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### Kohn-Sham Equations

Kohn and Sham [9] introduced a method based on the Hohenberg-Kohn theorem that enables one to minimise the functional by varying over all densities containing N electrons. This constraint is introduced by the Lagrange multiplier, , chosen so that ,

Kohn and Sham chose to separate into three parts, so that becomes

where is defined as the kinetic energy of a non-interacting electron gas with density ,

Eq.() also acts as a definition for the exchange-correlation energy functional, . We can now rewrite Eq.() in terms of an effective potential, , as follows

where

and

Now, if one considers a system that really contained non-interacting electrons moving in an external potential equal to , as defined in Eq.(), then the same analysis would lead to exactly the same Eq.(). Therefore, to find the groundstate energy and density, and all one has to do is solve the one-electron equations

As the density is constructed according to

these equations (-) must be solved self-consistently with Eq.().

The above derivation assumes that the exchange-correlation functional is known. At present numerical exchange-correlation potentials have only been determined for a few simple model systems, and so most current density functional calculations use the Local Density Approximation (LDA). The LDA approximates the XC functional to a simple function of the density at any position, . The value of this function is the XC energy per electron in a uniform homogeneous electron gas of density . The LDA expression for is

The LDA is remarkably accurate, but often fails when the electrons are strongly correlated, as in systems containing d and f orbital electrons.

Next: Quantum Monte Carlo Calculations Up: Density Functional Methods Previous: Density Functional Methods

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996