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Physical Review A

Necessary conditions for the mapping ofγintoρ

Eduardo V. Ludeña, Aníbal Sierraalta

Published Jul 1, 1985• 24 citations
Publisher

Abstract

We have obtained necessary conditions for the mapping of the reduced first-order density operator \ensuremath{\gamma} into the density \ensuremath{\rho}. These conditions are given as restrictions on the elements of \ensuremath{\Gamma}, the reduced first-order density matrix. The implications of these conditions for a density-functional theory based on \ensuremath{\gamma}, with built-in pure-state n-representability conditions, are discussed.

DOI
10.1103/physreva.32.19

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D. Reidel Pub. Co. eBooks•Jan 1, 1987•19 citations
We have obtained necessary conditions for the mapping of the reduced first-order density operator \ensuremath{\gamma} into the density \ensuremath{\rho}. These conditions are given as restrictions on the elements of \ensuremath{\Gamma}, the reduced first-order density matrix. The implications of these conditions for a density-functional theory based on \ensuremath{\gamma}, with built-in pure-state n-representability conditions, are discussed.
We establish necessary and sufficient conditions for the N-representability of the universal one-electron reduced density matrix functional. Functionals satisfying these conditions are guaranteed to yield variational upper bounds on the true energy in one-electron reduced density matrix functional theory, regardless of the strength of the interparticle repulsion. Conversely, any functional violating these conditions will necessarily underestimate the true energy for certain systems. These exact constraints impose a stringent restriction on density matrix functional approximations, as many existing functionals-including the Hartree-Fock functional-appear to violate them. This mathematical formalism, therefore, can guide the development of new approximate functionals and numerical algorithms.
The N-representability conditions on the reduced second-order reduced density matrix (2-RDM), impose restrictions not only in the context of reduced density matrix theory (RDMT), but also on functionals advanced in one-matrix theory such as natural orbital functional theory (NOFT), and on functionals depending on the one-electron density such as those of density functional theory (DFT). We review some aspects of the applications of these N-representability conditions in these theories and present some conclusions.
The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x1, x2) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x1, x2) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θn(x1, x2) converging at n → ∞ to ρ(x1, x2)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x1, x2). The results obtained may open new possibilities for using ρ(x1, x2) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 127–142, 1997
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.