Matrix elements of unitary group generators in many-fermion correlation problem. II. Graphical methods of spin algebras

Abstract

This second instalment continues our survey inter-relating various approaches to the evaluation of matrix elements (MEs) of the unitary group generators, their products, and spin-orbital U(2n) generators, in the basis of the electronic Gel’fand–Tsetlin (G–T) [or Gel’fand–Paldus (G–P) or P-] states spanning the carrier spaces of the two-column irreducible representations (irreps) of the unitary group approach (UGA) to the many-electron correlation problem. Exploiting an intimate relationship between the U(n), $${\mathcal {S}}_N$$ , and SU(2) groups we rely here on the SU(2) formalism within the confines of UGA rather than on an earlier approach relying on the permutation group $${\mathcal {S}}_N$$ . The key is a close relationship between the P-states of UGA and the Yamanouchi–Kotani (Y–K) states of SU(2) that differ only in a phase. This enables one to exploit the powerful graphical techniques of spin (or angular momentum) algebras as first developed by Jucys. We show the usefulness and power of these techniques in the problem of UGA ME evaluation which, moreover, leads automatically to their segmentation, first derived for single generators by Shavitt and for generator products by Boyle and Paldus. We also show how the same technique can be successfully employed for MEs of spin-dependent U(2n) generators in the UGA P-basis.