Coulson–Fischer theory

Theory

Coulson–Fischer theory is an extension of modern valence bond theory that uses localized atomic orbitals as the basis for VBT structures. In Coulson-Fischer Theory, orbitals are delocalized towards nearby atoms. This is described for H₂ as follows:

${\displaystyle {\varphi }_1=a+\lambda b}$

${\displaystyle {\varphi }_2=b+\lambda a}$

where a and b are atomic 1s orbitals, that are used as the basis functions for VBT, and λ is a delocalization parameter from 0 to 1. The VB structures then use ${\displaystyle {\varphi }_1}$ and ${\displaystyle {\varphi }_2}$ as the basis functions to describe the total electronic wavefunction as

${\displaystyle {\mathrm{\Phi }}_{CF}=\left|{\varphi }_1\overline{{\varphi }_2}\right|-\left|\overline{{\varphi }_1}{\varphi }_2\right|}$

in obvious analogy to the Heitler-London wavefunction. However, an expansion of the Coulson-Fischer description of the wavefunction in terms of a and b gives:

${\displaystyle {\mathrm{\Phi }}_{CF}=(1+{\lambda }^2)(\left|a\overline{b}\right|-\left|\overline{a}b\right|)+(2\lambda )(\left|a\overline{a}\right|-\left|b\overline{b}\right|)}$

A full VBT description of H₂ that includes both ionic and covalent contributions is

${\displaystyle {\mathrm{\Phi }}_{VBT}=ϵ(\left|a\overline{b}\right|-\left|\overline{a}b\right|)+\mu (\left|a\overline{a}\right|-\left|b\overline{b}\right|)}$

where ε and μ are constants between 0 and 1.

As a result, the CF description gives the same description as a full valence bond description, but with just one VB structure.