Clusters molecular orbital theory

Only the structures of di- and trisulfane have been determined experimentally. For a number of other sulfanes structural information is available from theoretical calculations using either density functional theory or ab initio molecular orbital theory. In all cases the unbranched chain has been confirmed as the most stable structure but these chains can exist as different ro-tamers and, in some cases, as enantiomers. However, by theoretical methods information about the structures and stabilities of additional isomeric sul-fane molecules with branched sulfur chains and cluster-like structures was obtained which were identified as local minima on the potential energy hypersurface (see later). 

The location of electrons linking more than three atoms cannot be illustrated as easily. The simple, descriptive models must give way to the theoretical treatment by molecular orbital theory. With its aid, however, certain electron counting rules have been deduced for cluster compounds that set up relations between the structure and the number of valence electrons. A bridge between molecular-orbital theory and vividness is offered by the electron-localization function (cf p. 89). 

Although Ni(CO)4 was discovered many years ago, no neutral Ni2(CO)x compound has ever been synthesized in macroscopic amounts. However, several communications report ionic species such as [Ni2(CO)8l+, [Ni2(CO)7], and [Ni2(CO)6]+, where structures with one or two bridging carbonyls are proposed.2418 Plausible structures for neutral Ni2(CO)x (x = 5, 6, 7) have been investigated by theoretical methods, and decomposition temperatures well below room temperature have been predicted.2419,2420 Tetra-, penta-, and hexanuclear nickel carbonyl clusters have been investigated by means of molecular orbital theory. It is found that the neutral forms are more stable than the corresponding anionic forms but the anionic forms gain in stability as the nuclearity rises.2421 Nickel carbonyl cluster anions are manifold, and structural systematics have been reviewed.2422,2423 An example includes the anion [Ni9(CO)i6]2- with a close-packed two-layer metal core.2424... 

However, in sulphides and related minerals, the effects of covalent bonding predominate and orbital overlap must be taken into account. Thus, concepts of molecular orbital theory are described in chapter 11 and applied to aspects of the sulfide mineralogy of transition elements. Examples of computed energy diagrams for molecular clusters are also presented in chapter 11. There, it is noted that the fundamental 3d orbital energy splitting parameter of crystal field theory, A, receives a similar interpretation in the molecular orbital theory. 

The present author found a non-empirical method to describe the antiferromagnetic state of transition metal oxides and hydrogen clusters with a relatively long H-H distance [1-3]. The study used the discrete variational (DV)-Xa molecular orbital theory, which has been successfully applied to analyze properties of... 

The most widely used semiempirical quantum chemistry technique for theoretical chemisorption studies is the Extended Hiickel Theory (EHT). The method was first proposed by Hoffmann/95/ in its nonrelativistic form, and by Lohr and Pyykko/96/ and also Messmer/97/ in its relativistic form, based on the molecular orbital theory for calculating molecular electronic and geometric properties. For a cluster the molecular orbitals are expanded as linear combinations of atomic orbitals... 

The electronic structures of Group lA and IB metal clusters have been determined using two theoretical methods ah initio molecular orbital theory and the semi-empirical diatomics-in-molecules (DIM)... 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

The transition from the atom to the cluster to the bulk metal can best be understood in the alkali metals. For example, the ionization potential (IP) (and also the electron affinity (EA)) of sodium clusters Na must approach the metallic sodium work function in the limit N - . We previously displayed this (1) by showing these values from the beautiful experiments by Schumacher et al. (36, 37) (also described in this volume 38)) plotted versus N". The electron affinity values also shown are from (39), (40) and (34) for N = 1,2 and 3, respectively. A better plot still is versus the radius R of the N-mer, equivalent to a plot versus as shown in Figure 1. The slopes of the lines labelled "metal sphere" are slightly uncertain those shown are 4/3 times the slope of Wood ( j ) and assume a simple cubic lattice relation of R and N. It is clear that reasonably accurate interpolation between the bulk work function and the IP and EA values for small clusters is now possible. There are, of course, important quantum and statistical effects for small N, e.g. the trimer has an anomalously low IP and high EA, which can be readily understood in terms of molecular orbital theory (, ). The positive trimer ions may in fact be "ionization sinks" in alkali vapor discharges a possible explanation for the "violet bands" seen in sodium vapor (20) is the radiative recombination of Na. Csj may be the hypothetical negative ion corresponding to EA == 1.2 eV... 

Bonding Energetics of OrganometaUic Compounds Electronic Structure of Main-group Compounds Electronic Structure of Solids Electronic Structure of Clusters Ligand Field Theory Spectra Molecular Orbital Theory Photoelectron Spectroscopy of Transition Metal Systems. 

Cluster Compounds Inorganometalhc Compounds Containing Transition Metal Main Group Elements Electronic Stmcture of Main-group Compounds Molecular Orbital Theory Phosphorus Inorganic Chemistry. 

Fig. 5.2. Bond overlap populations, n(Si-O), calculated using extended Hiickel molecular-orbital theory for the (SiOJ tetrahedron of forsterite and plotted against Si-O bond distance (A). Inset tetrahedron at left shows dimensions observed for SiO cluster and one at the right shows overlap populations with maximum range of (Si-0) in square brackets for the estimated standard deviations in the bond length (after Louisnathan and Gibbs, 1972b reproduced with the publisher s permission).

With any type of molecular modeling, there is generally a tradeoff between cost and reliability, and one typically shuns models that cost more without increasing reliability. In practice, this cost is usually expressed as computational effort, or computer time. In gas phase modeling, one typically finds molecular mechanics and semiempirical molecular orbital theory at the low-cost end and multireference configuration interaction or coupled-cluster theory at the other, with the choice dictated by the size of the system. System size also influences the choice of solvation model. We consider first the least expensive models, those that take no account of the quantum mechanical nature of the solute. 

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