Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed

Abstract

Some fundamental issues in ground-state density functional theory are discussed without equations: (1) The standard Hohenberg-Kohn and Kohn-Sham theorems were proven for a Hamiltonian that is not quite exact for real atoms, molecules, and solids. (2) The density functional for the exchange-correlation energy, which must be approximated, arises from the tendency of electrons to avoid one another as they move through the electron density. (3) In the absence of a magnetic field, either spin densities or total electron density can be used, although the former choice is better for approximations. (4) "Spin contamination" of the determinant of Kohn-Sham orbitals for an open-shell system is not wrong but right. (5) Only to the extent that symmetries of the interacting wave function are reflected in the spin densities should those symmetries be respected by the Kohn-Sham noninteracting or determinantal wave function. Functionals below the highest level of approximations should however sometimes break even those symmetries, for good physical reasons. (6) Simple and commonly used semilocal (lower-level) approximations for the exchange-correlation energy as a functional of the density can be accurate for closed systems near equilibrium and yet fail for open systems of fluctuating electron number. (7) The exact Kohn-Sham noninteracting state need not be a single determinant, but common approximations can fail when it is not. (8) Over an open system of fluctuating electron number, connected to another such system by stretched bonds, semilocal approximations make the exchange-correlation energy and hole-density sum rule too negative. (9) The gap in the exact Kohn-Sham band structure of a crystal underestimates the real fundamental gap but may approximate the first exciton energy in the large-gap limit. (10) Density functional theory is not really a mean-field theory, although it looks like one. The exact functional includes strong correlation, and semilocal approximations often overestimate the strength of static correlation through their semilocal exchange contributions. (11) Only under rare conditions can excited states arise directly from a ground-state theory.

Spin in density‐functional theory

“…One important observation is that it is impossible to set up a KS-DFT formalism such that for the noninteracting reference system one obtains both the correct spin density and a wavefunction that is an eigenfunction of Ŝ 2 (see also the discussion of this issue in Refs. [88,112]). …”

“…[88,112] Thus, it is not possible to define a spin-state specific analog of T ðuÞ s [q, Q]. Consequently, the spin-state dependence only enters the exchange-correlation functional, which becomes…”

Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists

“…27,29,30 Such functionals are competitive with their parametrized counterparts, 31 and allow to define a whole family of non-empirical functionals which spans over the highest three rungs of the Perdew ladder. 1 The aim of this paper is to illustrate such a contribution to the field of non-empirical DHs through a critical discussion of their performance in the modeling of chemically relevant systems. In particular, the behavior of non-empirical DH density-functionals will be compared to that of a parameterized DH (B2-PLYP), 16 and to that of standard functionals, nowadays in common use for chemical applications.…”

Comparison of the performance of exact-exchange-based density functional methods

“…[4][5][6][7][8] Directly related to this is the failure of describing bond dissociation, mainly due to the lack of left-right correlation in these models. The latter is the dominant part of the nondynamic correlation in polyatomic systems.…”