A Simple Example Thomas-Fermi Theory

After these abstract considerations, let us now consider one way in which one can make practical use of DFT. Assume we have specified our system (i.e. w(r) is known). Assume further that we have reliable approximations for C/[n] and T[ri. In principle, all one then has to do is to minimize the sum of kinetic, interaction and potential energies 

For a noninteracting system (specified by subscript s, for single particle ), the function ° (n) is known explicitly. 

The Thomas-Fermi approximation consists in combining (39) with (42) and minimizing the resulting energy functional 

The Thomas-Fermi approximation to the dielectric constant, which provides a simple description of static screening, is obtained by minimizing E [n with respect to n and linearizing the resulting relation between w(r) and n(r). It thus involves one more approximation (the linearization) compared with what is called the Thomas Fermi approximation in DFT. In two dimensions no linearization is required and both uses of the name Thomas-Fermi become equivalent.  

---