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Journal of Mathematical Physics

Expansion of the Density Matrix of an N-Fermion System in Terms of the Correlation Densities of Fermion Clusters

Tai‐ichi Shibuya, Oktay Si̇nanoğlu

Published Jun 1, 1969• 4 citations
Publisher

Abstract

The N-fermion density operator D = |Ψ〈〉Ψ| is decomposed into the densities of clusters of correlating fermions. The m-body cluster densities γ̂(m) are ``orthogonal'' to the one-body ``Fock-Dirac''-type densities ρ1, i.e., Γr1(ρ1γ̂1,2,(m)…,m)=0. The reduced-correlation-density matrices obtained differ from the conventional reduced-density-matrices and are particularly convenient for the treatment of fermion-fermion correlations.

DOI
10.1063/1.1664931

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arXiv:cond-mat/0508750•Physical Review B•Oct 31, 2006•4 citations
The N-fermion density operator D = |Ψ〈〉Ψ| is decomposed into the densities of clusters of correlating fermions. The m-body cluster densities γ̂(m) are orthogonal'' to the one-body Fock-Dirac''-type densities ρ1, i.e., Γr1(ρ1γ̂1,2,(m)…,m)=0. The reduced-correlation-density matrices obtained differ from the conventional reduced-density-matrices and are particularly convenient for the treatment of fermion-fermion correlations.
The reduced kkk-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of kkk-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of kkk-body operators for all kkk.
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.]
A Tribute to A. John Coleman - The Tame Mathematician.- Reduced Density Matrices: 1929-1989.- Some Aspects on the Development of the Theory of Reduced Density Matrices and the Representability Problem.- Representability Conditions.- On the Diagonal N-Representability Problem.- Fermion N-Representability Conditions Generated by a Decomposition of the 1-Particle Identity Operator onto Mutually Orthogonal Projection Operators.- The Unitarily Invariant Decomposition of Hermitian Operators.- Building Up N-Electron States with Symplectic Symmetry.- Time Dependent Antisymmetrized Geminal Power Theory Using a Coherent State Formulation.- Griffiths Inequalities for Fermion Systems.- Entropy of Reduced Density Matrices.- A Lower Bound to the Ground State Energy of a Boson System With Fermion Source.- Reduced Density Operators, Their Related von Neumann Density Operators, Close Cousins of These, and Their Physical Interpretation.- Theory and Practice of the Spin-Adapted Reduced Hamiltonians (SRH).- Variational Principle with Built-In Pure State N-Representability Conditions. The N-Electron Case.- Wigner Distributions as Representations of the Density Matrix.- Inter-Relationships Between Various Representations of One-Matrices and Related Densities: A Road Map and an Example ..- Current Problems in Density Functional Theory.- The Interface Between Reduced Density Matrices and Density Functional Theory.- The Physics Underlying the Langreth-Mehl Scheme for Non-Uniform Systems.- Understanding Energy Differences in Density Functional Theory.- Density Functional Calculations of Molecular Bond Energies.- Non-Local Effects on Atomic and Molecular Correlation Energies Studied with a Gradient-Corrected Density Functional.- An Evaluation of Local Electron Correlation Corrections and Non-Local Exchange Corrections to the Hartree-Fock-Slater Method from Calculations on Bond Energies and Electronic Spectra of Molecular Systems.- Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities.- Some Remarks on Scaling Relations in Density Functional Theory.- Deduction of Semiempirical MO Methods from Density Functional Theory.- Charge and Spin Densities in Molecular Solids: Local Density Functional Calculations Versus Experiment.- A Functional of the Two-Particle Density Matrix for the Approximate Calculation of the Electronic Correlation Energy.- Extracules, Intracules, Correlation Holes, Potentials, Coefficients and All That.- The Exact Schroedinger Equation for the Electron Density.- Adiabatic Separation, Broken Symmetries and Geometry Optimization.- Asymptotic Results for Density Matrices and Electron Density in Atoms and Nearly Spherical Molecules.- An Algorithm for Calculating Isoelectronic Changes in Energies, Densities, and One-Matrices.- Atoms and Ions in the Limit of Large Nuclear Charge.- Improved Thomas-Fermi Theory for Atoms.- A Bond Energy from Quantum Mechanics.- Measured Electron Densities and Band Structure Calculations.- X-ray Orthonormal Orbital Model for Crystallography.
A division of the 2-matrix Γ, for a system of N identical fermions, into 2 parts is proposed in such a way that Γ has the same 1-matrix γ as does Γ and Γ has a vanishing 1-matrix (and therefore vanishing trace). This is accomplished by evaluating the natural spin orbital (NSO) occupation numbers Γ of Γ from (where the ni are NSO occupation numbers of γ); and letting γ0 have the same NSO's as does γ. The physical interpretation of Γ should be easier with this decomposition, since Γ is completely determined by γ and has a Hartree-Fock-like form, while Γ corrects the pair density of Γ without disturbing its 1-density.
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction n‾\overline{n}n, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.