Category:Exchange-correlation functionals

In the Kohn-Sham (KS) formulation of density-functional theory (DFT)^[1]^[2], the total energy is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle E_{\rm tot}^{\rm KS-DFT} = -\frac{1}{2}\sum_{i}\int\psi_{i}^{*}({\bf r})\nabla^{2}\psi_{i}({\bf r})d^{3}r - \sum_{A}\int\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert}n({\bf r})d^{3}r + \frac{1}{2}\int\int\frac{n({\bf r})n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}rd^{3}r' + E_{\rm xc} + \frac{1}{2}\sum_{A\ne B}\frac{Z_{A}Z_{B}}{\left\vert{\bf R}_{A}-{\bf R}_{B}\right\vert} }

where the terms on the right-hand side represent the non-interacting kinetic energy of the electrons, the electrons-nuclei attraction energy, the classical Coulomb electron-electron repulsive energy, the exchange-correlation energy, and the nuclei-nuclei repulsion energy, respectively. The KS orbitals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \psi_{i}} and the electronic density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle n=\sum_{i}\left\vert\psi_{i}\right\vert^{2}} that are used to evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle E_{\rm tot}^{\rm KS-DFT}} are obtained by solving self-consistently the KS equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle \left(-\frac{1}{2}\nabla^{2} -\sum_{A}\frac{Z_{A}}{\left\vert{\bf r}-{\bf R}_{A}\right\vert} + \int\frac{n({\bf r'})}{\left\vert{\bf r}-{\bf r'}\right\vert}d^{3}r' + v_{\rm xc}({\bf r})\right)\psi_{i}({\bf r}) = \epsilon_{i}\psi_{i}({\bf r}). }

The only terms in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle E_{\rm tot}^{\rm KS-DFT}} and in the KS equations that are not known exactly are the exchange-correlation energy functional Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle E_{\rm xc}} and potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle v_{\rm xc}=\delta E_{\rm xc}/\delta n} . Therefore, the accuracy of the calculated properties depends strongly on the approximations used for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle E_{\rm xc}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\displaystyle v_{\rm xc}} .

Several hundreds of approximations for the exchange and correlation have been proposed^[3]. They can be classified into families: the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid. There is also the possibility to include van der Waals corrections or an on-site Coulomb repulsion using DFT+U on top of another functional. More details on the different types of approximations available in VASP and how to use them can be found in the pages and subcategories listed below.

How to

Semilocal functionals:
- LDA and GGA: GGA
- Meta-GGA: METAGGA
Hybrids: LHFCALC, AEXX, HFSCREEN and list of hybrid functionals
DFT+U: LDAU and LDAUTYPE
Atom-pairwise and many-body methods for van der Waals interactions (selected with the IVDW tag):
- Methods from Grimme et al.:
  - DFT-D2^[4]
  - DFT-D3^[5]^[6]
  - DFT-D4^[7] (available as of VASP.6.2 as external package)
- Methods from Tkatchenko, Scheffler et al.:
- dDsC dispersion correction^[18]^[19]
- DFT-ulg^[20]
Nonlocal vdW-DF functionals for van der Waals interactions: LUSE_VDW and IVDW_NL

References

Subcategories

This category has the following 5 subcategories, out of 5 total.

Pages in category "Exchange-correlation functionals"

The following 118 pages are in this category, out of 118 total.

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IVDW NL

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ZAB VDW

[hohenberg:pr:1964-1] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).

[kohn:pr:1965-2] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).

[libxc_list-3] ttps://libxc.gitlab.io/functionals/

[grimme:jcc:06-4] S. Grimme, J. Comput. Chem. 27, 1787 (2006).

[grimme:jcp:10-5] S. Grimme, J. Antony, S. Ehrlich, and S. Krieg, J. Chem. Phys. 132, 154104 (2010).

[grimme:jcc:11-6] S. Grimme, S. Ehrlich, and L. Goerigk, J. Comput. Chem. 32, 1456 (2011).

[caldeweyher:jcp:2019-7] E. Caldeweyher, S. Ehlert, A. Hansen, H. Neugebauer, S. Spicher, C. Bannwarth, and S. Grimme, J. Chem. Phys. 150, 154122 (2019).

[tkatchenko:prl:09-8] A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009).

[bucko:jctc:13-9] T. Bučko, S. Lebègue, J. Hafner, and J. G. Ángyán, J. Chem. Theory Comput. 9, 4293 (2013)

[bucko:jcp:14-10] T. Bučko, S. Lebègue, J. G. Ángyán, and J. Hafner, J. Chem. Phys. 141, 034114 (2014).

[tkatchenko:prl:12-11] A. Tkatchenko, R. A. DiStasio, Jr., R. Car, and M. Scheffler, Phys. Rev. Lett. 108, 236402 (2012).

[ambrosetti:jcp:14-12] A. Ambrosetti, A. M. Reilly, and R. A. DiStasio Jr., J. Chem. Phys. 140, 018A508 (2014).

[gould:jctc:2016_a-13] T. Gould and T. Bučko, C6 Coefficients and Dipole Polarizabilities for All Atoms and Many Ions in Rows 1–6 of the Periodic Table, J. Chem. Theory Comput. 12, 3603 (2016).

[gould:jctc:2016_b-14] T. Gould, S. Lebègue, J. G. Ángyán, and T. Bučko, A Fractionally Ionic Approach to Polarizability and van der Waals Many-Body Dispersion Calculations, J. Chem. Theory Comput. 12, 5920 (2016).

[libmbd_1-15] ttps://libmbd.github.io/

[libmbd_2-16] ttps://github.com/libmbd/libmbd

[hermann:jcp:2023-17] J. Hermann, M. Stöhr, S. Góger, S. Chaudhuri, B. Aradi, R. J. Maurer, and A. Tkatchenko, libMBD: A general-purpose package for scalable quantum many-body dispersion calculations, J. Chem. Phys. 159, 174802 (2023).

[steinmann:jcp:11-18] S. N. Steinmann and C. Corminboeuf, J. Chem. Phys. 134, 044117 (2011).

[steinmann:jctc:11-19] S. N. Steinmann and C. Corminboeuf, J. Chem. Theory Comput. 7, 3567 (2011).

[kim:jpcl:2012-20] H. Kim, J.-M. Choi, and W. A. Goddard, III, J. Phys. Chem. Lett. 3, 360 (2012).

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