Multistate density functional theory (MSDFT) generalizes Kohn–Sham density functional theory to a finite subspace of interacting states through a Hamiltonian matrix functional of the matrix density D(r). A central challenge is to construct matrix functionals that preserve subspace unitary invariance while encoding nontrivial state coupling. Here we show that any local, unitary-covariant matrix functional of D(r) must be codiagonalizable with D(r), and is therefore completely specified by a scalar generator acting on its eigenvalues. This establishes a one-to-all mapping from a scalar generator to the full matrix functional. As a consequence, the construction of N2 matrix elements is reduced to a single scalar mapping evaluated on the eigenvalue spectrum at each spatial grid point. The formalism is illustrated using a four-state Hubbard model, where exact reconstruction is achieved with a known scalar function, and deviations from this mapping can be systematically corrected within the spectral framework. The results provide a rigorous foundation for constructing local matrix exchange–correlation functionals with computational complexity comparable to Kohn–Sham DFT, and offer a practical route toward scalable MSDFT approximations.
Multistate density functional theory (MSDFT) generalizes Kohn–Sham density functional theory to a finite subspace of interacting states through a Hamiltonian matrix functional of the matrix density D(r). A central challenge is to construct matrix functionals that preserve subspace unitary invariance while encoding nontrivial state coupling. Here we show that any local, unitary-covariant matrix functional of D(r) must be codiagonalizable with D(r), and is therefore completely specified by a scalar generator acting on its eigenvalues. This establishes a one-to-all mapping from a scalar generator to the full matrix functional. As a consequence, the construction of N2 matrix elements is reduced to a single scalar mapping evaluated on the eigenvalue spectrum at each spatial grid point. The formalism is illustrated using a four-state Hubbard model, where exact reconstruction is achieved with a known scalar function, and deviations from this mapping can be systematically corrected within the spectral framework. The results provide a rigorous foundation for constructing local matrix exchange–correlation functionals with computational complexity comparable to Kohn–Sham DFT, and offer a practical route toward scalable MSDFT approximations.
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.
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.
A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.
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.
Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
We present analytic expressions for the exact density functional and Kohn-Sham Hamiltonian of simple tight-binding models of correlated electrons. These are the single- and double-site versions of the Anderson, Hubbard, and spinless fermion models. The exact exchange and correlation potentials keep the full nonlocal dependence on electron occupations. The analytic expressions allow us to compare the Kohn-Sham eigenstates of exact density functional theory with the many-body quasiparticle states of these correlated-electron systems. The exact Kohn-Sham spectrum describes correctly many of the nontrivial features of the many-body quasiparticle spectrum such as, for example, the precursors of the Kondo peak. However, we find that some pieces of the quasiparticle spectrum are missing because the many-body phase space for electron and hole excitations is richer.