Understanding Valence Bond Theory

Abstract

This article describes the basic principles of VB theory. It defines VB structures for various situations such as electron-pair bonds, odd-electron bonds, Pauli repulsions and so on, and it teaches the reader how to write them in terms of Slater determinants constructed with localized or semi-localized atomic orbitals. The various types of hybridized atomic orbitals are defined. Formulas are given for computing overlaps between VB determinants and weights of VB structures in a multi-structure wave function. In this latter case, the way to choose a complete and linearly independent set of VB structures is given by the Rumer rules. The VB formalism using the exact Hamiltonian is described, allowing one to compute the energies of VB structures and the Hamiltonian matrix elements between them. A qualitative VB theory is outlined, using an effective Hamiltonian in terms of overlaps (S) and reduced resonance integrals (β) just like in the framework of extended Hückel theory. This qualitative VB model can be used to calculate the relative energies of various VB structures and their Hamiltonian matrix elements without the need of a computer. As such, the various guidelines can be applied to get insight into the nature of the ground state and excited states of molecules. This qualitative VB model is applied to understand the spin density patterns in the 11Ag ground state and 21Ag excited state of butadiene, and in polyenyl radicals. It is also applied to the Valence Ionization Spectrum of CH4 and to the electronic structures of the ground state and three lowest lying states of dioxygen. In the latter two cases, which are often alleged to be failures of VB theory, it is shown that application of the qualitative model yields results in perfect agreement with sophisticated ab initio VB calculations and with experiments. As such, the allegations of VB failures are found to be mythical, at any level of VB theory, both the accurate ab initio level as well as the simple qualitative level in terms of β and S integrals.