Harris-Foulkes functional

Recently there was an increased interest in the so called Harris-Foulkes (HF) functional. This functional is non selfconsistent: The potential is constructed for some "input" charge density, then the band-structure term is calculated for this fixed non self-consistent potential. Double counting corrections are calculated from the input charge density. The functional can be written as

$E_{\mathrm{HF}} [\rho_{\mathrm{in}} ,\rho] = \mathrm{ band-structure for } (VH_{\mathrm{in}} + V^{xc}_{\mathrm{in}}) + \mathrm{Tr}[(-V^{H}_{\mathrm{in}}/2 -V^{xc}_{\mathrm{in}}) \rho_{\mathrm{in}} ] + E^{xc}[\rho_{\mathrm{in}}+\rho_{c}].$

It is interesting that the functional gives a good description of the binding-energies, equilibrium lattice constants, and bulk-modulus even for covalently bonded systems like Ge. In a test calculation we have found that the pair-correlation function of l-Sb calculated with the HF-function and the full Kohn-Sham functional differs only slightly. Nevertheless, we must point out that the computational gain in comparison to a self-consistent calculation is in many cases very small (for Sb less than $20~\%$). The main reason why to use the HF functional is therefore to access and establish the accuracy of the HF-functional, a topic which is currently widely discussed within the community of solid state physicists. To our knowledge VASP is one of the few pseudo-potential codes, which can access the validity of the HF-functional at a very basic level, i.e. without any additional restrictions like local basis-sets etc.

Within VASP the band-structure energy is exactly evaluated using the same plane-wave basis-set and the same accuracy which is used for the self-consistent calculation. The forces and the stress tensor are correct, insofar as they are an exact derivative of the Harris-Foulkes functional. During a MD calculation or an ionic relaxation the charge density is correctly updated at each ionic step.

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