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

Journal of Mathematical Physics

Variational Theorem for Reduced Density Matrices

Hsiao S. Kiang

Published Mar 1, 1967• 4 citations
Publisher

Abstract

This paper discusses a theorem concerning the variational description of the eigenfunctions and eigenvalues of the two complementary reduced density matrices for a many-particle system in a bound state.
DOI
10.1063/1.1705212

Related papers

Some Properties of the Reduced Density Matrix

Chongming Yang

Two conjectures made in a previous paper are proved. A further remark is made concerning the largest eigenvalue of the reduced density matrices.

Journal of Mathematical Physics•Mar 1, 1963•10 citations

Eigenvalues of Density Matrices

B. C. Carlson, Joseph M. Keller

For a system of $N$ identical particles in a pure bound state, the density matrices of positive orders $p$ and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.

Physical Review•Feb 1, 1961•134 citations

Necessary conditions for the N -representability of the second-order reduced density matrix: Upper bounds on the P and Q matrices

Dimitri Van Neck, Paul W. Ayers

Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``$P$'' and ``$Q$'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for $N$-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''

Physical Review A•Mar 5, 2007•13 citations

Conditions on the Expansion Coefficients of the Reduced Two-Particle Density Matrix

Louis J. Kijewski

An explicit reduction in the number of terms of the expansion of the reduced two-particle density matrix for atomic systems is shown to occur when the expansion functions are eigenfunctions of ${L}_{z}(1)+{L}_{z}(2)$ and ${S}_{z}(1)+{S}_{z}(2)$.

Physical Review A•Jan 1, 1970•1 citation

Contraction relations for Grassmann products of reduced density matrices and implications for density matrix reconstruction

John M. Herbert, John E. Harriman

We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.

Physical Review A•Jan 16, 2002•27 citations

A variational principle on certain operators in general and the density matrix in particular

L. L. Van Zandt

Physics Letters A•Apr 1, 1969•0 citations
For a system of NNN identical particles in a pure bound state, the density matrices of positive orders ppp and $N\ensuremath{-}p$ have the same nonzero eigenvalues with the same multiplicities. If the number of nonzero eigenvalues is finite, these density matrices are unitarily equivalent.
Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``PPP'' and ``QQQ'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for NNN-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''
An explicit reduction in the number of terms of the expansion of the reduced two-particle density matrix for atomic systems is shown to occur when the expansion functions are eigenfunctions of Lz(1)+Lz(2){L}_{z}(1)+{L}_{z}(2)Lz​(1)+Lz​(2) and Sz(1)+Sz(2){S}_{z}(1)+{S}_{z}(2)Sz​(1)+Sz​(2).
We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.