Kohn–Sham equations

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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.[1][2] 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,[3] to yield the Kohn – Sham potential as

,

where the last term

is the exchange-correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn – Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory.

The Kohn – Sham orbital energies εi, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as

.

Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).

References[edit]

  1. ^ Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
  2. ^ Parr, Robert G.; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford University Press. ISBN 978-0-19-509276-9.
  3. ^ http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html