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Electronic Structure of Clusters Electronic Structure of Clusters David J. Wales Volume III, pp. 1506–1525 in Encyclopedia of Inorganic Chemistry, Second Edition (ISBN 0-470-86078-2) Editor-in-Chief : R. Bruce King John Wiley & Sons, Ltd, Chichester, 2005 material, but more than a single molecule or even two Electronic Structure of atoms or molecules’.2 Here, we must recognize that the domain of cluster chemistry extends beyond systems such Clusters as boranes, carboranes, and transition metal carbonyls, which are generally considered to be the stuff of inorganic cluster David J. Wales chemistry.1,3−5 These are certainly the molecules of primary University of Cambridge, Cambridge, UK interest here, but it would be shortsighted to ignore the connections between these species and clusters bound, for example, by van der Waals forces. There is often much to be learned from making such comparisons: for instance, we 1 Introduction 1 may find analogous structures and rearrangement mechanisms 2 Computational Approaches 1 in clusters bound by quite different forces. The converse is 3 Empirical Structure–Electron Count Correlations 2 also true, and a better understanding of the dynamics of small 4 Models of Cluster Bonding 3 inert gas clusters has been achieved partly by the application 5 Further Development of TSH Theory 8 of ideas that originated in inorganic chemistry concerning 6 Cluster Rearrangements 15 rearrangement mechanisms.6 7 Related Articles 18 To appreciate the ubiquity of clusters in chemistry and 8 References 18 physics, one need only consult the proceedings of one of the biennial International Symposia on Small Particles and Inorganic Clusters (ISSPIC).7 These volumes include species 8 Glossary ranging from hydrocarbon polyhedra, such as dodecahedrane (see also Dodecahedral), to large carbon clusters,9 and from small inert gas clusters to colloidal metal particles containing Ab initio calculations: attempt to calculate from first thousands of atoms. These systems are of interest for a variety principles, without use of empirical data, solutions to of reasons, such as the investigation of fundamental processes Schrodinger’s¨ equation including melting and freezing, and can sometimes provide Deltahedron: a polyhedron with only triangular faces insight into the behavior of larger systems at a more tractable level. Furthermore, small clusters may exhibit properties that are unique to the intermediate size regime, which may be of Abbreviations technological importance.10 The principal subject of this contribution is the electronic DFT = Density Functional Theory; DSD = Diamond- structure of clusters, and so we shall mainly be concerned Square-Diamond; HOMO = Highest Occupied Molecular with molecules in which the interatomic forces cannot Orbital; IR = Irreducible Representation; LCAO = Linear be well described by simple empirical potential energy Combination of Atomic Orbitals; Ph = Phenyl; PSEPT = functions. Hence, the following section provides an overview Polyhedral Skeletal Electron Pair Theory; SCF = of the progress that has been made in performing quantum Self-Consistent-Field; SDDS = Square-Diamond, Diamond- mechanical calculations for clusters. Major difficulties remain Square; TSH = Tensor Surface Harmonic. to be solved in this field, particularly with regard to systems containing Transition Metals. However, although accurate calculations are certainly important, there are various patterns concerning the correlation between structure and electron count, which require a more general explanation. The latter 1 INTRODUCTION are considered in Section 3, and the various models that have been developed to explain them are reviewed in Section 4. The first problem encountered in writing an article of this One of these, Stone’s Tensor Surface Harmonic (TSH) theory, kind lies in deciding what a Cluster actually is, and which is then developed in Section 5, and some further applications sorts of cluster are relevant to the audience in question. to cluster rearrangements are described in Section 6. The former problem may be addressed in terms of size or number of constituent atoms; for example, Sugano1 considers a classification in terms of molecules, microclusters, and fine particles. On the other hand, Berry has distinguished clusters 2 COMPUTATIONAL APPROACHES as ‘finite aggregates whose composition can be changed by adding or taking away units of the species that comprise It is certainly not the purpose of this section to describe them...they are finite bits of what, in composition but not the mechanics of ab initio quantum chemical calculations necessarily in structure, constitute a tiny sample of bulk on clusters.11 However, some understanding of the problems 2 ELECTRONIC STRUCTURE OF CLUSTERS and achievements in this field is necessary to appreciate the calculations on boranes, carboranes, metal, and silicon clusters simpler general models described in subsequent sections. Even will be considered in the following sections. if extensive and accurate calculations were possible for all the inorganic clusters of interest, the latter models would still be needed to explain the structural patterns that have been deciphered. 3 EMPIRICAL STRUCTURE–ELECTRON COUNT Most applications of ab initio quantum chemistry involve CORRELATIONS expanding the wavefunction using a basis set of atomic orbitals. As the many-body problem is not analytically tractable in either classical or quantum mechanics, we can With the explosion in size of the database of known cluster never obtain the exact wavefunction. However, the variation compounds, it is reassuring to note that a number of patterns were soon recognized and exploited by inorganic chemists. principle12 tells us that the more basis functions we use, the The most important relationships between geometry and the more accurate our calculation should become. Unfortunately, number of electrons in a cluster (‘electron-counting rules’, see the computational cost of the most commonly employed self- Counting Electrons) are summarized in the Debor Principle21 consistent-field12 (SCF) method increases roughly as N 4, (or Wade’s Rules), Polyhedral Skeletal Electron Pair Theory22 where N is the number of basis orbitals. Hence, for clusters (PSEPT), and the Isolobal Principle.23 These ‘rules’ (to which containing Transition Metals, for which even a minimal basis there are various exceptions) include three-connected clusters set of atomic orbitals requires many functions, the calculations (By convention, the connectivity’ of a vertex is the number of rapidly become impossible. There is another problem as nearest-neighbors in the cluster cage, not counting terminally well: the usual Hartree–Fock SCF procedure neglects bound ligands.), like prismane, deltahedral clusters (which the instantaneous electron–electron interaction, replacing it have only triangular faces), such as closo-boranes; ‘naked’ instead with an average field12,13 (see Hartree–Fock Theory). clusters24 like Sn 2− and Pb 2−; clusters with interstitial The error incurred by this treatment of the electron correlation 5 5 atoms, fused molecules, which can be decomposed into problem generally becomes more significant for systems deltahedra that share vertices, edges, or faces, and even with low-lying electronic states, as are commonly found large multispherical transition metal clusters. Much of the in transition metal compounds. Density functional theory14 rest of this article is devoted to rationalizing these patterns, (DFT) can help deal with the correlation problem, but the whose importance can perhaps be appreciated by the example reliability of such calculations is often subject to debate. This of Au Cl (PMe Ph) 3+. This compound was successfully is particularly true for molecules containing heavy elements, 13 2 2 10 synthesized several years after the stability of gold clusters where relativistic effects are also important.15 Au (PR ) 5+ was predicted on the basis of simple bonding Given this background, it is not surprising that many 13 3 12 considerations.25 different approaches, all introducing further approximations,16 Some of the historical development of the electron- have been applied to inorganic clusters. The two most counting rules is summarized in a review by Mingos commonly encountered approaches in this area are the and Johnston;26 here, we will concentrate on the patterns Extended Huckel¨ 17 and Fenske–Hall18 methods (see also themselves, beginning with the simplest cases – more details Extended Huckel¨ Molecular Orbital Theory). The former can be found elsewhere.27 The most straightforward clusters is the simplest approach, making use of empirical data, commonly encountered in inorganic chemistry are those that while the latter uses no experimental information, but instead can be described by essentially localized two-center, two- approximates the quantum mechanical Hamiltonian so as to electron bonds, where all the atoms obey an effective atomic avoid many of the most time-consuming integrals. Despite number rule EAN Rule. For main group atoms, this means the sweeping assumptions used in these methods,19 they have that each one is associated with 8 valence electrons to give an often been found to give valuable insights, although DFT is ‘inert gas configuration’; for Transition Metals, 18 electrons now commonly used instead. are usually required to do this. Hence, such clusters are The early DFT studies of main group clusters by Jones associated with and coworkers should be mentioned here.20 However, as these are primarily concerned with clusters that are describable
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