Motivation for wave functions built from electron pairs
In traditional one-electron wave function-based electronic structure methods, we start from a set of spinor single-particle functions \(\require{physics} \set{\phi_P(\vb{r})}\), which are also loosely called orbitals. We then adopt a second quantized formalism, in which we create and annihilate electrons in these spinors, such that we then describe the wave function (model) for \(N\) electrons in an \(N\)-electron Fock space, whose basis vectors are the occupation number vectors (ONVs).
According to Kutzelnigg (Miller et al. 1977), an electron pair theory can then be described as follows. For a system with an even number of electrons, we use as many two-electron functions \(\set{\omega_{RS}(\vb{r})}\) as there are electron pairs, where some or all of these pair functions are written in terms of products two one-electron functions. A geminal-type wave function should thus be able to account for intra-pair correlation effects. (P. R. Surján 2016) We effectively reduce to a one-electron theory if every pair function is solely written as a product of two one-electron functions. (Miller et al. 1977)
In this thesis, we explore wave function models based on geminals, i.e. electron pairs, from the Latin word , meaning twin. A more proper definition for geminals, especially suited for this thesis would be electron pairs.
Electron pairs occur in many models in chemistry. For example, in organic chemistry, we find them in Lewis structures (Lewis 1916) and in materials chemistry, we encounter them in Bardeen-Cooper-Schieffer (Bardeen, Cooper, and Schrieffer 1957) pairs used to describe super conductivity.
The molecular Hamiltonian contains at most two-electron interactions, so it is natural to expect that an electron pair theory can work to describe \(N\) electrons. (Miller et al. 1977) Furthermore, due to the Pauli principle, only two electrons can come close to occupying the same point in space, so three-particle contributions are expected to have only neglegible short-range contributions. (Miller et al. 1977)
If we can describe two-electron pair subunits, it would be possible to qualitatively describe chemical processes in which a single bond is broken. (P. R. Surján 2016) Geminals would then take into account intra-pair correlation, but would describe inter-pair correlation only in a mean-field way. (P. R. Surján 2016)
In order to aid the convergence of the configuration interaction expansion, it seems that including inter-electron coordinates \(\vb{r}_{12}\) is important: this can be described with pair functions. (P. R. Surján 2016)
Analogously, it is also possible to construct triplets or quartets, but bigger ‘quasi-particles’ are not always resembling physical or chemical pictures. (P. R. Surján 2016) The reason why geminals are such a natural building block is because two-electron pair functions can be put together without breaking the molecular symmetry: think of the classical \(\sigma\)-bonds in methane, for example. (P. R. Surján 2016)