Thomas-Fermi kinetic functional derivation

Following the success of the von Weizsacker approach in improving the Thomas-Fermi kinetic functional. Sham showed in 1971 that an analogous correction to the Dirac exchange functional can be derived, Kleinman later demonstrated that the Sham derivation was flawed and that his correction was too small by exactly 10/7, It is now agreed that the correct second-order alpha exchange functional is... 

For a homogeneous electron density in a given volume V, the kinetic energy can be derived form the electron gas theory. For a pair of homogeneous electron densities Pa t) = phA and Pb t) = phg) the analytic form of T ad pA, Pb] obtained using Thomas-Fermi kinetic energy functional applied to Ts[pA + pb], Ts[pa], and Ts[pb] reads ... 

This argument shows that the locality hypothesis fails for more than two electrons because the assumed Frechet derivative must be generalized to a Gateaux derivative, equivalent in the context of OEL equations to a linear operator that acts on orbital wave functions. The conclusion is that the use by Kohn and Sham of Schrodinger s operator t is variationally correct, but no equivalent Thomas-Fermi theory exists for more than two electrons. Empirical evidence (atomic shell structure, chemical binding) supports the Kohn-Sham choice of the nonlocal kinetic energy operator, in comparison with Thomas-Fermi theory [288]. A further implication is that if an explicit approximate local density functional Exc is postulated, as in the local-density approximation (LDA) [205], the resulting Kohn-Sham theory is variation-ally correct. Typically, for Exc = f exc(p)p d3r, the density functional derivative is a Frechet derivative, the local potential function vxc = exc + p dexc/dp. 

This kinetic energy functional was the first local density approximation (LDA). In the next year (1928), Fermi independently derived the same kinetic energy functional as Thomas s functional using Fermi statistics at the absolute zero point, completing what is now known as the Thomas-Fermi method, on the basis of the Hartree method (see Sect. 2.1) (Fermi 1928). 

This approach to derive kinetic energy functionals was pioneered by Anderson et al., Ayers et al., Ghosh and Berkowitz, and Lee and Parr. The Thomas-Fermi model and generalizations thereto have been derived using this approach. 

The first kinetic energy density functional was derived, independently, by Fermi and Thomas" in 1928 and 1927, respectively. The Thomas-Fermi functional is the simplest local density approximation. 

The conventional WDA for the kinetic energy results when the effective Fermi vector is determined by substituting the asymmetric one-matrix (Equation 1.110) into the diagonal idempotency condition (Equation 1.115). This functional is also exact for the uniform electron gas, but the kinetic energies of atoms and molecules are still predicted to be far too high. Indeed, this functional is only slightly more accurate than the Thomas-Fermi functional. This is surprising, since the WDA and the TF functional were derived from the same formula for the one-matrix, but the WDA adds an additional exact constraint. 

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