Density Functional Theory: From The Many-Body Schrödinger Equation to the Kohn–Sham Methodology

Abstract

Density functional theory (DFT) describes the ground-state electron density and energy of matter. This paper traces the theoretical path from the many-body Schrödinger equation, which describes a system of nuclei and electrons completely but cannot be solved for any real system, to the Kohn-Sham methodology. We review the Born-Oppenheimer approximation, which fixes the nuclei and solves only for the motion of the electrons; the Hartree-Fock approximation, which uses a Slater determinant of individual spin-orbitals but neglects electron correlation; and the Hohenberg–Kohn theorems, which show that the electron density alone suffices to describe the system. The Kohn–Sham construction, based on a fictitious non-interacting system moving in an effective potential, confines all the unknown physics into the exchange–correlation energy, and the self-consistent field (SCF) loop used to solve the Kohn-Sham equations is described. Finally, approximations to the exchange–correlation energy are presented, from the local density approximation (LDA) through generalized gradient approximations (GGA, PBE), meta-GGA (SCAN) and hybrid functionals, including range-separated hybrids and the self-consistent choice of the mixing fraction, together with Jacob’s ladder and a comparison of accuracy and computational cost. The assumptions underlying the Kohn–Sham scheme and the main limitations of current approximations — self-interaction and delocalization errors, strongly correlated systems and dispersion — are also discussed.