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Physical Review A

Spin-adapted reduced Hamiltonian in view of the spectral-distribution method

Masao Nomura

Published Apr 1, 1988• 16 citations
Publisher

Abstract

Recently, Karwowski, Duch, and Valdemoro have developed a method for constructing a matrix representation of a spin-adapted reduced Hamiltonian based on evaluation of traces of occupation number operators over finite spaces of many electrons being coupled to a definite spin S. We show that the spectral-distribution method, used in nuclear physics, leads to a unified and general reduction formula for the fixed-S traces.

DOI
10.1103/physreva.37.2709

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Physical Review A•Sep 1, 1995•11 citations
Recently, Karwowski, Duch, and Valdemoro have developed a method for constructing a matrix representation of a spin-adapted reduced Hamiltonian based on evaluation of traces of occupation number operators over finite spaces of many electrons being coupled to a definite spin S. We show that the spectral-distribution method, used in nuclear physics, leads to a unified and general reduction formula for the fixed-S traces.
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 formulas for matrix elements of the pth-order spin-adapted reduced Hamiltonian (p-SRH) are derived based on the theory of generalized reduced density operators. The method is illustrated by a diagrammatic technique developed in detail for p=4. The p-SRH is a matrix representing a p-particle operator in a finite-dimensional p-particle Hilbert space. It is related to the N-particle (N\ensuremath{\ge}p) Hamiltonian defined in an N-particle spin-adapted, finite-dimensional Hilbert space. The p-SRH's, apart from their importance in theory of atomic and molecular structure, may find applications in statistical theories of spectra and in nuclear shell theory.
A formalism based on the unitary- and symmetric-group approaches to configuration-interaction methods has been applied to an evaluation of matrix elements of a spin-adapted reduced Hamiltonian (SRH). It has been shown that there is a simple closed-form expression for the matrix elements of the SRH matrices. This formula has been expressed as a linear combination of two-electron integrals with coefficients given as explicit functions of the parameters defining the finite-dimensional Hilbert space (number of orbitals, number of electrons, and the total spin quantum number).
In our previous studies with the evaluation of traces of p-order replacement operators calculated in finite-dimensional, antisymmetric, and spin-adapted N-electron spaces (p\ensuremath{\le}N), we described a technique for the calculation of those expressions based on the reduction of the opertor order [A. Torre, L. Lain, and J. Millan, Phys. Rev. A 47, 923 (1993)]. Now, we report a general formula which, condensing all the reduction steps, leads to the direct evaluation of those traces. Some examples for illuminating the usefulness of that formula are reported.