Tensor surface harmonic, theory

The topological homeomorphism of a deltahedron to a sphere used in the graph-theory-derived models is also the basis of the tensor surface harmonic theory devel- 

The tensor surface harmonic theory defines the vertices of a 

The twin internal orbitals are uninodal (i.e., n-type)and lead to surface bonding described by the vector surface harmonics. Two vector surface harmonic functions can be generated from each Yu, as follows  

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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... 

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. ... 

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

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

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. 

A complementary approach is the Tensor Surface Harmonic Theory [19], based on the linear combination of atomic orbitals (LCAO) model, which explicitly incorporates the atomic positions. A set of atomic cores on the surface of a sphere are considered, and a basis set of s atomic orbitals used. If only these s orbitals are used, then the results are identical to the spherical jellium model. The three most stable orbitals are respectively 1,3 and 5 fold degenerate, leading to closed shells at 2, 8 and 18 electrons. 

The core and surface orbitals defined above by tensor surface harmonic theory can be related to the following aspects of the graph-theory-derived model for the skeletal bonding in boranes with the deltahedral structures depicted in Figure 4 ... 

Survey of cluster chemistry/Closed-shell electronic requirements for cluster com-pounds/Introduction to tensor surface harmonic theory/Clusters where radial... 

Recently, an elegant approach to the bonding in clusters has been developed by Stone [35], whose Tensor Surface Harmonic Theory derives cluster skeletal molecular orbitals as expansions of vector surface harmonic functions. The skeletal molecular orbitals are generated from basis sets with 0(o) and l(jt) nodes with respect to a radial vector passing through the atoms. 

Stone [35] showed that for deltahedral clusters the L" MOs are generally bonding. The L" MOs are, therefore, generally antibonding, since the parity inversion operation leads to a reversal of the bonding properties of L" and L . He used the methodology of Tensor Surface Harmonic theory to derive the n + 1 skeletal electron pair rule for... 

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