Coulson–Fischer theory

Coulson–Fischer theory is an extension of modern valence bond theory that uses localized atomic orbitals as the basis for VBT structures.[4] In Coulson-Fischer Theory, orbitals are delocalized towards nearby atoms. This is described for H₂ as follows:[1]

${\displaystyle \phi _{1}=a+\lambda b}$

${\displaystyle \phi _{2}=b+\lambda a}$

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 ${\displaystyle \phi _{1}}$ and ${\displaystyle \phi _{2}}$ as the basis functions to describe the total electronic wavefunction as

${\displaystyle \Phi _{CF}=\left\vert \phi _{1}{\overline {\phi _{2}}}\right\vert -\left\vert {\overline {\phi _{1}}}\phi _{2}\right\vert }$

in obvious analogy to the Heitler-London wavefunction.[5] However, an expansion of the Coulson-Fischer description of the wavefunction in terms of a and b gives:

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

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

${\displaystyle \Phi _{VBT}=\epsilon (\left\vert a{\overline {b}}\right\vert -\left\vert {\overline {a}}b\right\vert )+\mu (\left\vert a{\overline {a}}\right\vert -\left\vert b{\overline {b}}\right\vert )}$

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.[4]

• Stephen Wilson. "The Coulson-Fischer theory of molecular electronic structure". Retrieved 2020-11-20.