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

arXiv:2604.06127

Necessary and sufficient conditions for the N-representability of functionals of the one-electron reduced density matrix

Jannis Erhard, Paul W. Ayers

quant-phSubmitted Apr 7, 2026• Announced Apr 8, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

Related papers

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

Jerzy Ciosłowski

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.

The Journal of Chemical Physics•Oct 22, 2005•20 citations

Appropriate constraints for variational optimization of electronic density matrices and electron densities

Philip W. Payne

Euler equations characteristic of the electronic reduced density matrix have previously been obtained by minimizing an energy functional subject to trace constraints P(K) = N, in which N is the number of particles. It is shown that these constraints are inadequate to exclude variations that violate N-representability conditions. However, a first-order reduced density matrix that does satisfy N-representability conditions is formulated as a functional of the orbital densities through use of angular momentum recoupling. Modified variational constraints are proposed.

Proceedings of the National Academy of Sciences•Oct 1, 1982•3 citations

Reference-state one-particle density-matrix theory

James P. Finley

A density-matrix formalism is developed based on the one-particle density-matrix of a single-determinantal reference state. Unlike traditional density-functional-theory approaches, the $v$-representable problem does not appear in the proposed method, nor the need to introduce functionals defined by a constrained search. The correlation-energy functionals are nonuniversal, in the sense that they depend on the external potential. Nevertheless, model systems can still be used to derive universal energy functionals. Variational and nonvariational energy functionals are introduced that yield the target-state energy when the reference state---or its corresponding one-particle density matrix---is constructed from Brueckner orbitals. Nonvariational energy functionals yield generalized Hartree-Fock equations involving a nonlocal correlation potential and the Hartree-Fock exchange operator; these equations are obtained by imposing the Brillouin-Brueckner condition. The same equations---for the most part---are obtained from variational energy functionals using functional minimization, yielding the (kernel of the) correlation potential as the functional derivative of correlation-energy functionals. Approximations for the correlation-energy functions are introduced, including a one-particle-density-matrix variant of the local-density approximation, a variant of the Colle-Salvetti functional, and a linear combination of the two that is a variant of the correlation-energy functional within the hybrid, three-parameter, Becke-Lee-Yang-Parr density functional (B3LYP).

arXiv:physics/0308056•Physical Review A•Apr 1, 2004•4 citations

Density matrices and density functionals : proceedings of the A. John Coleman symposium

Alan Coleman, Robert Erdahl, Vedene H. Smith

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.

D. Reidel Pub. Co. eBooks•Jan 1, 1987•19 citations

Pure- N -representability conditions of two-fermion reduced density matrices

David A. Mazziotti

We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure $N$-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure $N$-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure $N$-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble $N$-representability conditions. Constraints on higher-order $p$-RDMs where $p>2$ are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.

Physical Review A•Sep 29, 2016•67 citations

Functional N-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

Eduardo V. Ludeña, Fernando Javier Torres, Cesar Costa

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.

Journal of Modern Physics•Jan 1, 2013•35 citations
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.
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.
Euler equations characteristic of the electronic reduced density matrix have previously been obtained by minimizing an energy functional subject to trace constraints P(K) = N, in which N is the number of particles. It is shown that these constraints are inadequate to exclude variations that violate N-representability conditions. However, a first-order reduced density matrix that does satisfy N-representability conditions is formulated as a functional of the orbital densities through use of angular momentum recoupling. Modified variational constraints are proposed.
A density-matrix formalism is developed based on the one-particle density-matrix of a single-determinantal reference state. Unlike traditional density-functional-theory approaches, the vvv-representable problem does not appear in the proposed method, nor the need to introduce functionals defined by a constrained search. The correlation-energy functionals are nonuniversal, in the sense that they depend on the external potential. Nevertheless, model systems can still be used to derive universal energy functionals. Variational and nonvariational energy functionals are introduced that yield the target-state energy when the reference state---or its corresponding one-particle density matrix---is constructed from Brueckner orbitals. Nonvariational energy functionals yield generalized Hartree-Fock equations involving a nonlocal correlation potential and the Hartree-Fock exchange operator; these equations are obtained by imposing the Brillouin-Brueckner condition. The same equations---for the most part---are obtained from variational energy functionals using functional minimization, yielding the (kernel of the) correlation potential as the functional derivative of correlation-energy functionals. Approximations for the correlation-energy functions are introduced, including a one-particle-density-matrix variant of the local-density approximation, a variant of the Colle-Salvetti functional, and a linear combination of the two that is a variant of the correlation-energy functional within the hybrid, three-parameter, Becke-Lee-Yang-Parr density functional (B3LYP).
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.
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure NNN-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure NNN-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure NNN-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble NNN-representability conditions. Constraints on higher-order ppp-RDMs where p>2p>2p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.
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.