Z. Kristallogr. 220 (2005) 399–457 399 # by Oldenbourg Wissenschaftsverlag, Mu¨nchen Chemical bonding in crystals: new directions Carlo Gatti* CNR-ISTM Istituto di Scienze e Tecnologie Molecolari, via Golgi 19, 20133 Milano, Italy Received November 8, 2004; accepted January 3, 2005 Chemical bonding / Crystals / Topological descriptors / Introduction QTAIMAC Quantum Theory of Atoms in Molecules and Crystals / ELF Electron Localization Function / Studies of chemical bonding in solids have experienced a Computational crystallography true blossoming over the past decade. The situation has clearly changed since when, in 1988, Roald Hoffman pro- Abstract. Analysis of the chemical bonding in the posi- vocatively observed [1] that “many solid chemists have tion space, instead of or besides that in the wave function isolated themselves from their organic or even inorganic (Hilbert) orbital space, has become increasingly popular for crystalline systems in the past decade. The two most colleagues by choosing not to see bonds in their materi- frequently used investigative tools, the Quantum Theory of als”. Many are the reasons behind this change and many Atoms in Molecules and Crystals (QTAIMAC) and the are the grafts from other scientific disciplines that have Electron Localization Function (ELF) are thoroughly dis- contributed to renovating the interest towards a local de- cussed. The treatment is focussed on the topological pecu- scription of bonding in solids beyond the tremendously liarities that necessarily arise from the periodicity of the successful, though empirical, Zintl–Klemm concept [2–4]. crystal lattice and on those facets of the two tools that A reason, on the one hand, is the continuously increasing have been more debated, especially when these tools are complexity (and reduced size) of new materials and the applied to the condensed phase. In particular, in the case ensuing necessity to understand their properties at an ato- of QTAIMAC, the physical and chemical significance of mistic level, if the materials’ potentialities are to be fully the bond paths for the very weak or the supposedly repul- exploited and the materials’ performances significantly im- sive interactions, the distinctive features and the appropri- proved. A decisive scientific graft is, on the other hand, ateness of the several schemes that have been proposed to the technical developments that are still continuing and classify chemical bonds, and, finally, the relative impor- that have made X-ray diffraction a unique tool for map- tance of the local and integrated electron density properties ping the charge density in crystals [5–7]. The accessibility for describing intermolecular interactions. In the case of to intense short-wavelength synchrotron sources, the avail- the ELF, particular attention is devoted to how this func- ability of commercial devices for low-temperature experi- tion is formulated and to the related physical meaning, and ments, the advent of area detectors, all have led to a sig- to how can the ELF be chemically interpreted and prop- nificant increase of the redundancy of measurements and erly analysed in crystals. Several examples are reported to hence of the X-ray data quality one may attains [5–8]. illustrate all these points and for critically examine the an- Parallel progresses have also been made in the refinement swers obtained and the problems encountered. The dis- of these data, by using improved aspherical-atom multi- cussed examples encompass the case of molecular crystals, pole models to describe the distortion of atomic clouds Zintl phases, intermetallic compounds, metals, supported due to bonding, and by adopting suitable formalisms to and unsupported metal-metal bonds in organometallics, ionic deconvolute thermal motion effects from the X-ray ampli- solids, crystal surfaces, crystal defects, etc. Whenever pos- tudes [5–8]. Presently, the final outcome is a static crystal- sible joint ELF and QTAIMAC studies are considered, line electron density, which most often represents a faith- with particular emphasis on the comparison of the bond ful extrapolation to infinite resolution from the finite set of description afforded by the ELF and the Laplacian of the experimental data and which may be so compared with electron density. Two recently proposed functions, the the corresponding density from theory. Deconvolution of Localized Orbital Locator (LOL) and the Source Function the thermal motion has become quite effectual as recently in its integrated or local form are also presented, in view testified by a multi-temperature experiment showing that of their potential interest for studies of chemical bonding identical atomic multipole parameters can be obtained, ir- in crystals. The use of approximated ELF and LOL, as respective of the acquisition temperature [9], provided the derived from the density functional form of the positive data are collected to high resolution and below 200 K. All kinetic energy density, is also discussed. these progresses have made electron densities derived from X-ray diffraction data of comparable [8] or, in some cases, of even better quality [10] than those obtained from * e-mail: [email protected] the ab-initio periodic approaches, whose capabilities have 400 C. Gatti nevertheless also largely been improved over the last dec- have been originally proposed for the gas phase systems ade [11]. This has provided a unique opportunity for a and, later on only, has their use been extended to the con- comparison and mutual validation of the theoretical and densed phase. For this reason, presentation of each ap- experimental techniques [6, 10], a comparison that goes proach is far from being complete. It is rather focused on far beyond the atomic positions or the visual inspection of those aspects which make each approach particularly im- electron and deformation densities [5, 6, 8–10]. Indeed, portant for the study of chemical bonding in crystals or on being the electron density the observable common to the those facets that are still a matter of debate and, occasion- two approaches, it has become a natural consequence and ally, of misuse or misinterpretation. Several examples are a common practice to adopt density-based topological reported to illustrate all these points and to critically ex- tools for confronting them [5–8]. This allows for a real amine the answers one obtains and the problems one is space quantitative description of the chemical bonding in facing when applying the various tools. Whenever possi- crystals instead of the conventional bonding analysis in ble, a comparison between the descriptions provided by the space of the specific and different mathematical func- different approaches is outlined. The presented applica- tions used to expand the wavefunction or the multipolar tions encompass the case of bonding in molecular crystals, model. If one would simply stop at this conventional level Zintl phases, intermetallic compounds, metals, organome- of analysis, neither a proper comparison between experi- tallics, ionic solids, crystal surfaces, crystal defects, etc. In ment and theory, nor an unbiased description of bonding spite of the over 290 bibliographic citations, many other within either approaches could be achieved, since the nat- important studies could have been certainly reported in ure of functions that are chosen to represent the system this review. The choices I made reflect the kind of topics I would “determine the flavour of the bonding model to a have addressed, the intrinsic didactic character of the se- too large extent” [12]. lected examples and, quite often, simply my own knowl- The Quantum Theory of Atoms in Molecules [13] is edge of the literature. Needless to say, I’m more familiar certainly the most complete density-based topological tool with the studies in which I’ve been personally involved or for chemical bonding studies. Although extensively applied to some extent related and I sincerely apologize if this has to gas-phase systems since the early 1980’s, it was only in resulted in an excessive emphasis on my own work and/or the last decade that, thanks to the progresses in experimen- in a too biased exposition. tal and theoretical densities, has this tool been increasingly There are also important thematic omissions in this re- applied to crystalline systems. Indisputably, the Quantum view. The comparison between theoretical and experimen- Theory of Atoms in Molecules has nowadays been elected tal descriptions of bonding is only touched upon occasion- by the X-ray density community as the primary standard ally, since excellent reviews have already appeared theory to discuss bonding in crystals [6–8, 14]. covering this subject [6, 8]. The role of X-ray data quality However, the charge density alone does not describe as for a quantitative analysis of the bonding details is not bonding in its entirety, especially the mechanism of elec- discussed. An enlightening test case study has recently tron pairing, despite the famous Hohenberg–Kohn theo- been published [29]. rem [15] had established a one-to-one correspondence be- Traditional investigations of bonding, based on the tween the electron density and the wave function of a LCAO theory of periodic structures and on the breakdown system. The structure of the whole one-electron density into atomic orbital contributions of the ensuing band struc- matrix (ODM) has been shown to be crucially influenced ture or density of states, retain an undisputed importance by covalent chemical bonding [16, 17]. Yet, the