Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Matrix elements of the third-order spin-adapted reduced Hamiltonian

Luis Laín, Alicia Torre, Jacek Karwowski, Carmela Valdemoro

Published Sep 1, 1988• 28 citations
Publisher

Abstract

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 ${S}_{3}$.

DOI
10.1103/physreva.38.2721

Related papers

Matrix elements of a spin-adapted reduced Hamiltonian

Jacek Karwowski, Włodzisław Duch, Carmela Valdemoro

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).

Physical Review A•Apr 1, 1986•50 citations

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

Masao Nomura

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.

Physical Review A•Apr 1, 1988•16 citations

Symmetric-group approach to the study of the traces ofp-order reduced-density operators and of products of these operators

Josep Planelles, Carmela Valdemoro, Jacek Karwowski

In this work we give the values of traces of p-order reduced-density operators. These traces are obtained by application of the spin functions and of the symmetric-group properties. The relations obtained here will allow an easy and fast evaluation of the high-order spin-adapted reduced Hamiltonian matrix elements and high-order Hamiltonian moments.

Physical Review A•Mar 1, 1990•33 citations

Matrix elements of spin-adapted reduced Hamiltonians

Josep Planelles, Carmela Valdemoro, Jacek Karwowski

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.

Physical Review A•Apr 1, 1991•15 citations

Second-order spin-adapted reduced Hamiltonian in the coordinate representation

Jacek Karwowski, Carmela Valdemoro, Luis Laín

The second-order spin-adapted reduced Hamiltonian, usually represented as a matrix, is transformed to the form of a two-body operator in the coordinate representation. The new form of this Hamiltonian deepens our insight into its structure and physical meaning. An analysis of its asymptotic form for complete Hilbert spaces gives a formal proof of an earlier assumption that the spin-adapted reduced Hamiltonian describes independent electron pairs.

Physical Review A•May 1, 1989•6 citations

Spin-adapted reduced Hamiltonian. I. Elementary excitations

Carmela Valdemoro

The spin-adapted pth-order reduced Hamiltonians (SRH) are obtained for an m-electron system, within the formalism of second quantization. This transformation is applied to the usual many-body Hamiltonian and involves an augmenting mapping, a spin projection, and a reducing mapping into the p-body reduced space. These three operations are combined into a single algorithm, allowing the work to be carried out within the two-body space. The SRH matrices thus obtained (1) have the same trace as that of the corresponding full configuration-interaction (CI) matrix, and (2) can be written as expansions involving the p eigenvalues and the corresponding pth-order reduced density matrices (RDM) of the m-body system in the chosen spin symmetry. Both of these properties guarantee that all and only the relevant information about our system is contained in the SRH matrices (and the SRH operators). The excitations of the eigenstates of the 1-SRH and the 2-SRH can be interpreted as the normal modes (or elementary excitations) of our electronic system in a given symmetry. A preliminary calculation on the Be atom yields encouraging results when compared with experiment. In the following paper (paper II of this series) the theory is applied to obtain the total energy and the reduced density matrices.

Physical Review A•Apr 1, 1985•69 citations
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​.
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).
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.
The spin-adapted pth-order reduced Hamiltonians (SRH) are obtained for an m-electron system, within the formalism of second quantization. This transformation is applied to the usual many-body Hamiltonian and involves an augmenting mapping, a spin projection, and a reducing mapping into the p-body reduced space. These three operations are combined into a single algorithm, allowing the work to be carried out within the two-body space. The SRH matrices thus obtained (1) have the same trace as that of the corresponding full configuration-interaction (CI) matrix, and (2) can be written as expansions involving the p eigenvalues and the corresponding pth-order reduced density matrices (RDM) of the m-body system in the chosen spin symmetry. Both of these properties guarantee that all and only the relevant information about our system is contained in the SRH matrices (and the SRH operators). The excitations of the eigenstates of the 1-SRH and the 2-SRH can be interpreted as the normal modes (or elementary excitations) of our electronic system in a given symmetry. A preliminary calculation on the Be atom yields encouraging results when compared with experiment. In the following paper (paper II of this series) the theory is applied to obtain the total energy and the reduced density matrices.