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

The Journal of Chemical Physics

Function domains and the universal matrix functional of multi-state density functional theory

Yangyi Lu, Jiali Gao

Published Mar 11, 2025• 10 citations
Publisher

Abstract

On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.

DOI
10.1063/5.0249583

Related papers

Dyson Field: A Quasiparticle Mapping for Multi-State Density Functional Theory

Yangyi Lu, Jiali Gao

Multistate density functional theory (MSDFT) provides a rigorous variational framework for the simultaneous description of multiple electronic states through a matrix density, but practical implementations have relied on multiconfigurational many-body wavefunctions. Here, we introduce the Dyson field, ψ(r), an L×N matrix-valued function of the coordinate r that factorizes the matrix density as D(r) = ψ†(r)ψ(r). The elements of ψ(r) = {ψpA(r)} are shown to be Dyson orbitals connecting the n-electron states of interest to (n − 1)-electron ionization channels of size L, providing a direct physical interpretation of the matrix density in terms of electron removal amplitudes. Variational minimization of the MSDFT subspace energy with respect to the Dyson field yields a matrix Fock equation, in which the kinetic and external potential operators act locally, while Hartree and exchange–correlation effects appear as N×N matrix potentials, VH[D](r) and Vxc[D](r), that couple electronic states within the subspace. This formulation defines a quasiparticle reference system for MSDFT that is analogous to the Kohn–Sham orbitals in Hohenberg-Kohn DFT. In appropriate limits, the formalism reduces exactly to Kohn-Sham DFT for N = 1 and to the Tamm-Dancoff approximation of time-dependent DFT within the single-excitation manifold. More generally, the Dyson-field construction transforms MSDFT from a formally exact but wavefunction-dependent framework into an orbital-based theory, opening a practical route to self-consistent, density-based treatments of strongly correlated and electronically coupled states.

•Apr 20, 2026•0 citations

Extension of Kohn–Sham theory to excited states by means of an off-diagonal density array

Abraham Klein, Reiner M. Dreizler

Previous work extending the Kohn–Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding average density. We suggest and develop an alternative to this description of excited states that utilizes the matrix of the density operator taken between any two states of the included space. Such an approach provides more detailed information about the states included, for example transition probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space of states that we wish to describe, viewed, however, as a functional of the associated array of matrix elements of the density. This finds expression in a matrix version of the Kohn–Sham theory. To illustrate the formalism, we study a suitably defined weak-coupling limit, which is our equivalent of the linear response approximation. On this basis, we derive an eigenvalue equation that has the same form as an equation derived directly from the time-dependent Kohn–Sham equation and applied recently with considerable success to molecular excitations. We provide an independent proof, within the defined approximations, that the eigenvalues can be interpreted as true excitation energies.

arXiv:physics/0011039•Journal of Physics A Mathematical and General•Mar 8, 2002•0 citations

Foundation of One-Particle Reduced Density Matrix Functional Theory for Excited States

Julia Liebert, Federico Castillo, Jean‐Philippe Labbé +1 more

In Phys. Rev. Lett . 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w -RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone’s pioneering work on ground state RDMFT as the special case w = (1,0, ···). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional’s domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

arXiv:2106.03918•Journal of Chemical Theory and Computation•Dec 21, 2021•38 citations

ON THE N-REPRESENTABILITY AND UNIVERSALITY OF F[ρ] IN THE HOHENBERG-KOHN-SHAM VERSION OF DENSITY FUNCTIONAL THEORY

Eduardo V. Ludeña, Francesc Illas, Alejandro Ramírez-Solís

We discuss two basic problems in the Hohenberg-Kohn-Sham version of density functional theory, HKS-DFT: the first, the N-representability of the functional F[ρ] and, the second, the universality of F[ρ]. In relation to the first, we show that F[ρ] must satisfy N-representability conditions that follow from those on the 2-matrix D2(r1, r2; r′, r′2). In the case of the second, we provide arguments based on the equivalence between ab initio DFT and HKS-DFT to show that the functional F[ρ] is not universal.

International Journal of Modern Physics B•Oct 20, 2008•12 citations

Variational calculation of many-body wave functions and energies from density functional theory

K. Capelle

A generating coordinate is introduced into the exchange-correlation functional of density functional theory (DFT). The many-body wave function is represented as a superposition of Kohn–Sham (KS) Slater determinants arising from different values of the generating coordinate. This superposition is used to variationally calculate many-body energies and wave functions from solutions of the KS equation of DFT. The method works for ground and excited states, and does not depend on identifying the KS orbitals and energies with physical ones. Numerical application to the Helium isoelectronic series illustrates the method’s viability and potential.

arXiv:cond-mat/0209364•The Journal of Chemical Physics•Jul 3, 2003•18 citations

Geometry of Generalized Density Functional Theories

Chih-Chun Wang

Density functional theory (DFT) is an indispensable ab initio method in both quantum chemistry and condensed matter physics. Based on recent advancements in reduced density matrix functional theory (RDMFT), a variant of DFT that is believed to be better suited for strongly correlated systems, we construct a mathematical framework generalizing all ground state functional theories, which in particular applies to fermionic, bosonic, and spin systems. Within the special class of such functional theories where the space of external potentials forms the Lie algebra of a compact Lie group, the $N$-representability problem is readily solved by applying techniques from the study of momentum maps in symplectic geometry, an approach complementary to Klyachko's famous solution to the quantum marginal problem. The ``boundary force'', a diverging repulsive force from the boundary of the functional's domain observed in previous works but only qualitatively understood in isolated systems, is studied quantitatively and extensively in our work. Specifically, we present a formula capturing the exact behavior of the functional close to the boundary. In the case where the Lie algebra is abelian, a completely rigorous proof of the boundary force formula based on Levy-Lieb constrained search is given. Our formula is a first step towards developing more accurate functional approximations, with the potential of improving current RDMFT approximate functionals such as the Piris natural orbital functionals (NOFs). All key concepts and ideas of our work are demonstrated in translation invariant bosonic lattice systems.

quant-pharXiv:2511.14822•Nov 19, 2025•0 citations
On the basis of recent advancements in the Hamiltonian matrix density functional for multiple electronic eigenstates, this study delves into the mathematical foundation of the multistate density functional theory (MSDFT). We extend a number of physical concepts at the core of Kohn-Sham DFT, such as density representability, to the matrix density functional. In this work, we establish the existence of the universal matrix functional for many states as a proper generalization of the Lieb universal functional for the ground state. Consequently, the variation principle of MSDFT can be rigorously defined within an appropriate domain of matrix densities, thereby providing a solid framework for DFT of both the ground state and excited states. We further show that the analytical structure of the Hamiltonian matrix functional is considerably constrained by the subspace symmetry and invariance properties, requiring and ensuring that all elements of the Hamiltonian matrix functional are variationally optimized in a coherent manner until the Hamiltonian matrix within the subspace spanned by the lowest eigenstates is obtained. This work solidifies the theoretical foundation to treat multiple electronic states using density functional theory.
Multistate density functional theory (MSDFT) provides a rigorous variational framework for the simultaneous description of multiple electronic states through a matrix density, but practical implementations have relied on multiconfigurational many-body wavefunctions. Here, we introduce the Dyson field, ψ(r), an L×N matrix-valued function of the coordinate r that factorizes the matrix density as D(r) = ψ†(r)ψ(r). The elements of ψ(r) = {ψpA(r)} are shown to be Dyson orbitals connecting the n-electron states of interest to (n − 1)-electron ionization channels of size L, providing a direct physical interpretation of the matrix density in terms of electron removal amplitudes. Variational minimization of the MSDFT subspace energy with respect to the Dyson field yields a matrix Fock equation, in which the kinetic and external potential operators act locally, while Hartree and exchange–correlation effects appear as N×N matrix potentials, VH[D](r) and Vxc[D](r), that couple electronic states within the subspace. This formulation defines a quasiparticle reference system for MSDFT that is analogous to the Kohn–Sham orbitals in Hohenberg-Kohn DFT. In appropriate limits, the formalism reduces exactly to Kohn-Sham DFT for N = 1 and to the Tamm-Dancoff approximation of time-dependent DFT within the single-excitation manifold. More generally, the Dyson-field construction transforms MSDFT from a formally exact but wavefunction-dependent framework into an orbital-based theory, opening a practical route to self-consistent, density-based treatments of strongly correlated and electronically coupled states.
Previous work extending the Kohn–Sham theory to excited states was based on replacing the study of the ground-state energy as a functional of the ground-state density by a study of an ensemble average of the Hamiltonian as a functional of the corresponding average density. We suggest and develop an alternative to this description of excited states that utilizes the matrix of the density operator taken between any two states of the included space. Such an approach provides more detailed information about the states included, for example transition probabilities between discrete states of local one-body operators. The new theory is also based on a variational principle for the trace of the Hamiltonian over the space of states that we wish to describe, viewed, however, as a functional of the associated array of matrix elements of the density. This finds expression in a matrix version of the Kohn–Sham theory. To illustrate the formalism, we study a suitably defined weak-coupling limit, which is our equivalent of the linear response approximation. On this basis, we derive an eigenvalue equation that has the same form as an equation derived directly from the time-dependent Kohn–Sham equation and applied recently with considerable success to molecular excitations. We provide an independent proof, within the defined approximations, that the eigenvalues can be interpreted as true excitation energies.
In Phys. Rev. Lett . 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w -RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone’s pioneering work on ground state RDMFT as the special case w = (1,0, ···). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional’s domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.
We discuss two basic problems in the Hohenberg-Kohn-Sham version of density functional theory, HKS-DFT: the first, the N-representability of the functional F[ρ] and, the second, the universality of F[ρ]. In relation to the first, we show that F[ρ] must satisfy N-representability conditions that follow from those on the 2-matrix D2(r1, r2; r′, r′2). In the case of the second, we provide arguments based on the equivalence between ab initio DFT and HKS-DFT to show that the functional F[ρ] is not universal.
A generating coordinate is introduced into the exchange-correlation functional of density functional theory (DFT). The many-body wave function is represented as a superposition of Kohn–Sham (KS) Slater determinants arising from different values of the generating coordinate. This superposition is used to variationally calculate many-body energies and wave functions from solutions of the KS equation of DFT. The method works for ground and excited states, and does not depend on identifying the KS orbitals and energies with physical ones. Numerical application to the Helium isoelectronic series illustrates the method’s viability and potential.
Density functional theory (DFT) is an indispensable ab initio method in both quantum chemistry and condensed matter physics. Based on recent advancements in reduced density matrix functional theory (RDMFT), a variant of DFT that is believed to be better suited for strongly correlated systems, we construct a mathematical framework generalizing all ground state functional theories, which in particular applies to fermionic, bosonic, and spin systems. Within the special class of such functional theories where the space of external potentials forms the Lie algebra of a compact Lie group, the NNN-representability problem is readily solved by applying techniques from the study of momentum maps in symplectic geometry, an approach complementary to Klyachko's famous solution to the quantum marginal problem. The ``boundary force'', a diverging repulsive force from the boundary of the functional's domain observed in previous works but only qualitatively understood in isolated systems, is studied quantitatively and extensively in our work. Specifically, we present a formula capturing the exact behavior of the functional close to the boundary. In the case where the Lie algebra is abelian, a completely rigorous proof of the boundary force formula based on Levy-Lieb constrained search is given. Our formula is a first step towards developing more accurate functional approximations, with the potential of improving current RDMFT approximate functionals such as the Piris natural orbital functionals (NOFs). All key concepts and ideas of our work are demonstrated in translation invariant bosonic lattice systems.