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Physical Review A

Spin-adapted reduced Hamiltonian. I. Elementary excitations

Carmela Valdemoro

Published Apr 1, 1985• 69 citations

Abstract

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.

DOI: 10.1103/physreva.31.2114

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Spin-adapted reduced Hamiltonians. II. Total energy and reduced density matrices

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In the preceding paper (paper I of this series) the spin-adapted reduced Hamiltonian theory was developed and applied to the study of the elementary excitations of a test example; the Be atom. Here this theory is extended in order to obtain other observables: the total energy and the reduced density matrices for an m-electron system, always working within the two-body space. Preliminary applications of this formalism produce very satisfactory results.

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The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.

International Journal of Quantum Chemistry•Jan 1, 1997•20 citations

Traces of spin‐adapted reduced density matrices

Encarnación Pérez-Romero, Luis M. Tel, C. Valdemoro

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.

International Journal of Quantum Chemistry•Jan 1, 1997•5 citations

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.

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).

It is proposed that by evaluating pth-order spin-adapted reduced Hamiltonians and applying a model of independent groups of p electrons we can approximate the pth-order reduced density matrix (p-RDM). Subsequently this pth-order matrix is contracted to the corresponding 2-RDM. As a test example we calculate the total energy of the beryllium-atom isoelectronic sequence using an independent-trio model. The results, as expected, have a higher accuracy than those obtained with the independent-pair model.

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.

The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.