The VB Model for Conjugated Molecules

It is well known that the properties of conjugated molecules are principally determined by their rc-electrons. Furthermore, planar conjugated molecules are prototypical in that their n-electrons could be separately treated from the remaining o -electrons. Hence, semiempirical theoretical models developed mainly for conjugated molecules treated only the jr-electrons explicitly but incorporated the effects of the o -electrons and the nuclei into some adjustable parameters featuring these models. The VB model (17) could be such a model, which may be further simplified as 

Now we are concerned with how the VB model (18) can be solved for a given conjugated system. In fact, the model Hamiltonian (18) actually acts on the space of pure spin functions, either of the Weyl-Rumer (WR) form [34] or the simple product of one-electron spin functions. The matrix element between any two WR functions can be obtained by using Pauling s graphical rules [4], while the matrix element between two simple spin products is easily available using the following expression 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then 

Here aj and a, are the creation operator of the spin-down electron and the annihilation operator of the spin-up at site i, respectively. In the derivation, we have employed the relationships such as SJ+ = a] a, Sj =a jaJ, 

SA = /2(a+jaj-aja-j), and so on, which are easily confirmed by testing the effect of each side of these expressions on the two spin orbitals 0 (r,.)a(.s ) and being the 2pn orbital of the jth carbon atom. 

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