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{{TAGDEF|GGA|91 {{!}} PE {{!}} RP {{!}} PS {{!}} AM| type of exchange-correlation in accordance with the {{FILE|POTCAR}} file}}
{{TAGDEF|GGA|PE {{!}} RP {{!}} PS {{!}} AM {{!}} LIBXC {{!}} ...}}


Description: {{TAG|GGA}} specifies the type of generalized-gradient-approximation one wishes to use.
Default: The functional specified by {{TAG|LEXCH}} in the {{FILE|POTCAR}} if {{TAG|METAGGA}} and {{TAG|XC}} are also not specified.
 
Description: Selects a LDA or GGA [[Exchange-correlation_functionals|exchange-correlation functional]].
----
----
This tag was added to perform GGA calculation with pseudopotentials generated with conventional LDA reference configurations.
<!--
This tag was added to perform GGA calculations with pseudopotentials generated with conventional LDA reference configurations.
-->
{{NB| important| VASP recalculates the exchange-correlation energy inside the PAW sphere and corrects the atomic energies given by the {{FILE|POTCAR}} file. For this to work, the original LEXCH tag must not be modified in the {{FILE|POTCAR}} file.}}
{{NB|mind|
*When the OR, BO, MK, ML or CX GGA is used in combination with the nonlocal vdW-DF functional of Dion ''et al.''{{cite|dion:prl:2004}}, the GGA component of the correlation should in principle be turned off with {{TAG|AGGAC}}{{=}}0 (see {{TAG|nonlocal vdW-DF functionals}}).
*The {{TAG|XC}} tag, available since VASP.6.4.3, can be used to specify any linear combination of LDA, {{TAG|GGA}} and {{TAG|METAGGA}} exchange-correlation functionals.}}
<!--
*The options in the last table allow to select range-separated ACFDT calculations, where a short-range local (DFT-like) exchange and correlation kernel is added to the long-range exchange and RPA correlation energy.
-->


Possible options are:
==  Available functionals  ==


:{| border="1" cellspacing="0" cellpadding="5"
This table lists the LDA and GGA functionals available in VASP. The names of functionals which end with "_X" and "_C" correspond to exchange-only and correlation functionals, respectively.  
<!-- these first three options have been obsolete since VASP.4.X
 
|PB  || Perdew - Becke
{| class="sortable wikitable"
|-
|-
|PW  || Perdew - Wang 86
! style="text-align:center;" style=width:6em | GGA= !! style="text-align:center;" style=width:3.5em | Type !! class="unsortable" | Description
 
|-
|-
|LM  || Langreth-Mehl-Hu
| style="text-align:center;"| LIBXC (or LI) || style="text-align:center;"| LDA/GGA || Any LDA or GGA from the external library Libxc.{{cite|marques:cpc:2012}}{{cite|lehtola:sx:2018}}{{cite|libxc}} It is necessary to have [[Makefile.include#Libxc_.28optional.29|Libxc >= 5.2.0 installed]] and VASP.6.3.0 or higher compiled with [[Precompiler_options#-DUSELIBXC|precompiler options]]. The {{TAG|LIBXC1}} and {{TAG|LIBXC2}} tags (where examples are shown) are also required.
|- -->
 
|{{TAG|GGA}} || Description
|-
|-
|91  || Perdew - Wang 91<ref name="perdew1992"/>
| style="text-align:center;"| CA (or PZ)<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange{{cite|dirac:mpcps:1930}} + Perdew-Zunger parametrization of Ceperley-Alder Monte Carlo correlation data.{{cite|ceperley1980}}{{cite|perdewzunger1981}}
 
|-
|-
|PE  || Perdew-Burke-Ernzerhof<ref name="perdew1996"/>
| style="text-align:center;"| PW92<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange{{cite|dirac:mpcps:1930}} + Perdew-Wang parametrization of Ceperley-Alder Monte Carlo correlation data.{{cite|ceperley1980}}{{cite|perdew1992}}
 
|-
|-
|RP  || revised Perdew-Burke-Ernzerhof
| style="text-align:center;"| SL<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange only.{{cite|dirac:mpcps:1930}} Available since VASP.6.4.3.
 
|-
|-
|AM  || AM05<ref name="armiento:prb:05"/><ref name="mattson:jcp:08"/><ref name="mattson:prb:09"/>
| style="text-align:center;"| CA_C (or PZ_C) || style="text-align:center;"| LDA || Correlation-only Perdew-Zunger parametrization of Ceperley-Alder Monte Carlo correlation data.{{cite|ceperley1980}}{{cite|perdewzunger1981}} Available since VASP.6.4.3.
 
|-
|-
|HL  || Hendin-Lundqvist<ref name="hedin1971"/>
| style="text-align:center;"| PW92_C || style="text-align:center;"| LDA || Correlation-only Perdew-Wang parametrization of Ceperley-Alder Monte Carlo correlation data.{{cite|ceperley1980}}{{cite|perdew1992}} Available since VASP.6.5.0.
 
|-
|-
|CA  || Ceperley-Alder<ref name="ceperley1980"/>
| style="text-align:center;"| VW<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange{{cite|dirac:mpcps:1930}} + Vosko-Wilk-Nusair correlation (VWN5).{{cite|vosko1980}}
 
|-
|-
|PZ  || Ceperley-Alder, parametrization of Perdew-Zunger<ref name="perdewzunger1981"/>
| style="text-align:center;"| HL<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange{{cite|dirac:mpcps:1930}} + Hedin-Lundqvist correlation.{{cite|hedin1971}}
 
|-
|-
|WI || Wigner<ref name="wigner1937"/>
| style="text-align:center;"| WI<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| LDA || Slater exchange{{cite|dirac:mpcps:1930}} + Wigner correlation{{cite|Wigner:tfs:1938}} (Eq. (3.2) in Ref. {{cite|pines:ssp:1955}}).
 
|-
|-
|RP  || RPBE with Pade Approximation<ref name="hammer1999"/>
| style="text-align:center;"| PE || style="text-align:center;"| GGA || Perdew-Burke-Ernzerhof (PBE).{{cite|perdew:prl:1996}}
 
|-
|-
|VW  || Vosko-Wilk-Nusair<ref name="vokso1980"/> (VWN)
| style="text-align:center;"| PBE_X || style="text-align:center;"| GGA || Exchange-only Perdew-Burke-Ernzerhof.{{cite|perdew:prl:1996}} Available since VASP.6.4.3.
 
|-
|-
|B3  || B3LYP<ref name="b3lyp"/> (Joachim Paier), where LDA part is with VWN3-correlation
| style="text-align:center;"| PBE_C || style="text-align:center;"| GGA || Correlation-only Perdew-Burke-Ernzerhof.{{cite|perdew:prl:1996}} Available since VASP.6.4.3.
 
|-
|-
|B5  || B3LYP (Joachim Paier), where LDA part is with VWN5-correlation
| style="text-align:center;"| RE || style="text-align:center;"| GGA || Revised PBE from Zhang and Yang (revPBE).{{cite|zhang1998}}
 
|-
|-
|BF  || BEEF<ref name="beef2012"/>, xc (with libbeef)
| style="text-align:center;"| RP || style="text-align:center;"| GGA || Revised PBE from Hammer ''et al''. (RPBE).{{cite|hammer1999}}
 
|-
|-
|CO  || no exchange-correlation
| style="text-align:center;"| PS || style="text-align:center;"| GGA || Revised PBE for solids (PBEsol).{{cite|perdew:prl:2008}}
 
|-
|-
|PS  || Perdew-Burke-Ernzerhof revised for solids (PBEsol)<ref name="perdew:prl:08"/>
| style="text-align:center;"| AM || style="text-align:center;"| GGA || Armiento-Mattsson (AM05).{{cite|armiento:prb:05}}{{cite|mattson:jcp:08}}{{cite|mattson:prb:09}}
 
|-
|-
|for range-separated ACFDT: ||
| style="text-align:center;"| 91<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| GGA || Perdew-Wang (PW91).{{cite|perdew:prb:1991}}
 
|-
|-
|RA  || new RPA Perdew Wang (by Judith Harl)
| style="text-align:center;"| B3<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| GGA || B3LYP{{cite|stephens:jpc:1994}} with VWN3{{cite|vosko1980}} for LDA correlation.
 
|-
|-
|03  || range-separated ACFDT (LDA - sr RPA) <math>\mu=0.3 \AA^3</math>  
| style="text-align:center;"| B5<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| GGA || B3LYP{{cite|stephens:jpc:1994}} with VWN5{{cite|vosko1980}} for LDA correlation.
 
|-
|-
|05  || range-separated ACFDT (LDA - sr RPA) <math>\mu=0.5 \AA^3</math>
| style="text-align:center;"| OR<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| GGA || optPBE exchange{{cite|klimes:jpcm:2010}} + PBE correlation.{{cite|perdew:prl:1996}}
 
|-
|-
|03  || range-separated ACFDT (LDA - sr RPA) <math>\mu=1.0 \AA^3</math>  
| style="text-align:center;"| BO<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| GGA || optB88 exchange{{cite|klimes:jpcm:2010}} + PBE correlation.{{cite|perdew:prl:1996}} {{TAGBL|PARAM1}}=0.1833333333 for <math>\beta</math> and {{TAGBL|PARAM2}}=0.22 for <math>\mu</math> also need to be specified.
 
|-
|-
|05  || range-separated ACFDT (LDA - sr RPA) <math>\mu=2.0 \AA^3</math>
| style="text-align:center;"| MK<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| GGA || optB86b exchange{{cite|klimes:prb:2011}} + PBE correlation.{{cite|perdew:prl:1996}} The {{TAGBL|PARAM1}} and {{TAGBL|PARAM2}} tags can be used to modify the parameters <math>\mu</math> and <math>\kappa</math>, respectively.
 
|-
|-
|PL  || new RPA+ Perdew Wang (by Judith Harl)
| style="text-align:center;"| ML<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| GGA || PW86R exchange{{cite|lee:prb:2010}} + PBE correlation.{{cite|perdew:prl:1996}}
 
|-
|-
|for vdW (Jiri Klimes): ||
| style="text-align:center;"| CX<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| GGA || CX (LV-PW86r) exchange{{cite|berland:prb:2014}} + PBE correlation.{{cite|perdew:prl:1996}}
 
|-
|-
|RE  || revPBE<ref name="zhang1998"/>
| style="text-align:center;"| BF || style="text-align:center;"| GGA || BEEF (requires VASP compiled with [[Precompiler_options#-Dlibbeef|-Dlibbeef]]).{{cite|beef2012}}
|-
 
|OR  || optPBE<ref name="klimes2010"/>
|-
|BO  || optB88<ref name="klimes2010"/>
|-
|MK  || optB86b<ref name="klimes2010"/>
|}
|}


The tags AM (AM05) and PS (PBEsol) are only supported by VASP.5.X. The AM05 functional and the PBEsol functional are constructed using different principles, but both aim at a decent description of yellium surface energies. In practice, they yield quite similar results for most materials. Both are available for spin polarized calculations.
<span style="color:blue">(1)</span> The Slater LDA exchange includes relativistic effects.{{cite|macdonald:jpc:1979}}
 
<span style="color:blue">(2)</span> The exchange component was designed in particular to be used as the exchange component of {{TAG|Nonlocal vdW-DF functionals}} and with {{TAG|AGGAC}}=0 such that only LDA is used for the local correlation, see [[Nonlocal_vdW-DF_functionals#List_of_nonlocal_vdW-DF_functionals|list of nonlocal vdW-DF functionals]].
 
== Related tags and articles ==
{{TAG|LIBXC1}},
{{TAG|LIBXC2}},
{{TAG|ALDAX}},
{{TAG|ALDAC}},
{{TAG|AGGAX}},
{{TAG|AGGAC}},
{{TAG|METAGGA}},
{{TAG|XC}}
 
{{sc|GGA|Examples|Examples that use this tag}}


== References ==
== References ==
<references>
<references/>
<ref name="armiento:prb:05">[http://link.aps.org/doi/10.1103/PhysRevB.72.085108 R. Armiento and A. E. Mattsson, Phys. Rev. B 72, 085108 (2005).]</ref>
 
<ref name="mattson:jcp:08">[http://dx.doi.org/10.1063/1.2835596 A. E. Mattsson, R. Armiento, J. Paier, G. Kresse, J.M. Wills, and T.R. Mattsson, J. Chem. Phys. 128, 084714 (2008).]</ref>
<ref name="mattson:prb:09">[http://link.aps.org/doi/10.1103/PhysRevB.79.155101 A. E. Mattsson and R. Armiento, Phys. Rev. B 79, 155101 (2009).]</ref>
<ref name="perdew:prl:08">[http://link.aps.org/doi/10.1103/PhysRevB.79.155107 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).]</ref>
<ref name="hedin1971">[http://iopscience.iop.org/article/10.1088/0022-3719/4/14/022/meta;jsessionid=6F1B9F8BE588208D706AAD78E6F0E49A.c2.iopscience.cld.iop.org L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).]</ref>
<ref name="ceperley1980">[http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.45.566 D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).]</ref>
<ref name="perdewzunger1981">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.23.5048 J. P. Perdew and Alex Zunger, Phys. Rev. B 23, 5048 (1981).]</ref>
<ref name="wigner1937">[http://aip.scitation.org/doi/10.1063/1.1750108 E. Wigner, J. Chem. Phys. 5, 726 (1937).]</ref>
<ref name="hammer1999">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.59.7413 B. Hammer, L. B. Hansen and J. K. Nørskov, Phys. Rev. B 59, 7413 (1999).]</ref>
<ref name="vokso1980">[http://www.nrcresearchpress.com/doi/abs/10.1139/p80-159#.WJLvYmf950w S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980).]</ref>
<ref name="b3lyp">[http://aip.scitation.org/doi/10.1063/1.464913 A. D. Becke, J. Chem. Phys. 98, 5648 (1993).]</ref>
<ref name="beef2012">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.235149 Jess Wellendorff, Keld T. Lundgaard, Andreas Møgelhøj, Vivien Petzold, David D. Landis, Jens K. Nørskov, Thomas Bligaard and Karsten W. Jacobsen, Phys. Rev. B 85, 235149 (2012).]</ref>
<ref name="zhang1998">[http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.890 Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998).]</ref>
<ref name="klimes2010">[http://iopscience.iop.org/article/10.1088/0953-8984/22/2/022201/meta J. Klimeš, D. R. Bowler, and A. Michaelides, J. Phys.: Cond. Matt. 22, 022201 (2010).]</ref>
<ref name="perdew1992">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.45.13244 J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).]</ref>
<ref name="perdew1996">[http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.77.3865 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).]</ref>
</references>
----
----
[[The_VASP_Manual|Contents]]


[[Category:INCAR]]
[[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:GGA]]

Latest revision as of 15:20, 19 December 2024

GGA = PE | RP | PS | AM | LIBXC | ... 

Default: The functional specified by LEXCH in the POTCAR if METAGGA and XC are also not specified.

Description: Selects a LDA or GGA exchange-correlation functional.

Important: VASP recalculates the exchange-correlation energy inside the PAW sphere and corrects the atomic energies given by the POTCAR file. For this to work, the original LEXCH tag must not be modified in the POTCAR file.
Mind:
  • When the OR, BO, MK, ML or CX GGA is used in combination with the nonlocal vdW-DF functional of Dion et al.[1], the GGA component of the correlation should in principle be turned off with AGGAC=0 (see nonlocal vdW-DF functionals).
  • The XC tag, available since VASP.6.4.3, can be used to specify any linear combination of LDA, GGA and METAGGA exchange-correlation functionals.

Available functionals

This table lists the LDA and GGA functionals available in VASP. The names of functionals which end with "_X" and "_C" correspond to exchange-only and correlation functionals, respectively.

GGA= Type Description
LIBXC (or LI) LDA/GGA Any LDA or GGA from the external library Libxc.[2][3][4] It is necessary to have Libxc >= 5.2.0 installed and VASP.6.3.0 or higher compiled with precompiler options. The LIBXC1 and LIBXC2 tags (where examples are shown) are also required.
CA (or PZ)(1) LDA Slater exchange[5] + Perdew-Zunger parametrization of Ceperley-Alder Monte Carlo correlation data.[6][7]
PW92(1) LDA Slater exchange[5] + Perdew-Wang parametrization of Ceperley-Alder Monte Carlo correlation data.[6][8]
SL(1) LDA Slater exchange only.[5] Available since VASP.6.4.3.
CA_C (or PZ_C) LDA Correlation-only Perdew-Zunger parametrization of Ceperley-Alder Monte Carlo correlation data.[6][7] Available since VASP.6.4.3.
PW92_C LDA Correlation-only Perdew-Wang parametrization of Ceperley-Alder Monte Carlo correlation data.[6][8] Available since VASP.6.5.0.
VW(1) LDA Slater exchange[5] + Vosko-Wilk-Nusair correlation (VWN5).[9]
HL(1) LDA Slater exchange[5] + Hedin-Lundqvist correlation.[10]
WI(1) LDA Slater exchange[5] + Wigner correlation[11] (Eq. (3.2) in Ref. [12]).
PE GGA Perdew-Burke-Ernzerhof (PBE).[13]
PBE_X GGA Exchange-only Perdew-Burke-Ernzerhof.[13] Available since VASP.6.4.3.
PBE_C GGA Correlation-only Perdew-Burke-Ernzerhof.[13] Available since VASP.6.4.3.
RE GGA Revised PBE from Zhang and Yang (revPBE).[14]
RP GGA Revised PBE from Hammer et al. (RPBE).[15]
PS GGA Revised PBE for solids (PBEsol).[16]
AM GGA Armiento-Mattsson (AM05).[17][18][19]
91(1) GGA Perdew-Wang (PW91).[20]
B3(1) GGA B3LYP[21] with VWN3[9] for LDA correlation.
B5(1) GGA B3LYP[21] with VWN5[9] for LDA correlation.
OR(2) GGA optPBE exchange[22] + PBE correlation.[13]
BO(2) GGA optB88 exchange[22] + PBE correlation.[13] PARAM1=0.1833333333 for and PARAM2=0.22 for also need to be specified.
MK(2) GGA optB86b exchange[23] + PBE correlation.[13] The PARAM1 and PARAM2 tags can be used to modify the parameters and , respectively.
ML(2) GGA PW86R exchange[24] + PBE correlation.[13]
CX(2) GGA CX (LV-PW86r) exchange[25] + PBE correlation.[13]
BF GGA BEEF (requires VASP compiled with -Dlibbeef).[26]

(1) The Slater LDA exchange includes relativistic effects.[27]

(2) The exchange component was designed in particular to be used as the exchange component of Nonlocal vdW-DF functionals and with AGGAC=0 such that only LDA is used for the local correlation, see list of nonlocal vdW-DF functionals.

Related tags and articles

LIBXC1, LIBXC2, ALDAX, ALDAC, AGGAX, AGGAC, METAGGA, XC

Examples that use this tag

References

  1. ↑ M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Phys. Rev. Lett. 92, 246401 (2004).
  2. ↑ M. A. L. Marques, M. J. T. Oliveira, and T. Burnus, Comput. Phys. Commun., 183, 2272 (2012).
  3. ↑ S. Lehtola, C. Steigemann, M. J. T. Oliveira, and M. A. L. Marques, SoftwareX, 7, 1 (2018).
  4. ↑ https://libxc.gitlab.io
  5. ↑ a b c d e f P. A. M. Dirac, Math. Proc. Cambridge Philos. Soc. 26, 376 (1930).
  6. ↑ a b c d D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
  7. ↑ a b J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
  8. ↑ a b J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
  9. ↑ a b c S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).
  10. ↑ L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).
  11. ↑ E. Wigner, Trans. Faraday Soc. 34, 678 (1938).
  12. ↑ D. Pines, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1955), Vol. I, p. 367.
  13. ↑ a b c d e f g h J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett., 77, 3865 (1996).
  14. ↑ Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998).
  15. ↑ B. Hammer, L. B. Hansen, and J. K. Nørskov, Phys. Rev. B 59, 7413 (1999).
  16. ↑ J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Phys. Rev. Lett. 100, 136406 (2008).
  17. ↑ R. Armiento and A. E. Mattsson, Phys. Rev. B 72, 085108 (2005).
  18. ↑ A. E. Mattsson, R. Armiento, J. Paier, G. Kresse, J. M. Wills, and T. R. Mattsson, J. Chem. Phys. 128, 084714 (2008).
  19. ↑ A. E. Mattsson and R. Armiento, Phys. Rev. B 79, 155101 (2009).
  20. ↑ J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).
  21. ↑ a b P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98, 11623 (1994).
  22. ↑ a b J. Klimeš, D. R. Bowler, and A. Michaelides, J. Phys.: Condens. Matter 22, 022201 (2010).
  23. ↑ J. Klimeš, D. R. Bowler, and A. Michaelides, Phys. Rev. B 83, 195131 (2011).
  24. ↑ K. Lee, E. D. Murray, L. Kong, B. I. Lundqvist, and D. C. Langreth, Phys. Rev. B 82, 081101(R) (2010).
  25. ↑ K. Berland and P. Hyldgaard, Phys. Rev. B 89, 035412 (2014).
  26. ↑ J. Wellendorff, K. T. Lundgaard, A. Møgelhøj, V. Petzold, D. D. Landis, Jens K. Nørskov, T. Bligaard, and K. W. Jacobsen, Phys. Rev. B 85, 235149 (2012).
  27. ↑ A. H. MacDonald and S. H. Vosko, A relativistic density functional formalism, J. Phys. C 12, 2977 (1979).
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