Category:Exchange-correlation functionals: Difference between revisions
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In the KS formulation of DFT{{cite|hohenberg:pr:1964}}{{cite|kohn:pr:1965}}, the total energy is given by | In the Kohn-Sham (KS) formulation of density-functional theory (DFT){{cite|hohenberg:pr:1964}}{{cite|kohn:pr:1965}}, the total energy is given by | ||
:<math> | :<math> | ||
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} | 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} | ||
</math> | </math> | ||
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 orbitals <math>\psi_{i}</math> and the | 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 <math>\psi_{i}</math> and the electronic density <math>n=\sum_{i}\left\vert\psi_{i}\right\vert^{2}</math> that are used to evaluate <math>E_{\rm tot}^{\rm KS-DFT}</math> are obtained by [[:Category:Electronic minimization|solving self-consistently the KS equations]] | ||
:<math> | :<math> | ||
\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}) | \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}). | ||
</math> | </math> | ||
The only terms in <math>E_{\rm tot}^{\rm KS-DFT}</math> and in the KS equations that are not known exactly are the exchange-correlation energy functional <math>E_{\rm xc}</math> and potential <math>v_{\rm xc}=\delta E_{\rm xc}/\delta n</math>. Therefore, the accuracy of the calculated properties depends | The only terms in <math>E_{\rm tot}^{\rm KS-DFT}</math> and in the KS equations that are not known exactly are the '''exchange-correlation energy functional''' <math>E_{\rm xc}</math> and potential <math>v_{\rm xc}=\delta E_{\rm xc}/\delta n</math>. Therefore, the accuracy of the calculated properties depends strongly on the approximations used for <math>E_{\rm xc}</math> and <math>v_{\rm xc}</math>. | ||
Several hundreds of approximations for the '''exchange and correlation''' have been proposed{{cite|libxc_list}}. They can be classified into families: the local density approximation (LDA), generalized gradient approximation (GGA), [[:Category:Meta-GGA|meta-GGA]], and [[:Category:Hybrid functionals|hybrid]]. There is also the possibility to include [[:Category:Van der Waals functionals|van der Waals corrections]] or an on-site Coulomb repulsion using [[:Category:DFT+U|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 == | == How to == | ||
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**LDA and GGA: {{TAG|GGA}} | **LDA and GGA: {{TAG|GGA}} | ||
**Meta-GGA: {{TAG|METAGGA}} | **Meta-GGA: {{TAG|METAGGA}} | ||
*Hybrids: {{TAG|LHFCALC}} and [[List_of_hybrid_functionals|list of hybrid functionals]] | *Hybrids: {{TAG|LHFCALC}}, {{TAG|AEXX}}, {{TAG|HFSCREEN}} and [[List_of_hybrid_functionals|list of hybrid functionals]] | ||
*DFT+U: {{TAG|LDAU}} and {{TAG|LDAUTYPE}} | *DFT+U: {{TAG|LDAU}} and {{TAG|LDAUTYPE}} | ||
*Atom-pairwise methods for van der Waals interactions (selected with the {{TAG|IVDW}} tag): | *Atom-pairwise and many-body methods for van der Waals interactions (selected with the {{TAG|IVDW}} tag): | ||
**Methods from Grimme et al.: | **Methods from Grimme et al.: | ||
***{{TAG|DFT-D2}} | ***{{TAG|DFT-D2}}{{cite|grimme:jcc:06}} | ||
***{{TAG|DFT-D3}} | ***{{TAG|DFT-D3}}{{cite|grimme:jcp:10}}{{cite|grimme:jcc:11}} | ||
***[[DFT-D4]]{{cite|caldeweyher:jcp:2019}} (available as of VASP.6.2 as [[Makefile.include#DFT-D4_.28optional.29|external package]]) | |||
**Methods from Tkatchenko, Scheffler et al.: | **Methods from Tkatchenko, Scheffler et al.: | ||
***{{TAG|Tkatchenko-Scheffler method}} | ***{{TAG|Tkatchenko-Scheffler method}}{{cite|tkatchenko:prl:09}} | ||
***{{TAG|Tkatchenko-Scheffler method with iterative Hirshfeld partitioning}} | ***{{TAG|Tkatchenko-Scheffler method with iterative Hirshfeld partitioning}}{{cite|bucko:jctc:13}}{{cite|bucko:jcp:14}} | ||
***{{TAG|Self-consistent screening in Tkatchenko-Scheffler method}} | ***{{TAG|Self-consistent screening in Tkatchenko-Scheffler method}}{{cite|tkatchenko:prl:12}} | ||
***{{TAG|Many-body dispersion energy}} | ***{{TAG|Many-body dispersion energy}}{{cite|tkatchenko:prl:12}}{{cite|ambrosetti:jcp:14}} | ||
***{{TAG|Many-body_dispersion_energy_with_fractionally_ionic_model_for_polarizability}} | ***{{TAG|Many-body_dispersion_energy_with_fractionally_ionic_model_for_polarizability}}{{cite|gould:jctc:2016_a}}{{cite|gould:jctc:2016_b}} | ||
**{{TAG|dDsC dispersion correction}} | ***[[LIBMBD_METHOD|Library libMBD of many-body dispersion methods]]{{cite|libmbd_1}}{{cite|libmbd_2}}{{cite|hermann:jcp:2023}} | ||
**{{TAG|DFT-ulg}} | **{{TAG|dDsC dispersion correction}}{{cite|steinmann:jcp:11}}{{cite|steinmann:jctc:11}} | ||
*{{TAG|Nonlocal vdW-DF functionals}} for van der Waals interactions: {{TAG|LUSE_VDW}} | **{{TAG|DFT-ulg}}{{cite|kim:jpcl:2012}} | ||
*{{TAG|Nonlocal vdW-DF functionals}} for van der Waals interactions: {{TAG|LUSE_VDW}} and {{TAG|IVDW_NL}} | |||
== References == | == References == |
Latest revision as of 14:51, 1 July 2024
In the Kohn-Sham (KS) formulation of density-functional theory (DFT)[1][2], the total energy is given by
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 and the electronic density that are used to evaluate are obtained by solving self-consistently the KS equations
The only terms in and in the KS equations that are not known exactly are the exchange-correlation energy functional and potential . Therefore, the accuracy of the calculated properties depends strongly on the approximations used for and .
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:
- 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.:
- Methods from Tkatchenko, Scheffler et al.:
- Tkatchenko-Scheffler method[8]
- Tkatchenko-Scheffler method with iterative Hirshfeld partitioning[9][10]
- Self-consistent screening in Tkatchenko-Scheffler method[11]
- Many-body dispersion energy[11][12]
- Many-body_dispersion_energy_with_fractionally_ionic_model_for_polarizability[13][14]
- Library libMBD of many-body dispersion methods[15][16][17]
- dDsC dispersion correction[18][19]
- DFT-ulg[20]
- Nonlocal vdW-DF functionals for van der Waals interactions: LUSE_VDW and IVDW_NL
References
- ↑ P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
- ↑ W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
- ↑ https://libxc.gitlab.io/functionals/
- ↑ S. Grimme, J. Comput. Chem. 27, 1787 (2006).
- ↑ S. Grimme, J. Antony, S. Ehrlich, and S. Krieg, J. Chem. Phys. 132, 154104 (2010).
- ↑ S. Grimme, S. Ehrlich, and L. Goerigk, J. Comput. Chem. 32, 1456 (2011).
- ↑ E. Caldeweyher, S. Ehlert, A. Hansen, H. Neugebauer, S. Spicher, C. Bannwarth, and S. Grimme, J. Chem. Phys. 150, 154122 (2019).
- ↑ A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009).
- ↑ T. Bučko, S. Lebègue, J. Hafner, and J. G. Ángyán, J. Chem. Theory Comput. 9, 4293 (2013)
- ↑ T. Bučko, S. Lebègue, J. G. Ángyán, and J. Hafner, J. Chem. Phys. 141, 034114 (2014).
- ↑ a b A. Tkatchenko, R. A. DiStasio, Jr., R. Car, and M. Scheffler, Phys. Rev. Lett. 108, 236402 (2012).
- ↑ A. Ambrosetti, A. M. Reilly, and R. A. DiStasio Jr., J. Chem. Phys. 140, 018A508 (2014).
- ↑ 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).
- ↑ 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).
- ↑ https://libmbd.github.io/
- ↑ https://github.com/libmbd/libmbd
- ↑ 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).
- ↑ S. N. Steinmann and C. Corminboeuf, J. Chem. Phys. 134, 044117 (2011).
- ↑ S. N. Steinmann and C. Corminboeuf, J. Chem. Theory Comput. 7, 3567 (2011).
- ↑ H. Kim, J.-M. Choi, and W. A. Goddard, III, J. Phys. Chem. Lett. 3, 360 (2012).
Subcategories
This category has the following 5 subcategories, out of 5 total.
Pages in category "Exchange-correlation functionals"
The following 119 pages are in this category, out of 119 total.
B
D
L
- LASPH
- LDAU
- LDAUJ
- LDAUL
- LDAUPRINT
- LDAUTYPE
- LDAUU
- LEXCH
- LFOCKACE
- LFOCKAEDFT
- LHFCALC
- LIBMBD ALPHA
- LIBMBD C6AU
- LIBMBD K GRID
- LIBMBD K GRID SHIFT
- LIBMBD MBD A
- LIBMBD MBD BETA
- LIBMBD METHOD
- LIBMBD N OMEGA GRID
- LIBMBD PARALLEL MODE
- LIBMBD R0AU
- LIBMBD TS D
- LIBMBD TS SR
- LIBMBD VDW PARAMS KIND
- LIBMBD XC
- LIBXC1
- LIBXC1 Pn
- LIBXC2
- LIBXC2 Pn
- List of hybrid functionals
- LMAXFOCK
- LMAXTAU
- LMIXTAU
- LMODELHF
- LRHFCALC
- LSCALER0
- LSCSGRAD
- LSFBXC
- LSPIN VDW
- LTBOUNDLIBXC
- LTHOMAS
- LTSSURF
- LUSE VDW
- LVDW EWALD
- LVDWEXPANSION
- LVDWSCS