies obtained via multipole expansion with the exact values, obtained via six-dimensional integration. The latter is a simultaneous integration over two (three-dimensional) atomic basins. Although the QCT energies converge more slowly than the DMA ones, excellent agreement is obtained between the two methods at high rank (L = 6), both for geometry and energy. This study is the first direct, complete and explicit comparison between AIM and DMA energy convergence behaviour. Contrary to views expressed before in the literature this work opens an avenue to introduce the topological approach in the construction of an accurate intermolecular force field, especially for microstructure.

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We then deepened [146] our understanding of the difference between the DMA and the QCT approach. An essential idea (or even driving force) behind the DMA approach is that the convergence of multipolar expansion is accelerated by distributing the multipoles over a multitude of sites (i.e. expansion centres endowed with local axis systems). We proved it possible to “unpack” or distribute the topological multipole moments over extra sites, over and above the nuclear positions. This distribution also led to accelerated convergence in the QCT approach, without interfering with the fundamental way QCT partitions the electron density. As such, we preserved the advantages that QCT offers in terms of ambiguous partitioning and transferability. In order to assess the quality and the speed of this convergence we made systematic comparisons between QCT and DMA for a set of small van der Waals systems and a set of much larger DNA base pairs. Based on a clearly made distinction between partitioning and distribution we showed for the first time how topological multipole moments can be distributed to off-nuclear sites. In the QCT context the addition of extra sites is more beneficial to the convergence of the electrostatic interaction energy of small systems. However, in large systems excellent convergence was found for QCT without the introduction of extra sites, a very surprising result that further encourages the development of a topological intermolecular force field. Following the success of the topological electrostatic model on van der Waals complexes we assessed [147] its performance on the important biological problem of DNA base-pairing. Geometries and intermolecular interaction energies predicted by AIM multipole moments, supplemented with a hard-sphere or Lennard-Jones potential, have been compared with other methods in two stages. First with supermolecular HF, MP2 and B3LYP calculations at the 6-31G(d,p) level and then with other potentials such as Merz–Kollman (MK), Natural Population Analysis (NPA) and DMA at 6311+G(2d,p) level. The geometries for all 27 base pairs predicted by AIM and B3LYP differ by 0.08 ˚ A and 3.5◦ for 55 selected intermolecular geometrical parameters, while the energies show an average discrepancy of 6 kJ/mol. The B3LYP functional proves to be a reliable alternative to MP2 since their energies are in excellent agreement (∼ 1 kJ/mol). Globally, the AIM interaction energy curve follows the same pattern as that of MK, NPA and DMA. The MK model systematically underestimates the interaction energy and NPA shows undesirable fluctuations. Surprisingly, the convergence of the AIM multipole expansion is somewhat better than that of DMA, but both have similar basis set dependence. A test of AIM on a DNA tetrad suggests that it is able to predict geometries of more complex nucleic acid oligomers than base pairs. This work clearly demonstrates that the electrostatic description dominates DNA base pair patterns but more work is needed to predict the three most stable base pairs better. A current inadequacy of this AIM potential is that it is combined with empirical repulsive potentials, and hence not completely derived

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from ab initio calculations. Ultimately and ideally, the AIM potential should draw all its information from ab initio calculations on monomers. In a related study [148] we made a small contribution to the ambitious question of what causes the stability of a DNA duplex. Recent work [149] indirectly supports the view that (reliable) hydrogen-bonded (as opposed to stacked) base pairs are primarily stabilised by electrostatic interactions. In the area of molecular recognition and supramolecular chemistry researchers are in need of rules that predict the stability of a complex, beyond counting the number of hydrogen bonds. Jorgensen provided such a rule in 1990 [150], known as the secondary interaction hypothesis. This rule focuses on “crosscontacts” between frontier atoms. It favours “donor-acceptor” interactions and penalises “donor-donor” or “acceptor-acceptor” interactions on the basis of simple (qualitative) point charges (δ+ or δ– ). We investigated whether this secondary interaction hypothesis could be supported by QCT. After a careful and systematic analysis of 27 natural base pairs the answer proved to be negative. We also questioned the existence of subsets of atoms located in two different bases, forming a complex whose interaction energy parallels the total interaction energy. After a series of calculations designed to find support for the secondary interaction hypothesis we failed to find a physical basis for it. Only in comparisons between highly similar chemical environments could the secondary interaction be invoked for the right reason [151]. However, in general, simple rules for rationalising the pattern of energetic stability across naturally occurring base pairs in terms of subsets of atoms unfortunately remains elusive so far. This work cautioned against unjustified use of secondary interactions, which may lead to the same quandary that the hydrogen bond once introduced by its over-generalised use. Water clusters and hydrated amino acids are further important systems that were subsequently investigated [152] with QCT multipole moments. For pure water clusters up to the nonamer we compared the results of the QCT multipolar potential with B3LYP/6-311+G(2d,p) supermolecular calculations and the popular TIP4P and TIP5P potentials. We found that (a) multipole interaction at ranks L = 3 and 5 perform better than that at L = 4 and 6 respectively, (b) lower rank representations are more successful in geometry prediction for the larger water clusters, (c) the amino acids (Tyr and Ser) perturb the structure of the water clusters very little, compared to the pure water clusters, and (d) TIP4P performs well for interatomic distances, often better than L = 6 for geometries. However, the QCT potential truncated at L = 5 is better overall. From a systematic analysis of the QCT-partitioned supermolecular electron densities we learnt that (a) the number of H-bonds donated by a water molecule determines its total molecular dipole, (b) the molecular volume of each water molecule is determined by the number of H-bonds it is involved in, (c) the atomic charge of each hydrogen atom is affected by whether it is directly involved in a hydrogen-bond, and (d) all oxy-

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gen atoms in water clusters are found to be more negatively charged than if they were in a free water molecule. Molecular Dynamics simulations of liquid H2 O [153] and HF [154] at ambient conditions, were carried out with the electrostatic interaction represented up to L = 5, and repulsion and dispersion via a LJ potential. The initial purpose of these studies was to explore how very high multipole moments perform in simulations. An implementation (in the program DLMULTI) of Ewald summation, developed by Leslie for (spherical tensor) high-order multipoles, accounted for the long-range nature of the multipole interactions. After preliminary calculations monitoring energy stability we decided to perform the “NPT ensemble” simulations with 216 water molecules at 278, 288, 300, 308 and 318 K and pressure of 1 atm. The equations of motion were integrated using a time step size of 0.5 fs with total simulation times up to 150 ps (300 000 steps). Our results revealed the importance of including higher multipoles (L = 5) in the liquid simulation. For L < 5, the pair distribution function does not show a second peak in the region between 4 and A. In addition, the valley between the first and second peaks is too shallow. 5˚ For L = 5, the OO, OH and HH pair distribution function are almost identical to those generated using the TIP5P potential [155]. With only two adjustable A) and ε (0.753 kJ mol–1 ) in the Lennard-Jones potenparameters, σ (3.140 ˚ tial function, the density of the equilibrated system is 0.999 g cm–3 . Liquid simulations at four extra temperatures showed a maximum in the density at about 15 ◦ C, which is off by 11 ◦ C from the well-known experimental value of 4 ◦ C. The density of simulated water is within less than one percent of the experimental value, while the calculated energy of the liquid is within 3% of the experimental result. The experimental value of the self-diffusion coefficient, D, is underestimated by at least 32%. The value for Cp is overestimated by 28% and the thermal expansion coefficient α by 27%. This homogeneous error distribution is most encouraging and also somewhat intriguing given that simpler models such as TIP5P produced values for the same bulk properties (D, Cp and α) deviating from experiment by 14%, 62% and 145%, respectively. The calculated correlation coefficients between the calculated QCT profile and the experimental profile of gOO (r), gOH (r) and gHH (r) are 0.976, 0.970 and 0.972, respectively. 6.4 Coulomb Energy We also focused [156] on the Coulomb energy between atoms in supermolecules. We proposed an atom-atom partitioning of the Coulomb interaction, which should not be confused with the electrostatic component of the intermolecular interaction, defined within the perturbation approach. Instead, this atom-atom Coulomb interaction energy uses the total molecular (in the case of a single, covalently bound molecule) or the supermolecular (in

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the case of a complex) electron density as its input. Atom-atom contributions to the molecular intra- and intermolecular Coulomb energy were computed exactly, i.e. via a double integration (6D) over atomic basins, and by means of the spherical tensor multipole expansion, up to rank L = lA + lB + 1 = 5. The convergence of the multipole expansion was able to reproduce the exact interaction energy with an accuracy of 0.1–2.3 kJ/mol at L = 5 for atom pairs, each atom belonging to a different molecule in a given van der Waals complex, and for non-bonded atom-atom interactions in single molecules. The atom-atom contributions did not show a significant basis set dependence (3%) provided electron correlation and polarisation basis functions were included. The proposed atom-atom Coulomb interaction energy could be used both with post-Hartree–Fock wave functions, without recourse to the Hilbert space of basis functions, and with experimental charge densities, in principle. After providing computational details of this method we applied it to (C2 H2 )2 , (HF)2 , (H2 O)2 , butane, 1,3,5-hexatriene, acrolein and urocanic acid, thereby covering a cross section of hydrogen bonds, and covalent bonds with and without charge transfer. The Coulomb interaction energy between two molecules in a van der Waals complex was computed by summing the additive atom-atom contributions between the molecules. Our method is able to extract from the supermolecule wavefunction an estimate of the molecular interaction energy in a complex, without invoking the reference state of free non-interacting molecules. Provided quadrupole-quadrupole interactions were included the convergence was adequate between atoms belonging to different interacting molecules. Within a single molecule the convergence was reasonable except for bonded neighbours. These observations were again useful as a guide for force field design. 6.5 Auxiliary Insight The work described so far constitutes a vital guide in the design of an intra- and intermolecular QCT force field, and complements our other work [157–159], which we do not explicitly describe in this review. This work computed, for the first time, atom types, based on a statistical analysis of 760 atoms taken from amino acids and smaller derived molecules. In a tangent paper [160] a new algorithm was proposed that calculates QCT atomic charges using surface integrals only. The divergence theorem expresses the atomic charge as the flux of the total molecular electric field through the boundary of the atomic basin. Since the molecular electric field can be calculated analytically by very fast algorithms, and since the surface integration requires one to two orders of magnitude fewer quadrature points, an atomic charge can now be obtained must faster. This approach also contributed to the robustness and accuracy of atomic integration because it does not suffer from the cusp problem or the multiple intersection problem. Beside

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a reduction in the number of quadrature points the new algorithm should provide a more straightforward route to obtain atomic charges. The “Topological Partitioning of Electronic Properties (TPEP)” [161] approach at the Hartree–Fock level was used to investigate charge transfer response in a water dimer [162]. Distributed polarisability components were employed to calculate the change in electron density under external fields. It was found that charge flow between the water monomers was most significant along the direction of the hydrogen bond. The molecular polarisability of the molecules in the dimer was reduced owing to formation of the hydrogen bond. Calculation of distributed polarisabilities was desirable for practical applications. Calculation of distributed polarisabilities was also desirable for conceptual understanding of how the electron distribution responds to the external influence of electric fields, affording insights beyond those available from analysis of the ground-state electron distribution. The calculation of the atomic and molecular polarisability of the dimer (and by implication of other sets of molecules) also proved to be a powerful tool for providing insight into the effect of interaction on the electron density and its response to electric fields. In particular, the separate molecular polarisabilities in the dimer can be calculated from the distributed components of the molecular polarisability of the dimer, which was not possible using standard ab initio packages. The ab initio computational approach of distributed response analysis was used to quantify how electrons move across conjugated molecules in an electric field, in analogy to conduction [163]. The method promised to be valuable for characterising the conductive behaviour of single molecules in electronic devices. We have shown that the ab initio computational technique of distributed response analysis can, in principle, be used to quantify the conduction behaviour of a single molecule acting as a switching region in a nanoscale field effect transistor design. In an analogy to conduction, it is shown that once an electron is introduced on one side of the molecule, it has the ability to move across the molecule. The calculations demonstrated in detail how charge flow occurred in para-nitroaniline(pNA), meta-nitroaniline(mNA) and fluorobenzene(FB). On the basis of our calculations, we predicted that pNA has a more efficient conductive behaviour than mNA and FB. We established the empirical result that 1 unit of molecular total conduction number (CN) is approximately equal to 1/6 a.u. of (static) molecular polarisability. This result has been used to estimate the conductive behaviour of similar conjugated molecules by using the average molecular polarisability. It was predicted that 4, 4 -biphenyldithiol has a CN of 25.6 and therefore should have the most effective conduction behaviour of the molecules investigated. In summary, this careful and systematic work has laid the groundwork for a topological force field (both inter and intra) drawn from ab initio electron densities and perhaps the second order reduced density matrix. Topological potentials are extractable and valid irrespective of computational

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scheme (plane wave, Slater, Gaussian, basis set stability, Hartree–Fock or post-Hartree–Fock), are properly rooted in quantum mechanics and part of a wider context than many perhaps efficient but ad hoc methods. We believe that, now that convergence issues are properly understood, we are in a good position to reap the advantages of a quantum topological potential, especially for short-range interactions.

7 Future Perspectives on QCT Potentials 7.1 The Need for Anisotropy The design of new and better potentials for structure and dynamics seems to be a fairly haphazard activity, with results appearing scattered in the literature. At one end papers report isolated ideas, unfortunately leading to methods or implementations with poor take-up. At the other end, papers describe ad hoc fixes in the interest of improved modelling of a specific compound or class of them. However, most papers basically take their potential for granted and do not worry about how the (commercial) package used compares to an alternative one. This is why rare papers such [164] are vital to make solid progress. This paper compared the performance of six wellknown force fields on “trialanine”, including versions of AMBER, CHARMM, GROMOS and OPLS. In their paper Mu et al. used diplomatic phrases such as “this result is somewhat at variance” and “most interesting” to describe the alarming and shocking discrepancies between the results generated by the different force fields, even if they only differ in their parameter set (e.g. AMBER’s parm94 and parm96). The article concluded that “it is therefore not clear to what extent commonly used force fields are capable of correctly describing non-equilibrium dynamics such as the folding or unfolding of a peptide”. This state of affairs, our previous work on intermolecular potentials and atom types, and illuminating papers such as [165] on point charges and their defects, prompted us to set up a QCT peptide force field. One central idea driving this endeavour is getting the non-bonded part of the potential right first and then forcing the bonded part to adjust the potential towards ab initio energies of a training set of molecules. This procedure is opposite to the customary one, which historically focused on the bonded part first and then introduced the non-bonded part. The reason for this development is that hydrocarbons were modelled first (Allinger’s MM force fields) and only then peptides and nucleotides, which are polar molecules. Our “upside-down” approach is justified by the fact that the long-range Coulomb interaction is

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physically well understood and can be represented rigorously. Remaining inadequacies should then be absorbed by the fitting of the bonded parameters. With an eye on even more powerful computers being available in the very near future it is a good idea to aim at enhanced accuracy of force fields. Designing, testing and completing the latter is a huge effort, which is still ongoing, but we have very recently proven that the basic idea behind it works. We have also incorporated in this force field the recently resurrected Ligand Closed Packing (LCP) model [166–168]. This model, which originates in the early work [169] of Bartell, highlights the importance of ligand-ligand interactions. These interactions are ignored in current force field thinking because of its fixation with central atom-ligand interaction, which can be traced back to bonding patterns. 7.2 Short Range Interactions and Overlap The next important topic to investigate by means of the topological approach is the short-range interaction. As research progressed it became increasingly clear that one should take advantage of a unique feature that the quantum topological approach offers, namely that atoms and hence molecules never overlap. Instead, molecules are perceived as malleable units that respond to each other’s presence by distorting their own atoms. This ansatz, which is a radical departure from classical thinking in the field of intermolecular interaction, will eliminate the need for damping functions and penetration energies. Figure 10 shows a local minimum of the water...methanal van der Waals complex.

Fig. 10 A local energy minimum of the water (dark). . .methanal (light) complex. Bond paths and interatomic surfaces (both in bold) are superimposed on an electron density contour plot

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The QCT partitioning enables the ambiguous definition of a molecule in a complex and results in molecules not overlapping but distorting each other. When taken to the limit, this natural consequence of the topology of the gradient vector field alters the philosophy that dominates current thinking in intermolecular interactions. Perturbation theory assumes that molecules retain their identity as they interact. However, if two molecules are in close proximity their boundaries and hence their identities become blurred, unless one can resort to a robust partitioning scheme. QCT is such a scheme, enabling the clear delineation of a molecule in a complex, even at very short range. Interestingly, this view links molecular interaction to daily life perception, where macroscopic objects have sharp boundaries and deform each other. It is therefore tempting to introduce techniques from engineering, such as finite element analysis, in the modelling of intermolecular interactions. That there is a need to think about alternative ways to model shortrange interaction is clear from the following. For closed-shell molecules ordinary non-degenerate Rayleigh–Schrödinger(RS) perturbation theory yields well-defined energy contributions in its so-called “polarisation approximation” [170]. The 0th-order energy term is the sum of the monomer energies, the 1st-order energy term is the electrostatic interaction and the 2nd-order energy term consists of mutual induction and dispersion energy terms. However, at short-range molecules overlap considerably, which seriously challenges the perturbation approach due to the need for antisymmetrisation between the interacting molecules. As a result, we cannot unambiguously assign an order to a term anymore. A related difficulty is that there can be no 0th-order Hamiltionian that has antisymmetrised wavefunction products as eigenstates. Hence the standard RS perturbation theory can no longer be used and a multitude of perturbation treatments have been proposed [4]. Moreover, electron correlation is not included [171] or if so, leads to an extremely complicated formalism [172] and is found to strongly affect repulsion. The problem of overlap and identity of molecules goes deeper than intermolecular perturbation theory. It also affects valence-bond theory where “overlap” is a commonly used concept to rationalise interaction between atoms. Figure 11 goes to the heart of the problem. Two clouds can easily be perceived as separate objects provided they are each marked by a colour. It is because they are marked and separable that we can speak of overlap. Overlap only makes sense as a concept, provided we can identify the (overlapping) objects in the first place. Figure 11b shows how the idea of overlap is challenged if the clouds are not marked by different colours. This picture is closer to reality, whereas the illustration in Fig. 11a is actually a mental construction, when applied to molecules or atoms. The power of the topological approach lies in its ability to partition the object in Fig. 11b into its constituent parts, without prior knowledge of their identity at long range. Assuming for a moment that there are only two attractors in Fig. 11b the topology will identify two non-overlapping objects,

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Fig. 11 (a) Overlap between two clouds is easy to define if each cloud already has a clear identity. (b) Overlap is challenged as a concept if the overlapping objects are not properly defined

bounded by a sharp curve (separatrix). To illustrate the general nature of the problem of overlap I show two examples of colliding galaxies in Fig. 12. Let us imagine that “star density” replaces the electron density. The object in Fig. 12a is clearly separable visually and corresponds to the regime of long-range interaction. Fig. 12b depicts a situation where the topological partitioning method can help in identifying the two originally separated galaxies that have become blurred upon collision. The bright orange cores would be identified as attractors. Topological atoms are defined solely by using the gradient of the electron density, . As a result they are independent of how  is obtained. For example, they can be derived from Gaussian, Slater or plane wave basis functions. Since they are not formulated in Hilbert space, the inclusion of electron correlation does not lead to conceptual difficulties. We adhere to this philosophy and extend it to the use of (spinless) reduced density matrices in the formulation of atomic interaction energy. So far our work has only focused on the Coulomb part but the exchange-correlation part can also be treated in a similar but generalised framework, as I show now.

Fig. 12 Examples of two colliding galaxies (a) Material flowing between two battered galaxies that bumped into each other about 100 million years ago. (b) The cores of the twin galaxies are the orange blobs (Photographs NGC 5426/5427 and NGC 4038/4039, taken by the Hubble telescope, from http://hubblesite.org/newsdesk/archive/releases/ 1997/34)

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After integrating out spin the energy of a molecular system can be written as [173, 174].     1 2  E= – ∇r 1 (r, r ) dr + v(r)(r)dr 2 r =r    Zα Zβ 1 + 2 (r 1 , r2 )dr1 dr 2 + (1) r12 Rαβ α β>α

where the terms are in order: the electronic kinetic energy, the nuclearelectron potential energy, the electron-electron potential energy Epot,ee and the nuclear-nuclear potential energy. This master equation, which is valid at Hartree–Fock level and beyond, is the starting point for our atomic partitioning. In Eq. 1 we introduced 1 as the first order reduced density matrix (and its diagonal ), and 2 the (diagonal) second order reduced density matrix, written as 1 2 (r 1 , r2 ) = (r 1 )(r 2 )[1 + h(r 1 , r2 )] (2) 2 where h(r 1 , r2 ) is the pair correlation function, a function that incorporates all non-classical effects, such as exchange and repulsion between monomers in a complex. The first contribution to Epot,ee is the Coulomb interaction given by  1 1 (r1 )(r 2 )dr 1 dr 2 (3) J= 2 r12 The second contribution to Epot,ee is the exchange-correlation energy  1 1 (r 1 )xc (r 1 , r2 )dr 1 dr2 (4) K= 2 r12 In the Eq. 4 xc is Slater’s exchange-correlation hole and is defined via xc (r 1 , r2 ) = (r 2 )h(r 1 , r2 )

(5)



2) The Coulomb potential, Vcoul (r 1 ) = (r r 12 dr 2 , and the non-classical potential  xc (r1 ,r2 ) dr2 , yield Vnon-class (r1 ) = r12  1 Epot,ee = (6) (r 1 )[Vcoul (r 1 ) + Vnon-class (r 1 )]dr1 2 The “non-class(ical)” terms are a direct consequence of the very core nature of quantum mechanics and covalency, a matter upon which I reflect in the Appendix B. We are only interested in the interaction between atoms in different molecules, namely A, associated with r1 space and B, associated with r 2 space. This means that an integration over r 1 space is written as a sum
of volume integrals [156], each over one topological atom ΩA , or dr1 = dr1 and

ΩA ΩA

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similarly



dr 2 =


 ΩB ΩB

dr 2 . After elimination of the self-interaction terms (i.e.

intra-atomic A = B energy contributions that don’t appear in an atom-atom interaction potential) and after some rearranging, the interaction between two molecules MA and MB can be written as a sum over their constituent atoms ΩA and ΩB as  Epot,ee (ΩA , ΩB ), and (7) Epot,ee (MA , MB ) = ΩA <ΩB

 Epot,ee (ΩA , ΩB ) =

dr1 (r 1 )[Vcoul (r 1 ; ΩB ) + Vnon-class (r 1 ; ΩB )]

ΩA

= Ecoul (ΩA , ΩB ) + Exc (ΩA , ΩB ) (8)
By replacing (r) with tot (r) = ZA δ(r – RA ) – (r) the nuclear-electron A

term can be absorbed into the electron-electron term [156]. The atomic partitioning does not distinguish inter- and intramolecular interaction in a purely topological sense, so specifically intramolecular equations can easily be generated. Using the familiar multipole expansion, the Coulomb interaction between two atoms is given by  Ecoul (ΩA , ΩB ) = TlA lB mA mB (RAB )QlA mA (ΩA )QlB mB (ΩB ) (9) l A l B mA mB

The exchange-correlation energy can be subject to the same type of multipole expansion. For any wavefunction we can write the second order reduced density matrix as a function of molecular orbitals via  (r 1 , r2 ) = Cijkl ψi∗ (r 1 )ψj (r1 )ψk∗ (r 2 )ψl (r 2 ) (10) ijkl

where the only non-zero C-coefficients at Hartree–Fock level are Ciijj = 2 and Cijji =– 1. Hence we obtain   Exc (ΩA , ΩB ) = TlA lB mA mB (RAB ) Cijkl QlA mA ,ij (ΩA )QlB mB ,kl (ΩB ) l A l B mA mB

ijkl

(11)  where QlA mA ,ij (ΩA ) =

drA ψi∗ (rA )RlA mA (rA )ψj (r A )

(12)

ΩA

and similarly for ΩB . This expansion is expected to be useful at interpretative level rather than as a starting point from which to develop a practical potential from. Indeed, a topological bond order has been proposed before [161],

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which is constructed from exactly those matrices appearing in Eq. 12 but with RlA mA (r A ) = 1. Finally, I mention that work is underway addressing the polarisation of topological atoms, especially at short range. This research is meant to serve molecular simulation of liquids in an attempt to predict bulk properties more accurately. Modestly sized Linux PC clusters are now powerful enough to generate reasonably accurate electron densities of thousands of water oligomers in a few days, including the computation of the concomitant atomic multipole moments. Traditional neural networks, and perhaps even more so the recently heralded kernel methods [175], are ideal tools to predict multipole moments of atoms (and molecules) in previously unseen geometries. Analytical fits would be unfeasible due to the high number of required samples in the training set. It would then be possible to have each molecule in the simulation box respond to its specific environment and adjust its multipole moments. This approach does not introduce polarisability tensors and avoids the socalled “polarisation catastrophe”, in which induced multipole moments may become infinite at small distances. Acknowledgements I thank Andrew Winter for his work on B – Li compounds and Paul J Smith, Fiona Aicken and Sean O’Brien for their help with the brief investigation into citrinin. Many thanks to Michel Rafat for preparing Fig. 6, and to Michael Devereux and Steve Liem for writing the Graphical User Interface that made this possible.

Appendix A Proof of the Falsehood of Cassam-Chenaï and Jayatilaka’s Counterexample In Sect. 3 of their paper [132] the authors scrutinize the justification of the use of zero-flux surfaces to define atomic basins from a subsystem variational principle. This contention is based on Hilbert and Courant’s generalisation of variation calculus to the case of variable domains. A number of conditions have to be obeyed for this generalisation to be applicable. The authors claimed that one such condition is in general violated, by means of a counterexample. Of course, one counterexample suffices but I show here that the calculation in their counterexample is flawed. Let us denote two different wave functions by ψ and φ, and their respective zero flux boundaries by ∂ψ and ∂φ. The condition that needs to be obeyed states that for a neighbourhood of ψ, there exists φ such that ∂φ can be deformed continuously by a bijective mapping into ∂ψ. The counterexample involves the ground state of the hydrogen atom. The authors then (wrongly) showed that no matter how small the neighbourhood of ψ is, ∂φ could not be deformed continuously into ∂ψ. Instead of reproducing their calculation

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I point out that it violated elementary unit (or dimension) analysis. The calculation hinged on the introduction of a parameter a, which appeared in the perturbing wave function ψ  (see below) but not in ψ. In fact, they started with an incorrect wave function ψ in which their parameter a is lacking, i.e. ψ(r) = π –1/2 e–r , although it appeared in their 2s hydrogen wave function,   1 r ψ  (r) = ψ2s (r) = π –1/2 (2a)–3/2 [1 – 12 ar ] e– 2 ( a ) . The former equation suggests that r is dimensionless because an argument of the exponential function can never have a dimension. However, in the latter equation r can have the dimension of length, provided the parameter a adopts this dimension as well. In effect, if a is identified with the Bohr radius a0 then r should indeed have the dimension of length. Following the line of thought detailed in their paper I now present what I believe to be the correct calculation, which will lead to the opposite conclusion. A zero flux boundary ∂Ω is given by ∇(r) · n(r) = 0, ∀r ∈ ∂Ω. It is acceptable to base the counterexample on a special type of boundary ∂φ, namely one for which ∇(r) = 0 in every point, where (r) is the electron density corresponding to wave function φ. This type of boundary is a nodal surface. It is straightforward to prove that if φ(r) = 0 then ∇(r) = 0. We write φ as a perturbation of wave function ψ, via φ(r) = ψ(r) + εψ  (r), where ε is a small non-vanishing number. The condition that φ(r) = 0 then leads to ψ(r) =– εψ  (r) or 1 r  –1( r ) π –1/2 a–3/2 e–r/a =– επ –1/2 (2a)–3/2 1 – e 2 a 2 a

(13)

Since a = 0 we can rid (Eq. 13) of the factor a at both sides, and of course of π –1/2 too. The variable we have to solve for is (r/a) rather than a as in 1 r paper [132]. We can only rid (Eq. 13) of e– 2 ( a ) provided this factor is not zero, which is equivalent to r/a not being infinite. However, we are entitled to isolate r/a = ∞ as a solution of (Eq. 13) because then both its sides vanish. Indeed, the second term at the right also vanishes because lim r

a

→∞

r a

1 r

e– 2 ( a ) = 0 ,

(14)

according to de l’Hôpital’s rule. So we have one boundary ∂φ located at infinity. The crucial question is whether (Eq. 13) permits a second finite solution, corresponding to another boundary ∂φ. This extra boundary would also be a nodal surface, with finite radius, and it would not be connected to the boundary at infinity. I work under the assertion of Cassam-Chenaï and Jayatilaka that, if this finite boundary existed, one could not find a mapping that continuously deformed ∂φ into ∂ψ. They claim that there is such a finite boundary; I now prove that there is not. After having further rid (Eq. 13) of

Quantum Chemical Topology: on Bonds and Potentials

49

1 r

e– 2 ( a ) one obtains (Eq. 15)   1 r –3/2 – 12 ( ar ) e =ε 1– – (2) 2 a

(15)

It is convenient to locate a solution graphically, as an intersection of two curves, each curve representing one side of (Eq. 15). If the abscissa is marked by the variable r/a, then the right hand side of (Eq. 15) is a straight line going through the coordinates (2,0) and (0,ε). The left hand side is a monotonic curve, with strictly negative ordinate values, going through coordinate (0,– 2–3/2 ) and asymptotically reaching zero for r/a = ∞. If ε = 0 there the straight line and the curve do intersect at a finite value for r/a. However, as ε is decreased the line pivots through the fixed point (2,0) until its becomes the abscissa itself. When ε = 0, the only point where the line intersect the curve is at infinite r/a = ∞. Hence, as φ becomes ψ, the concomitant boundaries become identical, or ∂φ → ∂ψ. This is at variance with the conclusion in ref. [132] that “no matter how small ε is chosen there is always at least two values of the parameter ‘a’ such that...in addition to the surface at infinity a surface of finite radius satisfies ∇(r) · n(r) = 0, for all r ∈ ∂Ω”.

B Covalency and Quantum Mechanics Covalency is a quantum effect eluding classical explanation. It left early pioneers of quantum chemistry with the absorbing question of what keeps the hydrogen molecule bound. Feynman once claimed that if one understands the (in)famous double-slit experiment then one understands quantum mechanics. To put it loosely, the double-slit experiment forces one to come to grips with the fact that a quantum particle can be in two places at the same time. This insight has continued to defy many, in spite of the (questionable) authority of the Copenhagen interpretation, which is challenged in professional research papers (e.g. [176, 177]). Inspired by a selection of popular science books on the subject [178–180] I came across a new idea that has not been published before, to the best of my knowledge. It is perhaps the boldest and least detailed idea in this account but I hope it may motivate future research and real physical experiments. The essence of this new idea is postulating that quantum particles do not experience space and time at all. To put it stronger, asking where a quantum particle is located in space and/or time (“spacetime”) is an absurd question. An analogy with daily life, explained below, will make this point clearer. Spacetime itself is a high-level property that emerges from strongly interacting quantum particles. This alternative view turns around the question of the non-locality of quantum particles. We do not wonder how and why it

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is possible that a particle is in two locations at the same time, but rather, we wonder how spacetime becomes sharper (better delineated) as a structure, and this as a result of a strong interaction between quantum particles. It is as if spacetime crystallises as a coherent structure in order to accommodate the co-existence of strongly interacting quantum particles. I suggest that the mystery of quantum non-locality is shifted elsewhere (and perhaps partially resolved) by stating that a quantum particle itself does not experience space time. It is only when the particle is brought in contact with a huge non-quantum object, such as a ZnS screen that it has to “speak up and admit where it is and when”. The wavefunction that describes the single quantum particle is a conceptually elusive concept because it struggles to squeeze the particle into the corset of spacetime. Put more positively, the wavefunction acts as a practical bridge between the macroscopic world of spacetime and an underlying reality, closer to a quantum particle’s true nature. The act of measurement forces the wavefunction to collapse and for the particle to “state” exactly where it is located in spacetime. It is the confrontation between the quantum particle and the screen that forces it to become located. The bizarre phenomenon is not how a particle can be everywhere at once in spacetime (in the shape of a wave function) but rather how it is that spacetime emerges from a huge collection of quantum particles that have lost their quantum character by forming a macroscopic object. Let me bring in a powerful analogy at this point. Imagine a hermit living totally alone on a small island in the Pacific Ocean. This hermit is the analogue of the quantum particle and the island depicts its isolation from the rest of the universe, in this case the rest of human society. Imagine that, one day, we subject the hermit to a “measurement”. We ask him to which political party he is affiliated. This question is obviously nonsense because a hermit does not interact with other humans, and therefore political parties are superfluous and meaningless to him. The question of political affiliation is equivalent to asking where a given quantum particle is located in spacetime. Keeping up the analogy, I argue that this question is nonsense. It is a “nonquestion”, because according to our main assumption the quantum particle does not experience spacetime. Spacetime is the equivalent of a system of political parties. The latter emerge as an (energy-consuming) structure in order to help organise and guide the interaction between (strongly interacting) humans. Equally, I propose that spacetime is actually a structure that emerges as a product of strongly interacting quantum particles. The following scheme (Fig. 13) should help in making these thoughts more precise. This analogy helps us in exposing the absurdity of the question: why can we not determine the exact position of a particle? It is only when the hermit is forced to integrate into a society that he will vote for a political party, or at least feel attracted to broad left or right wing ideas. Equally, when a particle is forced to strongly interact with a macroscopic object (i.e. the society) it will be located in space. The mystery of “superposition of states” in quantum me-

Quantum Chemical Topology: on Bonds and Potentials

51

Fig. 13

chanics is equivalent to wondering how it is possible that the hermit may be simultaneously far-right and far-left, while living totally alone on his island. The question mark in the scheme above prompts us to think about the nature of the fabric in which a single quantum particle exists. Which kind of world has no space and time associated with it? One glimpse of this world reveals itself in the inherent spatial non-locality of the quantum world, as deduced from Bell’s inequality and experimentally confirmed by the work of Aspect and others. Another glimpse is perhaps the lack of temporal order at the very small quantum scale, where quantum particles are thought of as moving forward as well as backward in time. By the word “time” in spacetime I always referred to the macroscopic irreversible time, in the sense of Prigogine’s non-equilibrium thermodynamics or Eddington’s “arrow of time”. The second important question that these ideas leave us with is how quantum particles can “strongly interact” without reference to spacetime! What happens exactly when a quantum particle hits a huge macroscopic screen? This is a difficult question because one is inclined to invoke spacetime as a probe or theatre for “strong interaction”. Being “far away” and “weakly interacting” are almost synonymous. I am tempted to believe that Pauli’s (anti)symmetrisation principle can help here. It is an enormously deep notion, that appears curiously independent and unscathed in even the most modern theories on matter. Yet, even in chemistry this principle has a massive (but generally poorly recognised) impact, since it is the prime reason why molecules look the way they do. For example, lone pairs in water cannot be explained by electrostatics alone. Indeed, it has been amply documented that Fermi (electron) correlation, which is due to the Pauli principle, is dominant over Coulomb correlation. Roughly speaking, the Pauli principle is an organ-

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isation recipe for the coexistence of interacting particles, without invoking spacetime. In other words, it is always in operation and does not experience distance. From it follows the fact that two particles with the same quantum numbers cannot be located in the same place. This is a serious clue as to how spacetime crystallises from a set of strongly interacting quantum particles. Perhaps a thorough study of large Slater determinants could cast these raw thoughts into a precise and testable mathematical framework.

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Struc Bond (2005) 115: 57–79 DOI 10.1007/b135830 © Springer-Verlag Berlin Heidelberg 2005 Published online: 19 July 2005

Angular Momentum and Spectral Decomposition of Ring Currents: Aromaticity and the Annulene Model A. Soncini1 · P. W. Fowler1,2 (u) · L. W. Jenneskens3 1 Department

of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK [email protected], P.W.Fowler@sheffield.ac.uk 2 Department of Chemistry, University of Sheffield, Sheffield S3 7HF, UK [email protected], P.W.Fowler@sheffield.ac.uk 3 Debye Institute, Organic Chemistry and Catalysis, Utrecht University, Padualaan 8, 3584 Utrecht, The Netherlands 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

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[N] carbocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Xn Yn heterocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A perturbed perimeter model for XHXH-clamped carbocycles Model 0: predictions of the unperturbed perimeter model . . Model I: effect of full connectivity . . . . . . . . . . . . . . . . Neutral [4m] perimeter: HC = CH clamps . . . . . . . . . . . [4m2m± ] annulene, m even: HB – BH and HN – NH clamps on a [4n] carbocycle . . . . . . 4.2.3 [4m2m± ] annulene, m odd: HB – BH and HN – NH annelation of a [4n + 2] carbocycle . . . . . . . . . . . . . . . . . . . . . . 4.3 Model II: inclusion of heteroatoms . . . . . . . . . . . . . . . . 4.4 Model III: full connectivity and inclusion of heteroatoms . . .

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Abstract By a widely accepted criterion, an aromatic (anti-aromatic) π-conjugated system is one that sustains a global diatropic (paratropic) ring current when perturbed by a perpendicular external magnetic field. Calculation of induced current densities by the specific distributed-gauge method known as the ipsocentric approach offers practical and conceptual advantages. Physically realistic distributions of current density are obtained with modest basis sets. In the ipsocentric choice, molecular orbital contributions are free of unphysical occupied–occupied mixing and so form the basis of an interpretation of current patterns in terms of frontier orbital symmetries, energies and nodal patterns. Selection rules for the sense of ring current in planar carbocycles can be formulated in terms of angular momentum: a diatropic (paratropic) current arises when a virtual excitation connects orbitals whose angular momentum quantum numbers differ by one (zero). Here we explore the extension of this reasoning to other systems based on the monocycle: to heterocycles and clamped systems, where the symmetry lowering can produce significant angular momentum mixing. It is shown that simple arguments

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can account for the existence and sense of current also in these systems, and hence for their aromaticity on the magnetic criterion. It is possible to understand why, for example, ab initio calculations show that clamping of a benzene (1) or cyclooctatetraene (2) cycle with cyclobutadieno-groups destroys the ring current of the free planar monocycle, but clamping with HB – BH or HN – NH units allows the current to persist, in planar geometries.

1 Introduction Aromaticity is a fundamental term in chemistry, qualitatively associated with expectation reactivity, thermodynamic stability, and physical properties of molecules [1]. According to a widely accepted criterion, aromaticity (antiaromaticity) of (planar) π-conjugated systems can be defined by their ability to sustain a global diatropic (paratropic) ring current when perturbed by a perpendicular magnetic field [2–6]. Although the physical basis of magnetic response has been clear for many years [7], computations of magnetic properties were for a long time plagued by the problem of gauge dependence. Accurate ab initio maps of induced current density can now be computed by means of gauge-distributed approaches [8–12], such as the ipsocentric method [10, 11], thereby providing a direct visualisation of the aromaticity, anti-aromaticity or non-aromaticity of conjugated systems [13, 14]. One in particular of the new approaches to computation of ring currents – the ipsocentric method – also has a decisive conceptual advantage in that it leads to explanations of the calculated currents in terms of orbital energy, symmetry and nodal structure [14, 15]. Angular momentum analysis, as exploited elegantly in much of Stone’s work on molecular properties, can be used to bring insight into the connections between electronic structure, electron current and magnetic aromaticity. Within the ipsocentric approach, it has been shown that it is possible to formulate a non-redundant frontier-orbital model for the interpretation of the induced current density, and hence to define symmetry-based selection rules for the sense and magnitude of global ring currents [14]. According to the ipsocentric model, the total current in delocalised systems is dominated by frontier-orbital contributions, determined by virtual transitions from high-lying occupied to low-lying unoccupied orbitals. If the symmetry product of the occupied and empty orbital matches the symmetry of an in-plane translation (rotation), the contribution to the corresponding orbital current is diatropic (paratropic). When a molecule has high symmetry, these translational and rotational selection rules for diatropicity and paratropicity allow a clear-cut characterisation of the orbital origin of the global ring current, and lead to a sound criterion for rationalising and even predicting the aromatic or anti-aromatic

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character of conjugated species, simply based on the symmetry properties of the frontier orbitals [14]. However, where the molecular symmetry is low, the rules give few, if any, restrictions on magnetic behaviour, and the analysis of the induced current density cannot rely on the symmetry criteria alone. In the particular case of planar carbocycles in geometries with the highest symmetry consistent with their electronic shell closure, it has been shown that the symmetry-based selection rules for diatropic and paratropic ring currents can be reformulated in the language of orbital angular momentum [15]. Within pictorial Hückel theory, a translational (rotational) transition connects occupied and empty orbitals only if their angular momenta differ by one (zero) unit, thereby defining selection rules for orbital current diatropicity (paratropicity) in equilateral carbocycles [15]. Here we present a generalisation of these angular momentum arguments to systems that have less than the full rotational symmetry [16–18]. The approach relies on the expansion of Hückel molecular orbitals (MOs) in a basis of angular momentum eigenfunctions defined over appropriate circuits of the molecular graph. Consistent with the ipsocentric model, the proposed angular momentum spectral decomposition maps a partition of each active occupied-to-unoccupied orbital transition into diatropic (angular momentum shifting) and paratropic (angular momentum conserving) components. It is shown that even for systems with low symmetry, wherein active transitions are often both translationally and rotationally allowed on formal symmetry grounds, decomposition into angular momentum modes provides a semi-quantitative criterion for weighting translational and rotational character, leading to useful predictions about the global diatropic, paratropic or localised character of the induced π-current density. A wide range of molecular systems (see Scheme 1) can be seen as variations on the monocycle theme, and their ring-current properties can be understood with angular momentum ideas. We illustrate the techniques through the investigation of the ringcurrent aromaticity and anti-aromaticity of two general classes of planar π-conjugated molecules. The first class consists of equilateral heterocycles of Dnh symmetry with general formula Xn Yn , where X and Y represent atoms or groups isolobal with sp2 carbon. The second class consists of even carbocycles clamped by cyclobutadieno groups (HC = CH) or their heteroatomic analogues of formula HX – XH, with X = B and N. For the first class, the model potential describing the electronegativity alternation pattern can be fully diagonalised in a basis of angular momentum eigenfunctions and an essentially complete prediction of the induced current made. For the clamped systems the loss of full rotational symmetry induced by the connectivity of the graph can be efficiently represented by means of a perturbation theory approach that uses the structure of the perimeter cycle as a zeroth-order ansatz. Scheme 1 shows molecules representative of the two classes of systems.

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Scheme 1 Examples of molecules based on unsaturated monocycles treated in this work

By means of the angular momentum analysis it is shown that disruption of aromaticity and anti-aromaticity in planar π-conjugated systems, leading to a localised profile of the current density maps, occurs through two fundamentally different routes: whereas the quenching of diatropic ring current can be fully described in terms of angular momentum mixing of frontier orbitals, disruption of paratropicity occurs through a specific orbital energy effect, leading to a widening of the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) that weakens the dominant paratropic transition. Interestingly, angular momentum decomposition shows that, for both aromatic and anti-aromatic formal π

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electron counts, localisation effects induced by cyclic potentials not endowed with full rotational symmetry, are most efficient when the Fermi level of the molecule corresponds to half-filling of the system.

2 [N] carbocycles It has been shown [15] that the ring-current response of even [N] carbocycles, in the highest symmetry consistent with shell closure, is fully determined by their π-electron count, in agreement with the classical Hückel rules for aromaticity and anti-aromaticity. The argument goes as follows. Within Hückel theory each π energy shell of a D(4n+2)h [4n + 2]-carbocycle and a D2nh [4n]carbocycle is characterised by an angular momentum quantum number equal to the number of angular nodes of the corresponding MOs (see Scheme 2). In [4n + 2] carbocycles, π electron filling corresponds to full occupation of n + 1 angular momentum shells (λ = 0, 1, ..., n). As the angular momentum operator for the perpendicular direction, the generator of in-plane rotations, is diagonal in such a basis, any occupied-to-unoccupied rotational transition in the valence space corresponds to a zero off-diagonal matrix element. Accordingly, paratropic orbital current is strictly forbidden in this representation. The action of a linear momentum operator, a generator of in-plane translations, on an orbital belonging to a given angular momentum shell produces a mixture of functions differing from the parent by one unit of angular momentum, and hence by one in the angular node count. It is the component with increased node count that has a nonzero overlap with a virtual orbital,

Scheme 2 Schematic energy level diagrams for (a) benzene (1) and (b) planar 1,3,5,7cyclooctatetraene (COT) (2), showing the angular momentum quantum number (λ), symmetry labelling and nodal character of the occupied and empty π molecular orbitals

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and it follows that the only active occupied-to-virtual magnetic transition in [4n + 2] carbocycles occurs between the doubly degenerate HOMO, with angular momentum λ = n, and the doubly degenerate LUMO, with angular momentum λ = (n + 1). Hence, any [4n + 2] equilateral carbocycle perturbed by a uniform perpendicular magnetic field sustains a purely diatropic ring current, the signature of magnetic aromaticity, originating from the four electrons occupying the doubly degenerate HOMO. Within this model there is an angular momentum selection rule ∆λ = ±1 for diatropicity. On the other hand, closed-shell D2nh [4n] carbocycles are characterised by nondegenerate HOMO and LUMO resulting from the Jahn–Teller splitting of the λ = n angular momentum shell. Accordingly, there are three magnetically active transitions: the two angular-node-increasing diatropic transitions, HOMO – 1 (λ = n – 1) to LUMO (λ = n), and HOMO (λ = n) to LUMO + 1 (λ = n + 1), and one node-preserving transition HOMO to LUMO. The latter is purely rotational in character, and weighted by an energy denominator much smaller than that of the diatropic transitions. In the ipsocentric sum-over-states expression for the induced current density, the HOMO-toLUMO paratropic transition therefore dominates the ring-current response, and provides a magnetic characterisation of the anti-aromaticity of symmetrical [4n] carbocycles. We can define a paratropic transition as one that preserves the angular momentum character of the MO. The model gives an angular momentum selection rule ∆λ = 0 for paratropicity.

3 Xn Yn heterocycles We consider planar equilateral heterocycles of general formula Xn Yn and Dnh symmetry (e.g. B3 N3 H6 (4), C3 N3 H3 (5), S3 N3 – (6– )). One way to relate the magnetic response of these systems to that of their carbocycle analogues with full D2nh rotational symmetry, is to consider a continuous transformation that starts from the Hückel secular equations for a C2n H2n carbocycle with equal Coulomb integrals α and resonance integrals β, and changes each Coulomb integral associated with an odd (even) numbered carbon, α, to αX = α + ηβ(αY = α – ηβ), as a mimic of electronegativity alternation (see Scheme 3). Typical values of η for common heteroatoms are listed in [19]. If the electronegativity values of X and Y are not symmetrical with respect to that of carbon, α is taken to represent their mean value. Transformation of the Hamiltonian halves the symmetry group but maintains the bipartite character of the graph, and therefore gives rise to a well-defined pattern in the spectrum of the heterocycle. After the transformation, each bonding level in the carbocycle spectrum becomes equisymmetric with its conjugate antibonding partner, characterised by equal but opposite energy relative to α. From

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Scheme 3 Numbering system for the general alternating heterocycle Xn Yn

the one-to-one correspondence between energy and angular momentum in the D2nh carbocycle, it follows that in the heterocycle each molecular orbital can be expressed as a linear combination of two angular momentum eigenfunctions, corresponding to the bonding/antibonding pair of the bipartite carbocycle. This mixing pattern has important consequences for the ipsocentric analysis of induced ring current. For Xn Yn systems with aromatic π electron counts, 2n = 4m + 2, HOMO and LUMO represent a conjugate bonding/antibonding pair, and when η
= 0, they are of mixed angular momentum character. Since both HOMO and LUMO of the heterocycle combine λ = m and λ = (m + 1) angular momenta, the HOMO–LUMO transition contains terms allowed under both ∆λ = ±1 and ∆λ = 0 rules, giving rise to a cancelling paratropic component with the originally purely diatropic transition. In principle, this process can lead to disruption of the aromatic ring current and localisation of current on the electronegative centres, depending on the extent of mixing. Within the ipsocentric model the electronegativity alternation that characterises X2m+1 Y2m+1 heterocycles therefore plays a key role in the balance between aromaticity and localisation. This conclusion is supported by simple inspection of the symmetry product of HOMO and LUMO in the halved symmetry group. The properties of the product can be seen by descent in symmetry from the cylindrical group. In D∞h the pair of functions with angular momentum λ spans the representation Eλg/u with g for even λ, u for odd λ. In this group, the product for successive values of λ obeys Eλg × E(λ+1)u = E1u + E(2λ+1)u .

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E1u is the representation of the pair of in-plane translations, and so the HOMO–LUMO transition in all subgroups of D∞h always contains a translationally allowed diatropic contribution. In the heterocycle, the HOMO and LUMO are equisymmetric, spanning the representation descended from Eλg/u in D∞h (where λ = n for the neutral molecule, though the particular value of λ does not affect the argument). In D∞h , the product of an angular momentum representation with itself obeys Eλg × Eλg = A1g + A2g + E2λg , where A2g is the symmetry of the in-plane rotation, and so in those groups where HOMO and LUMO become equisymmetric, the HOMO–LUMO transition always contains a rotationally allowed, paratropic contribution. Both symmetry and angular momentum arguments give a mechanism for potential disruption of aromaticity in 4n + 2 systems, but do not predict the point at which the paratropic component in the HOMO–LUMO transition can be expected to lead to actual localisation. For instance, although both borazine B3 N3 H6 (4) and s-triazine C3 N3 H3 (5) are formally aromatic 6π electron D3h heterocycles with η
= 0, borazine (4) presents a completely localised profile of induced current density (Fig. 1a), whereas s-triazine (5) sustains a diatropic ring current of similar strength to that in benzene (1) [20] (Fig. 1b). For Xn Yn with anti-aromatic π electron counts, the symmetry group is the same as the group of the planar Jahn–Teller distorted [2n] carbocycle, and the angular momentum of both HOMO and LUMO remains well defined, re-

Fig. 1 π current-density maps computed at the ipsocentric RHF/6-31G∗∗ level for (a) borazine (4), (b) s-triazine (5) and (c) the sulphur–nitrogen heterocycle, S3 N3 – (6– ). Each map shows the current density induced by a perpendicular external magnetic field and plotted in a plane 1 bohr above that of the nuclei. Arrows indicate the relative strength of the in-plane projection of current, and contours show the magnitude of the full three-dimensional current, in all cases taking only π-electron contributions into account. Projections of nuclear positions are indicated by filled, barred, crossed, empty and dotted circles for carbon, nitrogen, boron, sulphur and hydrogen, respectively. Anticlockwise circulation in the diagram corresponds to a diatropic current

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taining the purely rotational character of the HOMO–LUMO transition. In these systems, potential for disruption/localisation of the paratropic ring current comes from changes in orbital energy denominators that reduce the main paratropic transition and increase the importance of other, diatropic transitions. Again, an explicit model of the effects of electronegativity alternation is needed to rationalise the ring current behaviour in particular systems. The η parameterisation provides this. To establish an analytical relation between the angular momentum mixing process and the electronegativity parameter η, we can perform a closer study of the Hückel secular equations for the heterocycle. If we number the atom positions so that atoms X (Y) occupy the odd (even) positions (see Scheme 3), with Coulomb integrals αX and αY , label the corresponding MO coefficients x2r–1 and y2r , with r = 1, ..., n, the typical secular equation can be written as the coupled pair
 (1) – z – η x2r+1 + y2r + y2r+2 = 0
 – z + η y2r + x2r+1 + x2r–1 = 0 where z = (ε – α)/β and ε are the MO energies. It can then be shown that the secular problem for the [2n] heterocycle can be separated into two copies of the [n]-carbocycle problem, leading to a rearrangement of Eq. 1 as        
2  y2r y2r–2 y2r+2 0 2 – z –η –2 + + = , r = 1, 2, ..., n . (2) x2r+1 x2r–1 x2r+3 0 where x2r+1 =

y2r + y2r+2 z–η

y2r =

x2r+1 + x2r–1 . z+η

(3)

This separation has two consequences. The first is the possibility of obtaining an analytical expression for the eigenvalues of Eq. 1, as functions of η and the eigenvalues of the [n] carbocycle, through the quadratic equation    2πΛ + n/2 (n even) 2 2 = 0 Λ = 0, ±1, ... z – η – 2 – 2 cos . (4) n ±(n – 1)/2 (n odd) In Eq. 4, the final term on the right-hand side represents an orbital energy associated with the angular momentum shell Λ in the [n] carbocycle. Each value of Λ generates two MO energies in the [2n] heterocycle spectrum according to  πΛ . (5) zΛ = ± 4 cos2 τΛ + η2 , τΛ = n The second consequence is that the eigenfunctions of the two [n]-cycles, respectively covering the odd- and even-numbered positions, provide an orthonormal basis in which to expand the eigenfunctions of the [2n] heterocycle. Denote by χq (r) the pz atomic orbital centred on the qth atom. The

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orthonormal basis set defined by the two [n]-cycles consists of the two disjointed sets 1  iτΛ (2r) e χ2r (r) and ξxΛ (r) = √ 2n r=1 n

1  iτΛ (2r+1) ξyΛ (r) = √ e χ2r+1 (r) . 2n r=1 n

As n-fold rotational symmetry remains in the presence of electronegativity alternation, we can expect the eigenfunctions of the [2n] heterocycle to be linear combinations of those [n] carbocycle eigenfunctions in the same angular momentum shell, i.e. with equal |Λ|. In particular, it is clear from Eq. 5 that each angular momentum shell  of the [n]-cycle generates conjugate b (r), ψ a (r) on the heterocycle, bonding/antibonding MO partners ψ+z –zλ λ still classified by the [n] carbocycle quantum number |Λ|. The degree of mixing will depend on both the degree of electronegativity alternation η, and the particular shell involved, so that if we express such dependencies through an η angle ϕΛ , we can write the orthogonal rotation as   b  

η η sin ϕΛ cos ϕΛ ξxΛ (r) ψ+zλ (r) . (6) η η a (r) = ξyΛ (r) ψ–z – sin ϕΛ cos ϕΛ λ Substitution of Eq. 6 into Eq. 3 leads to an analytical expression for the angle η ϕΛ η

cot ϕΛ =

2 cos τΛ . zΛ – η

(7)

 b (r)ψ a (r) in the It is then possible to expand the equisymmetric pairs ψ+z –zλ λ basis of eigenfunctions defined on the full [2n] cycle with angular
 momentum eigenvalues λb = Λ (bonding component), and λa = ± n – |Λ| (antibonding component) given by 1  i πΛ q 1  i ±π(n–|Λ|) q n e n χq (r) θλa (r) = √ e χq (r) , θλb (r) = √ 2n q=1 2n q=1 2n

2n

so that the expansion reads   

 b η η  sin ωλ cos ωλ θλb (r) ψ+zλ (r) . = η η a (r) θλa (r) ψ–z – sin ωλ cos ωλ λ η

(8)

By comparison of Eq. 6 and Eq. 8, ωλ , the angle describing the degree of mixing of the equisymmetric bonding/antibonding MO pair of the [2n] car-

Angular Momentum and Spectral Decomposition of Ring

bocycle when η
= 0 is   η 1 η –1 ωλ = tan , 2 2 cos τλ η

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(9)

where ωλ ranges from zero ([2n] carbocycle, η = 0) to π/4 (complete mixing of λa and λb , η → ∞). Equations 5 and 9 allow evaluation of the effects of electronegativity alternation η on both the energy gaps, and the degree of angular momentum mixing in the occupied-to-virtual transitions responsible for the induced ring current. As an example, let us reconsider the case of borazine (4) and striazine (5). Within simple Hückel theory, borazine (4) can be described by η = 1.25β, and s-triazine (5) by η = 0.25β [19]. As we have seen, remixing of the benzene (1) HOMO (λ = 1) and LUMO (λ = 2) occurs in both 6π electron heterocycles. However, from Eq. 9, the HOMO of borazine (4) already has 15% λ = 2 character, whereas the λ = 2 component represents less than 1% of the HOMO of s-triazine (5), which is therefore essentially indistinguishable from the benzene (1) HOMO. This picture [18] rationalises the ab initio results reported in [20, 21], and provides an explanation of why s-triazine (5) can be considered, on the magnetic criterion, ‘as aromatic as benzene (1)’, but borazine (4) has the localised current density response of a nonaromatic. In the previous study on 6π heterocycles [18], a value of η = 0.5β has been proposed as the threshold where the HOMO–LUMO paratropic component becomes significant, and starts to lead to appreciable weakening of the ring current. For this value the estimated HOMO–LUMO mixing is about 5%. It is seen from Eq. 9 that for larger [4m + 2] cycles, as the angular momentum of HOMO and LUMO increases, the value of η needed to cause significant mixing of λ = m + 1 into λ = m becomes smaller. For instance, according to this model, in the hypothetical triacontagonal analogue of s-triazine (5), C15 N15 H15 , the angular momentum HOMO–LUMO mixing already reaches 20%. Despite the small electronegativity alternation we can expect, on the basis of the borazine (4) results, mixing to this extent would be large enough to lead to complete localisation of the current density. Another consequence of the form of Eq. 9 is that, for any given value η of η, the mixing angle ωλ reaches a maximum with λHOMO = m and λLUMO = m + 1. This maximum implies that the localisation mechanism is most efficient at half-filling the Hückel spectrum, leading to the prediction that π-excessive and π-deficient heterocyclic compounds will be less susceptible than π-precise systems to aromaticity disruption. This conclusion is illustrated by an example. As shown in [17], the angular momentum decomposition of the spectrum of the 10π heterocycle S3 N3 – (6– ) is comparable to that of borazine (4). However, with 10π electrons in a six-membered ring, the HOMO and LUMO of S3 N3 – (6– ) do not cor-

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Scheme 4 Schematic energy level diagrams for the X3 Y3 cycle, illustrating the angular momentum mixing on descent in symmetry to D3h from the D6h group of the carbocycle, and the different symmetry characteristics for 6π and 10π electron counts

respond to a bonding/antibonding angular momentum pair derived according to Eq. 8 from the formal parent, benzene (1). As illustrated in Scheme 4, since the HOMO–LUMO gap does not occur at half-filling, no rotational (∆λ = 0) component can occur in the HOMO–LUMO transition. A rotational transition activated by angular momentum mixing is possible between HOMO–2 and LUMO, but this should be weak because of the η smaller ωλ and the larger energy denominator. Irrespective of the value of η, ring current survives in S3 N3 – (6– , see Fig. 1c). The model explains why π-excessive heterocycles, such as sulphur nitrogen heterocycles, should be aromatic despite electronegativity alternation [17]. Calculations on the hypothetical planar tetra-anion of borazine (44– ), B3 N3 H6 4– indicate that an excess of π electrons would restore an aromatic ring current to borazine (4) [17]. For heterocycles with a formally anti-aromatic π-electron count 2n = 4m, it can be argued from Eq. 5 that widening of the HOMO–LUMO gap provides a mechanism for disruption of the paratropic ring current. It is clear from Eq. 5 that the HOMO (LUMO) energy decreases (increases) linearly with η. The widening of the HOMO–LUMO gap weakens the dominant paratropic transition, thereby allowing more efficient cancellation by the purely diatropic transitions HOMO–1 to LUMO and HOMO to LUMO+1. These predictions of the model have been confirmed by full ab initio calculations on B2 N2 H4 and B4 N4 H8 [18], which show a completely localised pattern of induced current density: by η = 1.25β the widening of the HOMO– LUMO gap is apparently sufficient to quench completely the paratropic ring current.

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4 A perturbed perimeter model for XHXH-clamped carbocycles Annelation of [N] carbocycles (N even) by means of N/2 XHXH unsaturated clamps leads to polycyclic molecules of general formula CN XN HN (see Scheme 5). According to whether X is a 2π-electron donor (e.g. N), 1π-electron donor (e.g. C) or 0 π-electron donor (e.g. B), the final structure is characterised by a well-defined π-electron count, which provides a natural starting point for the investigation of its aromaticity. We will follow previous authors in considering the π systems of these molecules in planar geometries; in fact full geometry optimisation at the RHF/6-31G∗∗ level gives

Scheme 5 The clamped [2m]monocycle, showing the numbering scheme for the p centres and illustrating schematically the various levels of approximation in Models 0 to III

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planar optima only for X = C; when X = N the central ring remains essentially planar, but when X = B the optimal structure is considerably distorted from the plane. With X = N or X = B the clamped system retains the formal aromatic or anti-aromatic character of the parent cycle, if we judge by πelectron count, i.e., if the [N] carbocycle has N = 4n + 2, then the clamped analogue has a π-electron count of 4m + 2 (m = n for B, m = 3n + 1 for N). Likewise, if N = 4n, the clamped carbocycle has 4m π-electrons (m = n for B, and m = 3n for N). On the other hand, if X = C, the clamped system invariably has an anti-aromatic π-electron count of 4m (m = 2n + 1 if N = 4n + 2, m = 2n if N = 4n), regardless of the electron count of the parent carbocycle. Accordingly, straightforward reliance on Hückel’s electron counting rule as a probe of aromaticity leads to precise predictions for the magnetic response of XHXH-clamped carbocycles. In particular, cyclobutadieno (HC = CH) clamps should always lead to an anti-aromatic system, one able to sustain an induced paratropic ring current, regardless of the formal aromaticity or anti-aromaticity of the clamped central ring. [4n + 2] and [4n] carbocycles clamped by (HB – BH) or (HN – NH) groups should preserve their formal aromatic/anti-aromatic character, retaining therefore their ability to sustain induced diatropic/paratropic ring currents. In fact, the electron counting approach has only mixed success. Ipsocentric ab initio calculations of the induced current density show that, whereas simple π-electron counting rules lead to sound predictions for HB – BH and HN – NH clamped systems, they fail to rationalise the actual magnetic response of HC = CH clamped carbocycles. The diatropic (paratropic) central ring current characterising the magnetic response of benzene (1) (COT (2)) is in fact completely disrupted when the carbocycle is clamped by cyclobutadieno groups, and the global paratropic ring current predicted by the π-electron count is simply not observed. Reliance on simple classical electron counting rules corresponds to an approximate description of the clamped cycle in terms of its annulene perimeter, and this is clearly insufficient. We can propose three perturbed perimeter models, whereby the simple perimeter annulene analogy is considered as the zeroth-order solution (Model 0) in a perturbative treatment. Model I includes the non-perimeter bonds perturbatively, Model II includes the perimeter heteroatoms perturbatively, Model III includes both (see Scheme 5). By means of these pictorial perturbative models (I, II and III) it is shown below that, when first-order corrections to the angular momentum character and orbital energies of the perimeter annulene (Model 0) are taken into account, it is possible, within the ipsocentric model, to give a unified rationalisation for the survival of the original ring current in XHXH clamped monocycles, and its extinction in HC = CH-clamped monocycles, even at the simple Hückel level of theory.

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4.1 Model 0: predictions of the unperturbed perimeter model The graph representing the connectivity of a XHXH-clamped [N] carbocycle can be regarded as a modification of the [4m] perimeter cycle (2N = 4m), in which new connections are added between each pair of vertices labelled (4r – 3) and (4r), with r = 1, 2..., m (see Scheme 5). The simplest approximation to the π-electronic structure of a clamped carbocycle is then obtained by identifying its connectivity with that of its perimeter, a [4m] annulene, cast in the maximum symmetry compatible with a closed shell structure (i.e. D2mh ). Ignoring differences in electronegativity arising from different choices of the clamping atom X, the only distinction that this crude model makes between different kinds of clamps is in the filling of the energy levels of the [4m] annulene spectrum. A neutral [4m] carbocycle is therefore a perimeter model for a HC = CHclamped [N] cycle, and its ionic versions [4m]2m+ and [4m]2m– are models for HB – BH- and HN – NH-clamped carbocycles. It is evident that there is a one-to-one correspondence between the predictions for magnetic properties of a XHXH-clamped [N] carbocycle in the simple perimeter representation and the predictions deduced by pure counting of π-electrons. A HC = CH-clamped [N] carbocycle is represented by a neutral [4m] cycle whose frontier orbitals are rotational partners. It is therefore predicted in Model 0 that a large paratropic ring current will be sustained, which matches the predictions based on its anti-aromatic π-electron count. A [4n + 2] carbocycle clamped by either HB – BH or HN – NH groups is represented by an ionic [4m]2m± carbocycle (m = 2n + 1) with 4k + 2 π-electrons (k = n for the [4m]2m+ cation, and k = 3n + 1 for the [4m]2m– anion). Such occupancy always leads to frontier orbitals whose numbers of angular nodes differ by one, so that the formally aromatic [4k + 2] annulene embedded in a [4m] cyclic backbone is predicted to sustain a purely diatropic ring current. Likewise, a [4n] carbocycle completely clamped by either HB – BH or HN – NH groups is represented by an ionic [4m]2m± annulene (m = 2n) with 4k πelectrons (k = n for the [4m]2m+ cation, and k = 3n for the [4m]2m– anion). This occupancy always leads to frontier orbitals that share the same number of angular nodes, and the formally anti-aromatic [4k] annulene embedded in a [4m] cyclic backbone is predicted to sustain a purely paratropic ring current. In summary, the unperturbed perimeter model therefore implies that a global ring current will survive whichever clamps are added, but with inversion of the original sense of the flow when a [4n + 2] annulene is clamped by HC = CH groups. If we want to account for possible disruption of the ring current, we clearly have to enrich this basic picture. As we have seen in the case of heterocycles, current density disruption follows different mechanisms, according to whether [4n + 2] or [4n] annulenes are considered. Two different ele-

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mentary approaches, separately and together, are considered to incorporate symmetry-lowering effects on the magnetic response of clamped carbocycles: inclusion of the neglected bonds defining the structure of the central [N] annulene; inclusion of electronegativity alternation in the outer perimeter, to mimic HB – BH and HN – NH clamping. Improvements on the crude perimeter model are then obtained by means of perturbation theory. 4.2 Model I: effect of full connectivity In the Hückel model, the matrix representation of the first-order Hamiltonian describing insertion of bonds between positions 4r – 3 and 4r, with r = 1, 2, ..., m, is 
h(1) β,pq = δp,(4r–3) δq,(4r) + δp,(4r) δq,(4r–3) β p, q = 1, 2, ..., 4m . Using standard Rayleigh–Schrödinger perturbation theory, the first-order correction to the orbital energies of the unperturbed [4m] carbocycle caused by insertion of the new bonds can be shown to be 

 β 3πλ (1) ± δλ,(km/2) , (10) ε±λ = cos 2m 2 where λ is the angular momentum quantum number of the degenerate (0) molecular orbital rotational partners (±λ) with zeroth-order energy ε±λ = 2 cos[πλ/2m]β. As the effect of symmetry lowering on the energy balance of the transitions among frontier orbitals is decisive in determining the survival or extinction of the global paratropic ring current in [4n] annulenes, the general result (Eq. 10) is of particular relevance for i) HC = CH annelation of any [N] annulene (see Scheme 1, 7 and 10), and ii) HB – BH and HN – NH annelation of a [4n] annulene (see Scheme 1, 11 and 12). Consequences for angular momentum coupling, and consequences for HB – BH and HN – NH clamping of [4n + 2] annulenes are investigated in Sect. 4.2.3. 4.2.1 Neutral [4m] perimeter: HC = CH clamps The angular momentum values characterising the HOMO and HOMO–1 of a neutral [4m] annulene are |λHOMO | = m and |λ(HOMO–1) | = m – 1. From Eq. 10, the first-order corrections to the HOMO and HOMO–1 energies of the neutral [4m] carbocycle model for the HC = CH annelation of a [N] annulene (N = 2m), can be written as   3π β β (1) ε(1) ε sin . = = – HOMO (HOMO–1) 2 2 2m

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It is evident that, for any value of m, the effect of the new bond connections at first order in β is to stabilise the HOMO and destabilise the HOMO– 1 and in this way reduce the difference in their energies; this is true even though for large m the effect on the HOMO–1 tends to zero. As the perturbation preserves the bipartite character of the graph, the energies of LUMO and LUMO+1 move oppositely to those of HOMO and HOMO–1. It follows that the purely rotational HOMO–LUMO gap tends to become of comparable magnitude to the purely translational gaps from HOMO to LUMO+1 and HOMO–1 to LUMO. Contributions to the total current density arising from these transitions will tend to equalise in intensity, and hence lead to cancellation and disruption of the global paratropic ring current that was predicted by the unperturbed perimeter model. The predictions of the improved model match the results obtained from ab initio calculation of the induced current density for the HC = CH-clamped benzene [22] and COT [23] (see Fig. 2a and 2b). The original benzene (1) diatropic ring current and the COT (2) paratropic current are completely quenched when HC = CH clamps are applied. Despite the formally antiaromatic π-electron count of both clamped molecules, global paratropic current is not observed, the total π-current consisting of a set of diatropic vortices localised over the short bonds. 4.2.2 [4m]2m± annulene, m even: HB – BH and HN – NH clamps on a [4n] carbocycle The angular momentum values for the HOMO, HOMO–1, LUMO and LUMO+1 of a [4m]2m+ annulene, with even m, are |λ(HOMO–1) | = (m/2 – 1), |λHOMO | = (m/2), |λLUMO | = m/2 and |λ(LUMO+1) | = (m/2 + 1). From Eq. 10 it follows that the first-order corrections to the frontier-orbital energies of the [4m]2m+ annulene with m = N/2, the model for the HB – BH annelation of [4n] carbocycles, are: √ √     2 3π 3π 2– 2 (1) (1) β ε(HOMO–1) = sin – cos β εHOMO = 4 4 2m 2m √ √     2 3π 3π 2+ 2 (1) (1) εLUMO = – β ε(LUMO+1) = – sin + cos β 4 4 2m 2m As the perturbed graph is still bipartite, equal but opposite corrections apply to the HOMO–1, HOMO, LUMO and LUMO+1 of the [4m]2m– annulene with m = N/2, i.e., to the model for the HN – NH annelation of a [4n] carbocycle. Whereas the inserted bonds destabilise the LUMO+1 for any allowed value of m (2, 4, 6,...), the HOMO–1 is stabilised for m = 2 (clamped cyclobutadiene) and m = 4 (clamped COT), unaffected for m = 6 (clamped [12] annulene), and destabilised for any other allowed value of m (8, 10, 12,...). In this first-

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Fig. 2 π current density maps computed at the ipsocentric RHF/6-31G∗∗ level for clamped benzene and cyclooctatetraene systems. Benzene-based systems are shown on the left and COT-based systems on the right, with the three rows representing HC = CH clamps (7, (a) and 10, (b)), HB – BH clamps (8, (c) and 11, (d)) and HN – NH clamps (9, (e) and 12, (f)). Plotting conventions are as in Fig. 1. All systems are shown in planar geometries: these are local optima for the HC = CH-clamped systems 7 and 10, but not for the others

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order analysis the HOMO–LUMO gap is widened by a constant (1β) for all m, whereas the HOMO–1/LUMO and HOMO/LUMO+1 energy differences are increased by an amount that depends on m, and in both cases tends asymptotically to 1/2β in the limit of large m. On energy grounds, therefore, all contributions to π-current density are expected to be reduced in the perturbed monocycle. If we take into account the true connectivity of the clamped system, the contribution to current density of the HOMO–LUMO transition of the [4m]2m+ cation and the [4m]2m– anion remain more intense than those from the HOMO to LUMO+1 and from HOMO–1 to LUMO for any value of m, and give rise to an essentially paratropic HOMO contribution to the total current in any HB – BH- and HN – NH-clamped [4n] carbocycle. The predictions of this perturbed perimeter model are fully consistent with the maps obtained in full ab initio calculations. The model provides a rationalisation of the success of straightforward π-electron counting in HB – BH- and HN – NH-clamped COT (see Scheme 1, 11 and 12). Figure 2d and 2f show the induced π-current density maps for HB – BH- and HN – NHclamped COT. The global paratropic ring current characteristic of the flattened 8 π-electron carbocycle is seen to survive both HB – BH and HN – NH clamping. 4.2.3 4m2m± annulene, m odd: HB – BH and HN – NH annelation of a [4n + 2] carbocycle We have seen that the mechanism causing disruption of the ring current in [4n + 2] carbocycles is associated with angular momentum mixing between their HOMO and LUMO pairs. The unperturbed perimeter model is therefore not flexible enough to predict the possibility of quenching of diatropic ring current in [4n + 2] annulenes clamped by HB – BH or HN – NH groups (see Scheme 1, 8 and 9). According to standard perturbation theory applied to the Hückel solutions for the [4m] annulene problem, the first-order correction to the MO of angular momentum λ2 arising from the mixing with an MO of angular momentum λ1 caused by addition of new bonds between positions 4r – 3 and 4r, is proportional to the matrix element     sin π
λ2 – λ1  3π π β  i 3π λ1  (1)  –i λ 2  ei(m+1) m (λ2 –λ1 ) . π
λ1 hβ  λ2 = e 2m + e 2m 4m sin m λ2 – λ1 (11) In particular, we are interested in the mixing of HOMO and LUMO in [4m]2m± annulenes with m = 2n + 1, i.e, the zeroth-order model for the annelation of a [4n + 2] carbocycle by HB – BH and HN – NH clamps.

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The HOMO and LUMO angular momentum values in a [4m]2m+ annulene are |λHOMO | = (m – 1)/2 and |λLUMO | = (m + 1)/2. In a [4m]2m– annulene they are |λHOMO | = (3m – 1)/2 and |λLUMO | = (3m + 1)/2. Substitution into Eq. 11, nothing that the perturbation couples HOMO and LUMO orbitals with oppositely signed angular momentum components, leads to the following expressions       β 3 3π [4m]2m+  (1)  [4m]2m+ λLUMO hβ  λHOMO = cos e–i 4 π 2 4m       β 2m–  (1)  [4m]2m– 9 3π e–i 4 π λ λ[4m] = cos h  LUMO HOMO β 2 4m Hence, the inclusion of the new connections causes angular momentum mixing between the HOMO and LUMO in both the cation and the anion, and this mixing grows stronger with increasing ring size. Explicit computation of the mixing coefficients shows that for clamped benzene (m = 3) the HOMO–LUMO mixing corresponds to about 13%. We have seen in the case of heterocycles that this represents a rather substantial mixing, sufficient to lead to disruption of the diatropic ring current. However, according to full ab initio calculations (see Fig. 2c and Fig. 2e), HB – BH and HN – NH clamps have almost no effect on the benzene global diatropic ring current (see Scheme 1, 8 and 9). Apparently, therefore, connectivity alone does not suffice to explain the ab initio results. As we will see below, explicit account must be taken of electronegativity alternation; when this is done, the reason for the survival of ring current in HB – BH- and HN – NH-clamped benzene becomes apparent. 4.3 Model II: inclusion of heteroatoms As we have seen, within simple Hückel theory, the presence of heteroatoms, X, in the π-network can be parameterised by corrections to the Coulomb integral, αX , with respect to that of carbon, αC , so that αX = αC + ηX β, with ηX > 0 if X is more electronegative than carbon, and ηX < 0 if more electropositive. The different electronegativity of boron and nitrogen clamps with respect to the carbon atoms of the central [N] cycle can therefore be taken into account in the simple perimeter annulene model by investigation of the firstorder corrections to the zeroth-order [4m]2m± carbocycle solutions. Within Hückel theory, the matrix representation of the first order Hamiltonian describing a change ηβ in the Coulomb integrals of positions 4r – 2 and 4r – 1, reads
 (1) hη,pq = δp,(4r–2) δq,(4r–2) + δp,(4r–1) δq,(4r–1) ηX β p, q = 1, 2, ..., 4m .

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The first-order correction to the molecular orbital energy levels with angular momentum ±λ is 

√ √ 2 2 ηX β (1) . (12) 1± δ m± δ 3 ε±λ = 2 2 λ, 2 2 λ, 2 m It is evident from Eq. 12 that the general effect of the perturbation is to make a uniform shift in all the energy levels, regardless of the value of λ. The only exceptions occur for λ = m/2 and λ = 3m/2, i.e. for the HOMO– LUMO rotational pairs of the [4m]2m+ and [4m]2m– annulenes, with even m, where the HOMO–LUMO splitting is increased by the Kronecker terms. Thus the effect on the energy balance of frontier orbital transitions in HB – BHand HN – NH-clamped [4n] carbocycles is to weaken the global paratropic induced ring current arising from the purely rotational HOMO–LUMO transition. Another effect arising from the inclusion in the perimeter model of the changes in electronegativity of the clamping atoms is the coupling of HOMO and LUMO angular momentum in [4m]2m+ and [4m]2m– annulenes for odd m, modelling HB – BH and HN – NH clamping of [4n + 2] carbocycles. The general matrix element describing the coupling of two [4m] annulene MOs with angular momentum λ1 and λ2 is:     η β sin  π (λ2 – λ1 ) X    2m
 λ1 h(1) η  λ2 = π 4m sin 4m λ2 – λ1 3π sin [π(λ2 – λ1 )] i(m+1) π (λ2 –λ1 ) m π
 e , × ei 4m (λ1 –λ2 ) sin m λ2 – λ1 which, for the [4m]2m+ annulene case (HB – BH clamps), when |λ1 | = |λLUMO | = (m + 1)/2 and |λ2 | = |λHOMO | = (m – 1)/2, reads:    η β  2m+  B (1)  [4m]2m+ –i 34 π λ[4m] , λ h  η LUMO HOMO = √ e 2 2 and for the [4m]2m– annulene case (HN – NH clamps), when |λ1 | = |λLUMO | = (3m + 1)/2 and |λ2 | = |λHOMO | = (3m – 1)/2, reduces to:     2m–  ηN β –i 9 π (1)  [4m]2m– 4 . λ[4m] h λ   η LUMO HOMO = – √ e 2 2 Accordingly, inclusion of alternation in electronegativity along the [4m] perimeter, with consequent lowering of the ring symmetry, induces angular momentum coupling. As we have seen in Model I, the HOMO–LUMO angular momentum coupling, implying extinction of the induced ring current, is at variance with the results of ab initio calculations. Clearly, both Model I and Model II, considered separately, are insufficient to account for the survival of diatropic ring current in 4n + 2 HB – BH and HN – NH-clamped monocycles.

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Nevertheless, as we will see in the next section, when considered together the two models succeed in rationalising the magnetic behaviour of these systems. 4.4 Model III: full connectivity and inclusion of heteroatoms We can estimate the total coupling induced by the inclusion of both the new bonds and the heteroatoms along the perimeter. A reasonable choice for ηx is ηN = 1 and ηB = – 1, so that the total coupling is given by:

  √     β  2m+ 2m+ 3 2 3π   (1) (1) [4m] – cos e–i 4 π λ[4m] LUMO hβ + hη  λHOMO = 2 4m 2

 √       β 9 2 3π [4m]2m–  (1) (1)  [4m]2m– λLUMO hβ + hη  λHOMO = – cos e–i 4 π . 2 4m 2 Remarkably, these expressions show that the two perturbations tend to cancel. The departure from the naïve perimeter model caused by the connectivity of the HN – NH- and HB – BH-clamped [4n + 2] annulene is opposed by the effects of electronegativity alternation along the perimeter. The resulting weakening of angular momentum coupling makes the HOMO-to-LUMO transition for these systems almost purely translationally allowed, and the resulting global current diatropic in character. In particular, for HB – BH (8) and HN – NH (9) benzene (m = 3), with this reasonable choice of parameters, it turns out that the total coupling is exactly equal to zero, making the HOMO-to-LUMO transition for these systems purely diatropic to first order in β and ηβ. The inclusion in the crude (Model 0) perimeter model of the combined effects of connectivity and electronegativity alternation, therefore suffices to explain the ab initio results for benzene clamped by HB – BH (8) and HN – NH (9) discussed in the previous section.

5 Conclusions The ipsocentric approach to calculation of induced current density leads naturally to a picture in which selection rules based on angular momentum govern contributions to aromatic and anti-aromatic ring currents. Insights from the simple [N] annulenes can be applied to rationalise survival and extinction of ring currents to a wide range of carbocyclic, heterocyclic and clamped systems. Although the details of the arguments can become complex in individual cases, it is clear that the classic concepts of orbital symmetry, energy and nodal characteristics remain useful and powerful in this area, despite or even because of progress in computational methods.

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Acknowledgements P.W.F and A.S. thank the EU for financial support from the Framework V programme (RTN Contract HPRN-CT-2002–00136 “WONDERFULL”) and P.W.F. thanks the Royal Society/Wolfson Scheme for a Research Merit Award.

References 1. Minkin VI, Glukhovtsev MN, Simkin BY (1994) Aromaticity and anti-aromaticity. Electronic and structural aspects, 1st edn. Wiley, New York 2. Pauling L (1936) J Chem Phys 4:673 3. London F (1937) J Phys Radium 8:397 4. Pople JA (1956) J Chem Phys 24:1111 5. Elvidge JA, Jackman LM (1961) J Chem Soc 859 6. Schleyer PvR, Maerker C, Dransfeld A, Jiao H, van Eikema Hommes NJR (1996) J Am Chem Soc 118:6317 7. van Vleck JH (1932) The theory of electric and magnetic susceptibilities, 1st edn. Oxford University Press, Oxford 8. Ditchfield R (1974) Mol Phys 27:789 9. Kutzelnigg W (1980) Isr J Chem 19:193 10. Keith TA, Bader RFW (1993) Chem Phys Lett 210:223 11. Lazzeretti P, Malagoli M, Zanasi R (1994) Chem Phys Lett 220:299 12. Zanasi R (1996) J. Chem Phys 105:1460 13. Steiner E, Fowler PW (1996) Int J Quant Chem 60:609 14. Steiner E, Fowler PW (2001) J Phys Chem A 105:9553 15. Steiner E, Fowler PW (2001) Chem Comm 2220 16. Soncini A, Fowler PW, Jenneskens LW (2004) Phys Chem Chem Phys 6:277 17. Fowler PW, Rees CW, Soncini A (2004) J Am Chem Soc 126:11202 18. Soncini A, Domene C, Engelberts JJ, Fowler PW, Rassat A, van Lenthe JH, Havenith RWA, Jenneskens LW (2005) Chem Eur J 11:1257 19. Streitwieser A Jr (1961) Molecular orbital theory for organic chemists, 1st edn. Wiley, New York 20. Fowler PW, Steiner E (1997) J Phys Chem A 101:1409 21. Schleyer PvR, Jiao H, van Eikema Hommes NJR, Malkin VG, Malkina OL (1997) J Am Chem Soc 119:12669 22. Soncini A, Havenith RWA, Fowler PW, Jenneskens LW, Steiner E (2002) J Org Chem 67:4753 23. Fowler PW, Havenith RWA, Jenneskens LW, Soncini A, Steiner E (2002) Angew Chem Int Ed 41:1558

Struc Bond (2005) 115: 81–123 DOI 10.1007/b135616 © Springer-Verlag Berlin Heidelberg 2005 Published online: 19 July 2005

Modelling Intermolecular Forces for Organic Crystal Structure Prediction Sarah (Sally) L. Price (u) · Louise S. Price Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 1

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Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . Close-packing principle . . . . . . . . . . . . . . . . . . . . . . . . Specific interactions . . . . . . . . . . . . . . . . . . . . . . . . . . Early atom–atom model intermolecular potentials . . . . . . . . . Lattice energy minimisation as a basis for testing model potentials Early empirical fitted model potentials and the limitations . . . . .

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Crystal structure prediction . . . . . . . . . . . . . . . The search problem and range of structures . . . . . . Early crystal structure prediction methods and results The multiple-minima problem and the requirement for more accurate intermolecular potentials . . . . . . The problems in improving the relative energies of different hypothetical structures . . . . . . . . . . .

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Development of transferable intermolecular potentials for crystal structure prediction by empirical fitting . . Additional tests of model intermolecular potentials . . . Global optimisation . . . . . . . . . . . . . . . . . . . . Effect of molecular dynamics . . . . . . . . . . . . . . . Second-derivative properties . . . . . . . . . . . . . . .

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Other uses of crystal structure prediction . . . . . . . . . . . . . . . . . .

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S.L. Price · L.S. Price Alternatives to model atom–atom potentials for modelling organic crystal structures . . . . . . . . . . . . . . . . . . .

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Abstract Computational prediction of the crystal structures of an organic molecule requires sufficiently accurate models for the forces between the molecules to discriminate between the energies of alternative crystal structures. Such computational predictions are particularly valuable in understanding polymorphism, the ability of some molecules to crystallise in more than one structure. As methods of searching for the most energetically favourable crystal structures have been developed and applied to a wide range of organic molecules, reflecting the potential industrial utility of this emerging field of computational chemistry, it has become clear that the force-fields will have to encapsulate many subtleties of intermolecular interactions. We review the development of model potentials for crystal structure prediction and the design of molecular materials, and their role in quantitatively understanding the interplay of thermodynamics and kinetics in crystallisation. Keywords Crystal Structure Prediction · Intermolecular Forces · Polymorphism · Organic Crystal Structures

1 Introduction The quantitative study of intermolecular forces [1] brings fundamental satisfaction in being able to reconcile diverse phenomena at the level of the behaviour of the atoms, and increasingly at the level of the electrons. However, when pressed for the practical benefits of studying intermolecular forces, most would cite the potential of computational modelling to design new materials. Initially, much of this effort has been focussed towards the development of new pharmaceuticals by modelling the binding of molecules to proteins or nucleic acids. The early models for the intermolecular forces between peptides and protein side-chains were largely derived by analysing their ability to model organic crystal structures. It soon became clear that the computer-aided design of new molecular materials should include the prediction of their crystal structures. For an active nonlinear optical material, the molecule has to crystallise in a noncentrosymmetric space group. Electrical conduction and many other physical properties are extremely sensitive to the exact crystal stacking. A high-energy molecule [2] has to crystallise in a dense crystal to be an effective energetic material. The organic chemist can make small variations to a molecule that barely affect the important molecular properties, such as replacing a methyl group with an ethyl group, that make a qualitative change to the way the molecules pack in the crystal. Thus, a method of predicting what crystal

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structure a molecule would adopt, prior to synthesis, would be a useful tool in the design of organic materials. Simultaneously, the pharmaceutical industry became aware of the importance of crystal packing when major patent cases arose for certain pharmaceuticals that were polymorphic [2], i.e. could crystallise in more than one solid form. Polymorphism is now a major quality-control concern for the pharmaceutical industry, because solubility and dissolution rates (and hence optimum dosages) can differ between polymorphs, so marketing licences are for a specified solid form. Hence, when the manufactured capsules of Abbott’s anti-HIV drug Norvir (generically ritonavir) [3] suddenly started to contain a different, less soluble polymorph, it was a major crisis for the company. After establishing that it was no longer possible to continue manufacturing the original formulation in a controlled fashion, the drug had to be reformulated quickly, under considerable pressure from those on the treatment and the media. The sentiment expressed at a press conference that “unfortunately, there is nothing we can do today to prevent a hurricane from striking any community or polymorphism from striking any drug” emphasises the experimental unpredictability of polymorphism. There are examples of new, more stable polymorphs only being found after decades of research on a compound [4], as well as the discovery of a new polymorph leading to the “disappearance” [5, 6] of the previous form. (Fortunately for our belief in scientific reproducibility some disappeared polymorphs [7] have been obtained again, but only after exceptional precautions to prevent seeding. A contrasting type of polymorphism is when two or more polymorphs crystallise together in the same experiment [8]). Given the experimental difficulties in establishing that all polymorphs that could be manufactured in a controllable manner are known, a method for predicting the structures of new polymorphs that could be found (and preferably the required crystallisation conditions), has immense potential [9] for helping the development and quality control of pharmaceuticals. The emergence of these requirements for computational organic crystal structure prediction fortunately coincided with the increasing availability of the necessary computer power to consider the huge range of possible crystal structures. Such searches rely fundamentally on evaluating the intermolecular forces in alternative packing arrangements of the molecule.

2 Historical Perspective The crystal structures of organic molecules have always fascinated scientists and demanded understanding. This has resulted in innumerable studies, for example, the five volumes of analysis of the external forms (morphologies) of

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crystals by Groth [10–14] in the early 1900s. Today, the Cambridge Structural Database contains over a quarter of a million organic crystal structures [15] characterised by X-ray or neutron diffraction. Whilst it is not possible to review the huge volume of work that has been published in developing a qualitative or even semi-quantitative understanding of the factors that determine the crystal structures, some mention is required of the underlying principles that generally map onto the intermolecular potential as being a key feature in determining crystal structures [16]. 2.1 Close-packing principle Organic molecules crystallise so that bumps fit into hollows to give a closepacked structure. This close-packing principle has been the basis for understanding organic crystal structures following the pioneering work of Kitaigorodskii [17], which included using a mechanical structure-seeker with hard-sphere molecular model to determine the optimal dense packing. Organic crystal structures are quite dense [18], with a packing coefficient of 0.65 to 0.75, generally approaching the value for close-packed identical spheres of 0.74. Larger molecules, in which voids seem inevitable, will usually have the cavities occupied by solvents, such as water [19]. The longrange attractive forces between the molecules give a more stable crystal as it becomes more densely packed, until counterbalanced by the short-range repulsion. 2.2 Specific interactions The other major qualitative principles of crystal packing focus on specific interactions such as hydrogen bonds. “All good proton donors and acceptors are used in hydrogen bonding in the crystal structure” [20] is a very good rule for organic crystal structures. Although a few exceptions were mentioned when Etter formulated her hydrogen-bonding rules [20], organic crystal structures with obviously unused hydrogen-bond acceptors still provoke comment [21] and demand explanation [22]. Statistical surveys [23] of the growing numbers of organic crystal structures in the Cambridge Structural Database (CSD) [15] have been used to define the geometrical characteristics of various specific interactions. For example N-H· · · O=C hydrogen bonds are highly directional, with a statistically significant tendency [24] for hydrogen bonding to occur in the directions of the conventionally viewed oxygen sp2 lone pairs. Weaker C-H· · · O hydrogen bonds also have sufficient directionality to be implicated in supramolecular design [25]. Certain atoms appear to have anisotropic van der Waals radii [26, 27], including the larger halogens, giving rise to considerable debate as to

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whether the orientation dependence of Cl· · · Cl contacts was evidence for a specific directional attractive force [28], or anisotropy in the repulsion and electrostatic forces [29]. These, and various other contacts, such as Cl· · · O/N [30], have been analysed using intermolecular perturbation theory, complementing the statistical distributions of contacts between various functional groups [31]. These concepts of the directionality, range and relative frequency of occurrence [32] (as a rough guide to relative strengths) of different intermolecular contacts within organic crystal structures have been widely used in crystal engineering [33–35] – the design of crystals with specific structures and properties. Qualitative understanding of molecular recognition has certainly been helped by our tendency to partition intermolecular interactions into discrete phenomenological fields [36] that are classified, named (and usually divorced): aromatic interactions, cationπ interactions and hydrophobic interactions are just a few more that can be added to the specific interactions mentioned above. However, to progress, any field of supramolecular recognition [36] needs the quantitative approach that can be provided by quantifying the theory of intermolecular forces. 2.3 Early atom–atom model intermolecular potentials These two semi-quantitative principles, close packing and direction-specific interactions, would be quantified and combined in sufficiently realistic model intermolecular potentials for modelling the organic solid state. The development of model intermolecular potentials outlined in this review could be seen as starting from simple quantification of the close-packing principle, with continuing refinement by the addition of the contributions that determine the specific interactions. Thus, the earliest atom–atom potentials as reviewed by Pertsin and Kitaigorodsky [18] were mainly of the form
MN MN U MN = Urep + Udisp = Aικ exp(– Bικ Rik ) – Cικ R–6 (1) ik i∈M, k∈N

where atom i of type ι in molecule M and atom k of type κ in molecule N are separated by a distance Rik . The short-range repulsive effects are represented by the exponential term and the long-range attractive dispersion contribution by the leading term in R–6 . In some cases, a Lennard–Jones 12–6 or 9–6 power model was used for computational efficiency. A variety of combining rules, such as  1/2    1/2 Aικ = Aιι Aκκ , Bικ = Bιι + Bκκ /2 , Cικ = Cιι Cκκ , (2) are generally used to reduce the number of independent parameters that have to be determined. Such potentials, which only explicitly model the repulsion and dispersion terms, keeping the molecular structure rigid, certainly

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model the dominant forces for the close-packing principle, with a repulsive wall defining the close contact distances and the attractive dispersion terms leading to the binding. Such models have been used historically in the development of computer modelling of organic crystals. 2.4 Lattice energy minimisation as a basis for testing model potentials The key concept in organic crystal modelling, after the crystal structure, is the static lattice energy (or packing energy), which literally sums the intermolecular interaction energy between all the rigid molecules in the crystal: Ulatt =

∞


U MN ,

(3)

M
(we will not discuss the technical issues in doing this [18, 37], which have been resolved in the wide range of crystal modelling programs such as MPA [38], GULP [39], DMAREL [40] etc.). Hence, the thermodynamically stable crystal structure at 0 K should correspond to the minimum in the lattice energy, with respect to all variables defining the crystal structure. Since most crystal structures were determined at room temperature (though 150 K is now almost routine for new determinations), thermal expansion will produce differences between the experimental crystal structure and the corresponding lattice energy minimum [41]. This is usually of the order of a few percent in the lattice vectors of small organic molecules, though it is highly dependent on the crystal packing and can be sufficiently anisotropic for some cell lengths to decrease on heating. The lattice energy is the major contribution to the heat of sublimation of the crystal, and various discussions [18, 37] of the entropic and zero-point vibration energy terms conclude that ∆Hsub = – Ulatt – 2RT or ∆Hsub ≈ – Ulatt are reasonable approximations. It is probable that the combination of experimental errors and these approximations mean that differences of up to 3–4 kcal mol–1 [18] or 2 kcal mol–1 [37] are no cause for concern about the potential, unless there is independent thermodynamic data for the approximations involved. The phonon modes and elastic properties can be estimated from the second derivatives of the lattice energy [18]. Hence, the parameters of the model potential can be fitted to known crystal structures, heats of sublimation, and any available phonon modes and elastic tensor properties. Thus historically, a model potential was empirically considered “good enough” if it reproduced these quantities within the likely error associated with these theoretical approximations and experimental measurements. (We will review some of the more stringent criteria now used in trying to find model potentials that are good enough for crystal structure prediction in Sects. 4 and 6).

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2.5 Early empirical fitted model potentials and the limitations Thus, the computational modelling of organic crystal structures on the basis of the isotropic atom–atom potential and lattice energy minimisation became well established [18]. Gradually, further refinements in the model potential, such as the introduction of atomic charges for the electrostatic contribution and ad hoc models for specific interactions such as hydrogen bonding [42, 43], were introduced. During this period, perhaps the most important use of organic crystal structures was to parameterise and test transferable isotropic atom–atom intermolecular potentials for representing the non-bonded interactions in the force-fields being developed for biological simulations. Thus, the parameterisations of LJ 12–6 and 9–6 atom– atom models for proteins and polypeptides from the crystal structures of amides [44] and carboxylic acids [45–47] became used in the development of AMBER [48], CVFF [49] and other force-fields. Crystal structures have been used in the validation of other force-fields, such as COMPASS [50]. Crystal structures have the great advantage of being a form of experimental data that is available for a sufficiently wide range of organic molecules that transferable potentials can be developed and tested, and indeed some crystal structures can be very sensitive to the model intermolecular potential. However, it has to be appreciated that they only sample the potential over the range of intermolecular separations actually found in the set of crystal structures, which, as the statistical surveys of the Cambridge Structural Database show, can be quite limited. For example N· · · N van der Waals contacts are sufficiently rare in organic crystals that the N· · · N repulsion defined by combining rules (i.e. via Eq. 2 from the sampled N· · · H and H· · · H contacts) will appear very satisfactory until a crystal where such contacts are inevitable, such as s-triazine (C2 H2 N4 ) is considered [51]. Furthermore, it is the net effect of all the applicable force-field terms that is being measured, and so errors in individual terms may not be apparent without careful analysis. (For example, a poor reproduction of a crystal structure often arises from quite subtle errors in the molecular structure due to problems in the description of the intramolecular forces, as discussed in Sect. 5). Nevertheless, the modelling of known crystal structures is usually one of the more sensitive tests for model intermolecular potentials. For example, when the force-field OPLS [52–54], which was derived for simulating the liquid state, was used for studying the crystal structures of carboxylic acids, even after ad hoc adjustment of the torsional potentials, it was generally found [55] to give an “adequate to poor” reproduction of the crystal. Thus, the ability of force-fields to reproduce crystal structures is often quite a stringent test and useful means of developing force-fields that can also be used in other types of condensed-phase simulation.

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The investigation of specific interactions clearly demonstrates that getting such detailed directionality right in crystal structure modelling would require more advanced, theoretically based, intermolecular potentials, but such potentials were expected to be too expensive to be useful in modelling biochemical interactions. It was the advent of crystal structure prediction as a research field that caused the accurate modelling of intermolecular forces in organic crystal structures to become an important issue in its own right.

3 Crystal structure prediction The basic assumption is that a molecule will adopt the most thermodynamically stable crystal structure. We can further simplify this by assuming that the observed crystal structure will correspond to the global minimum in the lattice energy. Even with this zeroth-order model of the factors that determine the crystal structure, this leaves a huge search problem if we are to consider a worthwhile range of possible crystal structures. Thus, it was developing the search method that was initially seen as the major challenge in crystal structure prediction, with the hope that existing models for the intermolecular potential and intramolecular degrees of freedom could be used. 3.1 The search problem and range of structures Considering all the different methods that could be used to search for the global minimum in the lattice-energy potential-energy surface would be a review in itself. From various statistical surveys [56–59] of the crystal structures in the Cambridge Structural Database, it is clear that the most popular space group for organic molecules is P21 /c, though not by such a huge margin that other monoclinic, triclinic and orthorhombic space groups such as P1, P1¯, P21 21 21 etc. can be ignored in any reasonable search method. Thus, there can be up to six lattice parameters (the lengths a, b, c and angles α, β, γ ) to be optimised in the search. If the molecule is assumed to be rigid, then only three to six more variables are required to determine its position in the unit cell. This is assuming that there is just one molecule in the asymmetric unit (i.e. Z = 1), which is becoming less reasonable as a simplifying assumption as modern crystallographic methods are increasing the proportion of structures with more than one molecule in the asymmetric unit cell [57]. The increased number of degrees of freedom that need to be optimised in

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the search to generate Z > 1 structures is also required for searches with two distinct molecular entities in the asymmetric unit, such as organic salts, hydrates, solvates and co-crystals, and so this is a major factor in limiting the applicability of crystal structure prediction methods to many industrially important systems. Cases where a symmetrical molecule lies at a special position in the unit cell, giving Z < 1 in a higher-symmetry space group, are included by default in most searches. Crystal structure prediction work has led to new tabulations of crystallographic relationships, such as the space groups when internal molecular symmetry is ignored [60], or the Z = 1 space groups that could be found by searching with Z = 2 in the optimal five space groups [55]. Over the years, it has become clear that lattice energy minimisation within the constraints of space group symmetry often [61] results in transition states, which should then be reminimised without one or more symmetry elements until a true minimum is found, usually with Z > 1. However, even from the earliest days, it was apparent that any credible search would have to consider unit cells containing up to eight molecules, and that monoclinic and triclinic cells were central to the search for organic crystal structures. The various search methods used can be roughly subdivided into those that use crystallographic insights to reduce the search space and those that seek to be mathematically complete within well-defined limitations of space group and Z values. Recent developments include the use of suitably modified genetic algorithms [62], and the use of search optimisation theory using low-discrepancy sequences [63]. Distributed network parallel implementations of such methods [63] claim to produce effectively complete searches within a defined set of space groups and Z values. Such searches typically consider between 104 and 105 distinct minimisations. This often leads to many hundreds of distinct lattice energy minima within 10 kJ mol–1 of the global minimum. These techniques that require extensive computing resources can be contrasted with the early searches, some of which involved only several hundred lattice energy minimisations, albeit from starting points which were generated to be crystallographically sensible. Hence, although the search problem can be considered effectively solved, in that methods exist so that stable structures within the defined limits of space group and Z (currently ≤ 2) will not be missed, this only means that there is now a very high probability of the crystal structure being located. Structures with Z ≥ 4 are known, and indeed the more readily obtained polymorphs of pyridine [64] and 5-fluorouracil [65] fall into this category. Some crystals have a substantial degree of disorder, and obviously such structures, or amorphous solid forms, are not considered in the search, though there are cases where the lowenergy hypothetical crystal structures can be used to interpret disordered structures [66, 67].

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3.2 Early crystal structure prediction methods and results The model for the intermolecular forces, although in principle separate from the search method, was initially very much intertwined, as different research groups developed different search methods aimed at the prediction of different classes of molecules using established intermolecular potentials, and grappled with the compromise between the computational cost of the search and the evaluation of the lattice energies. The early PROMET [68] approach built up clusters of 2–4 molecules related by the common symmetry elements of inversion, glide and screw axes, and then built the clusters into structures in the most common space groups by adding translational symmetry. Both steps were guided by energy calculations and statistical analyses of known crystal structures. The resulting [68] sets of lattice energy minima for rigid hydrocarbons included the known structure as the global minimum in some favourable cases, but quantitatively demonstrated that many low-energy structures were possible for rigid non-polar molecules. A very similar conclusion was reached for ICE9 [69] which also concentrated on hydrocarbons, using a systematic search assuming close packing. For saturated hydrocarbons with distinctive shapes, simple model potentials reflecting close packing could predict the structure well. However, their consideration of various computationally efficient methods of modelling the electrostatic interactions, including using the molecular multipoles, led to the conclusion that “we need to do more than merely add electrostatic potentials on top of van der Waals’ pair potentials to obtain results accurate enough to distinguish the experimental structure from a number of reasonable possibilities on the basis of energy, particularly for planar aromatic molecules” [69]. Hydrocarbons may have been preferred on the academic basis of having the simplest intermolecular interactions, but the practical incentive is to predict functional crystal structures. Achieving very good close packing is particularly important for energetic materials, as the crystal density is very important in determining the effectiveness of an explosive. Thus the prediction of dense crystal structures is the fundamental idea behind the MOLPAK program [70], which systematically searches for dense crystal structures within a range of common packing types using a rigid pseudohard-sphere molecular probe. Most of the 193 structures considered within each packing type are far from close packed, but the densest 50–200 structures are then refined to optimise the fits of bumps into hollows, prior to being used as starting points for lattice energy minimisation. The original MOLPAK study [70] of nitrated molecules containing C, H, N, O and F used the experimental molecular structure as rigid, and a transferable exp-6 potential. Although the known structure was found as the global minimum in some cases, the majority of calculations indicated that there is relatively lit-

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tle difference energetically between the observed and other plausible crystal structures. Some preliminary work [70] with more specifically refined potentials with atomic charges and AM1-optimised molecular structures suggested that developing the methodology to genuine structure prediction with welldifferentiated lattice energies would not be trivial. Despite its origin in the search for densely packed energetic materials, MOLPAK has also been used very successfully to generate starting-point crystal structures for hydrogenbonded crystals [9], suggesting that there is no great conflict between the close-packing and hydrogen-bonding directionality in determining crystal structures. A methodology of crystal structure prediction primarily aimed at the pharmaceutical industry was the program Polymorph Predictor [71]. This was based on a Monte Carlo simulated-annealing search process, which therefore requires a few runs in each space group to ensure that no significant minima have been lost in the stochastic process. The first reports considered ethane and hexamethylbenzene [72] and then rapidly used CHARMM to consider [73] azobis(isobutyronitrile), 4,8-Dimethoxy-3,7-diazatricyclo[4.2.2.22,5 ]-dodeca-3,7,9,11-tetraene, cyclo-L-Alanyl-L-alanyl, cyclo-Bis(dehydroalanyl)-3,6-dimethylenepiperazine-2,5-dione and isoiridomyrmecin. It was designed [71] to be used with the force-fields available in the programming suite Cerius [74], which are mainly biological force-fields that incorporate molecular flexibility, and hence are capable of optimising the molecular conformation in response to the inter- and intramolecular forces. Nevertheless, a recent highly ambitious crystal structure prediction [75] on two diastereomeric salts consisting of a chlorinesubstituted cyclic phosphoric acid and the two enantiomers of ephedrine discusses how the search problem limits the practical range of utility of such programs for flexible molecules, even with significant computational resources. One group interested in the crystal structures of sugars, started by studying six hexopyranoses [76] by a systematic search of the P21 21 21 space group with Z = 1 (which is heavily populated by chiral sugars), optimising the nine lattice and rigid-body parameters and the six intramolecular torsion angles using the GROMOS force-field. This first version of the UPACK program was a systematic search method, which initially examined over eight million trial structures and used rigid molecules with no explicit hydrogens, and then rejected and clustered structures according to various criteria before the full energy minimisation. The authors were [76] “impressed by the astonishingly large number of structures within a few kcal mol–1 ” as they found the number of possible crystal structures within 10 kcal mol–1 of the global minimum to be of the order of 1000. This started an impressive research effort in refining the model potentials for crystal structure predictions for sugars and alcohols (c.f. Sect. 6) to improve the energy ranking of the known crystal structures.

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3.3 The multiple-minima problem and the requirement for more accurate intermolecular potentials Sugars are notoriously difficult to crystallise, and hydrocarbons are very weakly bound; nevertheless, the observation that there are many hypothetical low-energy crystal structures within the energy range of polymorphism clearly showed that crystal structure prediction on the basis of lattice energy minimisation was very dependent on the accuracy of the relative crystal energies. That this extended to hydrogen-bonded crystals was demonstrated by the results of a small study, based on the MOLPAK search [77], of uracil, 6azauracil and allopurinol, a set of rigid heterocycles that have a variety of distinct hydrogen-bond donors and acceptors. A key feature of this study was that the electrostatic contribution to the lattice energy was calculated from the sets of atomic multipole moments obtained by a distributed multipole analysis [1, 78, 79] (DMA) of a self-consistent field (SCF) charge density of the rigid molecule. Thus, the dominant contribution to the energy of the different possible hydrogen bonds was derived from the molecular charge distribution. The key result was that each molecule had hypothetical crystal structures using different combinations of hydrogen-bond donors and acceptors from the known structure within a small energy range. The known structure of uracil was at the global minimum [77], but a hypothetical structure with alternative hydrogen bonds was only 2 kJ mol–1 less stable, and 6-azauracil [77] had an alternative hydrogen-bonding motif within 0.2 kJ mol–1 of the global minimum (corresponding to the experimental structure) and crystal structures with all possible donor and acceptor combinations within 15 kJ mol–1. In the most favourable case [77], allopurinol, the known structure was the most stable by 1.8 kJ mol–1 , and the energy gap was over 10 kJ mol–1 to the lowest alternative involving different hydrogen bonds. (This was the result of an early search using only 500 starting crystal structures: a more modern search method [63] with a non-nuclear site atomic charge model [80] finds structures for allopurinol with Z = 2 that are slightly more stable than the minimum corresponding to the experimental structure). Thus, the early study [77] showed that, at least for these polar hydrogen-bonding molecules, the length and directionality requirements for the electrostatic stabilisation of hydrogen bonding could be satisfied by a variety of structures with relatively small energy differences. Two independent studies on acetic acid (coincidentally published at the same time) emphasised the emerging picture that searches for lattice energy minima produce a large number of structures within a relatively small energy range. An industrial group [81] had chosen acetic acid and its monohalogenated analogues to evaluate Polymorph Predictor software using the Dreiding 2.2.1 force-field, and concluded that, whilst the program was capable of finding the known structures, it was not capable of ranking

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them correctly in terms of the lattice enthalpy, probably because of deficiencies in the force-field, particularly in describing the hydrogen bonding and halogen–halogen interactions. The second study [82] developed the use of UPACK and contrasted the results of lattice energy minimisations using GROMOS, AMBER and an intermolecular potential which included a DMA electrostatic model. Around 100 crystal structures were found within 5 kJ mol–1, and although this number could be reduced considerably by removing symmetry constraints or a primitive Molecular Dynamics shake up, in all cases sufficient hypothetical structures remained that were of equal or lower energy than the known structures (though the different potentials gave different energy orderings) that it was obvious that factors other than lattice energy would be needed for genuine structure prediction. This even included predicting whether the structure would have the carboxylic acids groups hydrogen bonding to form a dimer or catemer. 3.4 The problems in improving the relative energies of different hypothetical structures Thus, during the first decade of crystal structure prediction studies, it became obvious that the zeroth-order assumption was usually inadequate because the lattice energy differences between significantly different hypothetical crystal structures were generally small compared with the likely errors in the calculated lattice energies. This was seen as primarily a problem with the accuracy of the available intermolecular potentials and force-fields, and hence gave considerable impetus to the development of better intermolecular potentials specifically for crystal structure prediction work, described in Sect. 4. However, for individual molecules, there is always the possibility that the known structure is not the most thermodynamically stable structure even under the conditions under which it was crystallised, let alone at 0 K. Energy differences between real polymorphs are often [2] in the range 0–10 kJ mol–1 . Comparisons of known polymorphic structures, albeit using a simple model potential and the harmonic approximation for rigid-molecule lattice modes, showed [83] that lattice-vibrational entropy differences seldom, if ever, exceed the enthalpy differences, and were generally about 15 J K–1 mol–1 [83]. Hence, unless there are reasons to suspect that kinetic factors could have stabilised a metastable polymorph, it is unlikely that a crystal structure that has been sufficiently long-lived for analysis without revealing its metastability, is going to be many kJ mol–1 above the global minimum in the lattice energy. Thus, a large energy difference between the known crystal structures and the global minimum in the lattice energy is a strong indication of inadequacies in the computational model.

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4 Development of transferable intermolecular potentials for crystal structure prediction by empirical fitting Hopes of improving on the transferable isotropic atom–atom potential in most areas of molecular modelling rested on the electrostatic contributions, with the hope that this would improve the modelling of polar molecules [84, 85]. Hence, it was an admittedly pragmatic step for Gavezzotti and Fillipini [86] to revisit the exp-6 potential to establish how far you could get in modelling organic crystal structures and energies by not explicitly modelling the electrostatic forces. (This saves considerable computer time in not having to perform the Ewald summations necessary to evaluate the electrostatic contribution to the lattice energy, and hence such potentials have considerable advantages in crystal structure prediction and molecular dynamics simulations.) The A, B, C parameters in A exp(– BR) – C/R6 were fitted for every atom–atom type to the crystal structures of 217 molecules containing C, H, N, O, Cl and S atoms without hydrogen bonds, and the heats of sublimation of 122 of these crystals. The fits were qualitatively constrained by reference to the atom–atom distance distributions obtained in a survey of 1846 crystal structures. The results were very reasonable, with, on average, the heats of sublimation being predicted within 2 kcal mol–1 (i.e. comparable to the probable experimental error) and the majority of crystal structures only showing small displacements on minimisation. When the study was extended to crystals containing X-H· · · Y (X,Y=N,O) hydrogen bonds [87], covering 173 carboxylic acids, 79 amides, 43 alcohols and 44 compounds with N-H· · · N hydrogen bonds, very different parameters were required for the polar hydrogen interacting with O in amides, acids, and alcohols, and for N atoms bonded to one or two hydrogens, all with very deep potentials. The lattice energies reproduced 54 heats of sublimation within 10% and the reproduction of the crystal structures and frequencies was generally satisfactory, though with some problems in the directionality of the hydrogen bonds. The resulting UNI potentials clearly showed that they had quite effectively absorbed the effects of the missing electrostatic model. This shows to what extent careful parameterisation can absorb the effects of missing terms in the model potentials, or conversely, how limited crystal structures are as a means of validating a potential model. The importance of the electrostatic interaction was emphasised by other work on developing intermolecular potentials for organic crystal structures. As far back as 1984 Williams had derived a set of exp-6 parameters for C, H and N by fitting to the heats of sublimations and crystal structures for the non-hydrogen-bonded azahydrocarbons (using the combining rules), and had found it necessary to use additional non-nuclear charges on the nitrogen atoms as well as atomic charges. This set of exp-6 parameters, along

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with those derived earlier for oxygen from crystal structures of oxohydrocarbons [88], chlorine from perchlorohydrocarbons [89] and fluorine from perfluorocarbons [90], have been quite widely used in crystal structure modelling with other models for the electrostatic forces, and form the basis of the non-bonded interactions in the Dreiding force-field [42]. With the addition of fitted polar hydrogen (HN ) parameters, Williams parameters were used in conjunction with a DMA electrostatic model to reproduce most of the crystal structures of 40 rigid C, H, N, O crystal structures, including nucleic acid bases, nonlinear optical materials and aza- and nitro-benzenes [91]. A large proportion of these crystal structures of predominantly essentially planar molecules were sensitive to the electrostatic interaction, in that qualitatively wrong crystal structures resulted when the anisotropic multipoles were removed. It is remarkable that the C, H, N, O repulsion–dispersion parameters that Williams derived from sets of non-hydrogen-bonding crystals without any N· · · O interactions were so effective at modelling a diverse set of N-H· · · O hydrogen-bonded crystals, (and later carboxylic acid crystal structures [92], with the parameterisation of a separate HO potential), that further refinement by optimising these parameters was not meaningful within the limitations of lattice energy minimisation. It had been established that the electrostatic forces around many molecules cannot be well modelled by an atomic point-charge model [84], even when the atomic charges were fitted to optimise the reproduction of the potential (i.e. potential-derived charges). The largest errors tend to be in regions where non-spherical features in the charge distribution, such as lone pair or π electron density, would be invoked to explain intermolecular interactions such as hydrogen bonds and π–π stacking qualitatively. The anisotropic electrostatic interactions arising from non-spherical features in the charge distribution are automatically modelled in more complete models for the ab initio charge distribution, such as distributed multipole models [1]. However, an alternative to avoid the computational expense and programming complexity of anisotropic electrostatic models is to introduce additional nonnuclear sites until the electrostatic potential around the ab initio charge distribution is reproduced with acceptable accuracy. Williams progressed his non-nuclear charge electrostatic models from the azabenzenes [93] to investigate a variety of electrostatic models [94], including establishing for even the most non-polar molecules, the n-alkanes, that non-nuclear angle bisector charges were required [95]. Throughout his work, Williams used chemical intuition to restrict the placing of non-nuclear interaction sites, whereas recently a method [80] of determining the optimal (both in terms of location and magnitude) set of point charges to reproduce the electrostatic potential around a molecule to a given accuracy has been implemented for use in crystal structure prediction calculations. Thus, it is probably safe to assume that the differences in the electrostatic potential around the models generated

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from DMA or such extended point-charge models are probably minor relative to the other errors in the potentials used. Williams determined, as his final tour de force, a revised set of empirically fitted molecule potentials for hydrocarbons [96], oxohydrocarbons including O-H· · · O hydrogen bonding [97], and finally, compatible sets of N atom parameters for azahydrocarbons [98], derived in conjunction with an electrostatic point-charge model fitted to the SCF 6-31G∗∗ potential including various lone-pair and methylene bisector sites. The resulting W99 force-field reproduced the crystal structures of carbohydrates, peptides and nucleosides well by lattice energy minimisation, with lattice length errors of usually less than 2%. A key feature of this force-field was the definition of atomic types by their intermolecular bonding, with separate sets of repulsion–dispersion parameters for 2-, 3- and 4-coordinated carbon atoms, carbonyl or 2-coordinated oxygens, nitrogens with 2, 1 or no bonded hydrogen atoms, (the last differentiated from triply bonded nitrogens), as well as different parameters for hydrogens bonded to carbon, nitrogen, oxygen in alcohols and oxygen in carboxylic acid groups. A second key feature was the explicit repositioning of the hydrogen interaction sites to be displaced A in from the nuclear site, to represent the displacement of the hydro0.1 ˚ gen charge density into the bond. Williams used rigid molecules taken from the experimental crystal structure (neutron or corrected X-ray) and hence avoided the problems caused by >N-H groups for example, which can readily distort under the influence of crystal packing to improve the hydrogen bonding. Thus, Williams’s potential set W99 probably represents the limit of accuracy of model potentials determined by empirical fitting to known crystal structures and some selected heats of sublimation, using transferability assumptions. 4.1 Additional tests of model intermolecular potentials 4.1.1 Global optimisation Using just the observed crystal structures and available heats of sublimation is only really sampling the intermolecular potential well depths and the forces at the atom–atom separations that are present in the observed crystal structures. One method of extending the sampling of the potential energy surface involved in the fitting is to target the lattice energy of the observed structure to be the global minimum in a crystal structure prediction search. This has been implemented in conjunction with a global optimisation method [99] adapted for crystal structures, the conformation-family Monte Carlo method [100], which was originally tested using both W99 and AMBER. The search performed well for the rigid molecules benzene, pyrim-

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idine, imidazole, 1,4-dinitrobenzene and 1,2,3,6-tetrahydrophthalamide, but less well for the flexible molecules 4-nitrophenol, benzylideneaniline, formamidioxime and 1,2-dimethyloxymethane, but the energy ordering showed the need for better potentials. The method proposed for optimising the exp-6-1-type potentials comprised optimising three components: the rank of the known crystal structure relative to all others found in the search, the deviations in the crystal structure on minimisation and the sublimation enthalpy [101]. It has been applied [101] to small saturated hydrocarbons and ethers, and produced an improvement on the W99 potential. However, they were unable to find a potential that correctly reproduced the structures and relative stability of the two polymorphs of dioxane, and concluded, in agreement with a previous study [102], that this polymorphic system appears to be incapable of being well-modelled with repulsion–dispersion parameters that are transferred from other molecules. 4.1.2 Effect of molecular dynamics A model potential that performs well in reproducing heats of sublimation and crystal structures (even relative to other hypothetical structures) is still limited in the aspects of the potential energy surface that are being sampled. Such potentials, for example, may badly estimate the thermal motions around the lattice energy minimum, and so transform to another structure when used in molecular dynamics (MD) studies. Since such detailed studies of thermal behaviour will aid the understanding and development of novel energetic materials, the criterion of stability in molecular dynamics simulations was considered in the development of the Sorescu–Rice–Thompson (SRT) transferable exp-6 plus atomic charges intermolecular potential for C, H, N, O atoms in the functional groups most commonly found in energetic materials [103]. Some of the heavily studied energetic materials used in its development are shown in Fig. 1. It started [104] by considering the

Fig. 1 Structures of some of the energetic materials used in the development of the SRT [103] CHNO intermolecular potential (a) RDX, (b) HMX and (c) CL-20

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explosive RDX, seeking a potential able to reproduce the crystal structure of the α form by both lattice energy minimisation, symmetry-constrained Monte Carlo and molecular dynamics simulations. The final potential provided excellent agreement with experiment, but the process showed that the dynamical modelling eliminated some models that would have been judged satisfactory by lattice energy minimisation. The potential also transferred well to give good molecular dynamics simulations of the α, β and δ polymorphs of HMX [105] and hydrostatic compression effects in NPT molecular dynamics simulations [106], and was successfully applied to study the three (partially) characterised polymorphs of CL-20 [107]. After building up confidence in the transferability of the potential through similar studies, it has recently been assessed [103] by considering the crystal structure predictions of 174 energetic molecules, assuming that the experimental molecular structure was rigid. The search using MOLPAK found the experimental crystal structure in 148 cases, with the failures generally being those where the crystal structures were not so well reproduced by lattice energy minimisation. Encouragingly, when the experimental structure was found in the search, it was at the global minimum in 75% of the cases, and only slightly higher in energy in all but two of the other cases, with an energy gap of 1.45 kcal mol–1 being regarded as exceptionally high [103]. This is impressive for a transferable exp-6 repulsion–dispersion potential, using the combining rules and hence only four sets of parameters in conjunction with the potential-derived atomic B3LYP/6-31G∗∗ charges. On the other hand, it might be thought that the close-packing principle should lead to large energy gaps as the majority of the molecules have quite complex three-dimensional shapes, which would be accurately represented using the experimental molecular structure. However, for the great majority of these crystals, the energy gap between the global and next minimum was less than 2 kcal mol–1 , with several cases giving hypothetical structures very close in energy, but not in structure, to the global minimum. Computer modelling of organic solids is increasingly moving towards the consideration of thermal effects, including a desire to simulate phase transformations. However, even facile phase transformations, for example between the two very closely related polymorphs of D,L-norleucine, prove [108] to be very demanding of the model potential. The conflict between the realism and computational efficiency of model intermolecular potentials that can be used in MD simulations is now becoming less of an issue. A new version of the MD simulation program DL_POLY [109] that can use distributed multipole electrostatic models, DL_MULTI [110], has recently been produced. DL_MULTI simulations [111] with a DMA and exp-6 potential were capable of reproducing the structure of imidazole at 100 K well, (unlike a potential that had been explicitly fitted to imidazole’s crystal structure and phonon frequencies [112]) and 5-azauracil at room temperature. The thermal expansions were physically reasonable, with far smaller expansion in the hydrogen-bonded directions

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than between the hydrogen-bonded sheets of 5-azauracil and between the hydrogen-bonded chains in imidazole. It is this type of marked anisotropy in the thermal expansion that complicates estimating whether the differences between the lattice energy minima and room-temperature crystal structures reflect errors in the force-field or the neglect of thermal effects. Accurate lattice energy surfaces and an effective way of estimating the effect of thermal motion could also improve crystal structure prediction by helping to eliminate from consideration minima that are sufficiently shallow that thermal motion would overcome the barrier and allow transformation to a related, more stable minimum. In order that the computer modelling of organic crystals can reliably simulate thermal motions, the potential should accurately describe the curvature around the lattice energy minima. Thus, the derivation of potentials using properties that depend on the second derivatives of the potential is clearly an important step forward. This is unfortunately limited by the availability of experimental data and the limitations of the models for predicting these properties. 4.1.3 Second-derivative properties 4.1.3.1 Mechanical properties The elastic constants of an organic crystal can be estimated from the second derivatives of the lattice energy [113]. A comparison [114] of the ability of various intermolecular potentials and a rigid molecular model to reproduce the elastic constants of urea, durene, m-dinitrobenzene, hexamethyltetramine, β-resorcinol and pentaerythritol [114] showed that potentials with a simplified function form, such as UNI, often performed very poorly for the elastic constants despite reproducing the structure and energy well, and that there was a marked improvement with the theoretically well-based models, such as those using distributed multipoles for the electrostatic term. However, the neglect of thermal effects generally led to the elastic stiffness constants being significantly overestimated, and any conformational flexibility makes the rigid modelling totally inappropriate. Unfortunately, measuring the elastic constants of a perfect organic crystal is extremely difficult, even when the problems associated with growing a suitably sized perfect crystal can be overcome, partly because there are 13 independent elastic constants for a monoclinic crystal and organic crystals can be extremely anisotropic. The mechanical properties of compactions of microcrystalline organic materials are important for pharmaceutical development; for example the metastable form II of paracetamol could be compacted [115] into a tablet because of the easy deformation of its hydrogen-bonded sheet structure, whereas binders

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are required for tablets of form I resulting in larger pills to be swallowed. Thus, computation of elastic constants of organic crystals is more useful as a predictive tool, to help rationalise the marked differences in mechanical properties between polymorphs, than as a method of validating potentials, except for specific molecules. 4.1.3.2 Phonon frequencies Similarly, phonon frequencies are a sensitive test of the intermolecular potential, but the frequencies for the purely lattice modes, typically in the region of 30–150 cm–1 , cannot currently be readily measured, though this may improve with new spectroscopic techniques [116]. Studies [117] of the rigid-body harmonic k = 0 frequencies computed for naphthalene, pyrazine, imidazole, hexamethylenetetramine and α-glycine showed that the frequencies that involved shearing close contacts between π systems or weak polar interactions were particularly improved by the use of more accurate distributed multipole electrostatic models, whereas atomic point-charge models generally underestimated the resistance to deforming hydrogen bonds. Nevertheless, even for very rigid molecules, the harmonic estimate of the phonon frequencies based on lattice dynamics neglects the anharmonicity in the lattice energy surface. However, a comparison [111] of the k = 0 phonon frequencies extracted from the motions within molecular dynamics simulations with the formal harmonic lattice estimates at 0 K showed good agreement when the two types of simulation were carried out with the same rigid molecular model and distributed multipole intermolecular potential. Most, but not all, modes decrease in frequency in the finite-temperature simulation, but generally by less than 5 cm–1 in the case of imidazole at 100 K, and by less than 20 cm–1 for 5-azauracil at 310 K. Thus, whilst the second-derivative properties of organic crystals can be used for estimating the differences in zero-point energy and entropy between different structures [117, 118], this is relying on various approximations being equally reasonable for the different structures. The need to model molecular motions in organic crystals sufficiently well to estimate entropic effects is important as many polymorphs are enantiotropically related (i.e. the relative stability of known forms changes with temperature). Whilst this challenge will need to be addressed in detailed studies of specific polymorphic systems, it does not appear to be a simple solution to the problem of a multitude of low-energy structures being found in many searches for minima in the lattice energy. Simple harmonic estimates of the entropy differences between known polymorphs show [83] that these are generally much smaller than the enthalpy differences, and it is found that adding simple estimates of thermal and zero-point energy effects to the lattice energy [64, 119] generally just reshuffles those

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structures that are already very close in lattice energy. For systems where denser structures have a lower lattice energy (because of the larger dispersion contribution), and less dense structures have lower energy motions (and hence are favoured by the entropy contribution), the two factors act in opposition so that the free energy discrimination would be worse than the lattice energy discrimination. However, in most searches the low-energy structures show a relatively small variation in density and so the entropy differences are dominated by structure-specific differences rather than any general trends [120]. Consideration of glycol and glycerol [118] and hydrocarbons [121] shows that consideration of free energies can help with structure prediction, but only once the lattice energy has been calculated with considerable accuracy.

5 Effect of molecular flexibility The existence of any degree of intramolecular flexibility (which gives isolatedmolecule vibration modes in the same region of the spectrum as lattice modes) clearly makes the use of the rigid-molecule approximation very suspect, even in the crystal structure prediction process, let alone the calculation of phonons and mechanical properties. However, most organic molecules have at least some degree of flexibility, where the molecular conformation could distort with a small energy penalty, ∆Eintra , which could be more than compensated for by the improved lattice energy. Conformational polymorphism [2], where the molecular conformation in the two polymorphs is significantly different, is quite common, and the disastrously late appearance of the more stable crystal structure of ritonavir [122] can be attributed to the difficulty it has in adopting the unusual conformation that allowed it to form strong hydrogen bonds in the crystal. Whilst many successful crystal structure prediction studies [123] have been carried out on molecules that show limited conformational flexibility, generally using Polymorph Predictor and all-atom force-fields, such combined inter- and intramolecular potentials can even fail to reproduce the known crystal structures satisfactorily [124]. Crystal structure predictions rely on the balance of the inter- and intramolecular forces and energies, and so, for example, a force-field that favoured the planar conformation of aspirin in the gas phase [125] predicted that aspirin ought to be planar in the crystal structure. However, when correlated ab initio calculations were used for the intramolecular energy penalty, the known crystal structure with a non-planar conformation was predicted [126]. The conclusion that ab initio intramolecular energies would be necessary was also demonstrated by an impressive series of crystal structure prediction studies using

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Fig. 2 The hydrogen-bonding motifs of the two conformational polymorphs of o-acetamidobenzamide [131]

UPACK [76, 127, 128] on highly flexible alcohols and sugars, which culminated in using [129] both ab initio conformational energies (∆Eintra ) and associated forces, within the final optimisation process for studies on glycol and glycerol. Thus predicting conformational polymorphism in such extreme cases as o-acetamidobenzamide, where one polymorph has an internal hydrogen bond and the other an additional intermolecular hydrogen bond (Fig. 2), will only be possible when both the intra- and intermolecular energies are calculated to the same high degree of accuracy [130]. Even molecules that are not generally considered as conformationally flexible can prove challenging to model computationally because of small, lowenergy conformational changes [132] such as the pyramidalisation of NH2 groups. Lattice energies can be very sensitive to the precise position of the protons, which unfortunately are not located accurately in X-ray structures. Thus there can be significant differences [91] between the lattice energy minima computed using the experimental molecular structure and with ab initio optimised or other idealised molecular models, and these can vary with the temperature and quality of the crystal structure determination [41]. This is a major problem for genuine structure predictions, which either have to use ab initio optimised gas-phase structures held rigid, or have a reliable model for small distortions of the molecule that can, for example, produce a shorter or more linear hydrogen bond. Thus, although this review is primarily about the intermolecular forces in organic crystal structures, the problems of balancing inter- and intramolecular forces is clearly a major aspect of the challenge that crystal structure prediction poses computational chemistry. It is also important to appreciate that tests of intermolecular potentials for crystal structure prediction and reproduction will be significantly more successful when the experimental molecular structure is used than when genuine predictions are made.

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6 Ab-initio-based intermolecular potentials for organic molecules The above account has focused on intermolecular potentials that all had at least the repulsion–dispersion component based on the assumption that the potentials were of a simple functional form that could be transferred between atoms of the same type, and derived by empirical fitting. These potentials are in marked contrast to the accurate potentials for specific small polyatomics described elsewhere in this volume. A first step in this direction was made by the derivation of a transferable potential from ab initio calculations on the methanol monomer, dimer and trimer [133], with the emphasis being on obtaining a potential with good accuracy in the relative energies for crystal structure prediction. A damped atom–atom R–6 dispersion model was fitted to the dispersion energies for 94 methanol dimer geometries calculated using London’s sum-over-state second-order perturbation theory with Møller–Plesset partitioning. Atomic dipole polarisabilities were fitted to the nonadditivity in the SCF interaction energy using 17 trimer geometries, and used to model the induction energy. Atomic multipoles up to dipole on the hydrogen atoms and quadrupole on the carbon and oxygen were fitted to the electrostatic potential around the monomer. Finally, an atom–atom repulsion model, with anisotropy on the oxygen atom, was fitted to the remainder of the MP2 interaction energies of the dimers, calculated using an interaction optimised basis set. The resulting model intermolecular potential was assumed to be transferable, except the atomic multipoles were redetermined for each molecule and each conformation that differed by more than 5◦ in a torsion angle, and used in conjunction with a separate intramolecular force-field, in crystal structure prediction studies [102] on methanol, ethanol, propane and 1,4-dioxane. The results using the ab-initio-based potentials were all superior to predictions based on standard force-fields, and very successful with the exception of the polymorphs of dioxane (c.f. Sect. 4.1.1). This ab initio-based approach to determining model intermolecular potentials is therefore very promising, but restricted to molecules where the interactions can be assumed transferable from molecules sufficiently small that a high-quality ab initio potential energy surface can be calculated. Thus, for organic molecules with larger functional groups, such as aromatic rings, a more practically feasible approach to accurate intermolecular potentials is to derive it from the ab initio charge density of the isolated molecule. The use of specific electrostatic models, derived to represent the potential around the ab initio charge density accurately by either a DMA or multi-site point-charge model, is now well established in crystal structure modelling (Sect. 4). It is also worth noting the sensitivity to the quality of the ab initio charge density, as well as how accurately it is represented.

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Scaling the distributed multipoles derived from SCF 6-31G∗∗ charge densities by a factor of 0.9, to approximate the tendency of SCF wavefunctions to overestimate the molecular dipole moment [134] and provide an appropriate scaling of electrostatic potential [135], generally improved the reproductions of the crystal structures of polar or hydrogen-bonded rigid molecules [91], though on average it affects the lattice vectors by less than 1%. This scaling of the electrostatic contribution to the lattice energy by (0.9)2 = 0.81 has a major effect on the total lattice energy, particularly for the more strongly bound crystals, but this is often within the large experimental and theoretical errors in comparing with the heats of sublimation. Now that correlated wavefunctions can generally be obtained for organic molecules, these should give a better estimate of the strictly defined electrostatic contribution to the lattice energy, excluding penetration effects. Unfortunately, the use of a theoretically better electrostatic model in conjunction with repulsion–dispersion parameters empirically fitted with an SCF electrostatic model, such as those due to Williams, does not necessarily imply a better total intermolecular potential. The short-range repulsion terms in the atom–atom model potential play a major role in determining the closest contact distances within the crystal. Unfortunately the close packing within crystals means that it is the balance of many types of short-range interactions that determine the contact distances. The frequent similarity of the geometry of many types of intermolecular contact within crystal structures compounds the difficulty of deriving parameters that meaningfully separate the repulsive contribution between atomic types. Thus, the empirically fitted potentials may not extrapolate correctly to other relative geometries of the functional groups. Oxalic acid provides one illustration of this problem [137], as many transferable potential schemes are unable to reproduce the crystal structures of both polymorphs satisfactorily, often predicting qualitatively distorted structures. This could be because oxalic acid polymorphs sample a wider range of relative contacts of the carboxylic acid group than other carboxylic acids (Fig. 3). In addition, the charge density of a carboxylic acid group in oxalic

Fig. 3 The hydrogen-bonding motifs in the α and β polymorphs of oxalic acid [136]

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acid is likely to differ from that in other carboxylic acids where charge density can be transferred from the bonded functional groups. Thus, the two polymorphs of oxalic acid were used as a first test of the overlap method for obtaining specific atom–atom repulsion potentials from the ab initio charge distribution of the molecule. 6.1 The overlap model for nearly nonempirical repulsion potentials The overlap model is based on the assumption that the short-range repulsive terms are proportional to the overlap,  MN Sρ = ρM (r)ρN (r) d3 r , of the molecular charge distributions ρM (r) and ρN (r) at a given relative orientation of the molecules M and N. Various tests have been made of this assumption on noble gas dimers, homonuclear diatomics [138] and molecules interacting by a O-H· · · N hydrogen bond [139], and it works reasonably well, although the refined model  y MN Urep = K SMN (4) ρ where the power y is slightly less than unity, is more accurate. The advantage of this assumption for organic molecules is that, unlike the short-range energies obtained from ab initio supermolecule methods, the molecular overlap can be partitioned into atom–atom contributions by first partitioning the charge density of the molecule between the atoms. Thus, GMUL [140] can be used to calculate the total overlap of the two molecules at hundreds of relative orientations to sample the van der Waals contact region adequately. Using the GMUL partitioning [141] of the molecular charge density between the atoms, which is analogous to a distributed multipole analysis but maintains the spatial extent of the charge distributions, the different contributions to the intermolecular overlap can be associated with an atom in each molecule. The relative separation of an intermolecular pair of atoms and the relative orientation of the intermolecular atom–atom vector and the covalent bond vectors, are known for each dimer geometry. Thus, a suitable atom–atom functional form can be fitted individually for each intermolecular pair of atoms to give an analytical atom–atom model for the overlap. Hence, there is no need to assume combining rules for any atom that contributes significantly to the short-range intermolecular repulsion, and of course, the final model can be tested against the total overlaps. This leaves just the proportionality parameter, K, and if possible, y, to be determined to provide a model for the short-range exponential terms in atom–atom form.

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In the case of oxalic acid, an isotropic exponential form fitted the individual overlaps well to give a repulsion potential [137] of the form:
MN Urep = KSMN Aικ exp(– Bικ Rik ) (5) ρ = i∈M,k∈N

with the functional types C, =O, -O- and H, once the proportionality constant K had been fitted to a small number of intermolecular perturbation theory (IMPT) [142] calculations of the exchange–repulsion energy. The derived model repulsion potentials, in conjunction with a DMA and a dispersion model estimated from atomic polarisabilities, were able to give a considerably better reproduction of the structures and relative energies of both polymorphs of oxalic acid than previous studies. Further systematic investigations of the overlap model for organic molecules considered [143] formamide, acetamide and trans N-methylacetamide, developing isotropic atom–atom potentials which were at least competitive with the best that had been empirically parameterised to the crystal structures of these molecules. In this study, the correlation of the overlap with exchange–repulsion, charge-transfer and penetration energies as calculated by IMPT were separately investigated. The advantage of the overlap model in allowing the investigation of the transferability of repulsion models was also raised. Since the sets of overlaps are initially calculated for every intermolecular atom pair (i.e. N2 /2 sets of data for a molecule of N atoms), the differences between the data sets for different similar atoms can then be compared and, if found sufficiently similar to warrant being considered the same atomic type, combined. 6.2 Development of anisotropic repulsion potentials A second assumption that has to be made for empirically fitted potentials, that the atom–atom repulsion is isotropic, can also be tested using the atom– atom overlaps, and if appropriate an anisotropic functional form fitted. Given that it had already been established from crystal structure analysis that the repulsive wall around organic chlorines was anisotropic [27, 29], such a development was clearly needed. The molecule chosen for developing a specific potential by the overlap model was cyanuric chloride [144], because it had already been established that the “fish-scale” effect of slightly overlapping layers seen in its crystal structure could not be reproduced with conventional model potentials for Cl· · · N interactions [145]. In this case, adequate fits to the Cl· · · Cl, Cl· · · N and N· · · N overlaps could be obtained with the expression
    MN Urep = KSMN Aικ exp – Bικ Rik – ρ Ωik , (6) ρ = i∈M, k∈N

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where the function       ρ Ωik = ρ0ικ + ρ1ι zi · Rik + ρ1κ – zk · Rik     2  2  + ρ2ι 3 zi · Rik – 1 /2 + ρ2κ 3 zk · Rik – 1 /2

(7)

varies the contour of constant atom–atom repulsion according to the relative orientation of the intermolecular atom–atom unit vector Rik and the unit vector zi along the C-Cl bond or bisector of the CNC angle. The overlap fitted form of anisotropy is consistent with the lone-pair density moving the repulsive wall out at the sides of the chlorine atom, allowing the “polar flattening” for linear C-Cl· · · Cl-C close contacts, whereas the N· · · N overlaps were well fitted with a form of anisotropy that is consistent with the lone-pair density. The resulting overlap model potential, when combined with a DMA electrostatic and isotropic dispersion, was able to reproduce the symmetry-breaking fish-scale effect well by lattice energy minimisation using a version of DMAREL which had been adapted to include the anisotropic repulsion [144]. A further step of developing the overlap model for organic molecules was the consideration of a series of molecules to produce a nearly nonempirical transferable intermolecular potential for the chlorobenzene [146] series of molecules. The overlap model was applied to monochlorobenzene and 1,2,3-trichlorobenzene, both separately and by finally combining the overlap data, once it had been established that the required atomic types were Cl, H, CCl and CH , i.e. the aromatic carbons bonded to chlorine and hydrogen had sufficiently different overlaps that they needed different parameters. The anisotropic functional form for all atoms was determined in the form of Eqs. 6 and 7, providing an anisotropic atom–atom potential consistent with lone-pair and π electron density within the molecule, and the proportionality constant K fitted to IMPT calculations. The dispersion, in the Cικ /R6ik form, used atomic polarisabilities derived from ab initio calculations on the chlorobenzene molecules. The electrostatic forces were calculated from the DMA of the wavefunction. Thus the final potential was derived purely from the ab initio charge density of the isolated molecule. However, the sensitivity of the reproductions of the 12 chlorobenzene crystal structures to various assumptions, such as atomic polarisabilities, and the crudeness of the model compared with small polyatomic intermolecular potentials, means that this nonempirical anisotropic atom–atom model is rather empirically justified. Further modelling of lattice energies, phonon frequencies and other properties provided satisfactory agreement with experiment, with the discrepancies being primarily due to approximations in the theoretical methods rather than the model intermolecular potential. When this potential was used in a crystal structure prediction for p-dichlorobenzene, the three polymorphs were found second, third and fourth in lattice energy within 0.16 kJ mol–1 of each other. The hypothetical crystal structure

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at the global minimum was less than 0.5 kJ mol–1 more stable than the α form and has a coordination sphere that was a combination of the α and β forms. This structure was predicted to grow less quickly from the vapour phase than the known forms, so it may not be observed for kinetic reasons, rather than necessarily implying errors in the relative lattice energies. The anisotropic chlorobenzene potential has since proved capable of reproducing the newly determined Z = 2 crystal structure of the low temperature α phase of tetracholorobenzene [147], and also a crystal structure prediction study had it as slightly lower in energy than the global minimum found in a Z = 1 search, which corresponded to the high-temperature β phase. The importance of Cl anisotropy in crystal structure prediction work has also been demonstrated by extension of this potential to chlorophenol, which was capable of predicting the structure of a new highpressure phase of 2-chlorophenol as one of the densest low-energy plausible structures [148]. Thus, there has been some progress in deriving nonempirical potentials for organic molecules using the theory of intermolecular potentials, but there is still a long way to go before there are potentials on as firm a theoretical basis as those available for water [149]. The lack of reliable models for the induction forces, as well as the poorly defined dispersion modelling and reliance on IMPT as the best theoretical validation of the model potentials, shows that there is huge potential for the improvement in model potentials for organic molecules using symmetry-adapted perturbation theory [150].

7 Evidence for the importance of the accuracy of intermolecular potentials for crystal structure prediction Whilst some crystal structures can be reasonably reproduced by a wide range of potentials, others are very sensitive to the potential model. There is often a correlation with the complexity of the three-dimensional (3D) shape – for example, many crystal structures of planar molecules allow considerable slippage of the molecules relative to each other, and so the correct structure is sensitive to the intermolecular potential, whereas crystal structures of rigid, more 3D molecules have sufficient fitting of bumps into hollows that virtually any distortion of the crystal structure is quickly restored by exponential repulsive forces. Unfortunately, for most 3D molecules, this fitting is very dependent on the correct molecular conformation: a recent study of the reproduction [124] of 48 crystal structures, chosen to represent those of pharmaceutical interest, found that the Dreiding force-field distorted the con-

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formation of many of the flexible molecules so consistently that a reasonable reproduction of the crystal structure was not possible with any electrostatic model. On the other hand, for the rigid molecules, there was a notable [124] improvement when the electrostatic forces were modelled by a distributed multipole model rather than atomic charges. Thus, getting the correct conformation and then reproducing the known crystal structure satisfactorily can be seen as prerequisites for any organic crystal structure modelling, whereas getting the relative energies correct is necessary for crystal structure prediction. Crystal structure prediction studies that have been done with more than one potential generally give better results, in that the known crystal structures become closer to the global minimum in the lattice energy, as the theoretical basis for the potential improves, as illustrated by the series of papers on alcohols and sugars [76, 102, 118, 128, 133] previously mentioned. A postevent study of the molecules used in the 1999 blind test of crystal structure prediction (c.f. Sect. 8.2) clearly demonstrated [151] that moving from potential derived atomic charges to distributed multipoles in conjunction with the Dreiding force-field would have significantly increased the success rate. A larger, more systematic consideration [152] of a number of molecules focused on 50 reasonably assumed rigid C, H, N, O molecules (which included many N-H, O-H and NO2 groups) corresponding to 62 appropriate crystal structures available from the CSD. A sample of 20 molecules was studied using the W99 and FIT exp-6 parameterisations and non-bonded parameters from the Dreiding, CVFF95 and COMPASS force-fields, in conjunction with the potential derived charges from the density functional theory (DFT) ab initio optimised molecular charge density or the bond increment charges of COMPASS and CVFF95. The electrostatic potential-derived (ESP) charges fitted to the molecular electrostatic potential were a clear improvement over those calculated from transferable bond increments, as might be expected. Overall the W99 potential gave the best reproductions with an overall r.m.s% change in the lattice parameters of 2.66% on lattice energy minimisation, followed by COMPASS with the same electrostatic model (2.96%). An extensive crystal structure prediction [152] was performed on each rigid DFT optimised molecule, using several runs of Polymorph Predictor to generate minima in the lattice energy described by the W99 plus ESP charges. All structures within 15 kJ mol–1 of the global minimum were reminimised using DMAREL, and the symmetry reduced to find true minima as judged by the elastic constants and k = 0 phonon frequencies. For these diverse 50 molecules, almost a third of the known crystal structures were found at the global minimum and approximately half within 1 kJ mol–1 of the global minimum [152], with the known crystals of almost all nonhydrogen-bonding molecules at, or within 1 kJ mol–1 of, the global minimum. When the ESP atomic point-charge model in the sets of low-energy structures was replaced by a DMA electrostatic model, the results improved [153]

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so that half of the experimentally observed crystal structures were either found to be the global minimum or to have calculated lattice energies within 0.5 kJ mol–1 of the global minimum. Furthermore, 69% of cases had five or fewer unobserved structures with lattice energies lower than the observed structure.

8 Changing success & scope of crystal structure prediction 8.1 Analysis of literature on crystal structure prediction studies In 2000, Theresa Beyer completed her Ph.D. thesis on the computer prediction of organic crystal structures by performing a literature survey of all the published work on crystal structure prediction by lattice energy minimisation without knowledge of the unit cell parameters. The 64 papers covered 253 searches on 189 molecules, of which 29 were known to be polymorphic. The number of studies was too small for any meaningful statistical analysis [154] but highlighted the difference between the molecules

Fig. 4 The types of molecules studied to date by crystal structure prediction, as derived from a survey of published lattice energy minimisation studies. Each molecule is included only once, although there is considerable overlap between categories; the category assigned to each molecule is dictated by the prime motivation for the study, where known. The main figure gives the cumulative total of 486 molecules, with the smaller figures showing the division into the 171 prior to the start of the century and the 352 after (including new studies on the same molecules)

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Fig. 5 The success rate of searches for a given crystal structure by lattice energy minimisation, as derived from a survey of published lattice energy minimisation studies. The results are distinguished by whether the known crystal structure was found as the global or local minimum, was similar to a not fully characterised structure, was found but no more information was given, or was not found in the search at all. The number of searches is given (a) for different categories of molecule (c.f. Fig. 4) and (b) by year of publication. Many molecules have been studied by more than one group or subjected to repeated attempts by the same group with improved methods and/or potentials, and so appear more than once on the graph. Where known polymorphic systems have been studied, only one result is given per molecule per search, i.e. if one form is found as the global minimum, it will count as a global minimum on the chart, regardless of whether/how the other forms were found in the search

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used in developing the computational methods of crystal structure prediction and the molecules considered in crystal engineering. Hydrocarbons, alcohols and sugars dominated the lists of molecules considered in crystal structure prediction in the 20th century (Fig. 4 inset), whereas crystal engineering tends to use molecules and molecular salts with complimentary multiple hydrogen bonds. Hence, it was clear that the industry of the Utrecht group in studying sugars and alcohols, and the love of hydrocarbons for which current potentials were expected to be adequate, may have distorted the picture of how readily organic crystal structures can be predicted. An analysis of the ongoing update of this survey [123] is presented in Fig. 4, which shows a considerable change in the types of molecules considered since it became clear that developing sufficiently realistic potentials for each type of molecule was a major part of the challenge. Experience is now more dominated by rigid C, H, N, O materials, from the large studies of crystal structure prediction methods, firstly for energetic materials (assumed rigid at the experimental structure) [103] (Sect. 4.1.2) and also for rigid polar molecules, using the ab initio optimised structure [152, 153] (Sect. 7). Apart from these large surveys, there are many more papers reporting crystal structure prediction studies on individual molecules performed as part of a multidisciplinary study on the solid-state behaviour of the molecule. Thus, Figs. 4 and 5 represent a survey of 98 papers covering 486 molecules, 70 of which have the crystal structure of more than one polymorph in the CSD. Figure 5 attempts to answer the question as to whether we have markedly improved our ability to predict organic crystal structures. The results of the published 705 searches show that 330 crystal structures were found at the global minimum, 289 as local minimum, 18 as an undefined minimum, and 57 structures were not found. There were also 11 searches carried which were aimed at helping characterise a structure which could not be fully determined by X-ray crystallography and so were counted separately. This is not a marked improvement on the previous survey, but some of this can be attributed to using fuller searches (and so finding more structures in the low-energy region), having more genuine searches that did not use the experimental molecular structure, and more ambitious studies of molecules which are difficult to crystallise or are known to have problems in defining good intermolecular potentials, such as the study of halogenated benzenes used to predict disorder in para-substituted benzenes [67]. There are also many factors involved in the sociology of publication, some of which are apparent in Fig. 5b, including, for example, that failures to locate the structure in the search are generally only reported as part of large surveys of the prediction method. Nevertheless, Fig. 5 does show that there is no class of molecule which with current potentials can be confidently predicted by searching for the global minimum in the lattice energy.

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8.2 The Cambridge Crystallographic Data Centre blind tests of crystal structure prediction An alternative method of evaluating the progress that has been made in crystal structure prediction is to hold blind tests. These [155–157] have been organised by the Cambridge Crystallographic Data Centre, and have provoked considerable interest amongst the experimental community as a genuine test of theoretical progress. In these tests, the groups active in the field are sent the chemical diagrams of molecules believed to be within the claimed capabilities of at least some of the computational approaches and invited to submit three predictions for each crystal structure by a given deadline, about six months later. The most valuable part of the test comes after the deadline, when the crystal structures are released to the participants for post-analysis of the accuracy of their predictions or reasons for lack of success, and then a meeting is held to pool experience. Looking at the summary of results in Fig. 6, also leads to the conclusion that there had not been much progress in the field over the five years since the first test. However, the three papers discussing the results are more positive, as the difficulty of the blind test has increased. Only part of this has been deliberate, for example, in 1999 and 2001 the crystal structures were restricted to Z = 1, and this was extended to Z ≤ 2 in 2004. However the Z = 2 structure for azetidine (XI) was found to be a transition state between Z = 4 structures of only slightly lower energy in many calculations. Similarly, the series of flexible molecules appear only to increase the number of torsion angles from one to three, but the crystal structure of X turned out to be sensitive to six torsion angles, including the rotation of the methyl groups. There has been considerable progress in the search methods, with a few of the more advanced methods being able to locate all the structures in 2004, in the sense that the minimum obtained starting from the experimental structure was found in the search, though not within the three structures submitted. However, increasing the quality of the search only adds to the problem that has been highlighted from the first. When the search gives a large number of different crystal structures within a small energy range of the global minimum, then choosing which three structures to submit becomes horribly dependent on the accuracy of the force-field if you assume that the crystal structure is the thermodynamically most stable, and fairly arbitrary if you acknowledge that the observed structure may not be the most stable. Indeed, it was the metastable polymorph of I that was successfully predicted in the first blind test, and no-one predicted the second polymorph which was found on all subsequent attempts to recrystallise I. Similarly, one participant was so convinced that the crystal structure of VI in 2002 ought to have an alternative hydrogen-bonding pattern that they performed a detailed

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Fig. 6 Summary of the results of the blind tests of crystal structure prediction organised by the Cambridge Crystallographic Data Centre. The success rate is given as x/y where x is the number of successful predictions within the three submitted by each of the y groups

experimental search [158] for it. After approximately 80 crystals had been analysed by single-crystal X-ray diffraction, an alternative polymorph was indeed found [158] with the predicted motif (though as a Z = 2 structure it was not within any of the submitted predictions). The theoreticians’ ideal that a polymorph screen should be carried out on the molecules prior to being used in the blind test is not exactly practical, given the difficulty in finding suitably small molecules whose crystal structures have only recently been determined by crystallographers who are willing to participate. The problem of having secure experimental information was illustrated by partial information about the structure of molecule VIII being found in a conference proceeding during the test. The independence of the choice of molecules is evident in that IX (the molecule intended as a test of methods of developing model intermolecular potentials) included iodine, an atom whose size currently precludes the ab initio-based methods of parameterising intermolecular potentials. Coin-

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cidentally, the few structures of related molecules in the Cambridge Structural Database which might have been used for empirical testing of O· · · I interactions etc. were poorly determined, some without even the hydrogen atoms being located. However, it is encouraging that the one successful truly blind prediction in the 2004 test [157] was for IX, using a DMA electrostatic model plus an anisotropic repulsion model derived by extrapolating from the chlorobenzene potential [146]. Thus, this success very much confirmed the conclusion from all the blind tests that progress in crystal structure prediction is very dependent on improving the accuracy of the intermolecular potentials for organic molecules.

9 Other uses of crystal structure prediction Even if crystal structure prediction methods are still far from their ultimate goal of predicting the crystal structures (and therefore the polymorphism) of organic molecules using only the chemical diagram, they can still provide a useful tool in developing our understanding of the solid state. Firstly, even if the relative lattice energies are not well predicted, provided that the crystal structures are accurately modelled, then the set of predicted structures could be used to help characterise structures where there is insufficient diffraction data for full characterisation [159]. A particular use is for materials where sufficiently large crystals for single-crystal diffraction cannot be grown, and the X-ray powder pattern cannot be analysed to give the unit-cell dimensions. Many pigments fall into this category, hence this industry’s particular interest in crystal structure prediction [160, 161] which has resulted in the successful structural characterisation of a yellow pigment from powder data [162]. Secondly, when a molecule crystallises in a form that is disordered, or otherwise complex, then the computation of energetically favoured ordered crystal structures by a lattice energy minimisation search, can provide insights into the disorder. Caffeine is a well-known example of a molecule that seems to have difficulty in crystallising in an ordered structure. A computational search found a thermodynamically ordered equivalent of the disordered structure of anhydrous caffeine as its global minimum [163], and rationalised the disorder by considering the strongest interactions of the caffeine dimer and caffeine-water complex. The agrochemical chlorothalonil, C6 Cl4 (CN)2 [66] had so many low-energy crystal structures in a search using a transferable potential that a specific anisotropic atom–atom potential was developed using the overlap model (Sect. 6.2) and used to refine the relative energies of crystal structures found in the search. Whilst form I was found as the global minimum, within 1.25 kJ mol–1 there were two Z = 1 crystal

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structures based on different stackings of a sheet which allowed a rationalisation of the disorder in form II, and two more based on different herringbone stackings, which can be combined to represent the Z = 3 structure of form III [66]. Thus the complexity of the landscape of low-energy crystal structures reflected [66] the complexity of the polymorphs of chlorothalonil. Perhaps the most exciting application of crystal structure prediction studies is to provide the structures of the unknown low-energy crystal structures, to suggest crystallisation experiments that may lead to new polymorphs [164]. For example, Polymorph Predictor searches on diflusinal [165] and 2-amino-4-nitrophenol [166] revealed that a variety of different hydrogen-bonding motifs could give low-energy crystal structures, and the results were used to suggest which solvent might target particular motifs. This strategy yielded two new polymorphs and two new solvates for diflusinal [165], and the non-solvated and two solvate structures for 2-amino-4nitrophenol [166]. Although the actual crystal structures were not predicted, the computational prediction that alternative hydrogen-bonding motifs were energetically feasible was certainly an aid to producing new polymorphs. For simpler molecules, without the flexibility and multipole donors and acceptors, the prediction that there should be alternative, more stable crystal structures has been influential in encouraging patient searches which were ultimately successful in finding new crystal structures. This is exemplified by studies of pyridine [64] and 5-fluorouracil [65], where the unusual Z = 4 structure of the known forms invited rationalisation. In both cases a new Z = 1 structure has been found, with that for 5-fluorouracil corresponding to the global minimum structure in the search [65]. New methods of crystallisation that find new polymorphs are being continually developed, such as the application of pressure [148], confinement in nanotubes [167], and templating with additives and surfaces, in addition to the almost infinite variations on solution crystallisation. For example, a new, more stable polymorph of dinitrobenzene [4] was found after 120 years of study, when templated by the structurally similar molecule trisinadine. Hence, there is a major role for accurate predictions of which crystal structures are thermodynamically feasible in developing our understanding of the kinetic factors that can lead to polymorphism.

10 Alternatives to model atom–atom potentials for modelling organic crystal structures The ability to compute the relative thermodynamic stability of different crystal structures of organic molecules sufficiently reliably for crystal structure prediction and understanding polymorphism represents a major challenge to

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computational chemistry. The above account has focussed on using increasingly realistic model intermolecular potentials, and it is worth considering other possibilities. The most obvious alternative is to use ab initio calculations on the lattice energies of crystal structures, which has the attraction of treating the intermolecular and intramolecular forces on the same level, and allowing for appropriate distortions of the molecular conformation. The main problem with this is the dispersion energy, which makes a significant attractive contribution to the lattice energy from the long-range interactions within the crystal. An early attempt to distinguish between the many alternative crystal structures for acetic acid by using Carr–Parrinello DFT calculations [168] found it necessary to add a term to adjust the dispersion energy, though arguably, when the structures have similar densities this may cancel. A recent study [169] of solid nitromethane, HMX, RDX and CL-20 (see Fig. 1) using the PW91 and Perdew–Burke–Ernzerhof (PBE) density functional theories found large errors in the lattice vectors, which suggest the need for new DFT methods to be developed to model the dispersion forces sufficiently accurately for organic crystals. A more novel alternative, clearly aimed at organic molecules, is the pixel semiclassical density sums (SCDS) [170–172] approach of Gavezzotti. In this method, the lattice energy is evaluated from the ab initio charge density of the isolated molecule within the crystal structure, using numerical integrations over the charge density. By definition, the electrostatic energy including penetration effects can be evaluated to an accuracy only limited by the quality of the charge density and the numerical integration. The model for the shortrange repulsion assumes the overlap model [172], and the polarisation and dispersion models assume an even distribution of the experimental atomic polarizability throughout the atoms. This model, which considers the entire spatial extent of the molecular charge distribution, gives a more realistic view of what determines crystal packing than the atom–atom directed specific interaction viewpoint. For example, it has shown that molecules in contact may not have particularly stabilising interactions between them, and that the contact may often just help complete the close packing. The method has been applied [173] (without minimisation) to some sets of low-energy crystal structures produced in lattice energy searches, and produces alternative orderings for the low-energy structures to those produced by various model intermolecular potentials, including a significant reranking of the structures of pyridine and parabanic acid relative to a DMA-based model. Since this method uses a completely different set of assumptions from those implicit in model potentials, it gives some feeling for the uncertainties in the model potentials.

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11 Future prospects The assumption that the crystal structures of organic molecules could be predicted by searching for the most thermodynamically stable structure by considering minima in the lattice energy has proved to be a useful starting point towards modelling solid-state behaviour. A significant proportion of crystal structures can be predicted as corresponding to the global minimum in the lattice energy, provided quite sophisticated model potentials are used, particularly for the electrostatic interactions of polar molecules capable of hydrogen bonding. However, it is relatively rare for the known structure to be so much more stable than any alternatives, allowing for the errors in the calculations, that polymorphism can be confidently ruled out. An exception is the 12 kJ mol–1 energy gap found [160] for pigment yellow 74, where the prediction of no polymorphs is consistent with observations over a long period of industrial manufacture and development. It is debatable [174–177] whether we can reasonably aspire to a computational method of predicting all polymorphs of any organic molecule of industrial or academic interest. Experimental and computational studies on polymorphism clearly show that the kinetics of nucleation [178], growth and transformation can be very important in determining the solid-state behaviour of organic molecules, and yet these can be very sensitive to the conditions of crystallisation, including impurities. However, when seen in the context of the experimental difficulty in establishing that all solid forms of a given molecule are known, then there is already clearly considerable value in performing a computational study in collaboration with experimental work to understand the solid-state behaviour, particularly for pharmaceuticals [9]. Computational crystal structure prediction is clearly a problem that presents a fundamentally important and practically worthwhile challenge to combine many fields of theoretical chemistry, particularly the accurate quantification of intermolecular forces for organic molecules. Acknowledgements SLP is particularly grateful to Anthony Stone for the training he gave her in the theory of intermolecular forces as a Ph.D. student, and for the help and encouragement that he has given her over the decades in developing the use, for organic molecules, of distributed multipoles and his other great contributions to our knowledge of intermolecular forces. Both authors acknowledge funding from the Basic Technology Programme of Research Councils UK and the help and support of other members of the CPOSS project and collaborators (particularly Graeme Day) in providing background to this review.

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Struc Bond (2005) 115: 125–148 DOI 10.1007/b135964 © Springer-Verlag Berlin Heidelberg 2005 Published online: 19 July 2005

Molecular Dynamics Simulations and Intermolecular Forces Claude Millot Equipe de Chimie et Biochimie Theoriques, UMR CNRS-UHP 7565, Université Henri Poincaré - Nancy 1, Faculté des Sciences et Techniques, Boulevard des Aiguillettes, BP 239, 54506 Vandoeuvre-lès-Nancy Cedex, France [email protected] 1 1.1 1.2 1.3 1.4 1.5

Calculation of Potential Energy and Molecular Dynamics . . . Two-Body Potentials . . . . . . . . . . . . . . . . . . . . . . . . . Potentials Including Polarizability . . . . . . . . . . . . . . . . . Tight-Binding Models . . . . . . . . . . . . . . . . . . . . . . . . Hartree–Fock ab initio and Semiempirical and DFT Techniques Hybrid Quantum Mechanical/Molecular Mechanical Techniques

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125 125 127 133 137 139

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Application to Selected Clusters . . . . . . . . . . . . . . . . . . . . . . . . Hydrated Halide Anions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silicon Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 140 143

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Abstract In this chapter, the main computational methods presently used in molecular modelling to compute the energy of an assembly of molecules and to perform a molecular dynamics simulation will be presented. Then, recent molecular dynamics and quantum chemical results for hydrated halide ions and silicon clusters will be reviewed.

1 Calculation of Potential Energy and Molecular Dynamics 1.1 Two-Body Potentials The simplest model to compute the interaction energy of an assembly of molecules is a pairwise additive potential: E = Evdw + Ees ,

(1)

where Ees is the electrostatic interaction energy and Evdw is the short range van der Waals interaction energy (essentially dispersion and repulsion energy). In many cases, intramolecular energy terms (bond stretching, valence angle bending, torsion, out-of-plane bending) are also included in the force

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field, but they will be omitted in the following presentation in order to simplify the discussion. Usually, the electrostatic interaction, when present, is modelled through Coulomb interactions between partial charges of atoms (sites) a and b:



qa qb Ees = , (2) rab A a∈A B
=A

b∈B

where qa is the electric charge of site a and rab is the distance between both sites a and b. When the partial charges included in a force field that does not include the polarizability are adjusted in order to reproduce the condensed phase properties, they are usually not efficient for modelling properties of the dimer or small clusters. To build accurate electrostatic models, several groups have proposed methods to obtain distributed multipoles from knowledge of the electronic wave function, the electron density or the electrostatic potential around a molecule [1–8]. These distributed multipole models should then be used in a force field with a polarizability model. The van der Waals interactions are usually modelled by a Lennard-Jones potential:   12 6



σab σab , (3) 4εab – 6 Evdw = 12 rab rab A B
=A a∈A b∈B

where εab is the well-depth and σab the sum of site radii for the pair (a,b); or by the Buckingham potential:



Cab Aa b e–Bab rab – 6 . (4) Evdw = rab A B
=A a∈A b∈B

Such simple potential functions are widely used to study the properties of molecular clusters. For example, sulfur hexafluoride clusters [9], methane and ethane clusters [10], water droplets [11, 12], methanol droplets [13], water/ethanol droplets [14], and acetonitrile clusters [15] have been investigated through molecular dynamics simulations using such potentials. Water clusters with ions have also been studied [16]. More sophisticated models based on ab initio calculations are also used to represent dispersion and exchange repulsion interactions in force fields, as well as other energy terms like explicit (intermolecular) charge transfer [17, 18]. Stone and Tong [19] describe the dispersion terms between the two molecules A and B by: Edisp =–

10

Cn (l1 , l2 , j, κ1 , κ2 ) κ κ Sl11l2 j2 (ωA , ωB , ω)fn (aR) , Rn n=6 l1 l2 jκ1 κ2

(5)

Molecular Dynamics Simulations and Intermolecular Forces

127 κ κ

where Cn (l1 , l2 , j, κ1 , κ2 ) are dispersion coefficients [20], Sl11l2 j2 (ωA , ωB , ω) are the normalized real components of Stone’s orientational S-functions depending on the Euler angles characterizing the orientation of molecule A (ωA ), of molecule B (ωB ) and of the vector from the centre of mass of molecule A to that of molecule B (ω). fn (aR) is a damping function [21], reducing the dispersion interaction at short range, arising from the fact that the multipolar expansion is not strictly valid when molecules A and B overlap. Hodges and Stone [22] have also proposed a simplified model for dispersion energy in which SAPT calculations are used to fit the dispersion energy, through the G(R) function, to a simple form: C6 G(R) . (6) R6 Williams and Stone [23] have proposed a scheme to obtain anisotropic distributed polarizabilities and dispersion coefficients by analyzing the molecular response to the point charge polarizing the molecule. Stone has proposed to describe the exchange-repulsion energy between the two molecules A and B by site-site exponential terms involving site anisotropy:

Erep = K e–αab (Rab –ρab (ωab )) , (7) Edisp =–

a∈A b∈B

with K = 0.001 hartree, Rab the distance between both sites a and b and
κκ κκ ρab (ωab ) = ρlaal jb Slaal bj (ωa , ωb , ω) . la lb jκa κb

b

b

(8)

Stone and collaborators fit the exchange-repulsion as the difference between the first-order perturbation energy obtained from the IMPT theory [24, 25] for a dimer and the electrostatic energy, including induction, obtained from the distributed multipole and polarizability models that will be used to model electrostatic and polarization effects in the force field [26–29]. 1.2 Potentials Including Polarizability For an accurate description of electrostatic interactions, it is necessary to take into account the polarization of the molecules due to the intermolecular interactions. Molecular polarizabilities and hyperpolarizabilities are introduced in the molecular mechanics for clusters (MMC) approach [30]. Several empirical or quantum chemical approaches exist to describe the molecular polarizabilities by atomic or site components [31–47]. The simplest model uses localized dipolar polarizabilities. Such a model can be extended to quadrupolar or higher order polarizabilities. Stone has developed the concept of distributed polarizabilities [36, 42]. In this model, each site of a molecule responds to the

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potential, the field and the successive derivatives at its position and at the other positions in the molecule. Thus each site of a molecule can get an induced charge as well as an induced dipole, induced quadrupole and so on. The charge-flow concept is in essence non-local: each site responds directly to the potential at the other sites in the molecules to preserve the total molecular charge. The local multipolar polarizabilities, the charge-flow polarizabilities and the non-local dipolar and higher order polarizabilities are also introduced in the model, providing a very detailed picture of polarizability effects. Different groups in quantum chemistry use different strategies to fit intermolecular potentials to ab initio calculations. Karlström et al. with their NEMO approach use ab initio calculation on monomers and SCF calculations on dimers to build potentials [39, 48]. Stone and coworkers use accurate monomer properties (distributed multipoles, distributed polarizabilities, anisotropic dispersion coefficients, anisotropic repulsion terms, charge transfer terms) and an IMPT perturbation calculation on the dimer, to fit as accurately as possible the different interaction energy terms [26, 28, 29, 49–51]. Others use perturbation calculations using SAPT theory [52] to fit different parts of the interaction energy. The perturbation theory partitions the interaction energy into several terms, each one having a well-defined physical meaning, as well as a specific distance and orientational dependence. Millié, Dognon and collaborators [53, 54] use a treatment of intermolecular interactions first proposed by Vigné-Maeder and Claverie [4]. In this framework, the different contributions of the interaction energy are fitted from second-order exchange perturbation treatment. Electrostatic interactions are computed using distributed charges, dipoles and quadrupoles on atoms and bonds. The polarization of the molecules is modelled using dipolar polarizabilities located on atoms and bonds. In most of the applications, the permanent electric multipoles and polarizabilities are modelled through a distribution of low-rank multipole and polarizability components on a selection of molecular sites. One considers that all or some species can be polarized by permanent multipoles (charges, dipoles, etc.) present in the cluster, and by induced multipoles. In such a situation, the potential energy calculation must start by the calculation of the induced multipoles on the polarizable particles of the system. Let us consider a cluster of N rigid molecules, i, bearing a set of distributed multipoles Qi (charges, dipoles, quadrupoles for instance: n components) on a selection of sites, and let us suppose that the polarizability is described by a set of distributed polarizabilities on the same sites and up to the second rank on each site (charge-flow, dipolar and quadrupolar and mixed polarizabilities: a maximum of n2 components, the molecular symmetry reducing the number of non-zero components), giving an n × n matrix Ai . For a given configuration of the cluster, the induced multipoles on molecule i are given by:
∆Qi =– Ai T ij (Qj + ∆Qj ) , (9) j

Molecular Dynamics Simulations and Intermolecular Forces

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where T ij is the electrostatic tensor whose components give rise to the potential, the field and field derivatives (with a minus or plus sign) at a given site of molecule i when they are multiplied by the permanent and induced multipole components at the sites of molecule j that create them. The total electrostatic energy is obtained from: 1
1

(Qi + ∆Qi )+ T ij (Qj + ∆Qj ) + ∆Q+i A–1 (10) E= i ∆Qi . 2 i 2 i j

The second term is the creation energy of the induced multipoles. The + superscript indicates a matrix transpose. Vesely [55] has shown for a set of polarizable dipoles that if the induced dipoles are given by Eq. 9, it corresponds to a minimum of the total electrostatic energy, the partial derivative of E with respect to the induced multipole components being zero. From Eqs.9 and 10 the total electrostatic energy of the cluster can be rewritten: 1

+ 1

+ E= Qi T ij Qj + Qi T ij ∆Qj = Ees + Eind . (11) 2 i 2 i j

j

The first term, Ees , interactions between permanent multipoles, is often called the electrostatic energy and the second one, Eind , which results from interactions between permanent and induced multipoles, is the induction energy. From Eq. 9, it can be seen that the induced multipoles can be obtained by inverting an Nn × Nn matrix, or by an iterative process. In molecular dynamics simulations, the iterative process initiated with the induced multipole components of the previous time step is usually used rather than matrix inversion, especially when large matrices are involved. Then the forces on the centres of mass can be obtained by derivation with respect to the centre of mass position:
∂T ij F i =– (Qi + ∆Qi )+ (Qj + ∆Qj ) . (12) ∂Ri j

Sometimes, the induced multipoles of the cluster are computed from the potential, field and derivatives created by permanent multipoles only.
∆Qi =– Ai T ij Qj . (13) j

This corresponds to a first-order approximation of the induction energy. In that case, the forces are given by:
∂T ij
∂T ij F i =– Q+i (Qj + ∆Qj ) – ∆Q+i Qj . (14) ∂Ri ∂Ri j

j

Such a first-order approximation has been used for example by Grégoire et al. [53] in a study of NaI(CH3 CN)n clusters. Formulae to compute the

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torques are similar, involving derivatives of the electrostatic tensor with respect to angular variables, and Price et al. [56] have given a practical scheme to compute the necessary matrix elements of the derivative of the electrostatic tensor. Moreover, the formulae for first and second derivatives of anisotropic potentials with respect to geometric parameters have been given by Popelier and Stone [57]. Derepas et al. [54] have analyzed the effect of iterating the induced multipole moments instead of using first-order induction in the case of cations (Na+ , Cs+ , Ca+ , Ba+ , La3+ ) and recommended to take into account the back polarization of the molecules due the induced multipole moments. An intermediate approximation (second-order induction) has been considered [58]. Kaminski et al. [59] have proposed a polarizable force field in which permanent charges and dipoles located on atoms and some extra sites and atomic dipolar polarizabilities are used. In the force fields, the partitioning of the polarizability can be done using an additive scheme or a non-additive one like the approach of Applequist [31] which includes explicitly the intramolecular polarization in the force field. Another model incorporating explicitly the polarization effect in the force field is the fluctuating charge model based on the chemical potential equalization technique. This model became quite popular for molecular dynamics simulations of clusters and bulk phases. Following Mortier et al. [60] and Rappé and Goddard [61], Stuart and Berne [62] write the energy E of an assembly of atoms in a cluster of N molecules as: N

1 0 2 E({Qai }, {ri }) = (Ea (0) + χa0 Qai + Jaiai Qai ) 2 ai

(15)

i=1

+

1



Jaibj (raibj )Qai Qbj 2 i ai j

bj
=ai





+ (Edisp (raibj ) + Erep (raibj )) . i

j > i ai

bj

0 are the atom’s electronegativity and twice the hardness respecχa0 and Jaiai tively. Jaibi and Jaibj are identified to the Coulomb repulsion between electron charges. ai and bj are the atomic sites in molecule i and j respectively. Finally,

χai =



∂E 0 = χa0 + Jaiai Qai + Jaibj (raibj )Qbj . ∂Qai j

(16)

bj
=ai

The optimal charges correspond to electronegativity equalization for the pairs of charges a and b: ∂E ∂E = , ∂Qa ∂Qb

(17)

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131

with one constraint (i) that the system remains globally neutral (constraint 1) or (ii) that each molecule remains neutral or keeps a fixed ionic charge δi (constraint 2). In molecular dynamics, the atomic net charges Qai are considered degrees of freedom of the system to which a fictitious mass MQ is associated. The Lagrangian of the system is then: L1 =

N

1 i=1 a∈i

2

mai r˙2ai ++

N

1 i=1 a∈i

2

˙ 2ai – E({Qai }, {rai }) – λ MQ Q

N



Qai

i=1 a∈i

for constraint 1, and

L2 =

N

1 i=1 a∈i

2

mai r˙2ai ++

– E({Qai }, {rai }) –

N

1 i=1 a∈i

N
i=1

2

˙ 2ai MQ Q


λi ( Qai – δi ) a∈i

for constraint 2. The atoms have mass mai and velocity r ai . The second case would hold for example for an ionic cluster composed of an ion M+ (δM+ =+ e) and N – 1 neutral molecules X (δi = 0) and the model is assumed to allow charge flow between the atoms of the ion, between the atoms of the same molecule, but prevents any charge transfer between the ion M+ and the solvent molecules X. The nuclear and electronic degrees of freedom evolve in time according to the equations: ∂E , ∂r ai ¨ ai =– ∂E – λi =– χai – λi . MQ Q ∂Qai mai ¨rai =–

(18)

λi is the Lagrange multiplier of molecule i in the case of constraint 2 and λ in the case of constraint 1. Using the fact that the total charge (constraint 1) or the molecular charge (constraint 2) are constants of the motion, one finds the following expressions for the Lagrange multipliers: λ =–

N 1

χai , Na a∈i

(19)

i=1

and λi =–

1
χai , Nai a∈i

(20)

where Nai is the number of atoms in molecule i and Na is the total number of atoms of the system. Inserting the Lagrange multipliers in the equations of

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motion leads to N 1

¨ MQ Qai =– (χai – χbj ), Na

(21)

j=1 b∈j

(constraint 1), and
¨ ai =– 1 (χai – χbi ) , MQ Q Nai

(22)

b∈i

(constraint 2). Different implementations of the model use different χai 0 and J definitions relying on different Jaa aibj (raibj ) parameterizations. Rick et al. [63] use the prescription of Rappé and Goddard and define  1 2 Φ2 Jaibi (raibi ) = dr 1 dr 2 ΦSTO, (r 2 ) . (23) ai (r 1 ) |r aibi + r2 – r 1 | STO, bi Φ STO, ai (r k ) is an s Slater-type orbital for atom a in molecule i characterized by a principal quantum number ni and exponent ξai . r1 and r2 are the electron positions with respect to the positions of atoms ai and bi respectively. 0 , which only depends on ξ . The intermolecular J For r = 0, Jaiai (r) is Jaa ai aibj functions are simply defined as the Coulomb interaction 1/raibj . The fictitious mass MQ is chosen to be sufficiently small to guarantee that the charges readjust very rapidly to the changes of nuclear positions. This choice corresponds to the Born–Oppenheimer adiabatic decoupling between the electronic and the nuclear degrees of freedom. However, one chooses a mass that is not too small in order to keep the time step as large as possible. The charge degrees of freedom have to stay at a temperature close to 0 K (since they must be close to the values minimizing the electrostatic energy). Using molecular dynamics, one can ensure this situation by thermostatting the charge degrees of freedom at a few kelvins, using the Nosé thermostat. This kind of fluctuating charge model has been used to model liquid water [63]. Liu et al. [64], exploring water properties, have examined practical features of the parametrization of such fluctuating charge models from ab initio calculations. Using ab initio calculations of the water trimer, the fluctuation charge model parameters have been fitted to obtain the best agreement between ab initio calculations and the model potential. Banks et al. [65] have coupled this approach to the OPLS-AA non-electrostatic terms to build polarizable force fields for peptides. Calvo [66] has parameterized such a model for modelling magnesium oxide clusters and Ribeiro and Almeida [67] for alkali cyanide crystals. Stern et al. [68] and Chelli et al. [69] have extended this approach to include atomic dipolar polarizabilities. For many systems, the induction term is the dominant many-body term in the interaction energy. However, dispersion and exchange-dispersion have non-zero three-body contributions, which are sometimes added in the force fields explicitly [70–73].

Molecular Dynamics Simulations and Intermolecular Forces

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1.3 Tight-Binding Models In the context of molecular dynamics simulations, this class of methods deals with the classical dynamics of nuclei using forces calculated from an approximate knowledge of the electronic distribution obtained from a tightbinding approach, reminiscent of the extended Hückel approach proposed by Hoffmann [74]. The electronic wave function is approximated by a set of molecular orbitals, linear combinations of atomic orbitals, obtained as the solution of a Schrödinger equation involving an effective one-electron Hamiltonian. Several different tight-binding schemes have been proposed, based on the use of an orthogonal or non-orthogonal basis set, and recently tightbinding models have been fitted to density functional approaches (DFTB and SCC-DFTB). Such TB models have modified the extended Hückel theory by modifying the matrix element calculation and introducing short-range repulsion terms. These methods can find applications to model clusters in which covalent and ionic-covalent bonding occurs in both solid state and liquid phase modelling. Let us describe the non-orthogonal tight-binding scheme of Menon and Subbaswamy [75]. The potential energy of a cluster of atoms is written: E = Eelec + Erep occ


Ei + Φrep (rab ) . = i

(24)

a b>a

The repulsive potential Φrep (rab ) varies exponentially with distance rab between atoms a and b. Ei is the energy of a molecular orbital, built as a linear combination of Nf non-orthogonal atomic orbitals: Ψi =




cµi Φµ .

(25)

µ

The molecular orbitals are optimized by a variational method and they are found as solutions to the following linear system of equations:
(Hµν – Ei Sµν )cνi = 0 (i = 1, ..., Nf ; µ = 1, ..., Nf ) . (26) ν

H is an effective one-electron Hamiltonian that takes into account electron kinetic energy, nuclear attraction and electron repulsion.   Hµν = Φµ  H |Φν  , (27)   Sµν = Φµ |Φν . (28)

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The electronic part of the force on an atom is obtained from the expression:  + ∂H – E ∂S C C i i i ∂r a ∂Ei ∂ra = , (29) ∂ra C+i SCi where Ci is a column vector containing the coefficients cµi , and H and S are the Hamiltonian and overlap matrices. The superscript + again denotes a matrix transpose. Initially, this tight-binding scheme has been designed to model materials with tetrahedral local order. Porezag et al. [76] have proposed a TB scheme based on the DFT theory (DFTB). In this approach, the pseudoatomic orbitals Φν (r) are developed as a linear combination of Slater-type orbitals and optimized to be solutions of the self-consistent modified atomic Kohn–Sham equations: psat

(T + V psat )Φν (r) = Eν Φν (r) , with T being the kinetic energy operator and the potential

N r psat LDA . V (r) = Vnuclear (r) + VHartree (n(r)) + VXC [n(r)] + r0

(30)

(31)

VXC is written in terms of local-density approximation with the parametrization of Perdew–Zunger. The last term, (r/r0 )N , has been introduced by Eschrig et al. to improve tight-binding calculations; it forces the wave function to avoid areas that are far from the nucleus, and the electron density is thus compressed with respect to the free atom. Then, the solutions of the DFT calculations give the basis set of the Φν functions, which is the input in the tight-binding calculation; this set is limited to valence orbitals. The electron wavefunctions of the system are written
cµi Φµ (32) Ψi = µ

and obtained as a solution of the Schrödinger equation with an effective one-electron potential Veff (r). To compute the matrix elements, the effective potential is written as
V0a (|r – ra |) , (33) Veff (r) = a

V0a

where is the Kohn–Sham potential of a neutral pseudoatom located at ra due to its compressed electron density, but not containing the (r/r0 )N term anymore. The overlap matrix elements Sµν are easily calculated   two-centre free atom if µ = ν, and Φ a  T + V a + integrals. The H elements are equal to E µν µ µ 0   V0b Φνb for elements involving two different atoms a and b, and 0 otherwise. The matrix elements only depend on the distance between atoms and are calculated analytically and tabulated.

Molecular Dynamics Simulations and Intermolecular Forces

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The tight-binding scheme allows us to compute the electronic energy ETB elec (r a ) and the short-range repulsive energy Erep (r a ) resulting in the difference between the energy obtained with a self-consistent LDA calculation

TB (r ) for many geometries. E (r ) and E = ESC a a rep a b > a Vrep (rab ) is then LDA elec fitted to a sum of atom-atom polynomials: Vrep (rab ) =

np


dn (Rc – rab )n

(34)

2

for rab < Rc and 0 otherwise and np lower or equal to 5. The DFTB scheme is suitable when the electron density of the many-atom structure can be represented to a good approximation by a sum of atom-like densities. If the chemical bonding is controlled by a subtle charge balance between different atomic constituents, the results obtained with this scheme are more uncertain. Elstner et al. [77] have extended the DFTB approach and proposed the selfconsistent charge-tight-binding scheme (SCC-DFTB). Starting from the DFT expression of the energy of an assembly of atoms: E=

occ
i

1 Ψi |T + Vnucl + 2



n(r  )  dr |Ψi  + EXC [n(r)] + ENN , |r – r |

(35)

where |Ψi  are the Kohn–Sham molecular orbitals, n(r) is the electron density, T is the one-electron kinetic energy operator, Vnucl is the nuclear attraction operator, EXC is the exchange-correlation energy functional, and ENN is the electrostatic Coulomb repulsion between nuclei. They calculate the total energy by replacing n(r) by a superposition of a reference density n0 (r) and a fluctuation δn(r):  occ
n0 (r )  Ψi | T + Vnucl + E= dr + VXC [n0 (r)] |Ψi  (36) |r – r  | i  1 n0 (r  )(n0 + δn)(r)dr  dr – 2 |r – r |  – VXC [n0 (r)](n0 + δn)(r)dr + EXC [(n0 + δn)(r)] 1 + 2



δn(r  )(n0 + δn)(r)  dr dr + ENN |r – r |

Using the relation:  EXC [(n0 + δn)(r)] = EXC [n0 (r)] +

1 δEXC δn dr + δn 2



δ2 EXC δnδn dr dr δnδn

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one obtains the expression:  occ
1 n0 (r  )n0 (r)dr dr Ψi | H0 |Ψi  – E= 2 |r – r  | i  – VXC [n0 (r)]n0 (r)dr + EXC [n0 (r)]    1 δ2 EXC 1 + + δnδn dr  dr + ENN , 2 |r – r | δnδn where

(37)



n0 (r  )  dr + VXC [n0 (r)] . (38) |r – r  | In the standard DFTB formalism, the last but one term (second-order term) in Eq. 39 is neglected. It is kept in the self-consistent charge DFTB scheme, and δn(r) is decomposed into atom-atom centred contributions that decay rapidly with the distance from the atomic centre. The second-order term in Eq. 39 then becomes:    1 1

δ2 EXC + (39) δna (r)δnb (r  )dr  dr . 2 a |r – r  | δnδn H0 = T + Vnucl +

b

One then decomposes δna (r) in a series of radial and angular functions: 



r – ra a (40) Klm Flm (|r – ra |)Ylm δna (r) = |r – r a | m l

When only retaining the first term in this multipolar expansion, this term is approximated by: a δna (r)  ∆qa F00 (|r – ra |)Y00 ,

(41)

where ∆qa is the charge transfered to atom a. The higher order terms in Eq. 42 decay more rapidly with the distance. The second-order term is now equal to: 1

∆qa ∆qb γab , (42) 2 a b

with 1 γab = 4π

 

 1 δ2 EXC a b + (|r  – r b |)dr  dr . F (|r – ra |)F00 |r – r | δnδn 00

(43)

In the limit of large interatomic distances, the XC contribution vanishes for LDA and the second-order energy is a pure Coulomb interaction between charges ∆qa and ∆qb . If the charges are located on the same atom, γaa  Ia – Aa = 2ηa with Ia , Aa and ηa the ionization potential, the electron affinity and the hardness of atom a respectively. Elstner et al. have proposed

Molecular Dynamics Simulations and Intermolecular Forces

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an analytical expression for γab in the general case, with the assumption of an exponential decay of δna , and the tight-binding equation is reformulated at second-order as: occ
1

Ψ | |Ψ  H + ETB = ∆qa ∆qb γab + Erep . (44) i 0 i 2 2 a i b

The atomic charges depend on Ψi and a self-consistent procedure is necessary 0 to find the minimum of ETB 2 . The charges ∆qa = qa – qa are estimated with the Mulliken scheme and the coefficients of the atomic orbitals in the molecular orbitals are obtained by solving an homogeneous set of linear equations similar to the non-SCC approach but with modified Hµν matrix elements:
  1 (γac + γbc )∆qc , (45) Hµν = φµ  H0 |φν  + Sµν 2 c 0 1 =Hµν + Hµν

∀µ ∈ a ,

∀ν ∈ b .

The second-order correction due to the fluctuation of charges is represented 1 matrix elements. by the non-diagonal Mulliken charge dependent Hµν 1.4 Hartree–Fock ab initio and Semiempirical and DFT Techniques For a given nuclear configuration, the Born–Oppenheimer dynamics can be performed in principle in a more rigorous way than the tight-binding approach by using Hartree–Fock ab initio or DFT approaches. The multielectronic wave function Ψ (ri , ra ) is obtained variationally for a given configuration of nuclear positions r a by solving numerically the Schrödinger equation Ψ (r i , ra ) = EΨ (r i , ra ) H

(46)

within the Hartree–Fock or DFT framework, leading to molecular orbitals Ψi . The forces are then obtained by deriving the energy with respect to the nuclear positions. Molecular dynamics simulations of silicon clusters have been done at the AM1 level by Mazzone [78] and at the MSINDO level by Nair, Bredow and Jug [79]. Implementation at the DFT level is usually performed using the Car–Parrinello framework in which the monoelectronic wave functions are considered as dynamical variables in a Lagrangian formulation of the equations of motion [80, 81]. One starts from a Lagrangian:


 1
   L=µ Ψ˙ i |Ψ˙ i + ma r˙a 2 – E + Λij Ψi |Ψj – δij , (47) 2 a i i j>i

in which µ is a fictitious mass associated with the electronic degrees of freedom Ψi , ma is a nuclear mass, r˙a is a nuclear velocity, Λij is a Lagrange multiplier arising from the constraints maintaining orthonormality of occupied Kohn–Sham molecular orbitals Ψi , and E is the DFT energy of the

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system, depending on the nuclear positions, ra , and the one-electron orbitals Ψi .
 Za 1
n(r)dr Ψi |∆|Ψi  + E= – 2 i |r – ra | a  1 n(r )n(r)dr dr + + EXC [n(r)] + ENN , (48) 2 |r – r  | in which n(r) is the electron density, EXC is the exchange-correlation energy, and ENN is the nuclear Coulomb repulsion. The Kohn–Sham orbitals are linear combinations of basis functions Φµ , which are usually plane-waves but can also be atom-centred functions:
Ψi = cµi Φµ . (49) µ

The equations of motion for the nuclear and electronic degrees of freedom are: ∂E
µc¨ki = – + Λil ckl , ∂cki l

∂E . (50) ma r¨a = – ∂ra The constraint forces ensuring that the orthonormality constraints are satisfied are determined by using a standard molecular dynamics constraint algorithm like SHAKE. Such a technique is used for condensed phases and for clusters. At each time step, the algorithm does not need to iterate the wave function to convergence, and the trajectory does not strictly follow the Born–Oppenheimer electronic surface, but remains close to it. An alternative has been proposed by Schlegel et al. [82–84] in which the density matrix elements rather than Kohn–Sham molecular orbitals are considered dynamical variables within the Lagrangian formalism. Their approach is based on atom-centred Gaussian orbitals rather than using planewaves which are used in the original Car–Parrinello method. Atom-centred basis functions are well-suited to deal with molecular systems, especially when bond-breaking and forming are involved. This approach, based on atom-centred basis functions and density matrix propagation (ADMP), is based on a DFT code scaling linearly with the system size N. The electronic degrees of freedom are the elements of the electronic density matrix P. The Lagrangian of the system is written: 1 1 L = Tr [V + MV] + Tr [(M1/4 WM1/4 )2 ] – E(R, P) – Tr [Λ(PP – P)] , 2 2 where R, V, M are the nuclei positions, velocities and masses, W is the density matrix velocity, and M is the (diagonal) matrix of fictitious mass for the electronic degrees of freedom.

Molecular Dynamics Simulations and Intermolecular Forces

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The Lagrange multiplier matrix Λ is used to impose the constraint on the total number of electrons and on the idempotency of the density matrix. When non-orthogonal Gaussian atomic-based functions are used, an othogonalization is performed to express the density matrix and the equations of motions in the orthonormalized basis set. The energy is computed at the Hartree–Fock or the DFT level using the McWeeny purification of the density ( P = 3P2 – 2P3 ) and the equations of motion are: ∂E , ∂R   ∂E P¨ = – M–1/2 + ΛP + PΛ – Λ M–1/2 . ∂P

¨ =– MR

(51)

These equations can be integrated by the velocity Verlet algorithm and the constraints are satisfied using an iterative scheme. 1.5 Hybrid Quantum Mechanical/Molecular Mechanical Techniques The sophistication of these techniques lies somewhere between the methods of molecular mechanics describing the energy by an effective two-body or n-body potential and the full quantum chemical approach in which the classical molecular dynamics trajectory of the nuclei is followed with a knowledge of the total electron structure of the system [85–87]. The system is divided into a quantum mechanical part (QM) and a molecular mechanical part (MM), so that the Hamiltonian is written as: H = HQM + HMM + HQM/MM ,

(52)

and the total energy is the sum of (i) the quantum energy of the QM subsystem calculated at semiempirical [87, 88], SCC-DFTB [89–92], ab initio Hartree–Fock or DFT level [88, 93, 94], (ii) the energy of the MM subsystem computed using an effective two-body or n-body potential, and (iii) the coupling term between the QM part and the MM part. The last term contains an electrostatic term, the interactions between QM nuclei and MM charge distributions and QM electron-MM charge interactions (including the QM Hamiltonian to modify the electronic wave function), as well as van der Waals interactions, usually Lennard-Jones terms between QM and MM atoms. Hybrid methods using an ab initio or DFT calculation on a subsystem (QM) and a semiempirical calculation on the environment (QM’) have also been used [95–97]. Thompson and Schenter [98] have proposed a QM/MM method at INDO/S QM level including explicitly the polarizability of the MM subsystem through atomic point dipolar polarizabilities. Several groups have proposed QM/MM techniques in which the MM solvent polarization is explicitly taken into account through a fluctuating charge model [61, 63] and

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the quantum mechanical treatment of the QM part is performed at the semiempirical [99–101] or DFT/B3LYP level [102, 103]. Rega et al. [104] have proposed the QM/MM approach ADMP/ONIOM which implements the atom-centred density matrix propagation within the QM/MM ONIOM scheme of Maseras and Morokuma [105]. The ONIOM approach consists of separating the system in parts described at different levels of theory, for example a model part treated at Hartree–Fock or DFT level (QM), embedded in surroundings that will be treated at a molecular mechanics level (MM). The real system is composed of the model system and of the surrounding system, and the ONIOM energy is: model model E = Ereal MM – EMM + EQM .

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2 Application to Selected Clusters 2.1 Hydrated Halide Anions Perera and Berkowitz [106–108] have performed molecular dynamics simulations for (H2 O)n Cl– and (H2 O)n F– (n = 2, ..., 15 and 20 for the chloride anion) using polarizable models for the water molecule and the ions. The water model is a polarizable version of the SPCE model, in which the polarizability is described by a dipolar polarizability distributed on the oxygen and hydrogen atoms, and a dipolar polarizability is assigned to the ion. Their models also include a three-body exchange repulsion ion-water term. Simulations of 1 ns have been performed in the range 225 K to 275 K. The results show that F– is solvated in water clusters with n < 4 and that Cl– is attached to the surface. These authors have shown that the intermolecular potentials they used were able to reproduce quite well the enthalpies of formation of these small clusters, as well as the electrostatic stabilization of Cl– , Br– and I– [107]. Stuart and Berne [62] have compared the effective two-body TIP4P and a polarizable version based on a fluctuating charge model TIP4P-FQ to solvate either a non-polarizable or a polarizable chloride ion in clusters (H2 O)n Cl– for n up to 255. The polarizability model of the ion is based on a Drude oscillator representation. Their simulations have shown that the chloride ion is solvated at the surface of the cluster for the polarizable model, but the ion is solvated inside the cluster for n ≤ 18 for the non-polarizable model. The explanation for this behaviour is mainly the stronger dipole moment that can be created on water molecules with the polarizable model. Chloride anion polarizability has not been found to have an important effect on the structure of the clusters.

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Yoo et al. [109] have performed 1 ns molecular dynamics simulations of (H2 O)60 Cl– and (H2 O)60 Br– clusters with the non-polarizable TIP4P water model and the polarizable Dang–Chang potential, along with polarizable or non-polarizable anions. With the non-polarizable TIP4P and polarizable anions, Cl– is fully solvated in the cluster with 60 molecules, whereas Br– is partially solvated at the surface. Switching off the bromide anion polarizability leads to internal solvation. With the polarizable Dang–Chang model, both anions are located at the surface of 60 or 500 molecule clusters. Switching off the ion polarizability, the chloride ion is fully internally solvated, while the bromide ion remains partially solvated at the surface of the cluster. Xantheas and Dang [110] have performed molecular dynamics simulations of (H2 O)n F– clusters for n = 4–9 at 300 K. They found that there is a competition between the interior and surface states of the ion, but that the interior states dominate for the clusters that have six or more water molecules. The structure and energetics of the cluster (H2 O)4 F– have been studied by Bryce et al. [102] using an hybrid QM/MM technique including a polarizable water model. The fluoride ion (QM) is treated at DFT B3LYP level with a 6-311++G(3d,2p) basis set. The water is treated at a molecular mechanics level (MM) with a fluctuating charge version of the SPC model (SPC(FQ)). Molecular dynamics simulations at 300 K with a fluctuating charge extended Lagrangian approach for the water polarizability revealed a preponderance of trisolvated configurations of the clusters, whereas a simulation with fixed charges on the water molecules led to a preponderance of interior solvated structures. A subsequent Monte Carlo simulation [111] of the same system using a complete quantum mechanical energy function (MP2/6-311++G(3d,2p)) at 300 K showed the occurrence of essentially solvated states with only traces of trisolvated structures. QM/MM NEV around 96 K and NVT at 300 K molecular dynamics simulations of the (H2 O)4 F– cluster at the B3LYP/6-311G(3d,2p) level for F– and using the same SPC(FQ) water model but reparameterized QM/MM interaction and water-water potentials have been done by Bryce et al. [103]. The simulation time was 30 ps after equilibration for each simulation. By considering the ion location and the type of bonding, they take into account four significant structures: a surface location (s), an interior location (i), a trisolvated structure (t, with three fluoride-water H-bonds and the fourth water molecule doubly-H-bonded) and another trisolvated structure (t  , with three fluoride-water H-bonds and the fourth water molecule triply-H-bonded). A predominance of i state has been found in the NEV simulation, whereas the NVT simulation at 300 K displayed all structures with a majority of t states. Truong and Stefanovich [112] have performed hybrid QM/MM perturbative Monte Carlo simulations of (H2 O)n Cl– clusters (n = 1–7) in which the chloride ion (QM) is treated at Hartree–Fock/6-31G∗ level and the wa-

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ter molecules (MM) with the effective TIP3P potential. These simulations have also shown a preferential solvation of the ion at the surface of the cluster. Kim et al. [113] and Lee et al. [114] have performed extensive ab initio calculations of (H2 O)n X– (n = 1 – 6, X = F, Cl, Br, I). They found a variety of minima on the potential energy surface, along with significant zero-point energy and entropy effects on the relative stability of these structures. They confirm that the fluoride ion tends to favour internal or semi-internal states, while the other halide ions favour surface states, with significant hydrogenbonding between water molecules. Tobias et al. [115] have performed a 5 ps Car–Parrinello molecular dynamics simulation of the (H2 O)6 Cl– cluster at 250 K with the BLYP gradientcorrected exchange functional. They found that the chloride ion is localized at the surface of the cluster. They also performed classical MM molecular dynamics simulation (i) with the polarizable water model of Caldwell et al. [116] and a polarizable ion, and (ii) with the effective water model SPC/E and a polarizable ion. The Car–Parrinello result is in agreement with the simulation using the polarizable water model, and in disagreement with the simulation using the non-polarizable SPC/E model, which predicts an internal hydration of the ion. They found also that the chloride ion is significantly polarized (induced dipole ∼ 0.8 debye) and in another study [117] these authors have analyzed the effect of hydration on the chloride ion polarizability and found 3 3 A for the free ion, to 4.5 ˚ A for that the value decreases from about 5.5 ˚ 3 A for surface and interior state of (H2 O)6 Cl– , (H2 O)3 Cl– and to 4.3 and 3.6 ˚ respectively. In liquid water, the ion polarizability is found to be around 3 A. 4˚ Schlegel et al. [84] have performed a molecular dynamics simulation of 1.2 ps duration for the cluster (H2 O)25 Cl– using the ab initio ADMP molecular dynamics approach at the DFT PBE/3-21G∗ level and they compared the result with a Born–Oppenheimer molecular dynamics where the electron density is converged at each time step. They have computed the vibrational frequencies of the cluster and reported a red shift of roughly 200 cm–1 for the hydroxyl stretching motion due to the presence of the ion. Obviously, such a short simulation time does not lead to the configuration sampling allowing us to answer most of the questions concerning the structure, the thermodynamics and the dynamics of the cluster. Using the hybrid ADMP/ONIOM technique, Rega et al. [104] have published the result of a molecular dynamics simulation at the B3LYP/631+G(d, p) level and with AMBER/TIP3P water model of a chloride anion embedded in a cluster of 256 water molecules. The time step was 0.25 fs and they have performed a 3 ps simulation after thermalization, allowing them to report the atom-atom radial distribution functions.

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2.2 Silicon Clusters Early Car–Parrinello molecular dynamics simulations of small silicon clusters Sin (n ≤ 10) have been carried out to obtain zero temperature structures and structural properties at finite temperature [118]. Ab initio and DFT calculations of small silicon clusters have been performed to locate optimal geometries and binding energies [119–121] and polarizabilities [122]. Raghavachari and Rohlfing report binding energies at the MP4/6-31G∗ level with geometries optimized at the HF/6-31G∗ level. Zhu and Zeng optimized the geometries at the MP2/6-31G∗ level and report MP2/6-31G∗ and CCSD(T)/6-31G∗ binding energies for the more stable clusters (for 7 ≤ n ≤ 11). The symmetry of the clusters can differ from one set of calculations to the other, highlighting the difficulty of accurately modelling such systems with many possible structures in the same energy range. Menon and Subbaswamy have used the non-orthogonal tight-binding molecular dynamics approach to obtain the equilibrium geometries, energies and vibrational frequencies of small silicon clusters [123]. They have shown the importance of taking into account the non-orthogonality to obtain a reasonable description of the vibrational frequencies. The method has been further refined to improve the agreement with ab initio calculations [124–126]. Jayanthi et al. [127] have implemented a linear scaling algorithm for this non-orthogonal tight-binding approach and have determined the equilibrium configuration of a cluster composed of 1000 atoms by a quenching and annealing technique based on molecular dynamics. Frauenheim et al. [128] have adapted the DFTB method for silicon and Chaudhuri et al. [129] have optimized the geometry of small silicon clusters within the SCC-DFTB molecular dynamics scheme to find the global minima. Nair et al. [79] have implemented Born–Oppenheimer dynamics in the semiempirical SCF Hartree–Fock method MSINDO [130] and have used this technique to get the optimized geometries of Sin clusters (n = 5–7, 45, 60) by simulating annealing. They found global minima of Si5 (D3h ), Si6 (C2v ) and Si7 (D5h ) with binding energies in good agreement with DFT PWGGA predictions. Using molecular dynamics simulations of 32 ps (80000 time steps) between 500 and 2500 K of the cluster Si7 , they observed a solid-liquid transition in the range 1600 to 1800 K. Mazzone [78] used the semiempirical SCF Hartree–Fock method AM1 to determine the binding energy and ionization potential of clusters Sin with n = 25–225 for columnar, spherical and geminate (two adjointed) clusters. The columnar clusters are found to be the more stable.

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3 Conclusion A brief overview of energy calculations and molecular dynamics approaches currently used to model the dynamics of atomic or molecular clusters has been presented. The discussion has been limited to methods in which the nuclear dynamics are classical. Much effort is currently devoted to the field of quantum dynamics, but such an approach is still limited to systems composed of a few nuclei and a small number of electrons. Methods to compute the interaction energy of a cluster of atoms and molecules have been developed at different levels of sophistication. The development of quantum chemical methods and computers has permitted us to accurately model the interaction energy of assemblies of molecules. During the last decades, Anthony Stone has played a prominent role in the endeavour of accurately modelling molecular properties and interactions from quantum chemistry. The IMPT theory, models of distributed multipoles, distributed polarizabilities, distributed dispersion coefficients, anisotropic repulsion, developed by Stone and collaborators, allow us to build accurate interaction potentials from quantum chemistry. Simple two-body potentials have been designed empirically or using some basis of quantum chemistry. This approach is cheap and allows one to simulate the dynamics of clusters on a microsecond time scale. Potentials including n-body effects, polarizability effects and also three-body repulsion and dispersion, allow us, nowadays, to perform molecular dynamics simulation of clusters composed of 102 –104 molecules for hundreds or thousands of ps. The accuracy of the intermolecular and intramolecular potentials is the cornerstone of the success of this approach. Molecular dynamics methods based on quantum chemical calculations to compute the energy of a cluster as a function of the nuclei positions avoid the difficult task of building an intermolecular potential. Tight-binding molecular dynamics, semiempirical and Hartree–Fock ab initio Born–Oppenheimer dynamics, Car–Parrinello DFT molecular dynamics, and ADMP molecular dynamics are becoming more and more popular. The more sophisticated techniques are still limited to the study of clusters over short simulation times (ps time scale). The tight-binding approach SCC-DFTB is cheap and appears to be quite accurate. The coupling of quantum mechanical approaches and molecular mechanics with accurate force fields adjusted from quantum chemical calculations is a strategy that is increasingly used to study condensed phases and clusters. A detailed quantum description of a subsystem in which important changes in electronic distribution occur is performed taking into account the effect of the environment. Finding a compromise between accuracy and the simplicity of the description of the quantum subsystem and of the environment inter-

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action potential is a challenge to allow us to perform the long simulations of large clusters and condensed phases necessary to answer many questions of physical and chemical interest.

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Struc Bond (2005) 115: 149–193 DOI 10.1007/b135618 © Springer-Verlag Berlin Heidelberg 2005 Published online: 19 July 2005

Interactions with Aromatic Rings Seiji Tsuzuki Research Institute of Computational Sciences (RICS), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, 305-8568 Ibaraki, Japan [email protected] 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermolecular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supermolecule Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermolecular Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . Effects of Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Electron Correlation . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of the CCSD(T)-Level Interaction Energy at the Basis Set Limit DFT calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Each Energy Term . . . . . . . . . . . . . . . . . . . . . . . Distributed Multipole Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Size of Electrostatic and Dispersion Energies . . . . . . . . . . . . . . . . .

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π/π Interactions . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . Benzene Dimer . . . . . . . . . . . . . . . . Naphthalene Dimer . . . . . . . . . . . . . . Toluene Dimer . . . . . . . . . . . . . . . . . Other Aromatic Hydrocarbon Complexes . . Aromatic Molecules Including Heteroatoms .

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OH/π and NH/π Interactions . . . . . Introduction . . . . . . . . . . . . . . . Benzene–Water Complex . . . . . . . . OH/π Interactions in Other Complexes Benzene–Ammonia Complex . . . . . . NH/π Interactions in Other Complexes Interactions with Ammonium Ion . . .

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CH/π Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Origin of Attraction . . . . . . . . . . . . . . . . . . . . . Magnitude of CH/π Interactions . . . . . . . . . . . . . . . . . . . Role of Electrostatic Interaction . . . . . . . . . . . . . . . . . . . CH/π Interactions Between Benzene and Hydrocarbon Molecules CH/π Interactions in Other Complexes . . . . . . . . . . . . . . . Interaction with Tetramethylammonium . . . . . . . . . . . . . .

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6 6.1 6.2

Cation/π Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Origin of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . .

184 184 185

7 7.1 7.2 7.3

Interactions of Fluorinated Benzene . . Hexafluorobenzene–Benzene Complex Fluorinated Benzene–Water Complexes Anion/π Interactions . . . . . . . . . .

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Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract Intermolecular interactions of aromatic molecules (π/π, OH/π, NH/π and cation/π interactions) are important in many fields of chemistry and biology. These interactions control the crystal structures of aromatic molecules, the stability of biological systems and their molecular recognition processes. The magnitude of the interactions and their physical origin are essential for understanding the structures and properties of molecular assemblies and are also important for improving material and drug design strategy. Although it is not easy to study the details of the weak interactions of aromatic molecules by experimental measurements alone, ab initio calculation is becoming a powerful tool for studying weak intermolecular interactions. Recent developments of computational methodologies and increasing computer performance enable us to study these interactions quatitatively by high-level ab initio molecular orbital calculations. This review attempts to summarize recent progress in the quantitative analysis of intermolecular interactions of aromatic molecules. Keywords High-level ab initio calculations · Intermolecular interaction · Aromatic molecules · Basis set · Electron correlation Abbreviations AIM Atoms in molecules B3LYP Becke’s three-parameter functionals and Lee, Yang and Parr’s correlation functionals BLYP Becke’s exchange and Lee, Yang and Parr’s correlation functionals BSSE Basis set superposition error CADPAC Cambridge Analytical Derivatives Package CCSD(T) Coupled-cluster calculations with single and double substitutions with inclusion of noniterative triple excitations CCSD(TQ) Coupled-cluster calculations with single and double substitutions with inclusion of noniterative triple and quadruple excitations ∆CCSD(T)) CCSD(T) correction term DFT Density functional theory DMA Distributed multipole analysis ECCSD(T)(limit) CCSD(T)-level interaction energy at the basis set limit EMP2(T)(limit) MP2-level interaction energy at the basis set limit GGA Generalized gradient approximation HF Hartree–Fock IMPT Intermolecular perturbation theory MP2 Second-order Møller–Plesset perturbation

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MP2-R12 MP2 calculations with a linear r12 term MP4(SDTQ) Full fourth-order Møller–Plesset perturbation method with single, double, triple and quadruple substitutions PBE Perdew, Burke and Ernzerhof ’s exchange and Perdew and Wang’s gradientcorrected correlation functionals PW91 Perdew and Wang’s 1991 gradient-corrected correlation functionals SAPT Symmetry-adapted perturbation theory TMA Tetramethylammonium ZPE Zero-point vibrational energy

1 Introduction Intermolecular interactions of aromatic molecules are important in many areas in chemistry and biology. The interactions play important roles in determining the structures of molecular crystals and biological systems, and in molecular recognition processes of artificial and biological systems. Crystal structures of aromatic hydrocarbon molecules suggest that the π/π interaction is important in determining their crystal packing. Crystal structures of protein complexes with small molecules suggest that interactions (π/π, OH/π, NH/π and cation/π interactions) of aromatic amino side chains of proteins with small molecules are important. Therefore, quantification of the interactions of aromatic rings is extremely important for understanding the structures of molecular assemblies and for improving material and drug design strategy. Intermolecular interactions of aromatic molecules have been studied extensively by crystal structure analysis and spectroscopic measurements [1–4]. Although the experimental measurements provide a variety of useful information on the nature of the interactions, it is still difficult to reveal the details of the interactions by experimental measurements alone. Crystal structure database mining does not provide direct information on the magnitude of individual interactions. An accurate measurement of intermolecular interaction energy by spectroscopy is not an easy task. It is especially difficult to determine the size of the interaction energy and its orientation dependence. Ab initio molecular orbital calculation is becoming a powerful tool for studying intermolecular interactions. Ab initio calculation provides a sufficiently accurate interaction energy, if an appropriate level of theory is applied [5]. Recently reported systematic coupled-cluster calculations with single and double substitutions with inclusion of noniterative triple excitations [CCSD(T)] calculations of small molecular clusters show that the CCSD(T) calculations using reasonably large basis sets reproduce the experimental binding energies quite well [6]. The calculated interaction energies of hydrogen-bonded systems and aromatic molecules at the CCSD(T) level are close to the experimental values, as shown in Table 1 [7–10]. Calculations

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Table 1 Intermolecular interaction energy (Ee ) for some small clusters (kcal/mol) Cluster

MP2

CCSD(T)

Exp.

H2 O – H2 Oa MeOH – MeOHa HCOOH – HCOOHa HF – HFa HCN – HFa C6 H6 – H2 Ob C6 H6 – NH3 b C6 H6 – C6 H6 c C6 H5 Me – C6 H5 Med

– 4.9 – 5.6 – 13.8 – 4.4 – 7.4 – 3.4 – 2.6 – 4.5 – 6.8

– 4.8 – 5.5 – 13.9 – 4.4 – 7.1 – 3.0 – 2.2 – 2.5 – 4.1

– 5.0 – 4.6 to – 5.9 – 13.2 – 4.6 ± 0.3 – 6.9 – 3.4 ± 0.1 – 2.0, – 2.4 ± 0.1 – 2.8 ± 0.4, – 2.0 ± 0.2 – 3.6 ± 0.2

a Reference

[7] [8] c Reference [9] d Reference [10] b Reference

also provide detailed information on the physical origin of the interactions. Accurate evaluation of the intermolecular interaction energies of aromatic molecules by ab initio calculations was not practical 10 years ago, as huge computational resources were required. For example, reliable CCSD(T)-level calculations of the benzene dimer were first reported in 1996 [11]. Recently, however, a number of high-level ab initio calculations of the interactions of aromatic molecules were reported, as summarized in recent reviews [12, 13]. Ab initio calculations are rapidly increasing our knowledge of the interactions. This review attempts to present recent progress in quantitative analysis of intermolecular interactions of aromatic molecules by computational methods. The review begins with an explanation of the computational methods (Sect. 2). The effects of basis set and electron correlation on the calculated intermolecular interaction energies are explained briefly. Problems with density functional theory (DFT) calculations are also discussed. The level of theory (basis set and electron correlation correction procedure) is important for quantitative analysis of weak intermolecular interactions, such as π/π, OH/π, NH/π and CH/π interactions, as the calculated interaction energy depends strongly on the choice of the approximation. Section 3 highlights the calculation of the π/π interactions. Difficulties in the accurate calculation of the π/π interaction energy are briefly explained. The size of the dimer interaction energies of small aromatic molecules (benzene, naphthalene, toluene, thiophene, etc.), the physical origin of the attraction and the directionality of the interactions are presented. Hydrogen bonds including aromatic rings (OH/π and NH/π interactions) are discussed in Sect. 4. In contrast to conventional hydrogen bonds, such as in the water dimer, where

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electrostatic interaction is mainly responsible for the attraction, the dispersion interaction is the major source of the attraction for OH/π and NH/π interactions. Although electrostatic interaction is weaker than dispersion interaction, highly orientation dependent electrostatic interaction mainly determines the directionality of these interactions. The CH/π interaction is discussed in Sect. 5. The CH/π interaction is sometimes described as a weak hydrogen bond, probably owing to its structural similarity to conventional hydrogen bonds. However, the physical origin and directionality of the CH/π interaction are considerably different from those of conventional hydrogen bonds. The cation/π interaction is a strong attraction between a cation and a π system. Electrostatic and induction interactions are mainly responsible for the attraction; therefore, even Hartree–Fock (HF) and DFT calculations provide sufficiently accurate interaction energies for the cation/π interactions. A number of calculations of the cation/π interaction have been reported. A comprehensive review of the cation/π interaction was published in 1997 [14]. Section 6 shows the recent progress of theoretical calculations for the cation/π interaction, especially investigations into the physical origin of the attraction. The interactions of fluorinated aromatic molecules are discussed in Sect. 7. The interactions of fluorinated aromatic molecules are considerably different from those of the unsubstituted counterparts.

2 Calculations 2.1 Intermolecular Forces There exist several intermolecular forces between an aromatic molecule and an interacting molecule [15]. Computational methods for their evaluation will be briefly explained in this section. Dispersion, electrostatic and exchangerepulsion interactions are the major intermolecular forces when the interacting molecules are both neutral. The dispersion contribution has paramount importance for the attraction in the π/π, OH/π, NH/π and CH/π interactions [8–10, 16]; therefore, accurate calculation of the dispersion energy is essential for the quantitative evaluation of these interactions. On the other hand, electrostatic and induction (induced polarization) interactions are the major source of the attraction in the cation/π interaction [17]. The contribution of the dispersion interaction is relatively small in the cation/π interactions. Intermolecular forces can be separated into two main types [15]. One is the long-range interaction, such as electrostatic, induction and dispersion terms, where the energy behaves as some inverse power of R (E ∼ R–n ; R is

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the intermolecular distance). The long-range contributions have their origin in Coulombic interaction between interacting molecules. Short-range interactions include exchange-repulsion and charge-transfer terms. Short-range interactions arise at distances where the molecular wave functions overlap significantly. The energies of short-range interactions decrease exponentially with distance (E ∼ e–αR ) [15]. 2.2 Supermolecule Method The supermolecule method is widely used for calculations of the intermolecular interaction energy. The total interaction energy (Etotal ) is calculated as the difference between the energy of the dimer [E(AB)] and the sum of the energies of monomers [E(A) and E(B)] as shown in Eq. 1: Etotal = E(AB) – [E(A) + E(B)] .

(1)

The calculated interaction energy for the supermolecule method includes basis set superposition error (BSSE) [18]. The BSSE is corrected by the counterpoise method [19]. The energies of both dimer and monomers are calculated using the dimer’s basis set in the counterpoise correction. The correction of BSSE is essential for accurate evaluation of weak intermolecular interactions, as the BSSE correction significantly changes the size of the calculated interaction energy. The effects of BSSE on the calculated interaction energy of the benzene dimer are illustrated in Fig. 1.

Fig. 1 Effects of basis set superposition error (BSSE) correction on the benzene dimer interaction energy

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2.3 Intermolecular Perturbation Theory Intermolecular perturbation theory (IMPT) is another method for calculating the intermolecular interaction energy [15, 20]. The IMPT method provides detailed information on the intermolecular interaction from direct calculation of each energy term (electrostatic, dispersion, etc.) in that interaction. The total interaction energy is the sum of the calculated energy terms. A molecular orbital program including an IMPT routine, such as the Cambridge Analytical Derivatives Package (CADPAC) [21], is required. An accurate evaluation of the total interaction energy by the IMPT method is not easy, as it requires the calculation of higher-order energy terms from the IMPT. 2.4 Effects of Basis Set Ab initio molecular orbital theory is a first principles method, which does not use any empirical parameters. However, ab initio molecular orbital calculations are an approximation. The accuracy of the calculated interaction energy depends strongly on the level of theory (basis set, electron correlation and BSSE correction). Each interaction energy term requires a different level of approximation for an accurate evaluation. Evaluation of the dispersion energy is the most computationally demanding, as it requires a very large basis set and an electron correlation correction. The dispersion interaction has its origin in electron correlation and molecular polarization. Small basis sets underestimate molecular polarizability and thereby underestimate the dispersion energy. Medium-sized basis sets such as 6-31G∗ , 6-311G∗∗ and cc-pVDZ are not large enough for the accurate evaluation of the dispersion energy [22, 23]. A very large basis set near saturation is necessary. On the other hand, electrostatic, induction and exchange-repulsion energies can be evaluated with moderate accuracy using medium-sized basis sets. An accurate evaluation of the π/π, OH/π, NH/π and CH/π interactions requires computationally demanding high-level ab initio calculations, as the dispersion interaction is the major source of the attraction in these interactions [8–10, 16]. On the other hand, a sufficiently accurate interaction energy for the cation/π interaction can be obtained using a medium-sized basis set, as electrostatic and induction interactions are mainly responsible for the attraction in the cation/π interaction [17]. Figure 2 shows the basis set dependence of the calculated HF and secondorder Møller–Plesset perturbation method (MP2) level interaction energies of the benzene dimer (π/π interaction) [9]. The weak basis set dependence of the calculated HF-level interaction energy (mainly exchange-repulsion and electrostatic interactions) shows that the basis set dependence of the electrostatic and exchange-repulsion energies is very weak, while the MP2-level in-

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Fig. 2 Basis set dependence of the Hartree–Fock (HF) and second-order Møller–Plesset perturbation (MP2) benzene dimer interaction energies

teraction energy depends strongly on the basis set. The medium-sized 6-31G∗ and 6-311G∗∗ basis sets underestimate the attraction considerably compared with the large cc-pVQZ basis set, as these medium-sized basis sets underestimate the dispersion interaction. Similar basis set dependence was reported in the calculated interaction energies of aliphatic hydrocarbon molecules and OH/π, NH/π and CH/π interactions [8, 16, 22, 23]. The calculated molecular polarizability of benzene with the 6-31G∗ basis set (αxx = αyy = 69.6 au and αzz = 20.9 au) is considerably smaller than the experimental one (83.1 and 42.9 au), as shown in Table 2 [22]. The underestimation of the polarizability is the cause of the underestimation of the dispersion energy with these medium-sized basis sets. Very large basis sets near saturation (cc-pVQZ, etc.) are necessary for accurate evaluation of the dispersion energy. Augmentation of diffuse functions (especially augmentation of diffuse polarization) is effective for improving calculations of the dispersion energy [23]. Basis set effects are not large in the calculation of the cation/π interaction. Table 3 shows the MP2-level interaction energies of the Li+ –benzene and Na+ –benzene complexes using several basis sets [17]. The MP2 calculations with medium-sized basis sets (6-31G∗ and 6-311G∗∗ ) provide sufficiently accurate interaction energies for the cation/π complexes. The calculated interaction energies with these basis sets are not significantly different from the experimental values [24]. The major source of the attraction in the cation/π interactions involves electrostatic and induction interactions [17, 25]. The

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Table 2 Polarizability of benzene Basis set

bf a

αxx , αyy

αzz

6-31 G∗ 6-311 G∗∗ 6-311 ++ G∗∗ 6-311 G(2d,2p) 6-311 G(3d,3p) cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ

102 144 174 192 240 114 264 510 192 414 756

69.9 73.0 76.4 74.8 77.2 72.6 76.0 77.9 79.8 80.2 80.3

20.9 28.5 42.3 31.8 40.1 24.6 34.4 40.2 45.7 45.7 45.7

83.1

42.9

Exp.

Polarizability in atomic units. See Ref. [22] a Number of basis functions used for calculations of benzene monomer Table 3 Interaction energy for Li+ –benzene and Na+ –benzene complexes Basis set

bf a

HF

MP2

Li+ 6-31 G∗ 6-311 G∗∗ 6-311 ++ G∗∗ 6-311 G(2d,2p) 6-311 G(3d,3p)

117 162 196 215 268

– 39.5 – 39.4 – 38.8 – 41.1 – 40.7

– 37.4 – 35.6 – 34.8 – 36.9 – 36.4

Na+ 6-31 G∗ 6-311 G∗∗ 6-311 ++ G∗∗ 6-311 G(2d,2p) 6-311 G(3d,3p)

121 170 204 223 276

– 26.2 – 24.0 – 23.5 – 25.4 – 25.3

– 24.8 – 21.6 – 21.0 – 22.8 – 22.7

Exp.

– 38.5b

– 22.1b

Energy in kilocalories per mole. See Ref. [17] a Number of basis functions used in the calculation b Reference [24]

contribution of the dispersion interaction is relatively small. The basis set dependence of electrostatic and induction energies is weaker than for the dispersion energy. The basis set dependence of the calculated charge distributions is not large, if basis sets including polarization functions are used.

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Therefore the basis set dependence of the electrostatic energy is not large, if these basis sets are used. However, small basis sets without polarization functions (3-21G and 6-31G) overestimate the electrostatic energy considerably [26]. In addition, these small basis sets underestimate the induction energy, as they underestimate the molecular polarizability, as shown in Table 2. 2.5 Effects of Electron Correlation Recently reported systematic calculations for small molecular clusters show that CCSD(T) calculations using very large basis sets provide accurate intermolecular interaction energies [6, 7]. The CCSD(T)-level electron correlation correction is highly computationally demanding. The CPU time for a CCSD(T) calculation is proportional to the seventh power of the number of basis functions. In early calculations, the MP2 method was used for the evaluation of the interactions of aromatic molecules. The CPU time for an MP2 calculation is proportional to the fifth power of the number of basis functions. However, recently reported CCSD(T) calculations of the benzene dimer show that the MP2 method overestimates the attraction considerably compared with the more reliable CCSD(T) calculations [9, 11]. A comparison of HF, MP2 and CCSD(T) calculations for the benzene dimer is shown in Fig. 3. The MP2 calculation substantially overestimates the attraction com-

Fig. 3 Effect of electron correlation on the benzene dimer interaction energy

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pared with the CCSD(T) calculation, which shows that electron correlation beyond MP2 is important for an accurate evaluation of the benzene dimer interaction. Although the effects beyond MP2 are smaller for interactions of benzene with nonaromatic molecules (OH/π, NH/π and CH/π interactions), the effects of CCSD(T)-level correction are not negligible even in these interactions [8, 16]. The large difference between the HF and CCSD(T) calculations shows that the dispersion interaction is significant for the attraction in the benzene dimer [9, 11]. The effects of electron correlation are not large for the cation/π interactions, as shown in Table 3 [17]. Electrostatic and induction interactions are the major source of the attraction in such systems. The contribution of the dispersion interaction is relatively small. The HF interaction energies obtained from using medium-sized basis sets are sufficiently accurate. HF calculations slightly overestimate the electrostatic energy [26]; therefore, the calculated HF interaction energy is slightly greater than the MP2 result. 2.6 Estimation of the CCSD(T)-Level Interaction Energy at the Basis Set Limit The accurate evaluation of the π/π interaction energy requires a highly computationally demanding CCSD(T)-level calculation using a large basis set near saturation. But such a calculation is still not practical even for the benzene dimer at present. Recently an appropriate method for estimating the CCSD(T)-level interaction energy of aromatic molecules at the basis set limit [ECCSD(T)(limit) ] was reported [9]. The ECCSD(T)(limit) was estimated according to Eq. 2: ECCSD(T)(limit) = EMP2(limit) + ∆CCSD(T) ,

(2)

where EMP2(limit) denotes the MP2-level interaction energy at the basis set limit and ∆CCSD(T) denotes the CCSD(T) correction term (the difference between the calculated CCSD(T)-level and MP2-level interaction energies). EMP2(limit) was obtained by Feller’s method [27]. In Feller’s method EMP2(limit) is obtained by an extrapolation from the calculated MP2 interaction energies using Dunning’s correlation-consistent basis sets (cc-pVXZ, X is D, T, Q, etc.) [28, 29]. ∆CCSD(T) is calculated using a medium-sized basis set, as the basis set dependence of ∆CCSD(T) is not large [9]. The CCSD(T) interaction energies of the benzene, toluene, naphthalene and thiophene dimers have been estimated with this method [9, 10, 30, 31]. 2.7 DFT calculations DFT calculations are often used for the evaluation of intermolecular interactions for aromatic molecules. However, DFT methods cannot accurately

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evaluate the dispersion energy. While DFT calculations with local exchange– correlation functionals (local density approximation) lead to overestimated binding energies in weakly bound systems, it was reported that nonlocal exchange–correlation functionals (generalized gradient approximation, GGA) very often underestimate the attraction [32–34]. Commonly used GGA functionals (Becke’s exchange and Lee, Yang and Parr’s correlation functionals, BLYP, and Becke’s three-parameter functionals and Lee, Yang and Parr’s correlation functionals, B3LYP) cannot reproduce the dispersion interaction [7]. The intermolecular interaction energy potential calculated with these BLYP functionals is close to that obtained by the HF method, as shown in Fig. 4. Some GGA calculations (Perdew and Wang’s 1991 gradient-corrected correlation functionals, PW91, and Perdew, Burke and Ernzerhof ’s exchange and Perdew and Wang’s gradient-corrected correlation functionals, PBE) give attractive potentials for rare-gas and hydrocarbon dimers. However, the size of the attraction is not accurate. The PW91 calculations considerably underestimate the attraction in the benzene dimer, as shown in Fig. 4. The attraction calculated by the PW91 method is probably not due to dispersion, since the basis set dependence is negligible, as shown in Fig. 5 [7]. The calculated attraction should depend significantly on the size of the basis set, as for the MP2 calculations shown in Fig. 2, if the physical origin of the attraction calculated by the PW91 method is dispersion. The negligible basis set dependence of the attraction calculated by the PW91 method shows that the calculated attraction is not dispersion energy.

Fig. 4 Benzene dimer interaction energy calculated by density functional theory

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Fig. 5 Basis set dependence of the benzene dimer interaction energy calculated by the PW91 method

DFT calculations are not suitable for evaluating the intermolecular interactions of aromatic molecules, as dispersion is the major source of the attraction in the interactions of aromatic molecules, with the exception of cation/π interactions. DFT calculations using basis sets with polarization functions provide sufficiently accurate intermolecular interaction energies for the cation/π interactions, as DFT calculations can reproduce electrostatic and induction energies sufficiently accurately. 2.8 Calculation of Each Energy Term The magnitude of each energy term clearly shows the physical origin of the interaction. The contribution of each energy term can be calculated by the IMPT method [15, 20] and by the energy decomposition method [35]. A reasonably large basis set must be used for accurate evaluation of each term, as the size of each contribution depends on the basis set, as in the case of the total interaction energy calculated by the supermolecule method. Medium-sized basis sets underestimate dispersion and induction energies. In addition, the size of charge-transfer energy depends strongly on the basis set. Medium-sized basis sets considerably overestimate the charge-transfer energy [36].

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2.9 Distributed Multipole Analysis Distributed multipole analysis (DMA) provides very accurate electrostatic and induction energies more easily than the IMPT method [15, 37]. The electrostatic energy was obtained from the interactions between multipoles on atoms of the interacting molecules (distributed multipoles). The distributed multipoles were obtained from the calculated density matrix of the monomer by ab initio methods. The induction energy was calculated from the electrostatic field produced by the distributed multipoles of monomers and atomic polarizabilities [38]. The accuracy of the calculated electrostatic energy was increased systematically by including higher-order multipoles for the calculation. The electrostatic interaction is the Coulombic interaction between the static charges of interacting monomers. The electrostatic energy is often estimated from the interaction between the atomic charges of monomers obtained by population analysis or by electrostatic potential fitting. However, the point-charge model is a rather crude approximation, as the charge of a molecule distributes around the molecule; therefore, the electrostatic energy obtained by the point-charge model sometimes involves a large error [15]. Accurate evaluation of the electrostatic energy is essential for understanding the role of the electrostatic interaction in the directionality of the intermolecular interaction. Simple force field calculations including only repulsion and electrostatic terms reproduce the structures of small molecular clusters quite well, if the electrostatic energy is accurately evaluated using DMA [39, 40]. On the other hand, force field calculations using a pointcharge model often fail to reproduce the structures of clusters. The good performance of the simple force field using DMA indicates the paramount importance of electrostatic interactions for the directionality of the intermolecular interaction. 2.10 Size of Electrostatic and Dispersion Energies Electrostatic and dispersion interactions are important for the attraction and directionality of the intermolecular interactions of aromatic molecules [8–10, 16]. Quantitative evaluation of electrostatic and dispersion energies is essential for understanding the intermolecular interactions of aromatic molecules. An accurate evaluation of electrostatic energy is not difficult, as DMA provides an accurate value (Ees ) [15]. On the other hand, an accurate evaluation of the dispersion energy is very difficult. An IMPT calculation using a large basis set is necessary. The contribution of the dispersion interaction can be estimated approximately from the size of the effect of electron correlation on the calculated

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total intermolecular interaction energy by the supermolecule method (Ecorr ) as shown in Eq. 3, since the dispersion interaction is the major contribution to Ecorr : Ecorr = ECCSD(T)(limit) – EHF ,

(3)

where EHF denotes the HF interaction energy. Ecorr also includes some other terms, such as the effects of intramolecular electron correlation. The intramolecular electron correlation changes molecular charge distributions and thereby changes Ees . The change of the electrostatic energy [Ees(corr) ] is included in the calculated Ecorr . Ees(corr) is negligible when the electrostatic energy is sufficiently small, as the size of Ees(corr) is usually 10–20% of the Ees value [26]. Ees is considerably smaller than Ecorr for π/π, OH/π, NH/π and CH/π interactions [8–10, 16]; therefore, the size of the dispersion energy for these interactions can be estimated approximately from the size of the Ecorr value. On the other hand, the electrostatic interaction is very strong in the cation/π interaction [17]. The size of the dispersion energy in the cation/π interaction cannot be estimated from Ecorr , as Ees(corr) is not negligible.

3 π/π Interactions 3.1 Introduction The interaction between π systems (π/π interaction) is important in many fields of chemistry, from molecular biology to material design. The π/π interaction influences the three-dimensional structures of biological systems, such as proteins and DNA, and is important for molecular recognition processes and crystal packing of organic molecules containing aromatic rings, including nonlinear optical materials. Information on the π/π interaction is essential for understanding the structures and properties of these systems and for simulating them. Recent progress of computational studies for π/π interactions in fundamental systems is reviewed in this section. 3.2 Benzene Dimer The intermolecular interaction of the benzene dimer has been studied extensively, both by experimental and by theoretical methods, as a prototype for the π/π interaction. From recently reported high-level ab initio calculations several conclusions were derived: (1) the benzene dimer has two nearly isoenergetic stable structures (T-shaped and slipped-parallel shown in Fig. 6), (2) the sandwich structure is unstable, (3) dispersion is the major source of

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Fig. 6 Structures of the benzene dimer

the attraction, (4) the electrostatic interaction stabilizes the T-shaped structure, while dispersion increases the relative stability of the slipped-parallel structure, and (5) the potential energy surface is very shallow, and the barrier height between the two stable structures is very small. The structure of the benzene dimer was a controversial issue. Early experimental measurements suggested the existence of a T-shaped structure, as the benzene dimer has a dipole moment [41, 42]. The T-shaped orientation is also observed in the crystal. However, the measurement of a dipole moment does not exclude the existence of other structures, as suggested by microwave and mass-selected hole-burning experiments [43, 44]. Ab initio calculations for the benzene dimer have been reported since the early 1980s [22, 45–49]. The MP2 method was mainly used for the electron correlation correction in these early calculations. The results show the importance of dispersion for the attraction: electron correlation drastically increases the attraction. MP2 calculations in 1994 using a medium-sized basis set showed that the slipped-parallel benzene dimer has larger (more negative) interaction energy (– 2.28 kcal/mol) than the T-shaped dimer (– 2.11 kcal/mol) [47]. However, comparison of the MP2 results with full fourth-order Møller–Plesset perturbation method with single, double, triple and quadruple substitutions [MP4(SDTQ)] calculations suggested that the MP2 approach overestimates the attraction [48]. More recent CCSD(T) calculations show that the T-shaped and slipped-parallel dimers are nearly isoenergetic (– 2.17 and – 2.01 kcal/mol, respectively) and that the sandwich dimer is unstable [11]. The MP2 calculations considerably (30–90%) overestimate the attraction compared with the CCSD(T) calculations. Similar overestimation of the attraction by the MP2 method was also reported for other aromatic molecules (toluene, naphthalene, tiophene, pyrrol, pyrimidine, triazine, aminotriazine, aminopyrimidine and 1-aminopyrimidine) [10, 30, 31, 50, 51]. These results show that electron correlation corrections beyond MP2 are essential for studying π/π interactions. Very recently reported coupled-cluster calculations with single and double substitutions and noniterative triple and quadruple excitations [CCSD(TQ)] calculations show that the effects of quadruple excitations are not negligible in the interaction of aromatic molecules [52]. Unfortunately CCSD(TQ) results for the benzene dimer were not reported. The size of the quadruple excitation

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correction for the furan dimer (0.2 kcal/mol) suggests that inclusion of quadruple excitations may slightly decrease the attraction in the benzene dimer. Recently reported CCSD(T) interaction energies of the T-shaped, slippedparallel and sandwich dimers at the basis set limit are – 2.46, – 2.48 and – 1.48 kcal/mol, respectively [9]. The experimental interaction energy of the benzene dimer (E0 ) is 2.4 ± 0.4 [53] and 1.6 ± 0.2 kcal/mol [54]. The calculated change of the zero-point vibrational energy (ZPE) by the formation of the complex (∆ZPE) is 0.4 kcal/mol. The E0 and ∆ZPE values imply that Ee lies between – 2.0 and – 2.8 kcal/mol, close to the calculated value. The CCSD(T)-level interaction energies of the three structures were also estimated using a similar scheme, but using the MP2 calculations with a linear r12 term (MP2-R12) interaction energies instead of the EMP2(limit) obtained by Feller’s method [55]. The calculated interaction energies of the T-shaped and slipped-parallel dimers are nearly isoenergetic. The interaction energies were also estimated from local MP2-based calculations [56]. Recently the interconversion path between T-shaped and slipped-parallel local minima was calculated [57]. The calculated CCSD(T)-level potential energy surface is very shallow, and the calculated barrier height between the two structures is very low (less than 0.2 kcal/mol), which suggests that large-amplitude oscillation occurs in the benzene dimer. Benzene has a significant quadrupole moment, and therefore it was pointed out that the electrostatic interaction between quadrupoles plays an important role in determining the stable benzene dimer structure [58]. The interacting quadrupoles prefer the T-shaped and slipped-parallel orientations. These orientations agree well with the two stable structures of the benzene dimer obtained by ab initio calculations. In the sandwich structure the quadrupole–quadrupole interaction is repulsive, which explains the instability of this structure. The importance of electrostatic interactions was also invoked to explain the geometries resulting from favorable π/π interactions in other systems [59]. Although electrostatics plays an important role in determining the geometry of the favorable π/π interaction, recent high-level ab initio calculations show that dispersion is important for attraction in the benzene dimer. The absolute value of Ecorr is always considerabaly larger than the Ees value, as shown in Table 4, which reveals that dispersion is the major source of attraction in the benzene dimer. Very recently reported symmetry-adapted perturbation theory (SAPT) calculations show that dispersion is the major source of attraction in the benzene dimer [60]. Similar results were also reported for other complexes of aromatic molecules (benzene–phenol, benzene–toluene, benzene–fluorobenzene and benzene–benzonitrile) [60]. Electrostatics stabilizes the T-shaped dimer, but destabilizes the sandwich and slipped-parallel dimers, as shown in Table 4. The electrostatic interaction in the slipped-parallel dimer becomes attractive when the molecules have

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Table 4 Intermolecular interaction energies of aromatic molecules Cluster C6 H6 – C6 H6 f Slipped-parallel T-shaped Sandwich C10 H8 – C10 H8 g Slipped-parallel Cross T-shaped Sandwich C6 H5 Me – C6 H5 Meh Cross Antiparallel Parallel T-shaped C4 H4 S – C4 H4 Si Perpendicular Parallel

Etotal a

Ees b

Eind c

Erep d

Ecorr e

– 2.48 – 2.46 – 1.48

0.90 – 0.55 1.24

– 0.25 – 0.17 – 0.21

3.01 1.74 3.23

– 6.14 – 3.48 – 5.74

– 5.73 – 5.28 – 4.34 – 3.78

1.38 1.67 – 0.70 1.60

– 0.40 – 0.39 – 0.20 – 0.35

5.29 5.27 3.20 5.24

– 12.00 – 11.83 – 6.64 – 10.26

– 4.08 – 3.77 – 3.41 – 2.62

0.82 0.86 1.05 – 0.61

– 0.32 – 0.33 – 0.31 – 0.14

3.93 3.98 3.75 1.81

– 8.50 – 8.27 – 7.89 – 3.68

– 3.12 – 1.71

– 1.14 1.46

– 0.29 – 0.30

3.55 5.02

– 5.24 – 7.89

Energy in kilocalories per mole. Geometries are shown in Figs. 6, 7, 8 and 9. a CCSD(T) interaction energy in the basis set limit b Electrostatic energy c Induction energy d Repulsion energy. E rep = EHF – Ees – Eind . EHF is the Hartree–Fock (HF) interaction energy. e Correlation interaction energy. E corr = Etotal – EHF . Ecorr is mainly dispersion energy. f Reference [9] g Reference [30] h Reference [10] i Reference [31]

a larger horizontal displacement [9]. Dispersion enhances the relative stability of the slipped-parallel dimers. The Ecorr value for the sandwich dimer is smaller (less negative) than that of the slipped-parallel dimer, which indicates that the smaller dispersion energy of the sandwich dimer is also a cause of A) its instability. The larger vertical separation in the sandwich dimer (3.8 ˚ A) is the cause of the compared with that in the slipped-parallel dimer (3.5 ˚ smaller dispersion energy. The calculated intermolecular interaction energy potentials of the benzene dimer show that substantial attraction still exists, even when the two molecules are well separated, as shown in Fig. 2 [9]. This result indicates that short-range interactions, such as charge-transfer interactions, are not the major source of the attraction in the benzene dimer, as the short-range interactions decrease exponentially with separation.

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3.3 Naphthalene Dimer The MP2 method was mainly used for ab initio calculations of the naphthalene dimer. Very recently CCSD(T) interaction energies of a few naphthalene dimers were estimated. The conclusions derived from the calculations are as follows: (1) the naphthalene dimer has two nearly isoenergetic stable structures (slipped-parallel and cross structures shown in Fig. 7), (2) the sandwich and T-shaped structures are unstable, (3) dispersion is the major source of the attraction, (4) electrostatics stabilizes the T-shaped structure, while dispersion increases the relative stability of the slipped-parallel and cross structures, and (5) the large dispersion energy is the cause of the stability of the slipped-parallel and cross structures. CCSD(T) calculations of the naphthalene dimer using a large basis set are extremely difficult; therefore, the MP2 method was used even in recently reported calculations [61–63]. The MP2 interaction energy depends significantly on the basis set (– 3.74 to – 10.69 kcal/mol); nevertheless, all the MP2 calculations concluded that the slipped-parallel dimer is stabler than the Tshaped dimer. Local MP2 calculations also show the same preference [64]. Very recently reported CCSD(T) interaction energies of the naphthalene dimers at the basis set limit show that the slipped-parallel and cross dimers are the stablest (– 5.28 and – 5.73 kcal/mol, respectively) and the T-shaped and sandwich dimers are substantially less stable (– 4.34 and – 3.78 kcal/mol, respectively) [30]. Table 4 shows that electrostatic interactions stabilize the T-shaped dimer, while dispersion increases the relative stability of the stacked dimers (slippedparallel, cross and sandwich dimers shown in Fig. 7) as in the case of the

Fig. 7 Structures of the naphthalene dimer

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benzene dimer [30]. Dispersion is always considerably larger than the electrostatic interaction. The T-shaped and slipped-parallel benzene dimers are nearly isoenergetic, while the slipped-parallel naphthalene dimer is substantially stabler than the T-shaped dimer. The dispersion interaction in the naphthalene dimer is substantially larger than in the benzene dimer. The larger dispersion interaction is the cause of the greater stability of the stacked naphthalene dimers. The Ecorr value of the sandwich dimer is smaller (less negative) than that of the slipped-parallel dimer, which indicates that the smaller dispersion energy is the cause of its instability. 3.4 Toluene Dimer Although the benzene dimer has been extensively studied as a prototype for the π/π interaction, its relevance to general properties of aromatic residues in proteins was questioned. The toluene dimer was instead proposed as a better prototype for the π/π interaction in proteins [65]. MP2 calculations show that the stacked dimers are substantially stabler than the T-shaped one [65, 66] and that the antiparallel dimer is 0.9 kcal/mol stabler than the parallel dimer [66]. Recently reported CCSD(T) calculations of the toluene dimer show that the cross dimer (– 4.08 kcal/mol) is slightly stabler than the antiparallel and parallel dimers (– 3.77 and – 3.41 kcal/mol, respectively), as shown in Table 4 [10]. These stacked dimers are substantially stabler than the T-shape dimer (– 2.62 kcal/mol). Spectroscopic measurements show that the toluene dimer consists of at least two isomers [67, 68]. The interaction energy of the toluene dimer (Ee ) obtained from a CCSD(T) calculation (– 4.1 kcal/mol) is not very different from the experimental Ee value (– 3.6 kcal/mol) [69]. Dispersion is again the major source of the attraction in the toluene dimer [10]. The dispersion energies of the stacked toluene dimers are substantially larger than the dispersion energy of the slipped-parallel benzene dimer. The larger dispersion interaction in toluene is the cause of the preference for the stacked orientation, as in the case of the naph-

Fig. 8 Structures of the toluene dimer

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thalene dimer. The electrostatic energies of the three stacked dimers are not very different, which shows that the stabilization of the antiparallel dimer by the dipole–dipole interaction is not large. The electrostatic interaction is repulsive in the three dimers owing to the quadrupole– quadrupole interaction between the two rings. The Ecorr value of the cross dimer is slightly larger (more negative) than the values of the parallel and antiparallel dimers, which indicates that the size of the dispersion interaction mainly determines the relative stability of the three stacked dimers. 3.5 Other Aromatic Hydrocarbon Complexes MP2 calculations of the benzene–naphthalene and naphthalene–anthracene complexes show that the slipped-parallel orientation is stabler than the T-shaped one [62, 70]. An interesting conclusion was reported from MP2 calculations on a set of aromatic complexes, namely that the energies of the T-shaped and slipped-parallel structures are quite comparable when the molecules are small, but the slipped-parallel structure becomes stabler than the T-shaped one as the molecules become larger [62]. The same conclusion was derived from the CCSD(T) interaction energies for the benzene and naphthalene dimers [9, 30]. 3.6 Aromatic Molecules Including Heteroatoms CCSD(T) interaction energies of the thiophene dimer show that the T-shaped dimer is stabler than the parallel dimer (Fig. 9), as shown in Table 4 [31]. Although dispersion is the major source of the attraction in the thiophene dimer, electrostatics plays an important role for stabilizing the T-shaped

Fig. 9 Structures of the thiophene dimer

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Fig. 10 Crystal structure of oligothiophene

dimer. The electrostatic interaction in the thiophene dimer is larger than that in the benzene dimer. The larger electrostatic interaction increases the relative stability of the T-shaped dimer. The calculated stable T-shaped dimer orientation is close to the orientation of neighboring thiophene rings in the crystal of oligothiophenes (α-quaterthiophene), as shown in Fig. 10, which suggests that the orientation dependence of the intermolecular interaction between the thiophene rings plays an important role for the crystal packing. The calculations of the pyrrole and indole dimers also show that these dimers prefer a T-shaped orientation [71, 72].

4 OH/π and NH/π Interactions 4.1 Introduction A hydrogen bond is the attraction between a proton donor (a proton attached to an electronegative atom: O – H, N – H, etc.) and a proton acceptor (another electronegative atom or an electronegative group). Aromatic hydrocarbons have proton-accepting ability and can play a role as proton acceptors. Spectroscopic measurements of the benzene–water and benzene–ammonia complexes show that the water and ammonia molecules are positioned above the benzene plane and that the benzene acts as a proton acceptor. The attractions of O – H and N – H bonds with a π system are denoted as OH/π and NH/π interactions [73, 74]. The NH/π interaction is observed in a variety of biological systems [75–77]. It is believed that the NH/π interaction plays an important role in stabilizing protein structures and for selective binding in molecular recognition processes of proteins.

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4.2 Benzene–Water Complex A few experimental measurements of the interaction energy of the benzene– water complex (E0 ) have been reported [78–80]. The most recently reported experimental E0 value is – 2.44 ± 0.09 kcal/mol. The calculated ∆ZPE value is 1.0 kcal/mol [81]. The E0 and ∆ZPE values lead to an Ee value of – 3.4 kcal/mol. The OH/π interaction is substantially weaker than the hydrogen bond of the water dimer (– 5 kcal/mol). From recently reported high-level ab initio calculations of the benzene– water complex a set of conclusions were derived: (1) the water molecule prefers to sit above the center of the benzene ring, (2) the monodentate structure shown in Fig. 11 is slightly stabler than the bidentate structure, (3) the structure where the lone pair points to the benzene ring is unstable, (4) dispersion is the major source of the attraction, (5) although the electrostatic interaction is smaller than the dispersion interaction, electrostatics mainly determines the orientation dependence of the interaction energy (short-range interactions such as charge-transfer interactions are not the major source of the directionality of the interaction), and (6) the binding energy of the benzene–water complex is substantially smaller than that of the water dimer. Ab initio calculations of the intermolecular interaction of the benzene– water complex have been reported since the early 1980s [45, 73, 82–84]. The relative stability of the monodentate and bidentate structures of the benzene– water complex was a controversial issue. A few MP2-level calculations were reported in the late 1990s. Although the calculated interaction energies scatter considerably depending on the basis set (– 1.8 to – 3.2 kcal/mol), all the MP2 calculations show that the monodentate structure is slightly stabler than the bidentate structure [85–87]. The MP2 interaction energy at the basis set limit (– 3.9 ± 0.2 kcal/mol) was estimated from the calculated MP2 interaction energies using Dunning’s correlation-consistent basis sets up to cc-pV5Z [81]. Recently reported CCSD(T)-level interaction energies of a few orientations of the benzene–water complex show that the MP2 calculations slightly

Fig. 11 Structures of the benzene–water complex

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Table 5 Intermolecular interaction energies of the benzene complex with water, ammnonia and methane Cluster

Etotal a

Ees b

Erep c

Ecorr d

C6 H6 – H2 Oe C6 H6 – NH3 e C6 H6 – CH4 f

– 3.02 – 2.22 – 1.45

– 1.86 – 1.01 – 0.25

1.07 1.14 1.10

– 2.23 – 2.36 – 2.30

Energy in kilocalories per mole. Geometries are shown in Figs. 11, 12 and 17. a CCSD(T) interaction energy in the basis set limit b Electrostatic energy c Repulsion energy. E rep = EHF – Ees . EHF is the calculated HF-level interaction energy. d Correlation interaction energy. E corr = Etotal – EHF . Ecorr is mainly dispersion energy. e Reference [8] f Reference [16]

overestimate the attraction compared with the more reliable CCSD(T) results [8]. The estimated CCSD(T) interaction energy at the basis set limit is – 3.2 kcal/mol, which is close to the experimental Ee value (– 3.4 kcal/mol). In contrast to conventional hydrogen bonds, dispersion is the major source of the attraction in the benzene–water complex, as shown in Table 5. SAPT calculations of the benzene–water and ethylene–water complexes also show the importance of dispersion for the attraction in the OH/π interaction [88]. Although the electrostatic interaction is weaker than the dispersion contribution, electrostatics is mainly responsible for the orientation dependence of the total interaction energy. 4.3 OH/π Interactions in Other Complexes The interaction energies of water complexes with benzene, phenol, indole and imidazole were calculated as models of the interactions of water with amino acids (Phe, Tyr, Trp and His) [89]. Although conventional hydrogen bonds are stronger, OH/π hydrogen bonds can also be formed. The strengths of the OH/π hydrogen bonds follow the trend Trp > His > Tyr ∼ Phe. B3LYP/631++ G∗∗ calculations for N-methylpyrrole complexes with hexafluoroisopropanol, trifluoroethanol, 2-chloroethanol and butanol show that the pyrrole ring acts as a proton acceptor in all cases [90]. The optimized geometries show that the O – H bonds point toward the aromatic rings. A large attraction was calculated for the complexes with the fluorinated alcohols (– 6.05 and – 4.10 kcal/mol, respectively). The electronegative fluorine atoms increase the positive charge on the hydrogen atom of the O – H bond and thereby increase the attractive electrostatic interaction. The larger electrostatic interaction is mainly responsible for the large attraction in the fluorinated alcohol com-

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plexes. In the indole–water complex, on the other hand, the indole N – H group acts as a proton donor and the water oxygen as a proton acceptor [91, 92]. MP2/6-31G∗∗ -level calculations for the pyridine–water complex also show that the hydrogen-bonded structure with the O – H bond directed toward the pyridine lone pair is stabler than the bifurcated structure [93]. The global minimum of the acetylene–water complex has a C2v structure, where water acts as a proton acceptor [94]. The calculated interaction energy of this complex is – 2.87 kcal/mol and a water donor arrangement lies 0.3 kcal/mol above the global minimum. The stablest acetylene–formic acid complex has a OH/π-bonded cyclic structure with a binding energy of – 4.9 kcal/mol [95]. 4.4 Benzene–Ammonia Complex Spectroscopic measurements for the benzene–ammonia complex show that the ammonia molecule is positioned above the benzene plane and that the benzene acts as a proton acceptor [74]. The estimated interaction energy (E0 ) from the experimental centrifugal distorsion constant is – 1.4 kcal/mol [74]. Very recently an E0 value of – 1.84 kcal/mol was reported [96]. The calculated ∆ZPE is 0.6 kcal/mol, which implies that Ee lies between – 2.0 and – 2.4 kcal/mol [8]. From recently reported high-level ab initio calculations of the benzene– ammonia complex a set of conclusions were derived: (1) the ammonia prefers to sit above the center of the benzene ring, (2) the monodentate structure shown in Fig. 12 is stabler than the bidentate and tridentate structures, (3) the structure where the lone pair points to the benzene is unstable, (4) dispersion is the major source of the attraction, (5) although the electrostatic interaction is smaller than the dispersion interaction, electrostatics mainly determines the orientation dependence of the interaction energy (short-range interactions such as charge-transfer interactions are not the major source of the directionality of the interaction), and (6) the binding energy of the benzene– ammonia complex is smaller than that of the benzene–water dimer (OH/π interaction), but larger than that of the benzene–methane dimer (CH/π interaction).

Fig. 12 Structures of the benzene–ammonia complex

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A few ab initio calculations for the benzene–ammonia complex have been reported since the late 1980s [82–84]. MP2/6-31G∗∗ -level results for the benzene–ammonia complex showed that the monodentate structure is slightly stabler than the bidentate structure [74]. Recently reported CCSD(T) interaction energies of a few orientations of the benzene–ammonia complex at the basis set limit show that the monodentate structure is stabler than the bidentate and tridentate structures [8]. The calculated interaction energy is – 2.22 kcal/mol, which agrees well with the experimental Ee value. Dispersion is the major contributor for the attraction, as shown in Table 5, while the weak electrostatic contribution mainly determines the directionality of the interaction. The calculated total interaction energy (EMP2 ) has a large A), as orientation dependence when molecules are in close contact (R = 3.6 ˚ shown in Fig. 13. The same orientation dependence is observed when the molecules are well separated (R = 4.6 ˚ A), which shows that long-range interactions mainly determine the directionality of the NH/π interaction. The orientation dependence of Ees is close to that of EMP2 , which shows that electrostatic interaction is mainly responsible for the directionality [8]. Similar

Fig. 13 Orientation dependence of the interaction energy for the benzene–ammonia complex. EMP2 is the total interaction energy. Ees is the electrostatic energy

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Fig. 14 Orientation dependence of the interaction energy for the benzene–water complex. EMP2 is the total interaction energy. Ees is the electrostatic energy

Fig. 15 MP2/cc-pVTZ intermolecular interaction energies of the benzene–ammonia, benzene–water and benzene–methane complexes

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orientation dependence is observed in the calculated interaction energy of the benzene–water complex, as shown in Fig. 14, which shows that electrostatic interaction also controls the directionality of the OH/π interaction in this complex [8]. CCSD(T) calculations for benzene complexes with water, ammonia and methane show that the strength of the interaction follows the trend OH/π > NH/π > CH/π (Table 5). The intermolecular potentials of the comA, plexes have their minima at intermolecular distances of 3.4, 3.6 and 3.8 ˚ respectively, as shown in Fig. 15. The size of the attraction contributes to the different equilibrium intermolecular distances [8]. SAPT calculations for the XH/π interaction in both the ethylene and benzene complexes with CH4 , NH3 , H2 O and HF also show that the strength of the XH/π interaction is enhanced as one progresses from CH4 to HF, and that this enhancement cannot be simply explained by the increase in electrostatic interaction or the electronegativity of the atom bound to the XH/π-bonded proton [97]. These calculations indicate that the contribution of dispersion is vital to obtain an accurate interaction energy. 4.5 NH/π Interactions in Other Complexes NH/π hydrogen-bonded structures were calculated for benzene complexes with amides, aniline and indole. Although data mining of X-ray protein structures suggested that the face-to-face orientation is the preferred configuration of the benzene–formamide complex, MP2 calculations showed that T-shaped NH/π hydrogen-bonded structures are stabler [98, 99]. The calculated binding energy is up to 4.0 kcal/mol over a wide range of conformational space. MP2 and B3LYP calculations for the benzene-N-methylformamide complex were also reported [100]. The calculated interaction energy of the complex at the MP2/6-31G∗∗ level is – 3.47 kcal/mol. The vibrational spectra of the jet-cooled benzene–aniline complex suggests a face-to-face structure with an NH2 /π hydrogen bond. An MP2 calculation also supported this structure [101]. The binding energy of the benzene–indole complex was calculated by the CCSD(T) method [102]. Although the MP2 calculations showed a preferential stability for the stacked structure, the CCSD(T) results favor the NH/π-bonded geometry. The calculated binding energy (5.3 kcal/mol) agrees well with the experimental value (5.2 kcal/mol). The NH/π interactions in other complexes were also reported. The MP2 calculations of aniline complexes with ethylene, propene and 1-butene suggest an NH/π interaction between the N – H bond of aniline and the double bond of the alkene for the propene and 1-butene complexes, with a CH/σ type interaction between the CH bond and the lone pair of nitrogen of aniline for the ethylene complex [103]. The MP2/6-311G∗∗ -level optimized geometry of the pyrrole dimer shows that the dimer prefers the tilted T-shaped orientation, in which one monomer points toward the ring plane of the

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other monomer and the angle formed by the planes of the two monomers is 63◦ [71]. The calculated interaction energy was – 5.11 kcal/mol. The indole dimer also prefers the T-shaped orientation [72]. MP2/6-31G∗∗ -level calculations of the interaction in the pyridine–ammonia complex have also been reported [93]. 4.6 Interactions with Ammonium Ion The interaction of an ammonium ion with a π system is sometimes described as a strong NH/π interaction. However, the physical origin of the attraction between the ammonium ion and a π system is completely different from that in the NH/π interaction between neutral molecules. B3LYP/6-311G∗∗ level calculations for the NH4 + complexes with benzene, phenol, pyrrole, imidazole, pyridine, indole, furane and thiophene were reported [104]. The optimized geometries could be divided into three types: NH4 + /π complexes, protonated-heterocyclic-NH3 hydrogen-bonded complexes, and heterocyclicNH4 + hydrogen-bonded complexes. Imidazole and pyridine form very stable hydrogen-bonded complexes with NH4 + . The calculated interaction energies of the complexes are – 45.9 and – 42.5 kcal/mol, respectively. Benzene, phenol, pyrrole, indole, furan and thiophene form NH4 + /π complexes. The calculated interaction energies of the NH4 + /π complexes are – 17.5, – 18.6, – 22.1, – 22.5, – 15.4 and – 16.7 kcal/mol, respectively. The interactions in these NH4 + /π complexes are significantly larger than the NH/π interaction between neutral molecules. The magnitude of the interactions in the NH4 + /π complexes is close to that in the K+ –benzene complex (– 17.0 kcal/mol) [17]. The electrostatic and induction interactions are the main contributions to the strong attraction in the NH4 + /π complexes, which is the same as the interaction in the K+ –benzene complex. The interaction of the ammonium ion with the π system is essentially a cation/π interaction (see Sect. 6).

5 CH/π Interactions 5.1 Introduction The attraction between a C-H bond and a π system is denoted as a CH/π interaction. The CH/π interaction was first proposed in the 1970s to explain the preference of conformations in which bulky alkyl and phenyl groups had close contact [105, 106]. Experimental measurements support the existence of this attraction. Close contacts were observed in stable conformations of

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many organic molecules [107, 108]. Statistical analysis of the crystal structure database showed that short contacts between a C – H bond and a π system are observed in crystals of organic molecules and proteins [109, 110]. It is sometimes claimed that the CH/π interaction is a crucial driving force for crystal packing and molecular recognition in both biological and artificial systems. Despite broad interests in the CH/π interaction in many areas of chemistry and biology, very little was known about the physical origin and magnitude of the CH/π interaction. It was difficult to study the size of the interaction energy of the CH/π interaction by experimental methods, as it is significantly smaller than for conventional hydrogen-bonded systems, such as the water dimer. Early ab initio calculations could not quantitatively determine the interaction energy, as evaluation of dispersion energy, which is the major source of the attraction in the CH/π interaction, was computationally too demanding. Recently reported high-level ab initio calculations provided a set of conclusions on the CH/π interaction: (1) the C – H bond prefers to point toward the π system, (2) dispersion (van der Waals attraction) is the major source of the attraction in the CH/π interaction, which is completely different from a conventional hydrogen bond, where electrostatic interaction is mainly responsible for the attraction, and (3) although the electrostatic interaction is substantially weaker than the dispersion interaction, electrostatics stabilizes the orientation where the C – H bond points toward the π system. 5.2 Physical Origin of Attraction Early energy decomposition analysis of the interaction energy of the ethylene–methane complex showed that electrostatic and charge-transfer terms were the main contributors to the attraction [111]; therefore, charge transfer was believed to be an important source of attraction in the CH/π interaction [106]. However, the basis sets used in these calculations were too small to evaluate the weak attractive interaction energy quantitatively. It was reported later that small basis sets significantly overestimate the chargetransfer energy [36]. Recently reported high-level ab initio calculations of the benzene–methane complex show that the contribution of charge transfer is actually negligible [16]. Electron correlation considerably increases the attraction, which shows that dispersion is the major source of the attraction in the complex [16]. The calculated intermolecular interaction potential shown in Fig. 16 indicates that substantial attraction still exists even when the molecules are well separated, which shows that short-range interactions such as charge transfer are not the major source of the attraction. The short-range interactions arise at distances where the molecular wave functions overlap significantly. The magnitude of the short-range interactions decreases exponentially with distance.

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Fig. 16 MP2/cc-pVTZ interaction energies of benzene complexes with some halogenated methanes

The dispersion interaction has its origin in molecular polarization [15]. The carbon atom of the C – H bond and the carbon atoms of the π system are mainly responsible for the dispersion interaction, as the atomic polarizability of a carbon atom is considerably larger than that of a hydrogen atom. This means that a large part of the attraction in the CH/π interaction has its origin not in the attraction between the hydrogen atom of the C – H bond and the π system, but in the attraction between the carbon atom of the C – H

Fig. 17 Structures of benzene complexes with hydrocarbon molecules

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Table 6 Intermolecular interaction energies of the complexes formed between benzene and some hydrocarbon and halogenated hydrocarbon molecules Cluster

Etotal a

Ees b

Erep c

Ecorr d

Re

C6 H6 – CH4 (A)f C6 H6 – CH4 (B)f C6 H6 – C2 H6 f C6 H6 – C2 H4 f C6 H6 – C2 H2 f C6 H6 – CH3 Clg C6 H6 – CH2 Cl2 g C6 H6 – CHCl3 g C6 H6 – CHF3 g

– 1.45 – 1.23 – 1.82 – 2.06 – 2.83 – 2.99 – 4.54 – 5.64 – 4.18

– 0.25 0.25 – 0.17 – 0.65 – 2.01 – 1.06 – 1.81 – 2.42 – 2.43

1.10 1.16 1.97 1.82 1.44 1.44 2.41 4.63 1.67

– 2.30 – 2.64 – 3.62 – 3.22 – 2.26 – 3.37 – 5.13 – 7.85 – 3.42

3.8 3.6 3.6 3.6 3.6 3.6 3.4 3.2 3.4

Energy in kilocalories per mole. Geometries of the complexes are shown in Fig. 17. a Calculated CCSD(T) level interaction energy at the basis set limit b Electrostatic energy c Repulsion energy. E rep = EHF – Ees . EHF is the HF interaction energy. d Correlation energy. E corr = Etotal – EHF . Ecorr is mainly dispersion energy. e Distance between centroid of benzene ring and carbon atom of C – H bond f Reference [16] g Reference [112]

bond and carbon atoms of the π system. The interaction energies of the monodentate and tridentate benzene–methane complexes shown in Fig. 17 are very close (– 1.45 and – 1.23 kcal/mol), respectively [16], as shown in Table 6, which indicates that the increased number of CH/π contacts in the tridentate complex does not enhance the attraction. This result is quite reasonable, as the dispersion interaction between carbon atoms is mainly responsible for the attraction. 5.3 Magnitude of CH/π Interactions The CH/π interaction is very weak in most cases. The interaction energies of benzene complexes with methane, ethane and ethylene are around – 2 kcal/mol, as summarized in Table 6. The attraction is enhanced when electronegative substituents (chlorine and fluorine atoms) are attached to the carbon atom of the C – H bond [112]. The interaction energy of the benzene– chloroform complex (– 5.6 kcal/mol) is considerably greater than that of the benzene–methane complex (– 1.5 kcal/mol). The enhancement of the attraction was explained by the increased electrostatic interaction. The substituent increases the positive charge on the hydrogen atom of the C – H bond and thereby increases the attractive electrostatic interaction. The electrostatic energy in the benzene–chloroform complex (– 2.4 kcal/mol) is 2.2 kcal/mol

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more negative than that in the benzene–methane complex (– 0.2 kcal/mol), but Table 5 shows that dispersion is mainly responsible for the larger attraction in the benzene–chloroform complex. Ecorr of the benzene–chloroform complex (– 7.9 kcal/mol) is 5.6 kcal/mol more negative than that of the benzene–methane complex (– 2.3 kcal/mol) [112]. The chlorine atoms, which have large polarizabilities, are the cause of the large dispersion contribution in the benzene–chloroform complex. The dispersion interaction between the chlorine atoms and the π system (this term is not a CH/π interaction) is the main source of the greater binding energy of the benzene–chloroform complex, which shows that the enhancement of the CH/π interaction (attraction between the C – H bond and the π system) by chlorination is not so large. It is sometimes claimed that the sum of the contributions of the CH/π interactions is large when a number of close CH/π contacts are observed. However, the size of the total interaction energy is not determined by the number of CH/π contacts, as the major source of the attraction in the CH/π interaction is the interaction between carbon atoms. The interaction energy of the tridentate complex is nearly the same as that of the monodentate complex. 5.4 Role of Electrostatic Interaction The sizes of the electrostatic interactions in benzene complexes with methane, ethane and ethylene (interactions with nonsubstituted sp3 and sp2 C – H bonds) are very small (absolute values are less than 1 kcal/mol), as shown in Table 6. The electrostatic interaction in the benzene–chloromethane complex (interaction with a monosubstituted sp3 C – H bond) is also around – 1 kcal/mol. Dispersion is the major source of attraction in these complexes. The only exceptions are the benzene–acetylene complex (interactions with an sp C – H bond) and dichloromethane and chloroform complexes (interactions with a dihalogenated or a trihalogenated sp3 C – H bond), where the electrostatic interaction is more negative than – 1 kcal/mol [16, 112]. Although dispersion is still the major source of the attraction in these complexes, the contribution of the electrostatic interaction is not negligible. The directionality of the CH/π interaction is considerably smaller than that of conventional hydrogen bonds such as in the water dimer. The highly orientation dependent electrostatic interaction is the major source of attraction in conventional hydrogen bonds, which exhibit significant orientation dependence. On the other hand, the contribution of electrostatics is small for the CH/π interaction. The small electrostatic component is the cause of the weak orientation dependence of the CH/π interaction. In supramolecular chemistry, it has been believed that the hydrogen bond is able to control and direct the structures of molecular assemblies because the hydrogen bond is sufficiently strong and directional [113]. It was some-

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times claimed that the CH/π interaction was also important in determining structures of molecular assemblies, as in the case of conventional hydrogen bonds. Probably the structural similarity between the CH/π interaction and the hydrogen bond is the reason for this claim. Despite the structural similarity, the physical origin of the attraction in the CH/π interaction is completely different from that of conventional hydrogen bonds. The magnitude of most CH/π interactions is very weak and their directionality is also very weak compared with that of conventional hydrogen bonds, as the contribution of electrostatic interaction is small in the CH/π interaction. The only exceptions are for interactions with an sp C – H bond and dihalogenated or trihalogenated sp3 C – H bonds. This suggests that most CH/π interactions, aside from these exceptions, do not play a crucial role in determining the structures of molecular assemblies. 5.5 CH/π Interactions Between Benzene and Hydrocarbon Molecules Ab initio calculations for the benzene–methane complex have been reported as the simplest model of a CH/π interaction [82, 114]. Early calculations employed the MP2 method for electron correlation. The large contribution of electron correlation suggests that dispersion is important [82, 114, 115]. Recently reported CCSD(T)-level interaction energies of the benzene– methane and benzene–ethane complexes at the basis set limit are – 1.45 and – 1.82 kcal/mol, respectively [16]. The MP2 calculations slightly overestimate the attraction compared with the more reliable CCSD(T) calculations, as in the case of the OH/π and NH/π interactions. In benzene–methane and benzene–ethane complexes dispersion is the major source of the attraction. The methane molecule prefers to sit above the center of the benzene ring. CCSD(T)-level interaction energies for the benzene–ethylene and benzene– acetylene complexes at the basis set limit are – 2.06 and – 2.83 kcal/mol, respectively [16]. The larger electrostatic energy in the acetylene complexes is the cause of the greater interaction energy. However, dispersion is still the major source of the attraction in the benzene–acetylene complex. MP2 and B3LYP calculations for the benzene–acetylene complex show the existence of two types of complexes: one where the acetylene is the proton donor and the other where the benzene is the proton donor. Only the complex where the acetylene acts as a proton donor was observed by matrix isolation IR spectroscopy [116]. The spectroscopic measurements and ab initio calculations for acetylene–ethylene complexes also reveal two types of stable geometry [117]. Atoms-in-molecules (AIM) calculations for the ethylene, acetylene and benzene dimers and the ethylene–acetylene, benzene–acetylene and benzene-ethylene complexes have also been reported [118]. Very recently CCSD(T) interaction energies of the benzene-halogenated methane complexes were reported to study the effects of halogenation on

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the CH/π interaction [112]. Halogenation increases the interaction energy considerably. The calculated interaction energies of the chloromethane, dichloromethane, chloroform and fluoroform complexes are – 3.0, – 4.5, – 5.6 and – 4.2 kcal/mol, respectively. The calculations show that the dispersion interaction between the chlorine atoms and benzene is the main reason for the very large interaction energy of the chloroform complex (– 5.6 kcal/mol). The effects of substituents on the CH/π interaction in the benzene–CH3 X (X is F, Cl, Br, I, CN, NO2 ) complexes were also studied by DFT calculations [119]. An AIM analysis of the benzene–C2 H3 X complexes was also reported [120]. 5.6 CH/π Interactions in Other Complexes A few ab initio calculations of the CH/π interaction in ethylene and acetylene complexes have been reported [121–123]. The interaction energies of the ethylene and acetylene complexes are weaker than those of the corresponding benzene complexes. The CCSD(T)-level interaction energy of the ethylene– methane complex at the basis set limit is only – 0.49 kcal/mol [123]. Ab initio calculations of acetylene oligomers show that the interaction energy is relatively insensitive to the position of the donating proton along the bond vector of the accepting triple bond as well as to the tilt angle of the major axis of the acetylene molecule, and that the gains in stabilization through cooperativity are not large [122]. The CCSD(T) calculations of the ethylene–methane complex show that dispersion is the major source of the attraction and that the electrostatic interaction stabilizes the orientation where a C – H bond points toward the C = C bond [123]. 5.7 Interaction with Tetramethylammonium The interactions between tetramethylammonium (TMA) and π systems may be described as strong CH/π interactions, as these systems have short contacts between the C – H bonds and the π system. However, the physical origin of the attraction in the TMA complex is completely different from that in the CH/π interaction between neutral molecules [124, 125]. The MP2-level interaction energies of TMA complexes with benzene, pyrrole, furan and imidazole are – 8.45, – 10.02, – 6.98 and – 16.37 kcal/mol, respectively. The calculations show that the electrostatic interaction is the major source of the attraction in these complexes, while dispersion is mainly responsible for the attraction in the CH/π interaction between neutral molecules. The physical origin of the attraction between the TMA and π systems is the same as that of other cation/π complexes; therefore, the interaction between TMA and π systems is essentially a cation/π interaction (see the next section).

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6 Cation/π Interactions 6.1 Introduction The strong attraction between a cation and a π system is denoted as a cation/π interaction. The size of the interaction energy for a cation/π interaction is considerably larger than that for interactions between aromatic molecules and neutral molecules (π/π, OH/π, NH/π and CH/π interactions), as shown in Table 7. The experimental interaction energies of benzene complexes with Li+ , Na+ and K+ shown in Fig. 18 are – 38.5, – 22.1 and – 17.5 kcal/mol, respectively [24]. The importance of the cation/π interaction in biological structures and molecular recognition processes has been

Table 7 Intermolecular interaction energies of benzene and cyclohexane complexes with cations Cluster

Etotal a

Ees b

Eind c

Rd

Li+ /C6 H6 e Na+ /C6 H6 e K+ /C6 H6 e Rb+ /C6 H6 e Cs+ /C6 H6 e Mg2+ /C6 H6 f Ca2+ /C6 H6 f Li+ /C6 H12 g

– 35.4 – 21.3 – 17.0 – 13.9 – 12.1 – 109.9 – 73.7 – 15.1

– 18.7 – 14.8 – 11.9 – 9.2 – 7.9 – 37.8 – 31.2 1.23

– 46.9 – 21.0 – 12.8 – 8.4 – 6.4 – 162.4 – 91.2 – 21.0

1.869 2.425 2.805 3.165 3.414 1.958 2.361 2.6

Energy in kilocalories per mole. Geometries are shown in Fig. 18. a Calculated interaction energy at the MP2/6-311G∗∗ level b Electrostatic energy c Induction energy d Distance between centroid of benzene or cyclohexane ring and carbon atom of C – H bond e Reference [17] f Reference [140]

Fig. 18 Structures of cation–benzene and cation–cyclohexane complexes

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stressed repeatedly [126–136]. A comprehensive review of the cation/π interaction was published in 1997 [14]. The physical origin of the cation/π interaction is mainly discussed in this section. 6.2 Physical Origin of Attraction Benzene has a significant quadrupole moment; therefore, it was believed that the electrostatic interaction (interaction between the quadrupole of the π system and the charge of the cation) was the major source of the attraction in the cation/π interaction [14, 137, 138]. This electrostatic model explained the substituent effects on the size of the cation/π interaction quite well. The substantially smaller binding energy of the cation–cyclohexane complex compared with that of the cation–benzene complex also supported this idea. The molecular polarizability of cyclohexane is slightly larger than that of benzene; therefore, it was believed that the cation–cyclohexane complex should have a larger binding energy, if the induction interaction (induced polarization) is important for the attraction. However, recently reported calculations show that the induction interaction is also an important source of attraction in cation/π complexes [17, 139, 140]. In particular, the induction interaction is the major source of attraction in benzene complexes with Li+ , Na+ and alkaline-earth-metal dications, as shown in Table 7. The size of the induction interaction is proportional to the square of the electric field produced by the cation. Therefore the complexes with dications (Mg2+ and Ca2+ ) have large induction energies. The electric field produced by a cation is proportional to the inverse square of the distance from the cation, therefore, the induction energy is proportional to the inverse fourth power of the distance. The cation/π complexes of small cations (Li+ and Na+ ) have large induction energies owing to the short separation, as shown in Table 7 and Fig. 19. Recently reported ab initio calculations show that the cyclohexane complex has a larger equilibrium intermolecular separation than the benzene complex. The steric repulsion with the axial hydrogen atoms of cyclohexane is the cause of the larger separation [17]. The larger separation in the cation–cyclohexane complex, which leads to a small induction energy, is the cause of the smaller interaction energy of the cyclohexane complex. The alkaline-earth-metal dication–benzene complexes have very large interaction energies [140–144]. The calculated interaction energies of the benzene complexes with Mg2+ and Ca2+ are – 109.9 and – 73.7 kcal/mol, respectively [140]. The magnitude of the binding energies of these complexes is close to the magnitudes of normal chemical bonds. Therefore, the interaction in the alkaline-earth-metal dication–benzene complexes was sometimes termed a cation/π bond [144]. It was claimed that the interaction was in nature a chemical bond and that the elongation of the C – C bonds in benzene by the complex formation also supported the existence of the cation/π

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Fig. 19 Total (MP2), electrostatic and induction energies of the Li+ –benzene complex

bond [141]. However, Table 6 shows that electrostatic and induction interactions are the major source of the attraction in the alkaline-earth-metal dication–benzene complexes [140, 141]. In addition, recent calculations for a benzene monomer in the electric field produced by a dication show that the strong electric field increases the equilibrium C – C bond distance [140]. These calculations show that the elongation of the C – C bond observed in the calculations of the alkaline-earth-metal dication–benzene complexes is not necessarily evidence to support the existence of chemical bonds between the dication and carbon atoms of the interacting benzene molecule. The interaction of ammonium ion with a π system may be described as a strong NH/π interaction and that of TMA may be described as a strong CH/π interaction. However, the physical origin of these interactions is the same as that of the cation/π interaction in the alkali-metal complexes, as mentioned in Sects. 4 and 5. The interactions in these complexes are essentially a cation/π interaction.

7 Interactions of Fluorinated Benzene 7.1 Hexafluorobenzene–Benzene Complex The melting point of a 1 : 1 hexafluorobenzene and benzene crystal is substantially higher than that of either of the pure crystals [145]. The higher

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melting point suggests that the attraction between hexafluorobenzene and benzene is stronger than that in the benzene dimer. It was believed that the quadrupole–quadrupole interaction is the cause of the strong attraction between hexafluorobenzene and benzene. The quadrupole moments of hexafluorobenzene and benzene are nearly equal in magnitude but are opposite in sign. The quadrupole–quadrupole interaction stabilizes the stacked hexafluorobenzene–benzene complex [146]. However, recent MP2 calculations for the hexaflurobenzene–benzene complex show that electron correlation increases the attraction considerably, which shows that dispersion is also important for the attraction in the complex, as in the case of the benzene dimer [147, 148]. The benzene dimer has two nearly isoenergetic structures (slippedparallel and T-shaped) [9, 11]. Dispersion stabilizes the slipped-parallel benzene dimer strongly, while the electrostatic interaction is repulsive in the slipped-parallel dimer. However, the electrostatic interaction stabilizes the T-shaped benzene dimer, while the stabilization due to dispersion is small in the T-shaped dimer [9]. On the other hand, both dispersion and electrostatic interactions stabilize the stacked hexafluorobenzene–benzene complex, producing a large attractive force between hexafluorobenzene and benzene. 7.2 Fluorinated Benzene–Water Complexes The stable structures of fluorinated benzene–water complexes are considerably different from that of the benzene–water complex [149]. The fluorobenzene–water and p-difluorobenzene–water complexes prefer conformations in which the water molecule is involved in the formation of a six-membered ring system with the F – C – C – H of the aromatic ring. DFT and MP2 calculations show that the fluorobenzene–methanol complex has an OH/π-bonded minimum [150]. The hexaflurobenzene–water complex prefers the geometry where the lone pairs of the oxygen atom point toward the π face of hexaflurobenzene, as shown in Fig. 20 [151–153], which is completely different from the case for the stable structure of the benzene–water complex, where a hydrogen atom of the water points toward the π face of benzene [85–87]. The quadrupole moments of hexafluorobenzene and benzene are opposite in sign. The stable structure for the hexaflurobenzene–water complex shows that the electrostatic interaction between the quadrupole of the π system and the negative charge on the lone pairs of the water oxygen atom stabilizes this orientation. The MP2-level interaction energies of the benzene–water and hexaflurobenzene–water complexes using a modified 6-31G∗ basis set are – 2.35 and – 2.69 kcal/mol, respectively [153].

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Fig. 20 Stable structure of the hexafluorobenzene–water complex

7.3 Anion/π Interactions Strong attraction between anion and fluorinated π systems was reported [154–156]. The quadrupole moment of hexafluorobenzene suggests that an attractive electrostatic interaction exists between the anion and the π system. The MP2 interaction energies of the hexafluorobenzene complexes with Cl– and Br– are – 13.2 and – 12.4 kcal/mol, respectively [154]. The binding energies of these anion/π complexes are smaller than those of the benzene complexes with Li+ , Na+ and K+ (– 35.4, – 21.3 and – 17.0 kcal/mol, respectively) [17]. Cl– and Br– are larger than Li+ , Na+ and K+ . The smaller electrostatic interaction due to the larger separation in the Cl– and Br– complexes is the cause of the smaller interaction energies. The calculated binding energies of the hexafluorobenzene complexes with Cl– and Br– are close to those of the benzene complexes with Rb+ and Cs+ (– 13.9 and – 12.1 kcal/mol, respectively) [17]. MP2 calculations for hexafluorobenzene complexes with Cl– and Br– show that electron correlation slightly increases the attraction, which suggests that dispersion also stabilizes these complexes [154]. The large polarizability of the anions is the cause of the dispersion interaction.

8 Summary Although intermolecular interaction between aromatic molecules plays an important role in many fields of chemistry and biology, the magnitude and the physical origin of the interaction were not well understood 10 years ago. Recent progress in computational methods and increasing computer per-

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formance have enabled quantitative analysis of weak intermolecular interactions of aromatic molecules. Ab initio calculation is now a powerful and practical tool for studying weak intermolecular interactions. It provides valuable information for understanding structures and properties of molecular assemblies, such as molecular crystals and biological systems. Quantitative analysis of intermolecular interactions by computational techniques is becoming increasingly important in many fields of chemistry and biology. Detailed information on the interactions is also important for industry, as it is essential for improving material design and drug development. Acknowledgements The author thanks K. Honda and T. Uchimaru for helpful discussions which helped to improve the quality of the presentation.

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110. Umezawa Y, Nishio M (1998) Bioorg Med Chem 6:493 111. Takagi T, Tanaka A, Matsuo S, Maezaki H, Tani M, Fujiwara H, Sasaki Y (1987) J Chem Soc Perkin Trans 2:1015 112. Tsuzuki S, Honda K, Uchimaru T, Mikami M, Tanabe K (2002) J Phys Chem A 106:4423 113. Desiraju GR, Steiner T (1999) The weak hydrogen bond in structural chemistry and biology. Oxford University Press, Oxford 114. Sakaki S, Kato K, Miyazawa T, Musahi Y, Ohkubo K, Ihara H, Hirayama C (1993) J Chem Soc Faraday Trans 89:659 115. Ribas J, Cubero E, Luque FJ, Orozco M (2002) J Org Chem 67:7057 116. Sundararajan K, Viswanathan KS, Kulkarni AD, Gadre SR (2002) J Mol Struct 613:209 117. Sundararajan K, Sankaran K, Viswanathan KS, Kulkarni AD, Gadre SR (2002) J Phys Chem A 106:1504 118. Novoa J, Mota F (2000) Chem Phys Lett 318:345 119. Ugozzoli F, Arduini A, Massera C, Pochini A, Secchi A (2002) New J Chem 26:1718 120. Takahashi O, Kohno Y, Saito K (2003) Chem Phys Lett 378:509 121. Fan M-F, Lin Z, McGrady JE, Mingos DMP (1996) J Chem Soc Perkin 2:563 122. Philp D, Robinson JMA (1998) J Chem Soc Perkin 2:1643 123. Tsuzuki S, Honda K, Uchimaru T, Mikami M, Tanabe K (1999) J Phys Chem A 103:8265 124. Liu T, Gu J, Tan X-J, Zhu W-L, Luo X-M, Jiang H-L, Ji R-Y, Chen K-X, Silman I, Sussman JL (2001) J Phys Chem 105:5431 125. Liu T, Gu J, Tan X-J, Zhu W-L, Luo X-M, Jiang H-L, Ji R-Y, Chen K-X, Silman I, Sussman JL (2002) J Phys Chem 106:157 126. Choi HS, Suh SB, Cho SJ, Kim KS (1998) Proc Natl Acad Sci USA 95:12094 127. Gallivan JP, Dougherty DA (1999) Proc Natl Acad Sci USA 96:9459 128. Minoux H, Chipot C (1999) J Am Chem Soc 121:10366 129. Gallivan JP, Dougherty DA (2000) J Am Chem Soc 122:870 130. Pletneva EV, Laederach AT, Fulton DB, Kostic NM (2001) J Am Chem Soc 123:6232 131. Shi ZS, Olson CA, Bell AJ, Kallenbach NR (2001) Biopolymers 60:366 132. Shi ZS, Olson CA, Kallenbach NR (2002) J Am Chem Soc 124:3284 133. Biot C, Buisine E, Kwasigroch JM, Wintjens R, Rooman M (2002) J Biol Chem 277:40816 134. Tsou LK, Tatko CD, Waters ML (2002) J Am Chem Soc 124:14917 135. Kim D, Hu S, Tarakeshwar P, Kim KS, Lisy JM (2003) J Phys Chem A 107:1228 136. Biot C, Buisine E, Rooman M (2003) J Am Chem Soc 125:13988 137. Mecozzi S, West P Jr, Dougherty DA (1996) J Am Chem Soc 118:2307 138. Mecozzi S, West P Jr, Dougherty DA (1996) Proc Natl Acad Sci USA 93:10566 139. Cubero E, Luque FJ, Orozco M (1998) Proc Natl Acad Sci USA 95:5976 140. Tsuzuki S, Uchimaru T, Mikami M (2003) J Phys Chem A 107:10414 141. Tan XJ, Zhu WL, Cui M, Luo XM, Gu JD, Silman I, Sussman JL, Jiang HL, Ji RY, Chem KX (2001) Chem Phys Lett 349:113 142. Zhu W, Liu T, Chem J, Luo X, Tan X, Puah CM, Jiang H, Chem K (2002) Chem Phys Lett 366:267 143. Cheng Y-H, Liu L, Fu Y, Chen R, Li X-S, Guo Q-X (2002) J Phys Chem A 106:11215 144. Zhu W, Tan X, Shen J, Luo X, Cheng F, Mok PC, Ji R, Chen K, Jiang H (2003) J Phys Chem A 107:2296 145. Patrick CR, Prosser GS (1960) Nature 187:1021 146. Williams JH (1993) Acc Chem Res 26:593

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147. Hernandez-Trujillo J, Colmenares F, Cuevas G, Costas M (1997) Chem Phys Lett 265:503 148. Vanspeybrouck W, Herrebout WA, van der Veken BJ, Lundell J, Perutz RN (2003) J Phys Chem B 107:13855 149. Tarakeshwar P, Kim KS, Brutschy B (1999) J Chem Phys 110:8501 150. Solimannejad M, Pejov L (2004) Chem Phys Lett 385:394 151. Danten Y, Tassaing T, Besnard M (1999) J Phys Chem A 103:3530 152. Gallivan JP, Dougherty DA (1999) Org Lett 1:103 153. Raimondi M, Calderoni G, Famulari A, Raimondi L, Cozz F (2003) J Phys Chem A 107:772 154. Quinonero D, Garau C, Rotger C, Frontera A, Ballester P, Costa A, Deya PM (2002) Angew Chem Int Ed Engl 41:3389 155. Garau C, Quinonero D, Frontera A, Ballester P, Costa A, Deya PM (2003) New J Chem 27:211 156. Mascal M, Armstrong A, Bartberger MD (2002) J Am Chem Soc 124:6274

Author Index Volumes 101–115 Author Index Vols. 1–100 see Vol. 100

The volume numbers are printed in italics Alajarin M, see Turner DR (2004) 108: 97–168 Aldinger F, see Seifert HJ (2002) 101: 1–58 Alfredsson M, see Corà F (2004) 113: 171–232 Aliev AE, Harris KDM (2004) Probing Hydrogen Bonding in Solids Using State NMR Spectroscopy 108: 1–54 Alloul H, see Brouet V (2004) 109: 165–199 Amstutz N, see Hauser A (2003) 106: 81–96 Anitha S, Rao KSJ (2003) The Complexity of Aluminium-DNA Interactions: Relevance to Alzheimer’s and Other Neurological Diseases 104: 79–98 Anthon C, Bendix J, Schäffer CE (2004) Elucidation of Ligand-Field Theory. Reformulation and Revival by Density Functional Theory 107: 207–302 Aramburu JA, see Moreno M (2003) 106: 127–152 Arˇcon D, Blinc R (2004) The Jahn-Teller Effect and Fullerene Ferromagnets 109: 231–276 Atanasov M, Daul CA, Rauzy C (2003) A DFT Based Ligand Field Theory 106: 97–125 Atanasov M, see Reinen D (2004) 107: 159–178 Atwood DA, see Conley B (2003) 104: 181–193 Atwood DA, Hutchison AR, Zhang Y (2003) Compounds Containing Five-Coordinate Group 13 Elements 105: 167–201 Autschbach J (2004) The Calculation of NMR Parameters in Transition Metal Complexes 112: 1–48 Baerends EJ, see Rosa A (2004) 112: 49–116 Barriuso MT, see Moreno M (2003) 106: 127–152 Beaulac R, see Nolet MC (2004) 107: 145–158 Bellandi F, see Contreras RR (2003) 106: 71–79 Bendix J, see Anthon C (2004) 107: 207–302 Berend K, van der Voet GB, de Wolff FA (2003) Acute Aluminium Intoxication 104: 1–58 Bianconi A, Saini NL (2005) Nanoscale Lattice Fluctuations in Cuprates and Manganites 114: 287–330 Blinc R, see Arcˇcon D (2004) 109: 231–276 Bohrer D, see Schetinger MRC (2003) 104: 99–138 Boulanger AM, see Nolet MC (2004) 107: 145–158 Boulon G (2004) Optical Transitions of Trivalent Neodymium and Chromium Centres in LiNbO3 Crystal Host Material 107: 1–25 Bowlby BE, Di Bartolo B (2003) Spectroscopy of Trivalent Praseodymium in Barium Yttrium Fluoride 106: 193–208

196

Author Index Volumes 101–115

Braga D, Maini L, Polito M, Grepioni F (2004) Hydrogen Bonding Interactions Between Ions: A Powerful Tool in Molecular Crystal Engineering 111: 1–32 Brouet V, Alloul H, Gàràj S, Forrò L (2004) NMR Studies of Insulating, Metallic, and Superconducting Fullerides: Importance of Correlations and Jahn-Teller Distortions 109: 165– 199 Buddhudu S, see Morita M (2004) 107: 115–144 Budzelaar PHM, Talarico G (2003) Insertion and β-Hydrogen Transfer at Aluminium 105: 141–165 Burrows AD (2004) Crystal Engineering Using Multiple Hydrogen Bonds 108: 55–96 Bussmann-Holder A, Keller H, Müller KA (2005) Evidences for Polaron Formation in Cuprates 114: 367–386 Bussmann-Holder A, see Micnas R (2005) 114: 13–69 Canadell E, see Sánchez-Portal D (2004) 113: 103–170 Cancines P, see Contreras RR (2003) 106: 71–79 Cartwright HM (2004) An Introduction to Evolutionary Computation and Evolutionary Algorithms 110: 1–32 Clot E, Eisenstein O (2004) Agostic Interactions from a Computational Perspective: One Name, Many Interpretations 113: 1–36 Conley B, Atwood DA (2003) Fluoroaluminate Chemistry 104: 181–193 Contreras RR, Suárez T, Reyes M, Bellandi F, Cancines P, Moreno J, Shahgholi M, Di Bilio AJ, Gray HB, Fontal B (2003) Electronic Structures and Reduction Potentials of Cu(II) Complexes of [N,N
-Alkyl-bis(ethyl-2-amino-1-cyclopentenecarbothioate)] (Alkyl = Ethyl, Propyl, and Butyl) 106: 71–79 Corà F, Alfredsson M, Mallia G, Middlemiss DS, Mackrodt WC, Dovesi R, Orlando R (2004) The Performance of Hybrid Density Functionals in Solid State Chemistry 113: 171–232 Crespi VH, see Gunnarson O (2005) 114: 71–101 Daul CA, see Atanasov M (2003) 106: 97–125 Day P (2003) Whereof Man Cannot Speak: Some Scientific Vocabulary of Michael Faraday and Klixbüll Jørgensen 106: 7–18 Deeth RJ (2004) Computational Bioinorganic Chemistry 113: 37–69 Delahaye S, see Hauser A (2003) 106: 81–96 Deng S, Simon A, Köhler J (2005) Pairing Mechanisms Viewed from Physics and Chemistry 114: 103–141 Di Bartolo B, see Bowlby BE (2003) 106: 191–208 Di Bilio AJ, see Contreras RR (2003) 106: 71–79 Dovesi R, see Corà F (2004) 113: 171–232 Egami T (2005) Electron-Phonon Coupling in High-Tc Superconductors 114: 267–286 Eisenstein O, see Clot E (2004) 113: 1–36 Fontal B, see Contreras RR (2003) 106: 71–79 Forrò L, see Brouet V (2004) 109: 165–199 Fowler PW, see Soncini A (2005) 115: 57–79 Frenking G, see Lein M (2003) 106: 181–191 Frühauf S, see Roewer G (2002) 101: 59–136 Frunzke J, see Lein M (2003) 106: 181–191 Furrer A (2005) Neutron Scattering Investigations of Charge Inhomogeneities and the Pseudogap State in High-Temperature Superconductors 114: 171–204

Author Index Volumes 101–115

197

Gàràj S, see Brouet V (2004) 109: 165–199 Gillet VJ (2004) Applications of Evolutionary Computation in Drug Design 110: 133–152 Golden MS, Pichler T, Rudolf P (2004) Charge Transfer and Bonding in Endohedral Fullerenes from High-Energy Spectroscopy 109: 201–229 Gorelesky SI, Lever ABP (2004) 107: 77–114 Gray HB, see Contreras RR (2003) 106: 71–79 Grepioni F, see Braga D (2004) 111: 1–32 Gritsenko O, see Rosa A (2004) 112: 49–116 Güdel HU, see Wenger OS (2003) 106: 59–70 Gütlich P, van Koningsbruggen PJ, Renz F (2004) Recent Advances in Spin Crossover Research 107: 27–76 Gunnarsson O, Han JE, Koch E, Crespi VH (2005) Superconductivity in Alkali-Doped Fullerides 114: 71–101 Habershon S, see Harris KDM (2004) 110: 55–94 Han JE, see Gunnarson O (2005) 114: 71–101 Hardie MJ (2004) Hydrogen Bonded Network Structures Constructed from Molecular Hosts 111: 139–174 Harris KDM, see Aliev (2004) 108: 1–54 Harris KDM, Johnston RL, Habershon S (2004) Application of Evolutionary Computation in Structure Determination from Diffraction Data 110: 55–94 Hartke B (2004) Application of Evolutionary Algorithms to Global Cluster Geometry Optimization 110: 33–53 Harvey JN (2004) DFT Computation of Relative Spin-State Energetics of Transition Metal Compounds 112: 151–183 Haubner R, Wilhelm M, Weissenbacher R, Lux B (2002) Boron Nitrides – Properties, Synthesis and Applications 102: 1–46 Hauser A, Amstutz N, Delahaye S, Sadki A, Schenker S, Sieber R, Zerara M (2003) Fine Tuning the Electronic Properties of [M(bpy)3 ]2+ Complexes by Chemical Pressure (M = Fe2+ , Ru2+ , Co2+ , bpy = 2,2
-Bipyridine) 106: 81–96 Herrmann M, see Petzow G (2002) 102: 47–166 Herzog U, see Roewer G (2002) 101: 59–136 Hoggard PE (2003) Angular Overlap Model Parameters 106: 37–57 Höpfl H (2002) Structure and Bonding in Boron Containing Macrocycles and Cages. 103: 1–56 Hubberstey P, Suksangpanya U (2004) Hydrogen-Bonded Supramolecular Chain and Sheet Formation by Coordinated Guranidine Derivatives 111: 33–83 Hutchison AR, see Atwood DA (2003) 105: 167–201 Iwasa Y, see Margadonna S (2004) 109: 127–164 Jansen M, Jäschke B, Jäschke T (2002) Amorphous Multinary Ceramics in the Si-B-N-C System 101: 137–192 Jäschke B, see Jansen M (2002) 101: 137–192 Jäschke T, see Jansen M (2002) 101: 137–192 Jaworska M, Macyk W, Stasicka Z (2003) Structure, Spectroscopy and Photochemistry of the [M(η5 -C5 H5 )(CO)2 ]2 Complexes (M = Fe, Ru) 106: 153–172 Jenneskens LW, see Soncini A (2005) 115: 57–79 Johnston RL, see Harris KDM (2004) 110: 55–94

198

Author Index Volumes 101–115

Kabanov VV, see Mihailovic D (2005) 114: 331–365 Keller H (2005) Unconventional Isotope Effects in Cuprate Superconductors 114: 143–169 Keller H, see Bussmann-Holder A (2005) 114: 367–386 Koch E, see Gunnarson O (2005) 114: 71–101 Kochelaev BI, Teitel’baum GB (2005) Nanoscale Properties of Superconducting Cuprates Probed by the Electron Paramagnetic Resonance 114: 205–266 Köhler J, see Deng (2005) 114: 103–141 van Koningsbruggen, see Gütlich P (2004) 107: 27–76 Lein M, Frunzke J, Frenking G (2003) Christian Klixbüll Jørgensen and the Nature of the Chemical Bond in HArF 106: 181–191 Lever ABP, Gorelesky SI (2004) Ruthenium Complexes of Non-Innocent Ligands; Aspects of Charge Transfer Spectroscopy 107: 77–114 Linton DJ, Wheatley AEH (2003) The Synthesis and Structural Properties of Aluminium Oxide, Hydroxide and Organooxide Compounds 105: 67–139 Lux B, see Haubner R (2002) 102: 1–46 Mackrodt WC, see Corà F (2004) 113: 171–232 Macyk W, see Jaworska M (2003) 106: 153–172 Mahalakshmi L, Stalke D (2002) The R2M+ Group 13 Organometallic Fragment Chelated by P-centered Ligands 103: 85–116 Maini L, see Braga D (2004) 111: 1–32 Mallia G, see Corà F (2004) 113: 171–232 Margadonna S, Iwasa Y, Takenobu T, Prassides K (2004) Structural and Electronic Properties of Selected Fulleride Salts 109: 127–164 Maseras F, see Ujaque G (2004) 112: 117–149 Micnas R, Robaszkiewicz S, Bussmann-Holder A (2005) Two-Component Scenarios for Non-Conventional (Exotic) Superconstructors 114: 13–69 Middlemiss DS, see Corà F (2004) 113: 171–232 Mihailovic D, Kabanov VV (2005) Dynamic Inhomogeneity, Pairing and Superconductivity in Cuprates 114: 331–365 Millot C (2005) Molecular Dynamics Simulations and Intermolecular Forces 115: 125–148 Miyake T, see Saito (2004) 109: 41–57 Moreno J, see Contreras RR (2003) 106: 71–79 Moreno M, Aramburu JA, Barriuso MT (2003) Electronic Properties and Bonding in Transition Metal Complexes: Influence of Pressure 106: 127–152 Morita M, Buddhudu S, Rau D, Murakami S (2004) Photoluminescence and Excitation Energy Transfer of Rare Earth Ions in Nanoporous Xerogel and Sol-Gel SiO2 Glasses 107: 115–143 Morsch VM, see Schetinger MRC (2003) 104: 99–138 Mossin S, Weihe H (2003) Average One-Center Two-Electron Exchange Integrals and Exchange Interactions 106: 173–180 Murakami S, see Morita M (2004) 107: 115–144 Müller E, see Roewer G (2002) 101: 59–136 Müller KA (2005) Essential Heterogeneities in Hole-Doped Cuprate Superconductors 114: 1–11 Müller KA, see Bussmann-Holder A (2005) 114: 367–386

Author Index Volumes 101–115

199

Nishibori E, see Takata M (2004) 109: 59–84 Nolet MC, Beaulac R, Boulanger AM, Reber C (2004) Allowed and Forbidden d-d Bands in Octa-hedral Coordination Compounds: Intensity Borrowing and Interference Dips in Absorption Spectra 107: 145–158 Ordejón P, see Sánchez-Portal D (2004) 113: 103–170 Orlando R, see Corà F (2004) 113: 171–232 Oshiro S (2003) A New Effect of Aluminium on Iron Metabolism in Mammalian Cells 104: 59–78 Pastor A, see Turner DR (2004) 108: 97–168 Patoˇcka J, see Strunecká A (2003) 104: 139–180 Petzow G, Hermann M (2002) Silicon Nitride Ceramics 102: 47–166 Pichler T, see Golden MS (2004) 109: 201–229 Polito M, see Braga D (2004) 111: 1–32 Popelier PLA (2005) Quantum Chemical Topology: on Bonds and Potentials 115: 1–56 Power P (2002) Multiple Bonding Between Heavier Group 13 Elements. 103: 57–84 Prassides K, see Margadonna S (2004) 109: 127–164 Prato M, see Tagmatarchis N (2004) 109: 1–39 Price LS, see Price SSL (2005) 115: 81–123 Price SSL, Price LS (2005) Modelling Intermolecular Forces for Organic Crystal Structure Prediction 115: 81–123 Rao KSJ, see Anitha S (2003) 104: 79–98 Rau D, see Morita M (2004) 107: 115–144 Rauzy C, see Atanasov (2003) 106: 97–125 Reber C, see Nolet MC (2004) 107: 145–158 Reinen D, Atanasov M (2004) The Angular Overlap Model and Vibronic Coupling in Treating s-p and d-s Mixing – a DFT Study 107: 159–178 Reisfeld R (2003) Rare Earth Ions: Their Spectroscopy of Cryptates and Related Complexes in Glasses 106: 209–237 Renz F, see Gütlich P (2004) 107: 27–76 Reyes M, see Contreras RR (2003) 106: 71–79 Ricciardi G, see Rosa A (2004) 112: 49–116 Riesen H (2004) Progress in Hole-Burning Spectroscopy of Coordination Compounds 107: 179–205 Robaszkiewicz S, see Micnas R (2005) 114: 13–69 Roewer G, Herzog U, Trommer K, Müller E, Frühauf S (2002) Silicon Carbide – A Survey of Synthetic Approaches, Properties and Applications 101: 59–136 Rosa A, Ricciardi G, Gritsenko O, Baerends EJ (2004) Excitation Energies of Metal Complexes with Time-dependent Density Functional Theory 112: 49–116 Rudolf P, see Golden MS (2004) 109: 201–229 Ruiz E (2004) Theoretical Study of the Exchange Coupling in Large Polynuclear Transition Metal Complexes Using DFT Methods 113: 71–102 Sadki A, see Hauser A (2003) 106: 81–96 Saini NL, see Bianconi A (2005) 114: 287–330 Saito S, Umemoto K, Miyake T (2004) Electronic Structure and Energetics of Fullerites, Fullerides, and Fullerene Polymers 109: 41–57 Sakata M, see Takata M (2004) 109: 59–84

200

Author Index Volumes 101–115

Sánchez-Portal D, Ordejón P, Canadell E (2004) Computing the Properties of Materials from First Principles with SIESTA 113: 103–170 Schäffer CE (2003) Axel Christian Klixbüll Jørgensen (1931–2001) 106: 1–5 Schäffer CE, see Anthon C (2004) 107: 207–301 Schenker S, see Hauser A (2003) 106: 81–96 Schetinger MRC, Morsch VM, Bohrer D (2003) Aluminium: Interaction with Nucleotides and Nucleotidases and Analytical Aspects of Determination 104: 99–138 Schmidtke HH (2003) The Variation of Slater-Condon Parameters Fk and Racah Parameters B and C with Chemical Bonding in Transition Group Complexes 106: 19–35 Schubert DM (2003) Borates in Industrial Use 105: 1–40 Schulz S (2002) Synthesis, Structure and Reactivity of Group 13/15 Compounds Containing the Heavier Elements of Group 15, Sb and Bi 103: 117–166 Seifert HJ, Aldinger F (2002) Phase Equilibria in the Si-B-C-N System 101: 1–58 Shahgholi M, see Contreras RR (2003) 106: 71–79 Shinohara H, see Takata M (2004) 109: 59–84 Sieber R, see Hauser A (2003) 106: 81–96 Simon A, see Deng (2005) 114: 103–141 Soncini A, Fowler PW, Jenneskens LW (2005) Angular Momentum and Spectral Decomposition of Ring Currents: Aromaticity and the Annulene Model 115: 57–79 Stalke D, see Mahalakshmi L (2002) 103: 85–116 Stasicka Z, see Jaworska M (2003) 106: 153–172 Steed JW, see Turner DR (2004) 108: 97–168 Strunecká A, Patoˇcka J (2003) Aluminofluoride Complexes in the Etiology of Alzheimer’s Disease 104: 139–180 Suárez T, see Contreras RR (2003) 106: 71–79 Suksangpanya U, see Hubberstey (2004) 111: 33–83 Sundqvist B (2004) Polymeric Fullerene Phases Formed Under Pressure 109: 85–126 Tagmatarchis N, Prato M (2004) Organofullerene Materials 109: 1–39 Takata M, Nishibori E, Sakata M, Shinohara M (2004) Charge Density Level Structures of Endohedral Metallofullerenes by MEM/Rietveld Method 109: 59–84 Takenobu T, see Margadonna S (2004) 109: 127–164 Talarico G, see Budzelaar PHM (2003) 105: 141–165 Teitel’baum GB, see Kochelaev BI (2005) 114: 205–266 Trommer K, see Roewer G (2002) 101: 59–136 Tsuzuki S (2005) Interactions with Aromatic Rings 115: 149–193 Turner DR, Pastor A, Alajarin M, Steed JW (2004) Molecular Containers: Design Approaches and Applications 108: 97–168 Uhl W (2003) Aluminium and Gallium Hydrazides 105: 41–66 Ujaque G, Maseras F (2004) Applications of Hybrid DFT/Molecular Mechanics to Homogeneous Catalysis 112: 117–149 Umemoto K, see Saito S (2004) 109: 41–57 Unger R (2004) The Genetic Algorithm Approach to Protein Structure Prediction 110: 153–175 van der Voet GB, see Berend K (2003) 104: 1–58 Vilar R (2004) Hydrogen-Bonding Templated Assemblies 111: 85–137

Author Index Volumes 101–115

201

Weihe H, see Mossin S (2003) 106: 173–180 Weissenbacher R, see Haubner R (2002) 102: 1–46 Wenger OS, Güdel HU (2003) Influence of Crystal Field Parameters on Near-Infrared to Visible Photon Upconversion in Ti2+ and Ni2+ Doped Halide Lattices 106: 59–70 Wheatley AEH, see Linton DJ (2003) 105: 67–139 Wilhelm M, see Haubner R (2002) 102: 1–46 de Wolff FA, see Berend K (2003) 104: 1–58 Woodley SM (2004) Prediction of Crystal Structures Using Evolutionary Algorithms and Related Techniques 110: 95–132 Zerara M, see Hauser A (2003) 106: 81–96 Zhang Y, see Atwood DA (2003) 105: 167–201

Subject Index

Acceptor–acceptor interactions 37 Acetamide 106 Acetic acid 92 AIM (atoms in molecules) 3 Alanine, Cα atoms, electrostatic potential 34 Alcohols, fluorinated, OH/π interactions 172 All-atom force-fields 101 Allinger’s MM force fields, hydrocarbons 41 AMBER 87 Amino acids 37 – OH/π interactions 172 Ammonium ion, NH/π interactions 177 Angular momentum 57 – quantum number 61 Aniline, NH/π interactions 176 Anisotropy 41 Annulene model, aromaticity 57 Antibinding 8, 21 Antisymmetrisation 43 Aromatic rings 149 Aromaticity 57 Atom–atom potentials 85 Atomic interactions 10 Atomic theorems 4 Atoms in molecules (AIM) 3 Attraction 9, 31 Azahydrocarbons, non-hydrogen-bonded 94 Azetidine 113 Basis set superposition error (BSSE) Benzene 112 – energy level diagrams 61 – fluorinated 186 – polarisability 157 – quadrupole moment 165

154

Benzene complexes, hydrocarbon molecules 179 Benzene dimer, density functional theory (DFT) 160 – electron correlations 158 – HF/MP2 156 – interaction energy, BSSE 154 – π–π interactions 163 – structures 164 Benzene-ammonia complex 173 Benzene-chloroform complex 180 Benzene-methane complex 175, 178 Benzene-N-methylformamide complex 176 Benzene-water complex, OH/π interactions 171 Binding 8 Bond critical points 9 – disappearance 25 Bond lines/path 9 Bonding 2, 6 Borazine 67 Born–Oppenheimer dynamics 137 Boron clamp 76 BSSE, counterpoise method 154 Buckingham potential 126 CADPAC 155 Caffeine 115 Cambridge Structural Database 84, 88 Carbocycles, planar 57 – XHXH-clamped 69 [N]Carbocycles 61 Catchment region 15 Cation-benzene complex 184 Cation-cyclohexane complex 185 CCSD(T) 151, 158 CH/π interactions 177 – magnitude 180

204

Subject Index

Charge-transfer interactions 166 Charges, electrostatic potential-derived 109 Chemical bond, definitions 6 Chlorobenzene 107, 115 Chlorothalonil 116 CHNO materials, rigid 112 Citrinin 22 Clashing groups 32 Closed-shell interaction 20 Close-packing principle 84 Clusters, intermolecular interaction energy 152 – molecular mechanics 127 Collision 44 COMPASS 87, 109 Conduction number 40 Conflict structure 11 Conformational flexibility 101 Continuous symmetry measures 26 Convergence 2, 34 Coulomb energy 38 Coulomb interactions 2, 46, 126, 162 Crystal structure prediction 81, 88 Current patterns, frontier orbital symmetries 57 CVFF 87 Cyclohexane complex 185 Cyclooctatetraene, energy level diagrams 61 Delocalised electron interactions Density functional theory (DFT) 109, 152, 159 DFTB 134 Diatropic current 57 Dispersion contribution 153 Dispersion energies, aromatic molecules 162 Distributed multipole analysis (DMA) 3, 92, 162 Donor–donor interactions 37 Dreiding force-field 109 Durene 99

Electron localisability indicator (ELI) 5 Electron localisation function 5 Electronegativity alternation 67 Electrostatic interaction 35 Electrostatic interaction energy 125 Electrostatic potential 2, 33 Electrostatic potential-derived charges 109 Energy, chemical bond 27 Energy of molecular system 45 Ethylene, NH/π interactions 176 F–F coupling 18 Flexibility, conformational/molecular 101 Fluorobenzene 40 Fluorobenzene-water complexes 187 Force fields 2 Formamide 106 Frontier orbital symmetries 57 Furan dimer 165 Galaxies, colliding 44 Global optimisation 96 Halide ions, hydrated 125, 140 Heats of sublimation 97 Hellmann–Feynman theorem 15 Hexafluorobenzene-benzene complexes 186 Hexafluorobenzene-water complexes 188 HF, molecular dynamics 38 H–H interaction 16 Hilbert space 44 Hydrocarbons 92 – Allinger’s MM force fields 41 – aromatic 149 Hydrogen bonding 84 Hyperpolarisability 127

16

Ehrenfest force 19 Elastic constants 99 Electron correlations, benzene dimer Electron density 2, 14 – gradient 44

Initial resistance 32 Intermolecular forces, aromatic molecules 153 Intermolecular interaction energies, aromatic molecules 166 Intermolecular perturbation theory (IMPT) 106, 128, 155 Ipsocentric approach 57 158 Kernel methods 47 Kohn–Sham potential

134

Subject Index Lattice energy minimisation 86, 110 Lennard–Jones potential 126 Li+ -benzene complex 186 Ligand closed packing 42 Ligand–ligand interactions 42 Lithium nuclei 11 Lysine, electrostatic potential 35 Merz–Kollman potential 36 Methyl hydrogen bond 23 Mezey’s catchment regions 31 Molecular dynamics (MD) studies 97 – calculation 125 Molecular flexibility 101 Molecular graphs 15 Molecular mechanics for clusters 127 Molecule, definition 8 Møller–Plesset perturbation method (MP2) 155, 164 Monte Carlo method, conformation-family 96 MP4(SDTQ) 164 Multiple moments 2 Multiple-mimina problem 92 Multipole expansion 35 Nanotubes 116 Naphthalene 100 – dimer, CCSD(T) 167 Natural population analysis 36 NH/π interactions 170, 176 Nitroaniline 40 Nitrogen clamp 76 Nodal patterns 57 Nonbonded interactions 30

205 Polarisability 40, 127 Polarisation approximation 43 Polymorphism 81, 92 Polymorphs, entropy differences 100 Potential energy, calculation 125 Potential energy surface 20 Propene, NH/π interactions 176 Protein folding, apolar side chains, steric steering 24 Proteins, force field 2 PW91 160 Pyramidalisation 102 Pyrrole ring 172 QCT force field 32 – peptide 41 Quadrupole–quadrupole interaction 187 Quantum chemical topology 1, 5 Quantum delocalisation 2 Quantum mechanics, total space 3 Quasi classical trajectory 5 Rayleigh–Schrödinger perturbation theory 43, 72 Reduced density matrix 2 Repulsion 9, 31 Repulsion potentials, nonempirical 105 Ring currents 57 Ritonavir 101

OH/π interactions 170 Oligothiophene, crystal structure 170 Organic crystal structure prediction 81 Organolithium compounds 11 Overlap, interactions 42 – model 105, 107 Oxalic acid 104

SAPT theory 128 Secondary interaction hypothesis 37 Self-consistent field (SCF) 92 Short-range interactions 42 Silicon clusters 125, 143 Sorescu–Rice–Thompson 97 Stability function 31 Star density 44 Sugars 92 Supermolecule method 154, 163 Supermolecules 38 Symmetry-adapted perturbation theory (SAPT) 165

Perdew–Burke–Ernzerhof density functional theories 117 Phonon frequencies 100 Pixel semiclassical density sums 117 Point-charge model 162 Polar molecules, modelling 94

Tensor formalism 32 Thiophene dimer, CCSD(T) 169 Tight-binding models 133 TIP4P/TIP5P potentials 37 Toluene dimer, CCSD(T) 168 Topological bond order 46

206 Topological partitioning of electronic properties 40 Topology 4 Torsion angle 23 Two-body potentials 125 Uracil 92 Urea 99 Valence-bond theory, overlap 43 van der Waals bonds 13, 18, 35

Subject Index van der Waals interaction energy Vibrational frequencies, silicon clusters 143 Virial fragments 3 Water, C2v symmetry 30 – liquid 132 – molecular dynamics 38 Water-methanal 42 Weak bonds 15

125