Atomic shell approximation kinetic energy

Calculations by Gryzinski and Kowalski (1993) for inner shell ionization by positrons also confirmed the general trend. Theirs was essentially a classical formulation based upon the binary-encounter approximation and a so-called atomic free-fall model, the latter representing the internal structure of the atom. The model allowed for the change in kinetic energy experienced by the positrons and electrons during their interactions with the screened field of the nucleus. 

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. 

The attempt to determine ELF directly fi om the experimental electron densities was realized in 2002 by Tsirelson and Stash [52]. Again, the ELF formulation of Savin was used with t replaced by second-order expansion of the kinetic energy density (Kirzhnits approximation [53]). The proposed modification of ELF is dependent only on the electron density and its derivatives. This modified ELF reveals the atomic shell structure. However, due to a deformation of the atomic shell shape toward the nucleus, a saddle point emerges at the position between the bonded atoms, where the original ELF displays a maximum. Several different... 

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