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 vs(r) or veff(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 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 vs(r) or veff(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, εi is the orbital energy of the corresponding Kohn – Sham orbital, φi, 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. where Ts is the Kohn – Sham kinetic energy which is expressed in terms of the Kohn – Sham orbitals as vext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), EH is the Hartree (or Coulomb) energy, and Exc is the exchange-correlation energy. The Kohn – Sham equations are found by varying the total energy expression with respect to a set of orbitals, subject to constraints on those orbitals, to yield the Kohn – Sham potential as