Metal clusters orbital analysis

The application of these rules may fail for the prediction of magnetic coupling in transition-metal clusters, where the molecular orbitals may be different from simple atomic d-orbitals and a simple ligand field analysis may not reflect the correct energy splittings. 

A detailed analysis of (S2) in DFT can be found in Refs. (129,130).) Note the change of meaning in the summation indices in Eq. (91) In the second line i,j label electronic (spin) coordinates, while they denote the indices of spin orbitals in the third line. For the study of open-shell transition-metal clusters, it is necessary to obtain an expression for the total spin expectation value, where the summation rims over the number of a- and / -electrons rather than over the total number of electrons N. Thus, the sum in Eq. (91) may be split into four sums over the various spin combinations,... 

The analysis of these molecules is complicated by the presence of the d orbitals, which contribute one orbital of CT symmetry (d 2 in local axes), two of n symmetry (d and Ayg), and two of 5 symmetry (d y and dj2 j,2). There are far more examples of deviations from the usual patterns in transition metal clusters than in main group clusters, and many of these must be treated specifically, although TSH theory may again provide a useful framework in which to perform the analysis. However, in this section the objective will simply be to understand the most common patterns, as set out in Section 3, which are typically found in clusters with u-acceptor ligands such as CO. 

Qualitative molecular-orbital analysis of computational results. Rather than employing purely qualitative, symmetry-based theory, we can also perform calculations on solids or clusters and then analyze them using qualitative MO arguments. We will consider two such studies (1) an EHMO study of SiOj in (3-quartz, stishovite, and hypothetical silica-w structures (Burdett and Caneva, 1985), and (2) an MS-SCF-Aa study of electron-rich transition-metal compounds (Tossell and Vaughan, 1981). 

Three-dimensional polyhedral molecules such as the borane anions, B Hn , are characterised by a total of n -I- 1 skeletal electron pairs or 4 n -I- 2 valence electrons in total [33]. The isolobal analogy between the BH and ML3 fragments has provided a theoretical relationship between the borane and the transition metal cluster compounds [34]. A detailed analysis of the molecular orbitals of deltahedral transition metal clusters showed that they also have a total of n -h 1 bonding skeletal molecular orbitals, which is the same as that in the borane clusters, and 6 n orbitals that are either non-bonding or involved in the metal-ligand bonding. Therefore, such clusters are characterised by 14 n -I- 2 valence electrons in total. 

Transition metal clusters have the virtue of bringing d-type orbitals into the stage, for which an analysis similar to the one already carried ont for p-type orbitals (see Section 16.1) can also be made. 

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