A preview of practical DFT

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., v(r) is known). Assume further that we have reliable approximations for U[n] and T[n. In principle, all one has to do then is to minimize the sum of kinetic, interaction and potential energies 

In theory it should be possible to calculate all observables, since the HK theorem guarantees that they are all functionals of no(r). In practice, one does not know how to do this explicitly. Another problem is that the minimization of Ev[n is, in general, a tough numerical problem on its own. And, moreover, one needs reliable approximations for T[n] and U[n] to begin with. In the next section, on the Kohn-Sham equations, we will see one widely used method for solving these problems. Before looking at that, however, it is worthwhile to recall an older, but still occasionally useful, alternative the Thomas-Fermi approximation. 

For a noninteracting system (specified by subscript s, for single-particle ) the function (n) is known explicitly, = 3fi2(37r2)2//3n5/3/(10m) 

A major defect of the Thomas-Fermi approximation is that within it molecules are unstable the energy of a set of isolated atoms is lower than that of the bound molecule. This fundamental deficiency, and the lack of accuracy resulting from neglect of correlations in (32) and from using the local approximation (34) for the kinetic energy, limit the practical use of the Thomas-Fermi 

16The Thomas-Fermi approximation for screening, discussed in many books on solid-state physics, is obtained by minimizing ETF[n] with respect to n and linearizing the resulting relation between v(r) and n(r). It thus involves one more approximation (the linearization) compared to what is called the Thomas-Fermi approximation in DFT [44]. In two dimensions no linearization is required and both become equivalent [44]. 

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