Modern valence bond theory is the application of valence bond theory (VBT) with computer programs that are competitive in accuracy and economy, with programs for the Hartree–Fock or post-Hartree-Fock methods. The latter methods dominated quantum chemistry from the advent of digital computers because they were easier to program. The early popularity of valence bond methods thus declined. It is only recently that the programming of valence bond methods has improved. These developments are due to and described by Gerratt, Cooper, Karadakov and Raimondi (1997); Li and McWeeny (2002); Joop H. van Lenthe and co-workers (2002); Song, Mo, Zhang and Wu (2005); and Shaik and Hiberty (2004) 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. Each of these valence bond structures can be described using linear combinations of either atomic orbitals, delocalized atomic orbitals (Coulson-Fischer theory), or even molecular orbital fragments. 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.
Theory
Bonding in H2 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. In VBT, wavefunctions are described as the sums and differences of VB determinants, which enforce the antisymmetric properties required by the Pauli exclusion principle. Taking H2 as an example, the VB determinant is
| a b ¯ | = N { a ( 1 ) b ( 2 ) [ α ( 1 ) β ( 2 ) ] − a ( 2 ) b ( 1 ) [ α ( 2 ) β ( 1 ) ] } {\displaystyle \left\vert a{\overline {b}}\right\vert =N\{a(1)b(2)[\alpha (1)\beta (2)]-a(2)b(1)[\alpha (2)\beta (1)]\}}
In this expression, N is a normalization constant, and a and b are basis functions that are localized on the two hydrogen atoms, often considered simply to be 1s atomic orbitals. The numbers are an index to describe the electron (i.e. a(1) represents the concept of ‘electron 1’ residing in orbital a). ɑ and β describe the spin of the electron. The bar over b in | a b ¯ | {\displaystyle \left\vert a{\overline {b}}\right\vert } indicates that the electron associated with orbital b has β spin (in the first term, electron 2 is in orbital b, and thus electron 2 has β spin). By itself, a single VB determinant is not a proper spin-eigenfunction, and thus cannot describe the true wavefunction. However, by taking the sum and difference (linear combinations) of VB determinants, two approximate wavefunctions can be obtained:
Φ H L = | a b ¯ | − | a ¯ b | {\displaystyle \Phi _{HL}=\left\vert a{\overline {b}}\right\vert -\left\vert {\overline {a}}b\right\vert }
Φ T = | a b ¯ | + | a ¯ b | {\displaystyle \Phi _{T}=\left\vert a{\overline {b}}\right\vert +\left\vert {\overline {a}}b\right\vert }
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. ΦT is a representation of the bond that where the electron spins are parallel, resulting in a triplet state. This is a highly repulsive interaction, so this description of the bonding will not play a major role in determining the wave function. Other ways of describing the wavefunction can also be constructed. Specifically, instead of considering a covalent interaction, the ionic interactions can be considered, resulting in the wavefunction
Φ I = | a a ¯ | + | b ¯ b | {\displaystyle \Phi _{I}=\left\vert a{\overline {a}}\right\vert +\left\vert {\overline {b}}b\right\vert }
This wavefunction describes the bonding in H2 as the ionic interaction between an H+ and an H−. Since none of these wavefunctions, ΦHL (covalent bonding) or ΦI (ionic bonding) perfectly approximates the wavefunction, a combination of these two can be used to describe the total wavefunction
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