Kohn-Sham equations conclusions

This paper gives a short overview of density functional calculations mainly based on the DV-Xa approach organized as follows. A short overview of Density Functional Theory, DFT, and Kohn-Sham equations is given in section II followed by a summary of different ways of solution of the Kohn-Sham equations in Sec. III. Comparisons of results from some old and some up-to-date density functional electronic structure calculations made by our group to show applications to clusters, surfaces, adsorbates on surfaces and Ceo are given in Sec. IV. Conclusions and outlook are summarized in Sec. V. 

Such functionals suffer, however, from numerical instabilities with respect to small density changes, which makes it practically impossible to obtain variational solutions of the Kohn-Sham equations [130]. Neumann and Handy [150] investigated the possibility of including terms of up to fourth order in V and also arrived at decidedly disappointing conclusions. 

The work Wnfr) is retained in the equation to ensure there is no self-interaction). In contrast to the Kohn-Sham equation, this differential equation can in practice be solved because the dependence of the Fermi hole p, (r, r ), and thus of the work W (r), on the orbitals is known. Furthermore, since the solution of this equation leads to the exact asymptotic structure of vj (r), and the fact that Coulomb correlation effects are generally small for finite systems, the highest occupied eigenvalue should approximate well the exact (nonrelativistic) removal energy. This conclusion too is borne out by results given in Sect. 5.2.2. 

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. 

There are yet other important consequences of the above conclusion. For asymptotic positions of the electron, the Kohn-Sham differential equation of Eq. (31) reduces to... 

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