Gradient-Corrected Thomas-Fermi Functionals

For the ground-state theory, the Thomas-Fermi functional can be made more accurate by adding the gradient correction to it. For ground states, the gradient correction up to the second order is given as [3-5]... 

Thomas-Fermi and gradient-corrected kinetic energy functionals. Response function analysis of HEG with split k-space shows that, to develop excited-state energy functionals, it is a good idea to work with densities corresponding to different regions of split -space rather than working with the total density. 

After reviewing the different approaches in the literature for the formulation of a relativistic Thomas-Fermi procedure for the study of complex electron systems, we will make contact between quantum mechanics with first-order relativistic corrections and the weak relativistic limit of quantum electrodynamics for finding explicit energy functionals that will be studied. In addition to this the possibility of using alternative near-nuclear corrections instead of gradient ones is discussed. 

The minimization of this functional, which includes second order gradient corrections leads to the relativistic analogous of the Thomas-Fermi-Dirac-Weizsacker model and constitutes the state of the art in relativistic semiclassical approaches for many-electron systems. 

Progress beyond this point becomes more difficult. The original expectation of Perdew was that c should be a number close to zero. Part of the motivation for this expectation was that the dependence of p upon R had been found extremely weak in variational density functional calculations using model electron densities [6,13]. Those variational calculations used an approximate form for the kinetic energy of the electrons the local Thomas-Fermi term plus the first density gradient correction (sec Ref. [14] for details)... 

Lee, H., Lee, C.,and Parr, R. G. (1991) Conjoint gradient correction to the Hartree-Fock kinetic-and exchange-energy density functionals. Phys. Rev., A44, 768-771. Yang, W. (1986) Gradient correcttion in Thomas-Fermi theory. Phys. Rev., A34, 4575-4585. 

This functional is found to be the exact LDA exchange functional. Furthermore, von Weizsacker proposed a correction term using the gradient of electron density for the Thomas-Fermi kinetic energy functional (von Weizsacker 1935),... 

In the LDA one exploits knowledge of the density at point r. Any real system is spatially inhomogeneous, i.e. it has a spatially varying density (r), and it would clearly be useful to also include information on the rate of this variation in the functional. A first attempt at doing this was the so-called GEAs. In this class of approximation one tries to systematically calculate gradient corrections of the form V (r), V (r)p, V n(r), etc. to the LDA. A famous example is the lowest-order gradient correction to the Thomas-Fermi approximation for Ts n],... 

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