Unitary group tensor operator algebras for many-electron systems: I. Clebsch-Gordan and Racah coefficients

Abstract

A basis for the Racah-Wigner algebra of irreducible representations of the unitary group U(n) that are pertinent to quantum chemical models of many-electron systems is developed. Standard Clebsch-Gordan coefficients and isoscalar factors (also called coupling factors or reduced Wigner coefficients) for both symmetric (S N ) and unitary [U(n)] groups are extended to transformation coefficients and corresponding isoscalar factors relating canonical Young-Yamanouchi or Gel'fand-Tsetlin bases to simple partitioned bases. All these different types of isoscalar factors are interrelated using the well-known reciprocity between the S N and U(n) tensor representations, and general expressions relating these different factors are given. For the two-column representations characterizing the many-electron theory, detailed explicit expressions are presented for both the above-mentioned relationships and for all relevant U(n) isoscalar factors. Finally, U(n) Racah coefficients are introduced and explicit expressions derived for certain special classes of these coefficients.