A Topological Semiempirical Hamiltonian

The spin-Hamiltonian VB theory rests on the same principles as the qualitative theory presented in Chapter 3, with some further simplifying assumptions. This chapter describes the method and focuses on its qualitative applications. 

A difference between the qualitative VB theory, discussed in Chapter 3, and the spin-Hamiltonian VB theory is that the basic constituent of the latter theory is the AO-based determinant, without any a priori bias for a given electronic coupling into bond pairs like those used in the Rumer basis set of VB structures. The bond coupling results from the diagonalization of the Hamiltonian matrix in the space of the determinant basis set. The theory is restricted to determinants having one electron per AO. This restriction does not mean, however, that the ionic structures are neglected since their effect is effectively included in the parameters of the theory. Nevertheless, since ionicity is introduced only in an effective manner, the treatment does not yield electronic states that are ionic in nature, and excludes molecules bearing lone pairs. Another simplification is the zero-differential overlap approximation, between the AOs. 

Let us now briefly describe the principles of the method and the rules for the construction of the Hamiltonian matrix. For the sake of consistency, rather than the original formulation (7-10), here we use a formulation that is in harmony with the VB theory in Chapter 3. The method can be summarized by the following principles (a) All overlaps between AOs are set to zero, (b) The energy of a VB determinant is proportional to the number of Pauli repulsions that exist between electrons with identical spins, which occupy adjacent AOs  

As already reasoned throughout this book, the lowest energy determinants are those possessing maximum spin-alternation. In a linear polyene or any 

FIGURE 8.1 Energies of the singlet ( Fo) and triplet excited ( P ) states of ethylene relative to the most spin-alternant determinant (MSAD). 

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