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

Variational principles for the determination of single-particle density matrices

Viraht Sahni, Joseph B. Krieger

Published Feb 1, 1975• 19 citations
Publisher

Abstract

A variational formalism for the determination of the spinless single-particle density matrix $\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})$ for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator $X=Y+iZ$, where $Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]$, $W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, ${T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})$ is a translation operator, and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator $W$ employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.

DOI
10.1103/physreva.11.409

Related papers

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

Quantum Theory of Many-Particle Systems. I. Physical Interpretations by Means of Density Matrices, Natural Spin-Orbitals, and Convergence Problems in the Method of Configurational Interaction

Per‐Olov Löwdin

In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

Physical Review•Mar 15, 1955•3,587 citations

Exact and variational investigations of Hubbard rings

William Hodge

We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.

•Jan 1, 2008•0 citations

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

Many-body density matrices for free fermions

Siew Ann Cheong, Christopher L. Henley

Building upon an analytical technique introduced by Chung and Peschel [Phys. Rev. B 64, 064412 (2001)], we calculated the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a finite block of B sites within an infinite system of free spinless fermions in arbitrary dimensions. In terms of the block Green function matrix G (whose elements are ${G}_{\ifmmode \bar{\imath}\else \={\i}\fi{}j}=〈{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}〉,$ where ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${c}_{j}$ are fermion creation and annihilation operators acting on sites i and j within the block, respectively), the density matrix can be written as ${\ensuremath{\rho}}_{B}=\mathrm{det}(1\ensuremath{-}G)\mathrm{exp}({\ensuremath{\sum}}_{\mathrm{ij}}[\mathrm{ln}G(1\ensuremath{-}{G)}^{\ensuremath{-}1}{]}_{\mathrm{ij}}{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}).$ Our results suggest that Hilbert space truncation schemes should retain the states created by a subset of the ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$'s (in any combination), rather than selecting eigenvectors of ${\ensuremath{\rho}}_{B}$ independently based on the eigenvalue.

Physical Review B•Feb 27, 2004•163 citations

Variational Theorem for Reduced Density Matrices

Hsiao S. Kiang

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.

Journal of Mathematical Physics•Mar 1, 1967•4 citations
A variational formalism for the determination of the spinless single-particle density matrix \ensuremathγ(r\ensuremath→,r\ensuremath→\ensuremath′)\ensuremath{\gamma}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, {\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}})\ensuremathγ(r\ensuremath→,r\ensuremath→\ensuremath′) for an interacting many-particle system, correct to second order, is developed. The method is based on variational principles for determining single-particle expectation values, correct to second order, without requiring highly accurate wave functions in order to represent the interacting system. The expectation value of the operator X=Y+iZX=Y+iZX=Y+iZ, where Y=(12)\ensuremathΣj[W(r\ensuremath→i)Tj(a\ensuremath→)+W(r\ensuremath→j\ensuremath′)Tj(\ensuremath−a\ensuremath→)]Y=(\frac{1}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})+W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]Y=(21​)\ensuremathΣj​[W(r\ensuremath→i​)Tj​(a\ensuremath→)+W(r\ensuremath→j\ensuremath′​)Tj​(\ensuremath−a\ensuremath→)], Z=\ensuremath−(i2)\ensuremathΣj[W(r\ensuremath→j)Tj(a\ensuremath→)\ensuremath−W(r\ensuremath→j\ensuremath′)Tj(\ensuremath−a\ensuremath→)]Z=\ensuremath{-}(\frac{i}{2}){\ensuremath{\Sigma}}_{j}[W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})\ensuremath{-}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}^{\ensuremath{'}}){T}_{j}(\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})]Z=\ensuremath−(2i​)\ensuremathΣj​[W(r\ensuremath→j​)Tj​(a\ensuremath→)\ensuremath−W(r\ensuremath→j\ensuremath′​)Tj​(\ensuremath−a\ensuremath→)], W(r\ensuremath→j)=\ensuremathδ(r\ensuremath→j\ensuremath−r\ensuremath→)W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j})=\ensuremath{\delta}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{j}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})W(r\ensuremath→j​)=\ensuremathδ(r\ensuremath→j​\ensuremath−r\ensuremath→), Tj(a\ensuremath→){T}_{j}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}})Tj​(a\ensuremath→) is a translation operator, and a\ensuremath→=r\ensuremath→\ensuremath′\ensuremath−r\ensuremath→\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}={\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{\ensuremath{'}}\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}a\ensuremath→=r\ensuremath→\ensuremath′\ensuremath−r\ensuremath→, is the reduced single-particle density matrix. For a given choice of a Slater-determinant-type trial wave function it is shown that the density matrix may be obtained from the charge density of the system where the latter too is determined via the same formalism to second order as the expectation value of the operator WWW employing the same system trial wave function. The applicability of the technique is then demonstrated by showing that for the model hydrogen-atom problem it leads to highly accurate results for the momentum density and the Compton profile in the impulse approximation, and that the virial theorem is closely satisfied, even when the calculations of the various properties employing an approximate trial wave function alone are substantially in error.
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.
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of NNN antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order kkk has two sets of indices of each kkk variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order NNN over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete kkk-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly NNN spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order MMM, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
We study the one-dimensional Hubbard model for interacting electrons in a crystal lattice, using two methods: solution of the Lieb-Wu equations in the atomic limit of no intersite hopping, and variational techniques to obtain a lower bound for the ground state energy. From the Lieb-Wu equations we derive an exact expression for the ground state energy in the atomic limit. We obtain our result by exploiting the spin symmetry of the Hubbard Hamiltonian. Our energy is lower than that obtained by previous published results, and we are able to explain why our results differ. Our method also gives information about the magnetic ordering of the ground state as a function of the number of electrons. Our results are valid for arbitrary electron density values. The variational techniques we use are applied to the two-particle reduced-density matrix (2-RDM), because this quantity suffices to obtain expectation values of all one- and two-particle operators, and because it is less computationally intensive than obtaining the complete N -particle wave function. Because the 2-RDM must be positive semidefinite, we use an efficient programming algorithm known as the Semidefinite Programming Algorithm. A naive search through the space of two-particle matrices encounters a difficult problem, known as the " N -representability problem", which is to assure that the trial density matrices actually come from a wave function that has the required exchange symmetry. This problem is ameliorated by imposing constraints on the variational search that eliminate trial matrices with the incorrect exchange symmetry. These constraints express known physical properties of the ground state (for example its total spin and its translational symmetry) and the so-called D, Q , and G two-positivity conditions that express the required exchange symmetry. To calculate matrix elements, we use the two-particle eigenstates of the reduced Hubbard Hamiltonian; this choice of basis significantly decreases the number of matrix elements needed for energy calculations. After an optimal two-particle density matrix has been obtained, we calculate additional properties of the ground state, including spin correlation functions, chemical potentials, and the momentum distribution. We compare all of these properties to values obtained by exact numerical solution of the Schrödinger equation.
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.]
Building upon an analytical technique introduced by Chung and Peschel [Phys. Rev. B 64, 064412 (2001)], we calculated the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a finite block of B sites within an infinite system of free spinless fermions in arbitrary dimensions. In terms of the block Green function matrix G (whose elements are ${G}_{\ifmmode \bar{\imath}\else \={\i}\fi{}j}=〈{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}〉,$ where ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$ and cj{c}_{j}cj​ are fermion creation and annihilation operators acting on sites i and j within the block, respectively), the density matrix can be written as ${\ensuremath{\rho}}_{B}=\mathrm{det}(1\ensuremath{-}G)\mathrm{exp}({\ensuremath{\sum}}_{\mathrm{ij}}[\mathrm{ln}G(1\ensuremath{-}{G)}^{\ensuremath{-}1}{]}_{\mathrm{ij}}{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}).$ Our results suggest that Hilbert space truncation schemes should retain the states created by a subset of the ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$'s (in any combination), rather than selecting eigenvectors of ${\ensuremath{\rho}}_{B}$ independently based on the eigenvalue.