An Introduction to the Theory of Geminals

Abstract

Two-electron functions, called also geminals, have been around in quantum chemistry for some time. They represent a generalization of one-electron orbitals accounting for intra-orbital correlation effects. Geminal-based methods can be tailored to be variational as well as size-consistent and size-extensive. In spite of these appealing features, geminals became somewhat eclipsed in modern quantum chemistry because of their relative complexity and because the associated energies do not always cover a sufficient fraction of the correlation energy. However, several recent investigations revisit geminals and advocate the use of some extended geminal models which may turn out to offer useful alternatives to conventional approaches. In this paper, the formalism of two-electron functions will be reviewed in a simple fashion, focusing mainly on qualitative and conceptual points rather than technical details. After a short historical survey, the basic notions of geminals will be reviewed both in first- and second-quantized notations, the latter being especially advantageous when dealing with geminals. A few important points about the optimization of geminal-based wave functions will then be discussed, followed by a discussion about the inherent connection between geminals and the localization problem. We shall close with a few remarks on the prospect of geminal theories.