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The density functional theory

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 $\rho_\sigma (r)$ in terms of which the total energy is

 
$\displaystyle E_{tot}(\rho_{\uparrow},\rho_{\downarrow})=$ $\textstyle T_s(\rho_{\uparrow},\rho_{\downarrow})+
E_{ee}(\rho_{\uparrow},\rho_{\downarrow})+$    
  $\textstyle E_{Ne}(\rho_{\uparrow},\rho_{\downarrow})+
E_{xc}(\rho_{\uparrow},\rho_{\downarrow})+E_{NN}$   (1)

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 $\mu_{xc}$ times the total (spin-up plus spin-down) electron density as


 \begin{displaymath}
E_{xc} = \int \mu_{xc} (\rho_{\uparrow} ,\rho_{\downarrow} ) *
[ \rho_{\uparrow} + \rho\downarrow ] d r
\end{displaymath} (2)

and ii), the particular form chosen for that $\mu_{xc}$. 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 $\chi_{ik}^\sigma$ constrained to construct the spin densities as


 \begin{displaymath}
\rho_{\sigma} (r) = \sum_{i,k}\rho_{ik}^\sigma
\vert \chi_{ik}^\sigma (r) \vert^2
\end{displaymath} (3)

Here, the $\rho_{ik}^\sigma$ are occupation numbers such that $0\leq \rho_{ik}^\sigma \leq 1/w_k$, where wk is the symmetry-required weight of point k. Then variation of Etot gives the Kohn-Sham equations (in Ry atomic units),


 \begin{displaymath}[- \nabla^2 + V_{Ne} +V_{ee} + V_{xc}^\sigma]
\chi_{ik}^\sigma (r) = \epsilon_{ik}^\sigma (r)
\chi_{ik}^\sigma (r)
\end{displaymath} (4)

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.


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Next: The Full Potential LAPW Up: The basic concepts of Previous: The basic concepts of

2000-04-11