Matrix elements of unitary group generators in many-fermion correlation problem. I. tensorial approaches

The objective of this series of papers is to survey important techniques for the evaluation of matrix elements (MEs) of unitary group generators and their products in the electronic Gel’fand–Tsetlin basis of two-column irreps of U(n) that are essential to the unitary group approach (UGA) to the many-electron correlation problem as handled by configuration interaction and coupled cluster approaches. Attention is also paid to the MEs of one-body spin-dependent operators and of their relationship to a standard spin-independent UGA formalism. The principal goal is to outline basic principles, concepts, and ideas without getting buried in technical details and thus to help an interested reader to follow the detailed developments in the original literature. In this first instalment we focus on tensorial techniques, particularly those designed specifically for UGA purposes, which exploit the spin-adapted tensorial analogues of the standard creation and annihilation operators of the ubiquitous second-quantization formalism. Subsequent instalments will address techniques based on the graphical methods of spin-algebras and on the Green–Gould polynomial formalism. In the “Appendix A” we then provide a succinct historical outline of the origins of the Lie group and Lie algebra concepts.