METAGGA: Difference between revisions

Latest revision as of 09:23, 20 December 2024

METAGGA = SCAN | LIBXC | MBJ | ...

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

Description: Selects a meta-GGA exchange-correlation functional.

Mind:

If you select a meta-GGA functional, make sure that you use POTCAR files that are suited for meta-GGA functionals. However, note that this requirement does not concern the deorbitalized meta-GGAs, i.e. those that do not depend on the kinetic-energy density, like SCAN-L.
Depending on the meta-GGA that is chosen, it may be recommended to use a PAW potential that is more accurate than the standard/recommended one. This is particularly the case with functionals (e.g., MBJ or the Minnesota functionals like M06-L) that are very different from the standard ones like PBE or SCAN. The reason is that for such special functionals, using a PAW potential that includes more states in the valence or that is harder may be required to obtain results that are closer to the results that would be obtained with an all-electron code. That also means that it may be a good idea to do test calculations with different PAW potentials.
For accuracy, it is strongly recommended to set LASPH=.TRUE. to account for aspherical contributions to the PAW one-centre terms.
Since VASP.6.4.0 it is possible to use hybrid functionals that mix meta-GGA and Hartree-Fock exchange (AEXX). Furthermore, two new tags, AMGGAX and AMGGAC, were created.
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.
The results obtained with the meta-GGA functionals that depend on the Laplacian of the density $\nabla^2n$ (e.g., SCAN-L) may not be reliable for large values of the energy cutoff ENCUT due to numerical instability. According to some tests, it is not recommended to use values of ENCUT above 800 eV.

Available functionals

This table lists the meta-GGA functionals available in VASP. There are essentially two types of meta-GGAs, that differ in the variable on which they depend (in addition to $n$ and $\nabla n$ ): the kinetic-energy density $\tau$ or the Laplacian of the density $\nabla^2n$ . The names of functionals which end with "_X" and "_C" correspond to exchange-only and correlation functionals, respectively.

METAGGA=	Variable	Description
LIBXC		Any MGGA from the external library Libxc.^[1]^[2]^[3] 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.
TPSS, TPSS_X or TPSS_C⁽¹⁾	$\tau$	TPSS.^[4]
RTPSS, RTPSS_X or RTPSS_C⁽¹⁾	$\tau$	revTPSS is a revised version of TPSS.^[5]
M06L, M06L_X or M06L_C⁽¹⁾	$\tau$	M06-L.^[6]
MS0, MS0_X or MS0_C⁽¹⁾	$\tau$	MS0 corresponds to $\kappa=0.29$ , $c=0.28771$ and $b=1.0$ .^[7]^[8] Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
MS1, MS1_X or MS1_C⁽¹⁾	$\tau$	MS1 corresponds to $\kappa=0.404$ , $c=0.18150$ and $b=1.0$ .^[8] Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
MS2, MS2_X or MS2_C⁽¹⁾	$\tau$	MS2 corresponds to $\kappa=0.504$ , $c=0.14601$ and $b=4.0$ .^[8] Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
SCAN, SCAN_X or SCAN_C⁽¹⁾	$\tau$	SCAN.^[9] May possibly lead to numerical instabilities. rSCAN or r $^{2}$ SCAN are more stable and should give similar results. Available since VASP.5.4.4.
RSCAN, RSCAN_X or RSCAN_C⁽¹⁾	$\tau$	rSCAN is a regularized version of SCAN that is numerically more stable.^[10]
R2SCAN, R2SCAN_X or R2SCAN_C⁽¹⁾	$\tau$	r $^{2}$ SCAN is a regularized version of SCAN that is numerically more stable.^[11] Available since VASP.6.2.0, or in version 5.4.4 by patch 4.
SREGTM1, SREGTM2 or SREGTM3	$\tau$	sregTM^[12] versions 1, 2 or 3 of a regularized Tao-Mo functional.^[13] Available since VASP.6.4.3.
TASK_X⁽²⁾	$\tau$	TASK exchange.^[14] Available since VASP.6.5.0.
LAK, LAK_X or LAK_C	$\tau$	LAK.^[15] Available since VASP.6.5.0.
MSPBEL, MSRPBEL or MSB86BL	$\tau$	MS-PBEl, MS-RPBEl or MS-B86bl.^[16] Available since VASP.6.5.0.
RMSPBEL, RMSRPBEL or RMSB86BL	$\tau$	rMS-PBEl, rMS-RPBEl or rMS-B86bl.^[17] Available since VASP.6.5.0.
SCANL	$\nabla^2n$	SCAN-L^[18]^[19] is a deorbitalized version of SCAN. Available since VASP.6.4.0.
RSCANL	$\nabla^2n$	rSCAN-L is a deorbitalized version of rSCAN. Available since VASP.6.4.0.
R2SCANL	$\nabla^2n$	r $^2$ SCAN-L is a deorbitalized versions of r $^2$ SCAN.^[20]^[21] Available since VASP.6.4.0.
OFR2	$\nabla^2n$	Orbital-free regularized-restored SCAN (OFR2).^[21] Available since VASP.6.4.0.
SREGTM2L	$\nabla^2n$	v2-sregTM-L is a deorbitalized versions of v2-sregTM.^[22] Available since VASP.6.4.0.
MBJ⁽³⁾	$\nabla^2n,\tau$	Modified Becke-Johnson potential.^[23]^[24] The CMBJA, CMBJB and CMBJE tags correspond to $\alpha$ , $\beta$ and the power $e=1/2$ (that can be modified) in Eq. (3) of Ref. ^[24], respectively. The default values are $\alpha=-0.012$ , $\beta=1.023$ bohr $^{1/2}$ and $e=1/2$ .^[24]
LMBJ⁽³⁾	$\nabla^2n,\tau$	The local MBJ (LMBJ) potential.^[25]^[26] The CMBJA, CMBJB, CMBJE, SMBJ, and RSMBJ tags correspond to $\alpha$ , $\beta$ , the power $e=1$ (that can be modified) of $\bar{g}$ , $\sigma$ and $r_{s}^{\mathrm{th}}$ in Eqs. (5)-(7) of Ref. ^[26], respectively. The default values are (see erratum of Ref. ^[26]) $\alpha=0.488$ , $\beta=0.5$ bohr, $e=1$ , $\sigma=2$ $\AA$ ( $=3.78$ bohr), and $r_{s}^{\mathrm{th}}=7$ bohr (which corresponds to $n_{\mathrm{th}}=6.96\times10^{-4}$ e/bohr $^{3}$ ).

(1) The exchange-only and correlation-only implementations are available since VASP.6.4.3.

(2) In Ref. ^[14] TASK exchange is combined with LDA-PW92 correlation.^[27] This can be done with XC=TASK_X PW92_C in INCAR.

(3) A few points about the MBJ and LMBJ potentials:

These are potential-only methods, i.e., there is no corresponding exchange-correlation energy $E_{xc}$ . The used expression for $E_{xc}$ is LDA, which is an arbitrary choice. This means that MBJ and LMBJ calculations can never be self-consistent with respect to the total energy, and thus we cannot compute Hellmann-Feynman forces (i.e., no ionic relaxation, etc.). Actually, these potentials aim solely at a description of the electronic properties, primarily the band gap, or magnetic moments.
MBJ and LMBJ calculations may converge very slowly, so the number of maximum electronic steps (NELM) should be set higher than usual.
In the presence of an extended vacuum region (e.g., surfaces) or an interface, the average of $|\nabla n|/n$ has no meaning. Therefore, MBJ calculations should be done with a fixed value of $c$ , which can be done with the CMBJ tag., or alternatively with the LMBJ that was proposed for the purpose to be applicable to systems with vacuum or interfaces.

POTCAR files: required information

Calculations with a meta-GGA that depends on the kinetic-energy density require POTCAR files that include information on the kinetic-energy density of the core electrons. Almost all recent POTCAR files do fulfill this requirement, but there are some notable exceptions like O_GW. To check whether a particular POTCAR contains this information, type:

grep kinetic POTCAR

This should yield at least the following lines (for each element on the file):

kinetic energy-density
mkinetic energy-density pseudized

and for PAW datasets with partial core corrections:

kinetic energy density (partial)

Mind: For POTCAR files without core electrons (H, He, Li_sv, Be_sv, and _GW variants thereof) the grep command given above will not return the line about pseudized kinetic energy-density, since all electrons are considered as valence. These potentials can nevertheless be used for all meta-GGA functionals.

Aspherical contributions related to one-center terms

LASPH =.TRUE. should be selected if a meta-GGA functional is selected. If LASPH =.FALSE., the one-center contributions are only calculated for a spherically averaged density and kinetic-energy density. This means that the one-center contributions to the Kohn-Sham potential are also spherical. Since the PAW method describes the entire space using plane waves, errors are often small even if the non-spherical contributions to the Kohn-Sham potential are neglected inside the PAW spheres (additive augmentation, as opposed to the APW or FLAPW method where the plane wave contribution only describes the interstitial region between the atoms). Anyhow, if the density is strongly non-spherical around some atoms in your structure, LASPH =.TRUE. must be selected. Non-spherical terms are particularly encountered in d- and f-elements, dimers, molecules, and solids with strong directional bonds.

Convergence issues

If convergence problems are encountered, it is recommended to preconverge the calculations using the PBE functional and start the calculation from the WAVECAR file corresponding to the PBE ground state. Furthermore, ALGO = A (conjugate gradient algorithm for orbitals) is often more stable than charge density mixing, in particular if the system contains vacuum regions.

References

[marques:cpc:2012-1] M. A. L. Marques, M. J. T. Oliveira, and T. Burnus, Comput. Phys. Commun., 183, 2272 (2012).

[lehtola:sx:2018-2] S. Lehtola, C. Steigemann, M. J. T. Oliveira, and M. A. L. Marques, SoftwareX, 7, 1 (2018).

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

[tao:prl:2003-4] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Climbing the Density Functional Ladder: Nonempirical Meta–Generalized Gradient Approximation Designed for Molecules and Solids, Phys. Rev. Lett. 91, 146401 (2003).

[perdew:prl:2009-5] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin, and J. Sun, Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry, Phys. Rev. Lett. 103, 026403 (2009).

[zhao:jcp:06-6] Y. Zhao and D. G. Truhlar, J. Chem. Phys. 125, 194101 (2006).

[sun:jcp:12-7] J. Sun, B. Xiao, and A. Ruzsinszky, J. Chem. Phys. 137, 051101 (2012).

[sun:jcp:13-8] J. Sun, R. Haunschild, B. Xiao, I. W. Bulik, G. E. Scuseria, and J. P. Perdew, J. Chem. Phys. 138, 044113 (2013).

[sun:prl:15-9] J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015).

[bartok:jcp:19-10] A. P. Bartók and J. R. Yates, J. Chem. Phys. 150, 161101 (2019).

[furness:jpcl:20-11] J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, and J. Sun, J. Phys. Chem. Lett. 11, 8208 (2020).

[francisco_a:jcp:2023-12] H. Francisco, A. C. cancio, and S. B. Trickey, Reworking the Tao–Mo exchange-correlation functional. I. Reconsideration and simplification, J. Chem. Phys. 159, 214102 (2023).

[tao:prl:2016-13] J. Tao and Y. Mo, Accurate Semilocal Density Functional for Condensed-Matter Physics and Quantum Chemistry, Phys. Rev. Lett. 117, 073001 (2015).

[aschebrock:prr:2019-14] T. Aschebrock and S. Kümmel, Ultranonlocality and accurate band gaps from a meta-generalized gradient approximation, Phys. Rev. Res. 1, 033082 (2019)

[lebeda:prl:2024-15] T. Lebeda, T. Aschebrock, and S. Kümmel, Balancing the Contributions to the Gradient Expansion: Accurate Binding and Band Gaps with a Nonempirical Meta-GGA, Phys. Rev. Lett. 133, 136402 (2024).

[smeets:jpca:2019-16] E. W. S. Smeets, J. Voos, and G.-J. Kroes, J. Phys. Chem. A 123, 5395 (2019).

[cai:jpcc:2024-17] Y. Cai, R. Michiels, F. De Luca, E. Neyts, X. Tu, A. Bogaerts, and N. Gerrits, J. Phys. Chem. C 128, 8611 (2024).

[mejia-rodriguez:pra:2017-18] D. Mejía-Rodríguez and S. B. Trickey, Deorbitalization strategies for meta-generalized-gradient-approximation exchange-correlation functionals, Phys. Rev. A 91, 052512 (2017).

[mejia-rodriguez:prb:2018-19] D. Mejia-Rodriguez and S. B. Trickey, Deorbitalized meta-GGA exchange-correlation functionals in solids, Phys. Rev. B 98, 115161 (2018).

[mejia-rodriguez:prb:2020-20] D. Mejía-Rodríguez and S. B. Trickey, Meta-GGA performance in solids at almost GGA cost, Phys. Rev. B 102, 121109(R) (2020).

[kaplan:prm:2022-21] A. D. Kaplan and J. P. Perdew, Phys. Rev. Mater. 6, 083803 (2022).

[francisco_b:jcp:2023-22] H. Francisco, A. C. cancio, and S. B. Trickey, Reworking the Tao–Mo exchange–correlation functional. II. De-orbitalization, J. Chem. Phys. 159, 214103 (2023).

[becke:jcp:06-23] A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006).

[tran:prl:09-24] F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009).

[rauch:jctc:2020-25] T. Rauch, M. A. L. Marques, and S. Botti, Local Modified Becke-Johnson Exchange-Correlation Potential for Interfaces, Surfaces, and Two-Dimensional Materials, J. Chem. Theory Comput. 16, 2654 (2020).

[rauch:prb:2020-26] T. Rauch, M. A. L. Marques, and S. Botti, Accurate electronic band gaps of two-dimensional materials from the local modified Becke-Johnson potential, Phys. Rev. B 101, 245163 (2020).

[perdew1992-27] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).

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@@ Line 1: / Line 1: @@
-{{TAGDEF|METAGGA|SCAN {{!}} RTPSS {{!}} MBJ {{!}} LIBXC {{!}} ...}}
+{{TAGDEF|METAGGA|SCAN {{!}} LIBXC {{!}} MBJ {{!}} ...}}
-Default: The functional specified by LEXCH in the POTCAR if also {{TAG|GGA}} is not specified.
+Default: The functional specified by {{TAG|LEXCH}} in the {{FILE|POTCAR}} if {{TAG|GGA}} and {{TAG|XC}} are also not specified.
-Description: selects a meta-GGA functional.
+Description: Selects a meta-GGA exchange-correlation functional.
 ----
 {{NB|mind|
-*If you select a meta-GGA functional, make sure that you use [[METAGGA#POTCAR files: required information | POTCAR files that are suited for meta-GGA functionals]].
+*If you select a meta-GGA functional, make sure that you use [[METAGGA#POTCAR files: required information | POTCAR files that are suited for meta-GGA functionals]]. However, note that this requirement does not concern the deorbitalized meta-GGAs, i.e. those that do not depend on the kinetic-energy density, like SCAN-L.
-*Depending on the meta-GGA that is chosen, it may be recommended to use a [[Available PAW potentials|PAW potential]] that is more accurate than the standard/recommended one. This is particularly the case with functionals (e.g., MBJ or the Minnesota functionals) that are very different from the standard ones like PBE or SCAN. The reason is that for such ''special'' functionals, using a PAW potential that includes more states in the valence or that is harder may be required to obtain results that are closer to the results that would be obtained with an all-electron code. Thus, it may be a good idea to do test calculations with different PAW potentials.
+*Depending on the meta-GGA that is chosen, it may be recommended to use a [[Available PAW potentials|PAW potential]] that is more accurate than the standard/recommended one. This is particularly the case with functionals (e.g., MBJ or the Minnesota functionals like M06-L) that are very different from the standard ones like PBE or SCAN. The reason is that for such ''special'' functionals, using a PAW potential that includes more states in the valence or that is harder may be required to obtain results that are closer to the results that would be obtained with an all-electron code. That also means that it may be a good idea to do test calculations with different PAW potentials.
-*It is strongly recommended to set {{TAG|LASPH}}{{=}}.TRUE. to [[METAGGA#Aspherical contributions related to one-center terms| account for aspherical contributions to the PAW one-centre terms]].
+*For accuracy, it is strongly recommended to set {{TAG|LASPH}}{{=}}.TRUE. to [[METAGGA#Aspherical contributions related to one-center terms| account for aspherical contributions to the PAW one-centre terms]].
-*Since VASP.6.4.0 it is possible to use hybrid functionals that mix meta-GGA and Hartree-Fock exchange ({{TAG|AEXX}}). Furthermore, two new tags, {{TAG|AMGGAX}} and {{TAG|AMGGAC}}, were created.}}
+*Since VASP.6.4.0 it is possible to use hybrid functionals that mix meta-GGA and Hartree-Fock exchange ({{TAG|AEXX}}). Furthermore, two new tags, {{TAG|AMGGAX}} and {{TAG|AMGGAC}}, were created.
+*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 results obtained with the meta-GGA functionals that depend on the Laplacian of the density <math>\nabla^2n</math> (e.g., SCAN-L) may not be reliable for large values of the energy cutoff {{TAG|ENCUT}} due to numerical instability. According to some tests, it is not recommended to use values of {{TAG|ENCUT}} above 800 eV.}}
 ==  Available functionals  ==
-*{{TAG|METAGGA}}=LIBXC
+This table lists the meta-GGA functionals available in VASP. There are essentially two types of meta-GGAs, that differ in the variable on which they depend (in addition to <math>n</math> and <math>\nabla n</math>): the kinetic-energy density <math>\tau</math> or the Laplacian of the density <math>\nabla^2n</math>. The names of functionals which end with "_X" and "_C" correspond to exchange-only and correlation functionals, respectively.
-:The LIBXC tag allows to use a meta-GGA functional from the library of exchange-correlation functionals Libxc{{cite|marques:cpc:2012}}{{cite|lehtola:sx:2018}}{{cite|libxc}}. Along with {{TAG|METAGGA}}=LIBXC, it is also necessary to specify the tags {{TAG|LIBXC1}} and {{TAG|LIBXC2}} that specify the particular functional. Note that 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]].
-*{{TAG|METAGGA}}=TPSS, RTPSS, or M06L
+{| class="sortable wikitable"
-:The implementation of the TPSS and RTPSS (revTPSS) self-consistent meta-generalized gradient approximation within the projector-augmented-wave method in VASP is discussed by Sun ''et al.''{{cite|sun:prb:11}}. For details on the M06-L functional, refer to the paper by Zhao and Truhlar{{cite|zhao:jcp:06}}.
+|-
+! style="text-align:center;" style=width:16em | METAGGA= !! style="text-align:center;"| Variable !! class="unsortable" | Description
-*{{TAG|METAGGA}}=MS0, MS1, or MS2
+|-
-:The MS (where MS stands for "made simple") functionals are presented in detail in references {{cite|sun:jcp:12}} and {{cite|sun:jcp:13}}. These functionals are believed to improve the description of noncovalent interactions over PBE, TPSS and revTPSS but not over M06L. The MS functionals are available as of VASP version ≥ 5.4.1.
+| style="text-align:center;"| LIBXC || style="text-align:center;"| ||
+Any MGGA 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|METAGGA}}=SCAN
+|-
-:The SCAN (Strongly constrained and appropriately normed) {{cite|sun:prl:15}} functional is a semilocal density functional that fulfills all known constraints that the exact density functional must fulfill. There are indications that this functional is superior to most gradient corrected functionals {{cite|sun:natm:16}}. This functional is only available as of VASP version ≥ 5.4.3.
+| style="text-align:center;"| TPSS, TPSS_X or TPSS_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+TPSS.{{cite|tao:prl:2003}}
-*{{TAG|METAGGA}}=RSCAN
+|-
-:The rSCAN (regularized SCAN) functional {{cite|bartok:jcp:19}}, introduces regularizations that improve the numerical sensitivity and convergence behavior. These regularizations break several of the exact constraints that the parent SCAN functional was designed to satisfy. However, testing has indicated that the accuracy of rSCAN can be inferior to SCAN in some cases {{cite|mejia-rodriguez:jcp:19}}. This functional is available as of VASP version ≥ 6.2.0.
+| style="text-align:center;"| RTPSS, RTPSS_X or RTPSS_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+revTPSS is a revised version of TPSS.{{cite|perdew:prl:2009}}
-*{{TAG|METAGGA}}=R2SCAN
+|-
-:The r<math>^2</math>SCAN (regularized-restored SCAN) functional {{cite|furness:jpcl:20}} modifies the regularizations introduced in rSCAN to enforce adherence to the exact constraints obeyed by SCAN. It fulfills all known constraints. However, it only recovers the slowly varying density-gradient expansion for exchange to second order, while SCAN recovers the expansion to 4th order. Testing indicates that r<math>^2</math>SCAN at least matches the accuracy of the parent SCAN functional but with significantly improved numerical efficiency and accuracy under low-cost computational settings. This functional is available as of VASP version ≥ 6.2.0, or in version 5.4.4 by [https://gitlab.com/dhamil/r2scan-subroutines/-/tree/master/vasp_patch_files patch 4].
+| style="text-align:center;"| M06L, M06L_X or M06L_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+M06-L.{{cite|zhao:jcp:06}}
-*{{TAG|METAGGA}}=SREGTM1, SREGTM2, or SREGTM3
+|-
-:The functionals v1-sregTM, v2-sregTM, and v3-sregTM from Francisco ''et al.''{{cite|francisco_a:jcp:2023}}
+| style="text-align:center;"| MS0, MS0_X or MS0_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+MS0 corresponds to <math>\kappa=0.29</math>, <math>c=0.28771</math> and <math>b=1.0</math>.{{cite|sun:jcp:12}}{{cite|sun:jcp:13}}
+Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
-*{{TAG|METAGGA}}=SCANL, RSCANL, or R2SCANL
+|-
-:The functionals SCAN-L{{cite|mejia-rodriguez:pra:2017}}{{cite|mejia-rodriguez:prb:2018}}, rSCAN-L, and r<math>^2</math>SCAN-L{{cite|mejia-rodriguez:prb:2020}}{{cite|kaplan:prm:2022}} are deorbitalized versions of SCAN, rSCAN, and r<math>^2</math>SCAN, respectively. They do not depend on the kinetic-energy density <math>\tau</math>, but on the Laplacian of the density <math>\nabla^{2}n</math>, instead.
+| style="text-align:center;"| MS1, MS1_X or MS1_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+MS1 corresponds to <math>\kappa=0.404</math>, <math>c=0.18150</math> and <math>b=1.0</math>.{{cite|sun:jcp:13}}
+Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
-*{{TAG|METAGGA}}=OFR2
+|-
-:The OFR2{{cite|kaplan:prm:2022}} functional depends on the Laplacian of the density <math>\nabla^{2}n</math>, but not on the kinetic-energy density <math>\tau</math>.
+| style="text-align:center;"| MS2, MS2_X or MS2_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+MS2 corresponds to <math>\kappa=0.504</math>, <math>c=0.14601</math> and <math>b=4.0</math>.{{cite|sun:jcp:13}}
+Note that the correlation component, called vPBEc or regTPSS in the literature, is a GGA. Available since VASP.5.4.1.
-*{{TAG|METAGGA}}=SREGTM2L
+|-
-:The functional v2-sregTM-L from Francisco ''et al.''{{cite|francisco_b:jcp:2023}}, which depends on the Laplacian of the density <math>\nabla^{2}n</math>, but not on the kinetic-energy density <math>\tau</math>.
+| style="text-align:center;"| SCAN, SCAN_X or SCAN_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
+SCAN.{{cite|sun:prl:15}} May possibly lead to numerical instabilities. rSCAN or r<math>^{2}</math>SCAN are more stable and should give similar results. Available since VASP.5.4.4.
-*{{TAG|METAGGA}}=MBJ
+|-
-:The modified Becke-Johnson (MBJ) potential{{cite|becke:jcp:06}}{{cite|tran:prl:09}} yields band gaps with an accuracy similar to hybrid functionals or GW methods, but is computationally less expensive. The exchange part of the MBJ potential (that is combined with the LDA correlation potential, <math>v_{xc}^{\rm MBJ}=v_{x}^{\rm MBJ}+v_{c}^{\rm LDA}</math>) consists of two terms whose relative weights are determined by a system-dependent constant <math>c</math>:
+| style="text-align:center;"| RSCAN, RSCAN_X or RSCAN_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
-:<math>
+rSCAN is a regularized version of SCAN that is numerically more stable.{{cite|bartok:jcp:19}}
- v_{x}^{\rm MBJ}(\mathbf{r}) = cv_{x}^{\rm BR}(\mathbf{r}) + (3c-2)\frac{1}{\pi}\sqrt{\frac{5}{6}}\sqrt{\frac{\tau(\mathbf{r})}{n(\mathbf{r})}}
-</math>
-:where <math>v_{x}^{\rm BR}</math> is the Becke-Roussel (BR) potential that mimics the Coulomb potential created by the exchange hole{{cite|becke:pra:1989}} and depends on <math>n</math>, <math>\nabla n</math>, <math>\nabla^{2}n</math> and <math>\tau</math>.
-:The system-dependent <math>c</math> is a function of the average of <math>|\nabla n|/n</math> in the unit cell (of volume <math>V_{\mathrm{cell}}</math>):
+|-
-:<math>
+| style="text-align:center;"| R2SCAN, R2SCAN_X or R2SCAN_C<span style="color:blue"><sup>(1)</sup></span> || style="text-align:center;"| <math>\tau</math> ||
-c=\alpha+\beta \left(\frac{1}{V_{\mathrm{cell}}}\int_{\mathrm{cell}}\frac{|\nabla n(\mathbf{r}')|}{n(\mathbf{r}')}d{\mathbf{r}'}\right)^{e}
+r<math>^{2}</math>SCAN is a regularized version of SCAN that is numerically more stable.{{cite|furness:jpcl:20}} Available since VASP.6.2.0, or in version 5.4.4 by [https://gitlab.com/dhamil/r2scan-subroutines/-/tree/master/vasp_patch_files patch 4].
-</math>
-:where <math>\alpha</math>, <math>\beta</math>, and <math>e</math> are free parameters that can be set by means of the {{TAG|CMBJA}}, {{TAG|CMBJB}}, and {{TAG|CMBJE}} tags, respectively. The default values are <math>\alpha=-0.012</math>, <math>\beta=1.023</math> bohr<math>^{1/2}</math>, and <math>e=1/2</math>{{cite|tran:prl:09}}. In Ref. {{cite|koller:prb:2012}}, the alternative values <math>\alpha=0.488</math>, <math>\beta=0.5</math> bohr, and <math>e=1</math> were proposed.
-:{{NB|mind|
+|-
-:*The MBJ functional is a ''potential-only'' functional, ''i.e.'', there is no corresponding MBJ exchange-correlation energy, instead <math>E_{xc}</math> is taken from LDA. This means that MBJ calculations can never be self-consistent with respect to the total energy, and thus we cannot compute Hellmann-Feynman forces (''i.e.'', no ionic relaxation, etc.). Actually, MBJ calculations aim solely at a description of the electronic properties, primarily band gaps.
+| style="text-align:center;"| SREGTM1, SREGTM2 or SREGTM3 || style="text-align:center;"| <math>\tau</math> ||
-:*MBJ calculations converge very slowly, so the number of maximum electronic steps ({{TAG|NELM}}) should be set higher than usual.
+sregTM{{cite|francisco_a:jcp:2023}} versions 1, 2 or 3 of a regularized Tao-Mo functional.{{cite|tao:prl:2016}} Available since VASP.6.4.3.
-:*In the presence of an extended vacuum region (e.g., surfaces), the average of <math>|\nabla n|/n</math> has no meaning. Therefore, MBJ calculations should be done with a fixed value of <math>c</math>, which can be done with the {{TAG|CMBJ}} tag.}}
-*{{TAG|METAGGA}}=LMBJ
+|-
-:The local MBJ (LMBJ) potential{{cite|rauch:jctc:2020}}{{cite|rauch:prb:2020}} is a variant of the MBJ potential that was modified such that it does not suffer from the problems related to the presence of vacuum mentioned above for MBJ. The LMBJ potential has the same analytical form as the MBJ potential:
+| style="text-align:center;"| TASK_X<span style="color:blue"><sup>(2)</sup></span> || style="text-align:center;"| <math>\tau</math> || TASK exchange.{{cite|aschebrock:prr:2019}} Available since VASP.6.5.0.
-:<math>
- v_{x}^{\rm LMBJ}(\mathbf{r}) = c(\mathbf{r})v_{x}^{\rm BR}(\mathbf{r}) + (3c(\mathbf{r})-2)\frac{1}{\pi}\sqrt{\frac{5}{6}}\sqrt{\frac{\tau(\mathbf{r})}{n(\mathbf{r})}}
-</math>
-:with the difference that <math>c</math> is now a position-dependent function:
-:<math>
-c(\mathbf{r})=\alpha+\beta(\tilde{g}(\mathbf{r}))^{e}
-</math>
-:where
-:<math>
-\tilde{g}(\mathbf{r})=\frac{1}{\left(2\pi\sigma^{2}\right)^{3/2}}\int g(\mathbf{r}')e^{-\frac{\left\vert\mathbf{r}-\mathbf{r}'\right\vert^2}{2\sigma^2}}d{\mathbf{r}'}
-</math>
-:with
-:<math>
-g(\mathbf{r})=\frac{1-\alpha}{\beta}\left(1-\mathrm{erf}\left(\frac{n(\mathbf{r})}{n_{\mathrm{th}}}\right)\right)+\frac{\left\vert\nabla n\right\vert}{n}\mathrm{erf}\left(\frac{n(\mathbf{r})}{n_{\mathrm{th}}}\right)
-</math>
-:The default values of the parameters in LMBJ are (see erratum of Ref. {{cite|rauch:prb:2020}}) <math>\alpha=0.488</math>, <math>\beta=0.5</math> bohr, <math>e=1</math>, <math>\sigma=2</math> <math>\AA</math> (<math>=3.78</math> bohr), and <math>n_{\mathrm{th}}=6.96\times10^{-4}</math> e/bohr<math>^{3}</math>. <math>\sigma</math> is the smearing parameter that determines the size of the region over which the average of <math>g</math> is calculated, and <math>n_{\mathrm{th}}</math> is the threshold density, which corresponds to the Wigner–Seitz radius <math>r_{s}^{\mathrm{th}}=((4/3)\pi n_{\mathrm{th}})^{-1/3}=7</math> bohr. <math>\alpha</math>, <math>\beta</math>, <math>e</math>, <math>\sigma</math>, and <math>r_{s}^{\mathrm{th}}</math> can be set by means of the {{TAG|CMBJA}}, {{TAG|CMBJB}}, {{TAG|CMBJE}}, {{TAG|SMBJ}}, and {{TAG|RSMBJ}} tags, respectively.
-:The first two points mentioned above for the MBJ potential also apply for the LMBJ potential.
+|-
+| style="text-align:center;"| LAK, LAK_X or LAK_C || style="text-align:center;"| <math>\tau</math> || LAK.{{cite|lebeda:prl:2024}} Available since VASP.6.5.0.
+|-
+| style="text-align:center;"| MSPBEL, MSRPBEL or MSB86BL || style="text-align:center;"| <math>\tau</math> ||
+MS-PBEl, MS-RPBEl or MS-B86bl.{{cite|smeets:jpca:2019}} Available since VASP.6.5.0.
+|-
+| style="text-align:center;"| RMSPBEL, RMSRPBEL or RMSB86BL || style="text-align:center;"| <math>\tau</math> ||
+rMS-PBEl, rMS-RPBEl or rMS-B86bl.{{cite|cai:jpcc:2024}} Available since VASP.6.5.0.
+|-
+| style="text-align:center;"| SCANL || style="text-align:center;"| <math>\nabla^2n</math> ||
+SCAN-L{{cite|mejia-rodriguez:pra:2017}}{{cite|mejia-rodriguez:prb:2018}} is a deorbitalized version of SCAN. Available since VASP.6.4.0.
+|-
+| style="text-align:center;"| RSCANL || style="text-align:center;"| <math>\nabla^2n</math> ||
+rSCAN-L is a deorbitalized version of rSCAN. Available since VASP.6.4.0.
+|-
+| style="text-align:center;"| R2SCANL || style="text-align:center;"| <math>\nabla^2n</math> ||
+r<math>^2</math>SCAN-L is a deorbitalized versions of r<math>^2</math>SCAN.{{cite|mejia-rodriguez:prb:2020}}{{cite|kaplan:prm:2022}} Available since VASP.6.4.0.
+|-
+| style="text-align:center;"| OFR2 || style="text-align:center;"| <math>\nabla^2n</math> ||
+Orbital-free regularized-restored SCAN (OFR2).{{cite|kaplan:prm:2022}} Available since VASP.6.4.0.
+|-
+| style="text-align:center;"| SREGTM2L || style="text-align:center;"| <math>\nabla^2n</math> ||
+v2-sregTM-L is a deorbitalized versions of v2-sregTM.{{cite|francisco_b:jcp:2023}} Available since VASP.6.4.0.
+|-
+| style="text-align:center;"| MBJ<span style="color:blue"><sup>(3)</sup></span> || style="text-align:center;"| <math>\nabla^2n,\tau</math> ||
+Modified Becke-Johnson potential.{{cite|becke:jcp:06}}{{cite|tran:prl:09}} The {{TAG|CMBJA}}, {{TAG|CMBJB}} and {{TAG|CMBJE}} tags correspond to <math>\alpha</math>, <math>\beta</math> and the power <math>e=1/2</math> (that can be modified) in Eq. (3) of Ref. {{cite|tran:prl:09}}, respectively. The default values are <math>\alpha=-0.012</math>, <math>\beta=1.023</math> bohr<math>^{1/2}</math> and <math>e=1/2</math>.{{cite|tran:prl:09}}
+|-
+| style="text-align:center;"| LMBJ<span style="color:blue"><sup>(3)</sup></span> || style="text-align:center;"| <math>\nabla^2n,\tau</math> ||
+The local MBJ (LMBJ) potential.{{cite|rauch:jctc:2020}}{{cite|rauch:prb:2020}} The {{TAG|CMBJA}}, {{TAG|CMBJB}}, {{TAG|CMBJE}}, {{TAG|SMBJ}}, and {{TAG|RSMBJ}} tags correspond to <math>\alpha</math>, <math>\beta</math>, the power <math>e=1</math> (that can be modified) of <math>\bar{g}</math>, <math>\sigma</math> and <math>r_{s}^{\mathrm{th}}</math> in Eqs. (5)-(7) of Ref. {{cite|rauch:prb:2020}}, respectively. The default values are (see erratum of Ref. {{cite|rauch:prb:2020}}) <math>\alpha=0.488</math>, <math>\beta=0.5</math> bohr, <math>e=1</math>, <math>\sigma=2</math> <math>\AA</math> (<math>=3.78</math> bohr), and <math>r_{s}^{\mathrm{th}}=7</math> bohr (which corresponds to <math>n_{\mathrm{th}}=6.96\times10^{-4}</math> e/bohr<math>^{3}</math>).
+|}
+<span style="color:blue">(1)</span> The exchange-only and correlation-only implementations are available since VASP.6.4.3.
+<span style="color:blue">(2)</span> In Ref. {{cite|aschebrock:prr:2019}} TASK exchange is combined with LDA-PW92 correlation.{{cite|perdew1992}} This can be done with {{TAG|XC}}=TASK_X PW92_C in {{FILE|INCAR}}.
+<span style="color:blue">(3)</span> A few points about the MBJ and LMBJ potentials:
+:*These are ''potential-only'' methods, ''i.e.'', there is no corresponding exchange-correlation energy <math>E_{xc}</math>. The used expression for <math>E_{xc}</math> is LDA, which is an arbitrary choice. This means that MBJ and LMBJ calculations can never be self-consistent with respect to the total energy, and thus we cannot compute Hellmann-Feynman forces (''i.e.'', no ionic relaxation, etc.). Actually, these potentials aim solely at a description of the electronic properties, primarily the band gap, or magnetic moments.
+:*MBJ and LMBJ calculations may converge very slowly, so the number of maximum electronic steps ({{TAG|NELM}}) should be set higher than usual.
+:*In the presence of an extended vacuum region (e.g., surfaces) or an interface, the average of <math>|\nabla n|/n</math> has no meaning. Therefore, MBJ calculations should be done with a fixed value of <math>c</math>, which can be done with the {{TAG|CMBJ}} tag., or alternatively with the LMBJ that was proposed for the purpose to be applicable to systems with vacuum or interfaces.
 == POTCAR files: required information ==
-Meta-GGA calculations require {{FILE|POTCAR}} files that include information on the kinetic energy density of the core-electrons. To check whether a particular {{FILE|POTCAR}} contains this information, type:
+Calculations with a meta-GGA that depends on the kinetic-energy density require {{FILE|POTCAR}} files that include information on the kinetic-energy density of the core electrons. Almost all recent {{FILE|POTCAR}} files do fulfill this requirement, but there are some notable exceptions like O_GW. To check whether a particular {{FILE|POTCAR}} contains this information, type:
   grep kinetic POTCAR
@@ Line 91: / Line 130: @@
 and for PAW datasets with partial core corrections:
   kinetic energy density (partial)
+{{NB|mind|For {{FILE|POTCAR}} files without core electrons (H, He, Li_sv, Be_sv, and _GW variants thereof) the <code>grep</code> command given above will not return the line about pseudized kinetic energy-density, since all electrons are considered as valence. These potentials can nevertheless be used for all meta-GGA functionals.}}
 ==  Aspherical contributions related to one-center terms  ==
@@ Line 114: / Line 154: @@
 {{TAG|LIBXC2}},
 {{TAG|GGA}},
+{{TAG|XC}},
 {{TAG|CMBJ}},
 {{TAG|CMBJA}},