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Journal of Physics A Mathematical and General

Young invariant decomposition of spin-interacting operators

Xiangzhu Li

Published Mar 7, 1995• 1 citation
Publisher

Abstract

The Young invariant decomposition of totally symmetric spin-interacting operators is examined from the viewpoint of outer-product isoscalar factors of the symmetric group. The coefficients required for the decomposition of oneand two-body operators are constructed and given explicitly. The formalism enables a separate treatment of the spatial and spin parts of the matrix elements of the spin-interacting operators.

DOI
10.1088/0305-4470/28/5/021

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The Young invariant decomposition of totally symmetric spin-interacting operators is examined from the viewpoint of outer-product isoscalar factors of the symmetric group. The coefficients required for the decomposition of oneand two-body operators are constructed and given explicitly. The formalism enables a separate treatment of the spatial and spin parts of the matrix elements of the spin-interacting operators.
Sums of matrix elements of spin-dependent two-body momentum-independent interactions and sums of their products are calculated analytically in the basis of many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. As in the first part of the series [V. A. Yurovsky, Phys. Rev. A 91, 053601 (2015)], the sum dependence on the many-body states is given by universal factors, which are independent of the Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra and to calculation of two-body spin-dependent local correlations.
A formalism recently developed to obtain a matrix representation of the second-order spin-adapted reduced Hamiltonian [J. Karwowski, W. Duch, and C. Valdemoro, Phys. Rev. A 33, 2254 (1986)] and derived from the symmetric-group approach to configuration-interaction methods has been generalized to the case of the third-order spin-adapted reduced Hamiltonian. As in the second-order case, the matrix elements are linear combinations of generalized two-electron integrals with coefficients explicitly related to traces of products of the orbital occupation-number operators. The third-order spin-adapted reduced Hamiltonian matrix has been block diagonalized according to the irreducible representations of the symmetric group S3{S}_{3}S3​.
General, closed-form expressions are presented for the matrix elements of ssSK{\mathit{ssS}}_{\mathit{K}}ssSK​(si{\mathbf{s}}_{\mathit{i}}si​,sj{\mathbf{s}}_{\mathit{j}}sj​) (rank K=0,1,2), tensors in the single-particle spin operators si{\mathbf{s}}_{\mathit{i}}si​, in the unitary-group approach. Factorization of the matrix elements in terms of 6-j and 9-j coefficients is achieved through methods of spin-graphical analysis. Results are also obtained for the rank-2 spin-spin operator.
We present a derivation of closed-form algebraic expressions for two-body operators in the context of the special-unitary-group approach to complex spectroscopy. The method involves the use of a graphical representation of the permutational symmetry-adapted irreducible-representation basis which facilitates the decomposition of the matrix elements into simple factors. The results afford significant reductions of computational complexity in cases pertaining to higher-than-spin-1/2 N-particle systems.