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Physical Review A

Nonuniqueness of algebraic first-order density-matrix functionals

Jian Wang, Peter J. Knowles

Published Jul 27, 2015• 24 citations
Publisher

Abstract

By explicit construction of counterexamples having the same eigenvalue spectrum of one-matrix, but different two-matrix, we show that density-matrix functionals for the electronic energy that are based solely on the eigenvalues of the one-matrix cannot be unique in functional representation of the two-matrix. The one-to-many mapping may be understood either through the number of independent parameters or the contraction relation.

DOI
10.1103/physreva.92.012520

Related papers

Reply to “Comment on “Nonuniqueness of algebraic first-order density-matrix functionals” ”

Jian Wang, Peter J. Knowles

It is shown that symmetry considerations do not alter the conclusions of our original work [Phys. Rev. A 92, 012520 (2015)], and that there exists an example of an electronic system for which at several geometries the one-matrix eigenvalues are identical, but the two-matrix spectrum is not. It is still therefore the case that JK and related functionals that depend on the one-matrix eigenvalues to model the two-matrix cannot be made arbitrarily accurate.

Physical Review A•Feb 14, 2018•5 citations

Comment on “Nonuniqueness of algebraic first-order density-matrix functionals”

O. V. Gritsenko

Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs) ${n}_{i}$. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.

Physical Review A•Feb 14, 2018•7 citations

Elementary properties of an energy functional of the first-order reduced density matrix

Robert A. Donnelly, Robert G. Parr

The Hohenberg–Kohn theorem implies the existence of an energy functional based solely on the first-order reduced density matrix of the ground state of an atomic or molecular system. Application of the variational principle for the functional EvM[γ] generates a set of coupled Euler equations for the representation coefficients and spin orbitals of a rank-M approximation to the exact ground-state density matrix. Defining the (assumed Hermitian) kernel FM[γ;x′,x]≡δEvM/δγ (x,x′), the equations in an arbitrary representation for the approximate density matrix, γ (x,x′) =ΣijMψ1(x) γijψj* (x′) are the following: ℱdξFM[γ;x′,ξ]ψi(ξ) =μψi(x′); i=1, 2,...,M; ℱdξFM[γ;x′,ξ] ΣMiψi(ξ) γ ij=ΣMiψi(x′) λij; j=1, 2, ...,M. The quantity μ is the chemical potential of the system of interest and the λij are a set of M2 Lagrange multipliers constraining the orthonormality of the spin orbital basis {ψ}. The coefficients γij must be chosen such that FM has the degenerate eigenvalue spectrum, FiiM=μ, i=1, 2,...,M, for all partially occupied orbitals. The spin orbitals, for a fixed set of coefficients, must be simultaneously determined so that λ is Hermitian. Provided all occupation numbers lie on the open interval (0,1), the stationary density matrix itself obeys the eigenvalue equation ℱdξFM[γ;x,ξ]γ (ξ,x′) =μγ (x,x′), and for any stationary density matrix the following commutation rules are valid: [FM[γ], γ]=0; [λ, γ]=0. The matrices γ and λ consequently can be brought simultaneously to diagonal form, and the canonical representation of the energy functional provides an eigenvalue equation determining the natural spin orbitals. The value of the chemical potential is μ=εi/ni, i=1, 2, ...,M; ε=diag (λ); from which follows the distribution function governing the occupation numbers of a stationary density matrix. Two limiting forms of the variational principle are examined, the exact and Hartree–Fock functionals, and related previous work by Gilbert is discussed. The physical content of the equations is illuminated by identification of the chemical potential as the negative of the electronegativity; Sanderson’s Principle of Electronegativity Equalization follows.

The Journal of Chemical Physics•Nov 15, 1978•285 citations

New constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix

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Restoration of Many Electron Wave Functions from One-Electron Density

A. I. Panin, Alexander N. Petrov

General theorem describing a relation between diagonal of one-electron density matrix and a certain class of many-electron ensembles of determinant states is proved. As a corollary to this theorem a constructive proof of sufficiency of Coleman's representability conditions is obtained. It is shown that there exist rigorous schemes for construction of energy of many-electron system as functionals of one-electron density.

arXiv:physics/0310043•Oct 9, 2003•0 citations

N representability of electron density and first-order density matrix

N. H. March

For noninteracting fermions, some explicit examples using first-order density matrices built from orthonormal orbitals like those of Harriman are first summarized, and the relation to the Thomas-Fermi statistical theory is emphasized. The relation of Harriman's work for interacting fermions to some exact properties of the first-order density matrix for atoms and molecules is considered.

Physical Review A•Oct 1, 1982•23 citations
By explicit construction of counterexamples having the same eigenvalue spectrum of one-matrix, but different two-matrix, we show that density-matrix functionals for the electronic energy that are based solely on the eigenvalues of the one-matrix cannot be unique in functional representation of the two-matrix. The one-to-many mapping may be understood either through the number of independent parameters or the contraction relation.
Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs) ni{n}_{i}ni​. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.
The Hohenberg–Kohn theorem implies the existence of an energy functional based solely on the first-order reduced density matrix of the ground state of an atomic or molecular system. Application of the variational principle for the functional EvM[γ] generates a set of coupled Euler equations for the representation coefficients and spin orbitals of a rank-M approximation to the exact ground-state density matrix. Defining the (assumed Hermitian) kernel FM[γ;x′,x]≡δEvM/δγ (x,x′), the equations in an arbitrary representation for the approximate density matrix, γ (x,x′) =ΣijMψ1(x) γijψj* (x′) are the following: ℱdξFM[γ;x′,ξ]ψi(ξ) =μψi(x′); i=1, 2,...,M; ℱdξFM[γ;x′,ξ] ΣMiψi(ξ) γ ij=ΣMiψi(x′) λij; j=1, 2, ...,M. The quantity μ is the chemical potential of the system of interest and the λij are a set of M2 Lagrange multipliers constraining the orthonormality of the spin orbital basis {ψ}. The coefficients γij must be chosen such that FM has the degenerate eigenvalue spectrum, FiiM=μ, i=1, 2,...,M, for all partially occupied orbitals. The spin orbitals, for a fixed set of coefficients, must be simultaneously determined so that λ is Hermitian. Provided all occupation numbers lie on the open interval (0,1), the stationary density matrix itself obeys the eigenvalue equation ℱdξFM[γ;x,ξ]γ (ξ,x′) =μγ (x,x′), and for any stationary density matrix the following commutation rules are valid: [FM[γ], γ]=0; [λ, γ]=0. The matrices γ and λ consequently can be brought simultaneously to diagonal form, and the canonical representation of the energy functional provides an eigenvalue equation determining the natural spin orbitals. The value of the chemical potential is μ=εi/ni, i=1, 2, ...,M; ε=diag (λ); from which follows the distribution function governing the occupation numbers of a stationary density matrix. Two limiting forms of the variational principle are examined, the exact and Hartree–Fock functionals, and related previous work by Gilbert is discussed. The physical content of the equations is illuminated by identification of the chemical potential as the negative of the electronegativity; Sanderson’s Principle of Electronegativity Equalization follows.
Three strict constraints upon the electron-electron repulsion energy functional of the one-electron reduced density matrix (the 1-matrix) are obtained by combining its invariance and stationary properties with the extended Koopmans' theorem. The constraints relate the quantities derived from the functional pertaining to an N-electron system with those of its (N-1)-electron congener. Together with the N-representability requirement for the 1-matrix of the congener, identities involving the electron-electron repulsion energies of the two systems and their derivatives with respect to the 1-matrices seriously narrow down the choices for potential approximate density-matrix functionals. This fact is well illustrated in the case of two-electron systems, where the validity of the new constraints is confirmed and found to originate from a nontrivial cancellation among different terms. Thus, the constraints provide a new tool for the construction and testing of new functionals that complements the previously known conditions such as the reproduction of the homogeneous gas energies and momentum distributions, convexity, and the N-representability of the associated 2-matrices.