MATRIX-ELEMENTS OF U(2N) GENERATORS IN A MULTISHELL SPIN-ORBIT BASIS - I - THE DEL-OPERATOR MES IN A 2-SHELL COMPOSITE GELFAND-PALDUS BASIS

This is the first in a series of three articles which aimed to derive the matrix elements of the U(2n) generators in a multishell spin-orbit basis. This is a basis appropriate to many-electron systems which hav e a natural partitioning of the orbital space and where also spin-depe ndent terms are included in the Hamiltonian. The method is based on a new spin-dependent unitary group approach to the many-electron correla tion problem due to Gould and Paldus [M. D. Gould and J. Paldus, J. Ch em. Phys. 92, 7394, (1990)]. In this approach, the matrix elements of the U(2n) generators in the U(n) x U(2)-adapted electronic Gelfand bas is are determined by the matrix elements of a single Ll(n) adjoint ten sor operator called the del-operator, denoted by Delta(j)(i) (1 less t han or equal to i, j less than or equal to n). Delta or del is a polyn omial of degree two in the U(n) matrix E = [E-j(i)]. The approach of G ould and Paldus is based on the transformation properties of the U(2n) generators as an adjoint tensor operator of U(n) x U(2) and applicati on of the Wigner-Eckart theorem. Hence, to generalize this approach, w e need to obtain formulas for the complete set of adjoint coupling coe fficients for the two-shell composite Gelfand-Paldus basis. The nonzer o shift coefficients are uniquely determined and may he evaluated by t he methods of Gould et al. [see the above reference]. In this article, we define zero-shift adjoint coupling coefficients for the two-shell composite Gelfand-Paldus basis which are appropriate to the many-elect ron problem. By definition, these are proportional to the correspondin g two-shell del-operator matrix elements, and it is shown that the Rac ah factorization lemma applies. Formulas for these coefficients are th en obtained by application of the Racah factorization lemma. The zero- shift adjoint reduced Wigner coefficients required for this procedure are evaluated first. All these coefficients are needed later for the m ultishell case, which leads directly to the two-shell del-operator mat rix elements. Finally, we discuss an application to charge and spin de nsities in a two-shell molecular system. (C) 1998 John Wiley & Sons.