Coulson–Fischer theory

Electronic structure methods
Valence bond theory
Coulson–Fischer theory Generalized valence bond Modern valence bond theory
Molecular orbital theory
Hartree–Fock method Semi-empirical quantum chemistry methods Møller–Plesset perturbation theory Configuration interaction Coupled cluster Multi-configurational self-consistent field Quantum chemistry composite methods Quantum Monte Carlo
Density functional theory
Time-dependent density functional theory Thomas–Fermi model Orbital-free density functional theory Linearized augmented-plane-wave method Projector augmented wave method
Electronic band structure
Nearly free electron model Tight binding Muffin-tin approximation k·p perturbation theory Empty lattice approximation GW approximation Korringa–Kohn–Rostoker method
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In theoretical chemistry and molecular physics, Coulson–Fischer theory provides a quantum mechanical description of the electronic structure of molecules. The 1949 seminal work of Coulson and Fischer^[1] established a theory of molecular electronic structure which combines the strengths of the two rival theories which emerged soon after the advent of quantum chemistry - valence bond theory and molecular orbital theory, whilst avoiding many of their weaknesses. For example, unlike the widely used Hartree–Fock molecular orbital method, Coulson–Fischer theory provides a qualitatively correct description of molecular dissociative processes.^[2] The Coulson–Fischer wave function has been said to provide a third way in quantum chemistry.^[3] Modern valence bond theory is often seen as an extension of the Coulson–Fischer method.

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]

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

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

\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:

\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

\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]

References

^ ^a ^b C.A. Coulson and I. Fischer, Notes on the Molecular Orbital Treatment of the Hydrogen Molecule, Phil. Mag. 40, 386 (1949)
^ S. Wilson and J. Gerratt, Calculation of potential energy curves for the ground state of the hydrogen molecule, Molec. Phys. 30, 777 (1975) https://doi.org/10.1080/14786444908521726
^ S. Wilson, On the Wave Function of Coulson and Fischer: A Third Way in Quantum Chemistry, in Advances in the Theory of Atomic and Molecular Systems, ed. P. Piecuch, J. Maruani, G. Delgado-Barrio and S. Wilson, Progress in Theoretical Chemistry and Physics 19, Springer (2009)
^ ^a ^b Shaik, Sason; Hiberty, Philippe C. (2007-11-16). A Chemist's Guide to Valence Bond Theory. Hoboken, NJ, USA: John Wiley & Sons, Inc. doi:10.1002/9780470192597. ISBN 978-0-470-19259-7.
^ "Heitler, W., & London, F. (1927). Wechselwirkung neutraler Atome und homopolare Bindung nach der Quantenmechanik. Zeitschrift für Physik, 44, 455-472. - References - Scientific Research Publishing"