next up previous contents
Next: The Exchange Correlation Up: Computational Techniques Previous: Density Functional Theory

The Kohn-Sham Equations

The proof of the Hohenberg-Kohn theorem is not constructive, hence the form of the functional in Equation 2.1 is not known. Kohn and Sham postulated [4] that could be written in the form

 

where is the Kinetic energy of a system of non-interacting electrons with density . Representing the electron density by a set of single particle orbitals gives

  

We must find the minimum of with respect to subject to the constraint that the system must contain a fixed number of electrons. Thus by the method of Lagrange multipliers

 

where is the effective potential given by

 

and is the exchange correlation potential .

Thus the difficult problem for a system interacting electrons has been mapped onto that of a system of non-interacting electrons moving in an effective potential given by Equation 2.6.



Mr. Matthew D. Segall
Fri Jul 21 15:33:30 BST 1995