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physica status solidi (b)

Tensorial Coupling and the Reduced Matrix Elements of One‐ and Two‐Body Operators

Jack A. Tuszyński, J.M. Dixon

Published Feb 1, 1985• 4 citations
Publisher

Abstract

Abstract The reduced matrix elements of products of tensor operators, coupled to form a non‐zero rank, are investigated to emphasize the importance of correctly incorporating all tensorial couplings. It has recently been realized that serious inaccuracies can arise, in some cases, unless the complete many body nature of the full operator and the states involved is properly incorporated. Exact results are presented for one‐ and two‐body operators which include all couplings and compare and contrast them with some approximate methods.

DOI
10.1002/pssb.2221270215

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Abstract The reduced matrix elements of products of tensor operators, coupled to form a non‐zero rank, are investigated to emphasize the importance of correctly incorporating all tensorial couplings. It has recently been realized that serious inaccuracies can arise, in some cases, unless the complete many body nature of the full operator and the states involved is properly incorporated. Exact results are presented for one‐ and two‐body operators which include all couplings and compare and contrast them with some approximate methods.
Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.
A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.