Pj. Burton et Md. Gould, MATRIX-ELEMENTS OF U(2N) GENERATORS IN A MULTISHELL SPIN-ORBIT BASIS .1. GENERAL FORMALISM, The Journal of chemical physics, 104(13), 1996, pp. 5112-5133
This is the first in a series of papers which derives the matrix eleme
nts of the spin-dependent U(2n) generators in a multishell spin-orbit
basis, i.e., a spin adapted composite Gelfand-Paldus basis. The advant
ages of such a multishell formalism are well known and well documented
. The approach taken exploits the properties of the U(n) adjoint tenso
r operator denoted by Delta(j)(i)(1 less than or equal to i,j less tha
n or equal to n) as defined by Gould and Paldus [IJ. Chem. Phys. 92, 7
394 (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 constructi
on of adjoint coupling coefficients for the zero-shift components of t
he U(2n) generators. The Racah factorization lemma may then be applied
to obtain the matrix elements of all the U(2n) generators. In this pa
per we investigate the underlying formalism of the approach and discus
s its advantages and its relationship to the shift operator method of
Gould and Battle [J. Chem. Phys. 99, 5961 (1993)]. The formalism is th
en 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 im
mediately yields the zero-shift adjoint coupling coefficients in such
a basis. The del-operator matrix elements are required for the calcula
tion of spin densities in a two-shell basis. In the third paper of the
series we derive the remaining nonzero shift adjoint coupling coeffic
ients all of which are required for the multishell case. We then use t
hese coupling coefficients to obtain formulas for the matrix elements
of the U(2n) generators in a two-shell spin-orbit basis. This result i
s 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 f
or a single-shell system. (C) 1996 American Institute of Physics.