Kohn-Sham Self-consistent Field Methodology

3 Kohn-Sham Self-consistent Field Methodology 

The crucial bit of cleverness, then, is to take as a starting point a fictitious system of non-interacting electrons that have for their overall ground-state density the same density as some real system of interest where the electrons do interact (note that since the density determines the position and atomic numbers of the nuclei (see Eq. (8.2)), these quantities are necessarily identical in die non-interacting and in the real systems). Next, we divide the energy functional into specific components to facilitate further analysis, in particular 

Note that, for a non-interacting system of electrons, the kinetic energy is just the sum of the individual electronic kinetic energies. Within an orbital expression for the density, Eq. (8.14) may then be rewritten as 

If we undertake in the usual fashion to find the orbitals / that minimize E in Eq. (8.15), we find that they satisfy the pseudoeigenvalue equations 

Vxc is a so-called functional derivative. A functional derivative is analogous in spirit to more typical derivatives, and Vxc is perhaps best described as the one-electron operator for which the expectation value of the KS Slater determinant is Exc- 

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KOHN-SHAM SELF-CONSISTENT FIELD METHODOLOGY 255... 

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