Three-orbital cluster functions

Stone applied similar reasoning to the problem of a three-dimensional cluster. Here, the solutions of the corresponding free-particle problem for an electron-on-a-sphere are spherical harmonics. These functions should be familiar because they also describe the angular properties of atomic orbitals. Two quantum numbers, L and M, are associated with the spherical harmonics, [Pg.1219]

Fhst, the number of valence functions (or frontier orbitals) and valence elechons (frontier orbital occupancy) determines the tendency toward cluster bonding. It is instructive to recall that the structural motif in elemental boron is the icosahedron with six-connected boron atoms see Borides Solid-state Chemistry), it is the tetrahedral carbon atom in the diamond form of elemental carbon with four-coimected carbon atoms and it is three-connected phosphorus atoms in the sheets of elemental black phosphorus (see Phosphides Solid-state Chemistry). Boron has more valence orbitals than valence elechons, naturally leading to orbitally rich cluster formation for example, BH has three orbitals and two elechons and forms... 

Silverstone and Sinanoglu [77] wrote the cluster expansion of fhe nonrela-f ivisfic N-elecf ron eigenfunction in terms of a zero-order reference wavefunc-fion fhaf is multicmfigurational, in accordance wifh the earlier suggestion of Wafson [31] and the study of H-ND in Be [75, 76]. In their formalism, the one-, two-, three-, etc. correlation functions (i.e., the virtual electron-excitations in the language of Cl) are linked fo spin orbitals from an extended zero-order set of occupied and unoccupied spin orbitals. This set was named the Hartree-Fock sea (H-F sea). Optimally, the H-F sea spin orbitals are supposed to be computed self-consistently. 

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). 

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