The Basic Machinery of Density Functional Programs

Recall the central ingredient of the Kohn-Sham approach to density functional theory, i. e., the one-electron KS equations, 

The term in square brackets defines the Kohn-Sham one-electron operator and equation (7-1) can be written more compactly as 

Note that the operator fKS differs from the Fock operator f that we introduced in Section 1.3 in connection with the Hartree-Fock scheme only in the way the exchange and correlation potentials are treated. In the former, the non-classical contributions are expressed via the - in its exact form unknown - exchange-correlation potential Vxc, the functional derivative of Exc with respect to the charge density. In the latter, correlation is neglected 

If we now multiply this equation from the left with an arbitrary basis function and integrate over space we get L equations 

The integrals on both sides of this equation each define a matrix. On the left hand side, 

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