Writing and Representing VB Wave Functions

VB Wave Functions with Localized Atomic Orbitals 

We illustrate the theory by using, as an example, the two-electron/two-center (2e/2c) bond. A VB determinant is an antisymmetrized wave function that may or may not also be a proper spin eigenfunction. For example, ab in Eq. [2] is a determinant that describes two spin-orbitals a and b having one electron each the bar over the b orbital indicates a p spin, while its absence indicates an a spin  

One inconvenience of using the expression vB-fuii (Eq- [-5]) is its relative complexity compared to the simpler HL function (Eq. [3]). Coulson and Fischer proposed an elegant way of combining the simplicity of Fhl with the accuracy of vB-fuii- In the Coulson-Fischer (CF) wave function, I cF, the two-electron bond is described as a formally covalent singlet coupling between two orbitals and (p, which are optimized with freedom to delocalize over the two centers. This is exemplified below for H2 (dropping once again the normalization factors)  

Here a and b are purely localized AOs, while cp and cp are delocalized AOs. In fact, experience shows that the Coulson-Fischer orbitals cp and cp, which result from the energy minimization, are generally not very delocalized (s 1). As such they can be viewed as distorted orbitals that remain atomic-like in nature. However minor this may look, the slight delocalization renders the Coulson-Fischer wave function equivalent to the VB-full (Eq. [5]) wave function with the three classical structures. A straightforward expansion of the Coulson-Fischer wave function leads to a linear combination of classical structures in Eq. [7]. 

the Coulson-Fischer representation keeps the simplicity of the covalent picture while treating the covalent-ionic balance by embedding the effect of the ionic terms in a variational way through the delocalization tails. The Coulson-Fischer idea was later generalized to polyatomic molecules and gave rise to the 

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