Iterative Minimization Technique for Total Energy Calculations

Iterative Minimization Technique for Total Energy Calculations 

Only the minimum value of the Kohn-Sham energy fimctional has physical meaning. At the minimum, the Kohn-Sham energy functional is equal to the groimd-state energy of the system of electrons with the ions in positions Ri. 

It is necessary to determine the set of eigenfimctions that minimize the Kohn-Sham energy fimctional E[ ipi ]. These are given by self-consistent solutions of the Kohn-Sham equations  

We have already noted that the difficulties of the many-body problem were overcome by Kohn and Scham, who showed that the task of finding the right electron density can be expressed in a way that involves solving of a set of equations in which each equation only involves a sin e electron. The main difference is that the Kohn-Sham equations are missing the summations that appear inside the Hamiltonian (8.4) of the full Schrodinger equation. This is because the solutions of the Kohn-Sham equations are single-electron eigenfimctions that depend only on three spatial variables and spin. 

The exchange-correlation potential is given formally by the functional derivative 

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