1 Density Functionals for Non-relativistic Coulomb Systems in the New Century John P. Perdew∗ and Stefan Kurth† ∗
Department of Physics and Quantum Theory Group, Tulane University, New Orleans LA 70118, USA
[email protected] †
Institut f¨ ur Theoretische Physik, Freie Universit¨ at Berlin, Arnimallee 14, 14195 Berlin, Germany
[email protected] John Perdew
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Introduction Quantum Mechanical Many-Electron Problem
The material world of everyday experience, as studied by chemistry and condensed-matter physics, is built up from electrons and a few (or at most a few hundred) kinds of nuclei . The basic interaction is electrostatic or Coulombic: An electron at position r is attracted to a nucleus of charge Z at R by the potential energy −Z/|r − R|, a pair of electrons at r and r repel one another by the potential energy 1/|r − r |, and two nuclei at R and R repel one another as Z Z/|R − R |. The electrons must be described by quantum mechanics, while the more massive nuclei can sometimes be regarded as classical particles. All of the electrons in the lighter elements, and the chemically important valence electrons in most elements, move at speeds much less than the speed of light, and so are non-relativistic. In essence, that is the simple story of practically everything. But there is still a long path from these general principles to theoretical prediction of the structures and properties of atoms, molecules, and solids, and eventually to the design of new chemicals or materials. If we restrict our focus to the important class of ground-state properties, we can take a shortcut through density functional theory. These lectures present an introduction to density functionals for nonrelativistic Coulomb systems. The reader is assumed to have a working knowledge of quantum mechanics at the level of one-particle wavefunctions ψ(r) [1]. The many-electron wavefunction Ψ (r1 , r2 , . . . , rN ) [2] is briefly introduced here, and then replaced as basic variable by the electron density n(r). Various terms of the total energy are defined as functionals of the electron density, and some formal properties of these functionals are discussed. The most widelyused density functionals – the local spin density and generalized gradient C. Fiolhais, F. Nogueira, M. Marques (Eds.): LNP 620, pp. 1–55, 2003. c Springer-Verlag Berlin Heidelberg 2003
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John P. Perdew and Stefan Kurth
approximations – are then introduced and discussed. At the end, the reader should be prepared to approach the broad literature of quantum chemistry and condensed-matter physics in which these density functionals are applied to predict diverse properties: the shapes and sizes of molecules, the crystal structures of solids, binding or atomization energies, ionization energies and electron affinities, the heights of energy barriers to various processes, static response functions, vibrational frequencies of nuclei, et