Matrix elements of U(2n) generators in a multishell spin-orbit basis .1. General formalism

摘要

This is the first in a series of papers which derives the matrix elements of the spin-dependent U(2n) generators in a multishell spin-orbit basis, i.e., a spin adapted composite Gelfand-Paldus basis. The advantages of such a multishell formalism are well known and well documented. The approach taken exploits the properties of the U(n) adjoint tensor operator denoted by Delta(j)(i)(1 less than or equal to i,j less than or equal to n) as defined by Gould and Paldus [IJ. Chem. Phys. 92, 7394 (1990)]. Delta is a polynomial of degree two in the U(n) matrix E=[E(j)(i)]. The unique properties of this operator allow the construction of adjoint coupling coefficients for the zero-shift components of the U(2n) generators. The Racah factorization lemma may then be applied to obtain the matrix elements of all the U(2n) generators. In this paper we investigate the underlying formalism of the approach and discuss its advantages and its relationship to the shift operator method of Gould and Battle [J. Chem. Phys. 99, 5961 (1993)]. The formalism is then applied, in the second paper of the series, to calculate the matrix elements of the del operator in a two-shell spin-orbit basis. This immediately yields the zero-shift adjoint coupling coefficients in such a basis. The del-operator matrix elements are required for the calculation of spin densities in a two-shell basis. In the third paper of the series we derive the remaining nonzero shift adjoint coupling coefficients all of which are required for the multishell case. We then use these coupling coefficients to obtain formulas for the matrix elements of the U(2n) generators in a two-shell spin-orbit basis. This result is then generalized, in the fourth paper, to the case of the multishell spin-orbit basis. Finally, we demonstrate that in the Gefand-Tsetlin limit the formula obtained is equivalent to that of Gould and Battle for a single-shell system. (C) 1996 American Institute of Physics.