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

International Journal of Quantum Chemistry

Use of invariants in bounding 2‐fermion reduced density matrix eigenvalues

Mark E. Casida

Published Jan 1, 1988• 3 citations
Publisher

Abstract

Abstract The known bounds on the eigenvalues of 2‐Fermion reduced density matrices may be improved by considering unitarily invariant properties of the associated natural geminals and of the system 1‐Fermion reduced density matrix. The polynomial invariants are particularly important, since they may be introduced systematically according to the degree of the polynomials.

DOI
10.1002/qua.560330106

Related papers

Eigenvalues of Fermion Density Matrices

Fukashi Sasaki

The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be $n$ for a system of $2n$ or $2n+1$ identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.

Physical Review•Jun 7, 1965•143 citations

Algebraic structure of fermion density matrices. I

G. G. Dyadyusha, Eugene S. Kryachko

Abstract In this paper we define the algebraic structure of a reduced fermion density matrix. We relate the algebraic structure to certain symmetry properties of the reduced density matrix.

International Journal of Quantum Chemistry•Feb 1, 1981•3 citations

The role of model systems in the few‐body reduction of the N‐fermion problem

Jerome K. Percus

Abstract The problem of upper and lower ground state energy bounds for many‐fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.

International Journal of Quantum Chemistry•Jan 1, 1978•147 citations

Note on the bounds and approximations to the ground state energy of N-fermion systems derived from density matrix theory

F. David Peat

Certain lower bounds to the ground state energy of N-fermion systems have been derived in the literature using the properties of reduced density matrices.It is indicated that while some good approximations to the energy may be obtained by similar consideration the rigorous lower bound expressions lie too far below the exact energy to prove generally useful.

Canadian Journal of Physics•Jan 15, 1970•4 citations

Calculation of lower bounds to energy eigenvalues by reduced density matrices and the representability problem

Per‐Olov Löwdin, T. K. Lim

The expectation value of the Hamiltonian H for a many-electron system may be expressed in terms of the reduced second-order density matrix and the “reduced” Hamiltonian K for a two-electron system. It may be shown that the antisymmetric component HAS of the original Hamiltonian is an “outer projection” of K with respect to the antisymmetrizer O, so that HAS = OKO. This result implies that the eigenvalues of K are, in order, lower bounds to the eigenvalues of HAS, i.e. to the eigenvalues of H associated with the antisymmetry requirement. The expectation values of K are hence often below the ground state energy of H, and the eigenvalues may be used for the calculation of lower bounds. Some numerical applications to atoms serve as an illustration, and it is shown that the lower bounds become worse as the number of electrons increases. The implications for the “representability problem” are discussed.

International Journal of Quantum Chemistry•Jun 18, 2009•3 citations

Eigenvalue bounds for multi-particle reduced density matrices of Coulombic wavefunctions

Peter Hearnshaw

For bound states of atoms and molecules we obtain decay estimates for the eigenvalues of the K-particle reduced density matrix. Previously, eigenvalue bounds were obtained by A.V. Sobolev in the K = 1 case and, by extension, the K = N − 1 case. The purpose of the current work is to obtain bounds for the remaining cases.

Journal of Mathematical Physics•Mar 1, 2026•0 citations
Abstract The known bounds on the eigenvalues of 2‐Fermion reduced density matrices may be improved by considering unitarily invariant properties of the associated natural geminals and of the system 1‐Fermion reduced density matrix. The polynomial invariants are particularly important, since they may be introduced systematically according to the degree of the polynomials.
The least upper bound of the eigenvalues of second-order reduced density matrices for a system of fermions is proved to be nnn for a system of 2n2n2n or 2n+12n+12n+1 identical fermions. It is also shown that this limiting state may be interpreted as a system of identical pairs behaving as quasibosons.
Abstract The problem of upper and lower ground state energy bounds for many‐fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.
The expectation value of the Hamiltonian H for a many-electron system may be expressed in terms of the reduced second-order density matrix and the “reduced” Hamiltonian K for a two-electron system. It may be shown that the antisymmetric component HAS of the original Hamiltonian is an “outer projection” of K with respect to the antisymmetrizer O, so that HAS = OKO. This result implies that the eigenvalues of K are, in order, lower bounds to the eigenvalues of HAS, i.e. to the eigenvalues of H associated with the antisymmetry requirement. The expectation values of K are hence often below the ground state energy of H, and the eigenvalues may be used for the calculation of lower bounds. Some numerical applications to atoms serve as an illustration, and it is shown that the lower bounds become worse as the number of electrons increases. The implications for the “representability problem” are discussed.