LDAUTYPE: Difference between revisions

Latest revision as of 08:53, 9 May 2023

LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2

Description: LDAUTYPE specifies the DFT+U variant that will be used.

The following variants of the DFT+U approach are available:

LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.^[1]

LDAUTYPE=2: The simplified (rotationally invariant) approach to DFT+U, introduced by Dudarev et al.^[2]

LDAUTYPE=3: Linear response ansatz of Cococcioni et al. ^[3] to compute U. See how to calculate U.

Mind: For LDAUTYPE=3, the LDAUU and LDAUJ tags specify the strength of the spherical potential acting on the spin-up and spin-down manifolds, respectively.

LDAUTYPE=4: Same as LDAUTYPE=1, but without exchange splitting.

A method to estimate the parameters for DFT+U is the constrained-random-phase approximation. Another method is the linear response ansatz with LDAUTYPE=3, mentioned above. On the other hand, in many applications, the DFT+U parameters are used as tuning parameters to fit experimental data.

Tip: For band-structure calculations, increase LMAXMIX to 4 (

d

elements) or 6 (

f

elements).

This is because the CHGCAR file contains only information up to angular momentum quantum number set by LMAXMIX for the on-site PAW occupancy matrices. When the CHGCAR file is read and kept fixed in the course of the calculations (ICHARG=11), the results will necessarily not be identical to a self-consistent run. The deviations are often large for DFT+U calculations.

Warning: The total energy will depend on the parameters

U

(LDAUU) and

J

(LDAUJ). It is, therefore, not meaningful to compare the total energies resulting from calculations with different

U

and/or

J

; or

U-J

in the case of Dudarev's approach (LDAUTYPE=2).

It is possible to use LDAUTYPE=1, 2, and 3 for a non–spin-polarized calculation with ISPIN=1.

References

[liechtenstein:prb:95-1] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).

[dudarev:prb:98-2] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).

[cococcioni:2005-3] M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
 {{TAGDEF|LDAUTYPE|1 {{!}} 2 {{!}} 4|2}}
-Description: {{TAG|LDAUTYPE}} specifies which type of L(S)DA+U approach will be used.
+Description: {{TAG|LDAUTYPE}} specifies the DFT+U variant that will be used.
 ----
-The L(S)DA often fails to describe systems with localized (strongly correlated) ''d'' and ''f''-electrons (this manifests itself primarily in the form of unrealistic one-electron energies). In some cases this can be remedied by introducing a strong intra-atomic interaction in a (screened) Hartree-Fock like manner, as an on-site replacement of the L(S)DA. This approach is commonly known as the L(S)DA+U method. Setting {{TAG|LDAU}}=.TRUE. in the {{FILE|INCAR}} file switches on the L(S)DA+U.
+The following variants of the [[DFT+U: formalism|DFT+U approach]] are available:
-*{{TAG|LDAUTYPE}}=1: The rotationally invariant LSDA+U introduced by Liechtenstein ''et al.''<ref name="liechtenstein:prb:95"/>
+*{{TAG|LDAUTYPE}}=1: The rotationally invariant DFT+U introduced by Liechtenstein ''et al.''{{cite|liechtenstein:prb:95}}
-:This particular flavour of LSDA+U is of the form
-::<math>
-E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}}
-(U_{\gamma_1\gamma_3\gamma_2\gamma_4} -
-U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat
-n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4}
-</math>
-:and is determined by the <span id="occmat">PAW on-site occupancies
-::<math>
-{\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle
-\langle m_1 \mid \Psi^{s_1} \rangle
-</math></span>
-:and the (unscreened) on-site electron-electron interaction
-::<math>
-U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid
-\frac{1}{|\mathbf{r}-\mathbf{r}^\prime|} \mid m_2 m_4 \rangle
-\delta_{s_1 s_2} \delta_{s_3 s_4}
-</math>
-:where |''m''&rang; are real spherical harmonics of angular momentum ''L''={{TAG|LDAUL}}.
-:The unscreened e-e interaction ''U''<sub>&gamma;<sub>1</sub></sub><sub>&gamma;<sub>3</sub></sub><sub>&gamma;<sub>2</sub></sub><sub>&gamma;<sub>4</sub></sub> can be written in terms of the Slater integrals <math>F^0</math>, <math>F^2</math>, <math>F^4</math>, and <math>F^6</math> (f-electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true e-e interaction, since in solids the Coulomb interaction is screened (especially <math>F^0</math>).
+*{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to DFT+U, introduced by Dudarev ''et al.''{{cite|dudarev:prb:98}}
-:In practice these integrals are therefore often treated as parameters, ''i.e.'', adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. They are normally specified in terms of the effective on-site Coulomb- and exchange parameters, ''U'' and ''J'' ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). ''U'' and ''J'' are sometimes extracted from constrained-LSDA calculations.
+*{{TAG|LDAUTYPE}}=3: Linear response ansatz of Cococcioni et al. {{cite|cococcioni:2005}} to compute U. See [[Calculate U for LSDA+U|how to calculate U]].
+{{NB|mind|For {{TAG|LDAUTYPE}}{{=}}3, the {{TAG|LDAUU}} and {{TAG|LDAUJ}} tags specify the strength of the spherical potential acting on the spin-up and spin-down manifolds, respectively.|:}}
-:These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
+*{{TAG|LDAUTYPE}}=4: Same as {{TAG|LDAUTYPE}}=1, but without exchange splitting.
-::{| cellpadding="5" cellspacing="0" border="1"
+A method to estimate the parameters for DFT+U is the [[Constrained-random-phase approximation|constrained-random-phase approximation]]. Another method is the linear response ansatz with {{TAG|LDAUTYPE}}=3, mentioned above. On the other hand, in many applications, the DFT+U parameters are used as tuning parameters to fit experimental data.
-| <math>L\;</math> || <math>F^0\;</math> || <math>F^2\;</math> || <math>F^4\;</math> || <math>F^6\;</math>
+{{NB|tip|For band-structure calculations, increase {{TAG|LMAXMIX}} to 4 (<math>d</math> elements) or 6 (<math>f</math> elements).}}
-|-
+This is because the {{FILE|CHGCAR}} file contains only information up to angular momentum quantum number set by {{TAG|LMAXMIX}} for the [[LDAUTYPE#occmat|on-site PAW occupancy matrices]]. When the {{FILE|CHGCAR}} file is read and kept fixed in the course of the calculations ({{TAG|ICHARG}}=11), the results will necessarily not be identical to a self-consistent run. The deviations are often large for DFT+U calculations.
-| <math>1\;</math> || <math>U\;</math> || <math>5J\;</math> || - || -
+{{NB|warning|The total energy will depend on the parameters <math>U</math> ({{TAG|LDAUU}}) and <math>J</math> ({{TAG|LDAUJ}}). It is, therefore, not meaningful to compare the total energies resulting from calculations with different <math>U</math> and/or <math>J</math>; or <math>U-J</math> in the case of Dudarev's approach ({{TAG|LDAUTYPE}}{{=}}2).}}
-|-
-| <math>2\;</math> || <math>U\;</math> || <math>\frac{14}{1+0.625}J</math> || <math>0.625 F^2\;</math> || -
-|-
-| <math>3\;</math> || <math>U\;</math> || <math>\frac{6435}{286+195 \cdot 0.668+250 \cdot 0.494}J</math> || <math>0.668 F^2\;</math> || <math>0.494 F^2\;</math>
-|}
-:The essence of the LSDA+U method consists of the assumption that one may now write the total energy as:
+It is possible to use {{TAG|LDAUTYPE}}=1, 2, and 3 for a non–spin-polarized calculation with {{TAG|ISPIN}}=1.
-::<math>
+== Related tags and articles ==
-E_{\mathrm{tot}}(n,\hat n)=E_{\mathrm{DFT}}(n)+E_{\mathrm{HF}}(\hat n)-E_{\mathrm{dc}}(\hat n)
-</math>
-:where the Hartree-Fock like interaction replaces the LSDA on site due to the fact that one subtracts a double counting energy <math>E_{\mathrm{dc}}</math>, which supposedly equals the on-site LSDA contribution to the total energy,
-::<math>
-E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
-\frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
-</math>
-*{{TAG|LDAUTYPE}}=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev ''et al.''<ref name="dudarev:prb:98"/>
-:This flavour of LSDA+U is of the following form:
-::<math>
-E_{\mathrm{LSDA+U}}=E_{\mathrm{LSDA}}+\frac{(U-J)}{2}\sum_\sigma \left[
-\left(\sum_{m_1} n_{m_1,m_1}^{\sigma}\right) - \left(\sum_{m_1,m_2}
-\hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right].
-</math>
-:This can be understood as adding a penalty functional to the LSDA total energy expression that forces the [[#occmat|on-site occupancy matrix]] in the direction of idempotency,
-::<math>\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}</math>.
-:Real matrices are only idempotent when their eigenvalues are either 1 or 0, which for an occupancy matrix translates to either fully occupied or fully unoccupied levels.
-:'''Note''': in Dudarev's approach the parameters ''U'' and ''J'' do not enter seperately, only the difference (''U''-''J'') is meaningful.
-*{{TAG|LDAUTYPE}}=4: same as {{TAG|LDAUTYPE}}=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).
-:In the LDA+U case the double counting energy is given by,
-::<math>
-E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
-\frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
-</math>
-'''Warning''': it is important to be aware of the fact that when using the L(S)DA+U, in general the total energy will depend on the parameters ''U'' and ''J'' ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different ''U'' and/or ''J'', or ''U''-''J'' in case of Dudarev's approach ({{TAG|LDAUTYPE}}=2).
-'''Note on bandstructure calculation''': the {{FILE|CHGCAR}} file contains only information up to angular momentum quantum number ''L''={{TAG|LMAXMIX}} for the [[LDAUTYPE#occmat|on-site PAW occupancy matrices]]. When the {{FILE|CHGCAR}} file is read and kept fixed in the course of the calculations ({{TAG|ICHARG}}=11), the results will be necessarily not identical to a selfconsistent run. The deviations are often large for L(S)DA+U calculations. For the calculation of band structures within the L(S)DA+U approach, it is hence strictly required to increase {{TAG|LMAXMIX}} to 4 (d elements) and 6 (f elements).
-== Related Tags and Sections ==
 {{TAG|LDAU}},
 {{TAG|LDAUL}},
@@ Line 89: / Line 27: @@
 {{TAG|LDAUJ}},
 {{TAG|LDAUPRINT}},
-{{TAG|LMAXMIX}}
+{{TAG|LMAXMIX}},
+{{TAG|DFT+U: formalism}}
+{{sc|LDAUTYPE|Examples|Examples that use this tag}}
 == References ==
-<references>
+<references/>
-<ref name="liechtenstein:prb:95">[http://link.aps.org/doi/10.1103/PhysRevB.52.R5467 A. I. Liechtenstein, V. I. Anisimov and J. Zaane, Phys. Rev. B 52, R5467 (1995).]</ref>
-<ref name="dudarev:prb:98">[http://link.aps.org/doi/10.1103/PhysRevB.57.1505 S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).]</ref>
-</references>
 ----
-[[The_VASP_Manual|Contents]]
-[[Category:INCAR]][[Category:LDA+U]]
+[[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:DFT+U]]