Hohenberg-Kohn Theorems and Kohn-Sham Equations

The Hohenberg-Kohn theorems prove that the electron density determines everything about a ground-state system, and the Kohn-Sham equations turn that proof into a practical computational scheme.

Definition

The theoretical foundation of density functional theory: a proof that ground-state properties are functionals of the electron density, together with an orbital-based scheme for computing them.

Scope

Covers the two Hohenberg-Kohn theorems establishing the density as the fundamental variable and the variational principle for the energy functional; the Kohn-Sham mapping to a non-interacting reference system; the meaning of Kohn-Sham orbitals and eigenvalues; and the self-consistent solution of the resulting equations.

Core questions

• Why does the ground-state density uniquely fix the external potential?
• How does the Kohn-Sham auxiliary system recover the bulk of the kinetic energy?
• What is the physical status of Kohn-Sham orbitals and their eigenvalues?
• How are the Kohn-Sham equations solved self-consistently?

Key theories

First Hohenberg-Kohn theorem

The external potential, and therefore the full Hamiltonian and all ground-state properties, is uniquely determined (up to a constant) by the ground-state electron density.

Kohn-Sham mapping

By introducing a non-interacting reference system with the same density as the real one, the unknown part of the energy is confined to the exchange-correlation functional while the kinetic energy is treated almost exactly.

Mechanisms

The Kohn-Sham equations are one-electron Schrödinger-like equations containing an effective potential that depends on the density; they are solved self-consistently in the same iterative manner as the Hartree-Fock equations.

Clinical relevance

These foundations justify why density functional calculations work and define exactly which quantity, the exchange-correlation functional, must be approximated, framing all practical DFT.

History

Hohenberg and Kohn published their existence theorems in 1964; Kohn and Sham provided the practical orbital scheme in 1965. Together these papers, recognized by Kohn's 1998 Nobel Prize, launched modern density functional theory.

Key figures

• Pierre Hohenberg
• Walter Kohn
• Lu Jeu Sham

Seminal works

• hohenberg1964
• kohn1965

Frequently asked questions

Do Kohn-Sham orbitals have physical meaning?

They belong to a fictitious non-interacting system and are not true wavefunctions, though in practice their shapes and energies are often used qualitatively to interpret bonding and excitations.

If the theory is exact, why are DFT results approximate?

The exchange-correlation functional that appears in the Kohn-Sham equations is not known in closed form and must be approximated, which is the sole source of error in principle.