An efficient and accurate scheme for solving the many-electron
problem of a crystal (with nuclei at fixed positions) is the
local spin density approximation (LSDA) within density functional
theory (Hohenberg and Kohn 64, Kohn and Sham 65). Therein the
key quantities are the spin densities
in terms of which
the total energy is
with ENN the repulsive Coulomb energy of the fixed nuclei and the
electronic contributions, labeled conventionally as, respectively, the
kinetic energy (of the non-interacting particles), the
electron-electron repulsion, nuclear-electron attraction, and
exchange-correlation energies. Two approximations comprise the LSDA,
i), the assumption that Exc can be written in terms of a local
exchange-correlation energy density
times the total
(spin-up plus spin-down) electron density as
and ii), the particular form chosen for that .
Several
forms exist in literature, for example by Hedin-Lundquist 72, Moruzzi,
Janak and Williams 78, or accurate fits to the Monte-Carlo simulations
by Ceperly and Alder e.g. Hedin-Lundqvist 71, Moruzzi, Janak, and
Williams 78, Perdew and Wang 92. Etot has a variational
equivalent with the familiar Rayleigh-Ritz principle. The most
effective way known to minimize Etot by means of the variational
principle is to introduce orbitals
constrained to
construct the spin densities as
Here, the
are occupation numbers such that
,
where wk is the symmetry-required weight of point k. Then variation
of Etot gives the Kohn-Sham equations (in Ry atomic units),
which must be solved and thus constitute the primary computational task. This Kohn-Sham equations must be solved self-consistently in an iterative process, since finding the Kohn-Sham orbitals requires the knowledge of the potentials which themselves depend on the (spin-) density and thus on the orbitals again.
Recent progress has been made going beyond the LSDA by adding gradient terms of the electron density to the exchange-correlation energy or its corresponding potential. This has led to the generalized gradient approximation (GGA) in various parameterizations, e.g. the one by Perdew et al 92 or Perdew et al 96.