Harris-Foulkes functional: Difference between revisions

Revision as of 14:41, 18 October 2023

The Harris-Foulkes (HF) functional is a non-self-consistent functional. 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{ bandstructure} \mathrm{ for }  (V^H_{\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}].

In our experience, 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 functional 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. To our knowledge, VASP is one of the few pseudopotential 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.

The band-structure energy is exactly evaluated using the same plane-wave basis-set and the same accuracy 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 molecular dynamics calculation or an ionic relaxation the charge density is updated at each ionic step to enforce the HF functional.

Ideas based on the Harris-Foulkes functional are applied when LCORR or ICHARG=12.

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-Recently there was an increased interest in the so called Harris-Foulkes (HF) functional. This functional is non selfconsistent:
+The '''Harris-Foulkes''' (HF) '''functional''' is a non-self-consistent functional.
-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
+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
-<math> 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}]. </math>
+::<math> E_{\mathrm{HF}} [\rho_{\mathrm{in}} ,\rho]  =   \mathrm{ bandstructure} \mathrm{ for }  (V^H_{\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}]. </math>
-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 <math>20~\%</math>). 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.
+In our experience, 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 functional 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 <math>20~\%</math>). The main reason why to use the HF functional is therefore to access and establish the accuracy of the HF functional. To our knowledge, VASP is one of the few pseudopotential 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 {{TAG|Harris Foulkes functional}}. During a MD calculation or an ionic relaxation the charge density is correctly updated at each ionic step.
+The band-structure energy is exactly evaluated using the same plane-wave basis-set and the same accuracy used for the self-consistent calculation. The forces and the stress tensor are correct, insofar as they are an exact derivative of the {{TAG|Harris-Foulkes functional}}. During a [[:Category:Molecular dynamics|molecular dynamics]] calculation or an [[:Category:Ionic minimization|ionic relaxation]] the charge density is updated at each ionic step to enforce the HF functional.
+Ideas based on the Harris-Foulkes functional are applied when {{TAG|LCORR}} or {{TAG|ICHARG}}=12.
+== Related tags and articles ==
+{{TAG|LCORR}},
+{{TAG|ICHARG}}
 {{sc|Harris Foules functional|Examples|Examples that use this tag}}
 ----
-[[The_VASP_Manual|Contents]]
+[[Category:Electronic minimization]][[Category:Theory]][[Category:Exchange-correlation functionals]]
-[[Category:Theory]]