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

Density Matrix and the Many-Body Problem

Richard H. Tredgold

Published Mar 1, 1957• 84 citations
Publisher

Abstract

The application of the two-particle density matrix to the calculation of approximate ground states for many-body systems is discussed. It is shown that the subsiduary conditions employed by Mayer in this context are insufficient to ensure that a valid result is obtained. The formulation of adequate subsidiary conditions presents considerable difficulty.

DOI
10.1103/physrev.105.1421

Related papers

Reduction of the N -Particle Variational Problem

Claude Garrod, Jerome K. Percus

A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]

Journal of Mathematical Physics•Dec 1, 1964•477 citations

Density Matrices in the Many-Body Problem

P. Gluck

The problem of correlations is considered in many-fermion systems, although with some modifications the methods are more general. The many-body problem is formulated in terms of reduced one- and two-particle density matrices, and the equations for these are treated by variational and iterative methods. The formalism is applied to a calculation of the energy per particle for a system interacting with Coulomb, screened Coulomb, and singular potentials. Comparisons with the ideas of Brueckner and others are made, and difficulties are discussed.

Physical Review•Dec 25, 1968•1 citation

Remarks on Mayer's Reduced Density Matrix Method

Yukio Mizuno, Takeo Izuyama

Mayer has proposed a variation method of finding the two-body reduced density matrix ρ2 for the ground state of a many fermion system. In this paper it is shown explicitly that his variation method is imperfect and sometimes leads to quite unsatisfactory results. Further, his trial form of ρ2 for electron gas is translated into that for the π electron system of benzene molecule and it is shown that such a trial form is unsatisfactory in several respects.

Progress of Theoretical Physics•Jul 1, 1957•24 citations

Reexamining the variational two-particle reduced density matrix for nuclear systems

J. G. Li, N. Michel, Zuo Wei +1 more

In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.

arXiv:2106.04103•Physical Review C•Jun 23, 2021•2 citations

Density-matrix formulation ofab initiomethods of nonrelativistic quantum mechanics

Antonio Redondo

A density-matrix formulation of the ab initio methods of nonrelativistic quantum mechanics is presented. The variational equations for the standard calculational techniques of molecular quantum mechanics are derived from a single expression for the total energy of an N-particle system. A generalization of the Hohenberg-Kohn theorem is proved, showing that the ground-state energy of a many-body Hamiltonian is a functional of the single-particle density matrix and hence of the first-order reduced-density function (the diagonal of which is the particle density of the standard theorem). The theorem is used to develop a new method to calculate the total energy and single-particle properties of the many-body problem.

Physical Review A•May 1, 1989•6 citations

Variational Approach to the Many-Body Problem

R. U. Ayres

Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.

Physical Review•Sep 15, 1958•39 citations
The application of the two-particle density matrix to the calculation of approximate ground states for many-body systems is discussed. It is shown that the subsiduary conditions employed by Mayer in this context are insufficient to ensure that a valid result is obtained. The formulation of adequate subsidiary conditions presents considerable difficulty.
A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]
The problem of correlations is considered in many-fermion systems, although with some modifications the methods are more general. The many-body problem is formulated in terms of reduced one- and two-particle density matrices, and the equations for these are treated by variational and iterative methods. The formalism is applied to a calculation of the energy per particle for a system interacting with Coulomb, screened Coulomb, and singular potentials. Comparisons with the ideas of Brueckner and others are made, and difficulties are discussed.
In most nuclear many-body methods, observables are calculated using many-body wave functions explicitly. The variational two-particle reduced density matrix method is one of the few exceptions to the rule. Ground-state energies of both closed-shell and open-shell nuclear systems can indeed be evaluated by minimizing a constrained linear functional of the two-particle reduced density matrix. However, it has virtually never been used in nuclear theory, because nuclear ground states were found to be well overbound, contrary to those of atoms and molecules. Consequently, we introduced new constraints in the nuclear variational two-particle reduced density matrix method, developed recently for atomic and molecular systems. Our calculations then show that this approach can provide a proper description of nuclear systems where only valence neutrons are included. For the nuclear systems where both neutrons and protons are active, however, the energies obtained with the variational two-particle reduced density matrix method are still overbound. The possible reasons for the noticed discrepancies and solutions to this problem will be discussed.
A density-matrix formulation of the ab initio methods of nonrelativistic quantum mechanics is presented. The variational equations for the standard calculational techniques of molecular quantum mechanics are derived from a single expression for the total energy of an N-particle system. A generalization of the Hohenberg-Kohn theorem is proved, showing that the ground-state energy of a many-body Hamiltonian is a functional of the single-particle density matrix and hence of the first-order reduced-density function (the diagonal of which is the particle density of the standard theorem). The theorem is used to develop a new method to calculate the total energy and single-particle properties of the many-body problem.
Certain problems connected with the choice of trial variational functions for the quantum many-body problem are discussed. It is convenient to make use of the reduced-density-matrix formulation developed by L\"owdin and Mayer, since a system whose Hamiltonian contains only 1- and 2-body interactions is characterized by the two simplest reduced density matrices ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. It is known that symmetry, Hermiticity, and normalizability conditions are not sufficient to ensure a physically realizable choice of the functions ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$. However an additional nontrivial restriction is imposed by the fact that such functions must be possible variational extrema. The implications of this condition are investigated in some detail and it turns out that all the nonphysical choices of ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$ which have been exhibited by various authors are eliminated. The condition is essentially nonquantum inasmuch as an analogous argument should be possible for a classical system. Moreover one of the consequences of this condition is that the system (classical or quantum) may exhibit macroscopic behavior, i.e., order-disorder transitions. The analysis leads to apparently reasonable choices of trial forms for ${\ensuremath{\Gamma}}^{(1)}$ and ${\ensuremath{\Gamma}}^{(2)}$, although it is still not known whether any further restriction must be imposed to complete the sufficiency argument.