Physical Review A

Hierarchy equations for reduced density matrices

Leon W. Cohen, C. Frishberg

Published Mar 1, 1976• 169 citations

Abstract

Starting with Schr\"odinger's time-independent equation for $N$ indistinguishable particles, an exact equation is derived for the reduced density matrix of order $s$, ${\ensuremath{\rho}}_{s}$. The equation contains terms dependent on ${\ensuremath{\rho}}_{s+1}$ and ${\ensuremath{\rho}}_{s+2}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\ensuremath{\rho}}_{2}$ and ${\ensuremath{\rho}}_{3}$ are particular functionals of ${\ensuremath{\rho}}_{1}$.

DOI: 10.1103/physreva.13.927

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Equation for the direct determination of the density matrix

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Physical Review A

Hierarchy equations for reduced density matrices

Leon W. Cohen, C. Frishberg

Published Mar 1, 1976• 169 citations

Publisher

Abstract

Starting with Schr"odinger's time-independent equation for

N

indistinguishable particles, an exact equation is derived for the reduced density matrix of order

s

{\ensuremath{\rho}}_{s}

. The equation contains terms dependent on

{\ensuremath{\rho}}_{s+1}

and

{\ensuremath{\rho}}_{s+2}

. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that

{\ensuremath{\rho}}_{2}

and

{\ensuremath{\rho}}_{3}

are particular functionals of

{\ensuremath{\rho}}_{1}

DOI: 10.1103/physreva.13.927

Related papers

Hierarchy equations for reduced transition operators

Herbert Schlosser

Physical Review AMar 1, 197715 citations

Equation for the direct determination of the density matrix

Hiroshi Nakatsuji

A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$ -representability conditions, although in such a case some unphysical solution may also occur in the non- $N$ -representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.A nonvariational equation, called the density equation, is proposed for the direct determination of the density matrix without using a wave function. It is connected with the Schr\"odinger equation by a necessary and sufficient theorem. The equation for the lowest order depends explicitly only on the fourth-order density matrix ${\ensuremath{\Gamma}}^{(4)}$ (or ${\ensuremath{\Gamma}}^{(3)}$ in a special case) and not on the higher-order density matrices. The equation always gives the density matrix and the associated energy which coincide with those obtained indirectly from the Schr\"odinger equation. This is true even if we solve the equation only with the known and tractable $N$-representability conditions, although in such a case some unphysical solution may also occur in the non-$N$-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.

Physical Review AJul 1, 1976257 citations

Hierarchy equations for reduced density matrices in different representations

Leon W. Cohen

International Journal of Quantum ChemistryFeb 1, 19861 citation

Exact reduced density matrices for a model problem

Leon W. Cohen, Chongmoon Lee

Journal of Mathematical PhysicsDec 1, 198558 citations

Hierarchy Bloch equations for the reduced statistical density operators in canonical and grand canonical ensembles

Liqiang Wei, Chia‐Chung Sun

arXiv:cond-mat/0306306Physica A Statistical Mechanics and its ApplicationsDec 2, 20034 citations

Linear scaling and the 1,2-contracted Schrödinger equation

David A. Mazziotti

A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.A contracted Schrödinger equation (1,2-CSE) is derived for the class of Hamiltonians without explicit interactions including those from Hartree–Fock and density functional theories. With cumulant reconstruction of the two-particle reduced density matrix (2-RDM) from the one-particle-RDM (1-RDM), the 1,2-CSE may be expressed solely in terms of the 1-RDM. We prove that a 1-RDM satisfies the 1,2-CSE if and only if it is an eigenstate of the N-particle Schrödinger equation. The 1,2-CSE is solved through the development and implementation of a reduced, linear-scaling analog of the ordinary power method for finding matrix eigenvalues. The power formula for updating the 1-RDM requires fewer matrix operations than the gradient procedure derived by Li et al. [Phys. Rev. B 47, 10891 (1993)] and Daw [Phys. Rev. B 47, 10895 (1993)]. Convergence of the contracted power method with purification is illustrated with several molecules. While providing a new tool for semiempirical, Hartree–Fock, and density functional calculations, the 1,2-CSE also represents an initial step toward a linear-scaling algorithm for solving higher CSEs which explicitly treat electron correlation.

The Journal of Chemical PhysicsNov 8, 200140 citations

Starting with Schr"odinger's time-independent equation for

N

indistinguishable particles, an exact equation is derived for the reduced density matrix of order

s

{\ensuremath{\rho}}_{s}

. The equation contains terms dependent on

{\ensuremath{\rho}}_{s+1}

and

{\ensuremath{\rho}}_{s+2}

. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that

{\ensuremath{\rho}}_{2}

and

{\ensuremath{\rho}}_{3}

are particular functionals of

{\ensuremath{\rho}}_{1}

N

-representability conditions, although in such a case some unphysical solution may also occur in the non-

N

-representable space. The equation is applicable to both fermion and boson systems, and to both ground and excited states. In contrast to the Schr\"odinger equation, the labor of solving the equation does not increase when the number of particles of the system is increased. When we have the Hartree-Fock solution, the equation is transformed such that the correlated density matrix and the correlation energy are the direct solution. The correlated density equation thus obtained is suitable for the study of electron correlations.