Modern valence bond theory
These developments are due to and described by Gerratt, Cooper, Karadakov and Raimondi (1997);[1] Li and McWeeny (2002); Joop H. van Lenthe and co-workers (2002);[2] Song, Mo, Zhang and Wu (2005); and Shaik and Hiberty (2004)[3] While molecular orbital theory (MOT) describes the electronic wavefunction as a linear combination of basis functions that are centered on the various atoms in a species (linear combination of atomic orbitals), VBT describes the electronic wavefunction as a linear combination of several valence bond structures.
[5] Although this is often overlooked, MOT and VBT are equally valid ways of describing the electronic wavefunction, and are actually related by a unitary transformation.
Assuming MOT and VBT are applied at the same level of theory, this relationship ensures that they will describe the same wavefunction, but will do so in different forms.
[4] Heitler and London's original work on VBT attempts to approximate the electronic wavefunction as a covalent combination of localized basis functions on the bonding atoms.
[6] In VBT, wavefunctions are described as the sums and differences of VB determinants, which enforce the antisymmetric properties required by the Pauli exclusion principle.
[5] However, by taking the sum and difference (linear combinations) of VB determinants, two approximate wavefunctions can be obtained: ΦHL is the wavefunction as described by Heiter and London originally, and describes the covalent bonding between orbitals a and b in which the spins are paired, as expected for a chemical bond.
This is a highly repulsive interaction, so this description of the bonding will not play a major role in determining the wave function.
[5] For more complicated molecules, ΦVBT could consider several possible structures that all contribute in various degrees (there would be several coefficients, not just λ and μ).
[1] In addition, the successful explanation of π-systems, pericyclic reactions, and extended solids further cemented MOT as the preeminent approach.
The relationship between MOT and VBT can be made more clear by directly comparing the results of the two theories for the hydrogen molecule, H2.
[7][5] It is known that O2 has a triplet ground state, but a classic Lewis structure depiction of oxygen would not indicate that any unpaired electrons exist.
[8] The photoelectron spectrum (PES) of methane is commonly used as an argument as to why MO theory is superior to VBT.
However, when one examines the VB description of CH4, it is clear that there are 4 equivalent bonds between C and H. If one were to invoke Koopman's Theorem (which is implicitly done when claiming that VBT is inadequate to describe PES), a single ionization energy peak would be predicted.
This allows from the distance between paired electrons to increase during variational optimization, lowering the resultant energy.
[8] The total wavefunction is described by a single set of orbitals, rather than a linear combination of multiple VB structures.
The difference is that the spin functions are allowed to adjust simultaneously with the orbitals during energy minimization procedures.
[5] Instead, this is a localization procedure that maps the full configuration interaction Hartree-Fock wavefunction (CASSCF) onto valence bond structures.
