Resonating valence bond expansion

There are many expansions of the total many-electron wave function which may be used in discussing the properties of an electronic system. For the present discussion, we assume a resonating valence bond expansion, which is represented by... 

Table 2 Results of Variational Localized-site Cluster expansions from either a Neel-state based ansatze or a Resonating Valence Bond ansatz. We notice that the lower level NSBA is unable of showing the dimerization of polyacetylene. rc is the critical bifurcation mean bond length, r and r2 are the optimized short and long bond distances (in A). E is the energy per carbon atom (in eV), taking the energy of the Neel state with 1.40 A equal bond lengths as zero of energy. NSB forth order perturbative and Dimer-covering second order perturbative (see Ref. 34), CEPA ab-initio estimate of Kpnig and Stollhoff [52], and the experimental results [46,47] for rx and r2 have been added for comparison.

Returning, then, to the expansion of Equation (2), we note that the terms represent different valence bond structures. Why should they all have the same amplitude and phase This situation is very similar to the problem of determining the "resonance energy" of ben-zenoid molecules (25,26,27). In that case, of all the possible valence bond structures which might contribute, only the Kekule structures are used. For large benzenoid systems this is only a small fraction of the total number of structures. Furthermore, it is assumed that they all enter with equal expansion coefficients (i.e., equal amplitude and phase). In addition, the matrix elements which convert one structure into another are set equal to a common value, determined empirically. Thus, the energy lowering associated with "resonance" in benzenoid molecules has a mathematical structure which maps onto the discussion in the Introduction. However, there are some important differences. 

A significant difference should be noted between Pauling s treatment of valence bonds and the corresponding NRT description of polar covalent or ionic bonding. In Pauling s formulation each polar bond requires two distinct resonance structures for its depiction, one covalent and one ionic. However, in the NRT framework each localized 2-center electron pair is represented as a bond-line of a single resonance structure, whether the bond is polar (ca /cb) or nonpolar (ca = cb)- Avoidance of covalent-ionic resonance in the NRT framework greatly reduces the number of NRT resonance structures required in expansions such as equation (26). 

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