Kohn–Sham equations

In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the one electron Schrödinger equation (more clearly, Schrödinger-like equation) of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles. The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as v_s(r) or v_eff(r), called the Kohn–Sham potential. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to

This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here, ε_i is the orbital energy of the corresponding Kohn–Sham orbital, φ_i, and the density for an N-particle system is

The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham (沈呂九), who introduced the concept at the University of California, San Diego in 1965.

In Kohn-Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as

where T_s is the Kohn–Sham kinetic energy which is expressed in terms of the Kohn–Sham orbitals as

v_ext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), E_H is the Hartree (or Coulomb) energy,

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