The Predictive Power of Exact Constraints and Appropriate Norms in Density Functional Theory.

Ground-state Kohn-Sham density functional theory provides, in principle, the exact ground-state energy and electronic spin densities of real interacting electrons in a static external potential. In practice, the exact density functional for the exchange-correlation (xc) energy must be approximated in a computationally efficient way. About 20 mathematical properties of the exact xc functional are known.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Simple hydrogenic estimates for the exchange and correlation energies of atoms and atomic ions, with implications for density functional theory.

Exact density functionals for the exchange and correlation energies are approximated in practical calculations for the ground-state electronic structure of a many-electron system. An important exact constraint for the construction of approximations is to recover the correct non-relativistic large-Z expansions for the corresponding energies of neutral atoms with atomic number Z and electron number N = Z, which are correct to the leading order (-0.221Z and -0.

On the best partitioning of the density functional energy.

This essay discusses special features for two different ways of partitioning the density functional energy expression. The contribution, which is part of the special issue for Pratim Chattaraj, was stimulated by a thought-provoking suggestion by him at a recent conference.

Approximating the Shifted Hartree-Exchange-Correlation Potential in Direct Energy Kohn-Sham Theory.

Levy and Zahariev [Phys. Rev. Lett.

Augmented potential, energy densities, and virial relations in the weak- and strong-interaction limits of DFT.

The augmented potential introduced by Levy and Zahariev [Phys. Rev. Lett.

Properties of Augmented Kohn-Sham Potential for Energy as Simple Sum of Orbital Energies.

A recent modification to the traditional Kohn-Sham method ( Levy , M. ; Zahariev , F. Phys.

Ground-state energy as a simple sum of orbital energies in Kohn-Sham theory: a shift in perspective through a shift in potential.

It is observed that the exact interacting ground-state electronic energy of interest may be obtained directly, in principle, as a simple sum of orbital energies when a universal density-dependent term is added to w([ρ];r), the familiar Hartree plus exchange-correlation component in the Kohn-Sham effective potential. The resultant shifted potential, w[over ¯]([ρ];r), actually changes less on average than w([ρ];r) when the density changes, including the fact that w[over ¯]([ρ];r) does not undergo a discontinuity when the number of electrons increases through an integer. Thus, the approximation of w[over ¯]([ρ];r) represents an alternative direct approach for the approximation of the ground-state energy and density.

Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number.

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities.

Tight constraints on the exchange-correlation potentials of degenerate states.

Identities for the difference of exchange-correlation potentials and energies in degenerate and nondegenerate ground states are derived. The constraints are strong for degenerate ground states, and suggest that local and semilocal approximations to the exchange-correlation energy functional are incapable of correctly treating degenerate ground states. For degenerate states, it is possible to provide both local (pointwise) equality and global inequality constraints for the exchange-correlation potential in terms of the Coulomb potential.

Average local ionization energies in the Hartree-Fock and Kohn-Sham theories.

We explore the connection between average local ionization energies computed within the Hartree-Fock (HF) and the Kohn-Sham (KS) frameworks, focusing on exchange-only KS theory. We find that they are connected through a local quantity for which good approximations exist; I(HF)(r) = I(KS)(r) + DeltaV(X)(r). This allows determination of HF local ionization energies from exchange-only KS calculations without utilizing a nonlocal potential.

Relation between exchange-only optimized potential and Kohn-Sham methods with finite basis sets, and effect of linearly dependent products of orbital basis functions.

Recently, Staroverov, Scuseria, and Davidson [J. Chem. Phys.

Generalizations of the Hohenberg-Kohn theorem: I. Legendre transform constructions of variational principles for density matrices and electron distribution functions.

Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional.