Tensor surface harmonics

More advanced mathematical aspects of the graph-theoretical models for aromaticity are given in other references [36, 48, 49]. Some alternative methods, beyond the scope of this chapter, for the study of aromaticity in deltahedral molecules include tensor surface harmonic theory [51-53] and the topological solutions of non-linear field theory related to the Skyrmions of nuclear physics [54]. 

The bonding in gold cluster molecules has been interpreted using free electron models based on Stone s tensor surface harmonic theory [48, 49]. High similarity has... 

DFT = Density Functional Theory DSD = Diamond-Square-Diamond HOMO = Highest Occupied Molecular Orbital IR = Irreducible Representation LCAO = Linear Combination of Atomic Orbitals Ph = Phenyl PSEPT = Polyhedral Skeletal Electron Pair Theory SCF = Self-Consistent-Field SDDS = Square-Diamond, Diamond-Square TSH = Tensor Surface Harmonic. 

The remainder of this article is largely concerned with describing how some of the above observations can be rationalized using Stone s Tensor Surface Harmonic theory, and with the further imphcations of this model for dynamical processes such as cluster rearrangements. The number of example systems and electron count rationalizations will be kept relatively small in favor of explaining the theoretical foundations that underhe the method. Tables of examples and more detailed analyses of the various cases may be found elsewhere. ... 

Table 3.14 The modulating coefficients, reduced to their simplest ratios, for the linear combinations over the u [here and v [here 5 ] from equations 3.22 to form the d5 group orbitals. Figure 3.12, of the Fe3 using the tensor surface harmonics of Table 3.13.

Spherical-Shell Technique, Tensor Surface Harmonic Theory... 

The theoretical models which have been used to describe the bonding in cluster compounds of the main group and transition metal elements are reviewed. The historical development of these models is outlined and special emphasis is placed on those studies which have led to the elucidation of structure-electron count correlations. Theoretical treatments of cluster bonding are based on localised, delocalised (molecular orbital) or free electron methods derived from the solution of the Schrodinger equation for a particle on a sphere. A detailed analysis of the Tensor Surface Harmonic method, as an example of a free electron model, is presented. Group theoretical consequences of the model are also presented. 

Application of Tensor Surface Harmonic Theory to 3- and 4-Connected 68... 

IV. Point Group Symmetry Theorems Within the Tensor Surface Harmonic... 

To study the bonding in transition metal cluster compounds, a new type of Spherical Harmonic, with tensor properties, is required. This is because the metal d orbitals have (in addition to 1 o and 2 jt components) 2 8 components (d and dx2 y2) which are doubly noded in the plane perpendicular to the radial vector (see Fig. 16c) and which, therefore, behave as tensors. Two Tensor Surface Harmonic functions may be obtained from each Scalar Spherical Harmonic as follows146 ... 

The tensor Surface Harmonics are a measure of the concavity of the Scalar Spherical Harmonics. Since the concavities of S° and P° are zero, there are no S6 or P8 functions. 

Several years before the formulation of the Tensor Surface Harmonic Theory, King presented a graph theoretical treatment of bonding in borane clusters563. It is instructive to compare this with Stone s methodology. 

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