Mixing of Degenerate Valence Bond Structures

Whenever a wave function is written as a normalized resonance hybrid between two VB structures of equivalent energies, for example, as in Equation 3.43, the energy of the hybrid is given by the normalized self-energies of the constituent resonance structures and the interaction matrix element, H 2, between the structures in Equation 3.44. 

A typical situation, where the VB wave function is written as a resonance hybrid, is odd-electron bonding (le or 3e bonds). For example, ale bond A B is a situation where only one electron is shared by two centers A and B (Eq. 3.46), while three electrons are distributed over the two centers in a 3e bond AB (Eq. 3.47)  

Simple algebra (see Exercise 3.3) shows that in both cases, the overlap between the two interacting VB structures is equal to S (the (a b) orbital overlap) and that resonance energy follows Equation 3.48  

Equation 3.48 also gives the bonding energy of a le bond. Combining Equations. 3.41 and 3.48, we get the bonding energy of the 3e bond, Equation 3.49  

These equations for odd-electron bonding energies are good for cases where the forms are degenerate or nearly so. In cases where the two structures are not identical in energy, one should use the perturbation theoretic expression (3). 

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SOME SIMPLE FORMULAS FOR ELEMENTARY INTERACTIONS 53 3.5.3 Mixing of Degenerate Valence Bond Structures... 

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