|
|
| (16 intermediate revisions by 2 users not shown) |
| Line 1: |
Line 1: |
| {{TAGDEF|LDAUTYPE|1 {{!}} 2 {{!}} 4|2}} | | {{TAGDEF|LDAUTYPE|1 {{!}} 2 {{!}} 4|2}} |
|
| |
|
| Description: {{TAG|LDAUTYPE}} specifies which type of DFT+U approach will be used. | | Description: {{TAG|LDAUTYPE}} specifies the DFT+U variant that will be used. |
| ---- | | ---- |
| Three types of DFT+U approaches are available in VASP. These are the following:
| | The following variants of the [[DFT+U: formalism|DFT+U approach]] are available: |
|
| |
|
| *{{TAG|LDAUTYPE}}=1: The rotationally invariant DFT+U introduced by Liechtenstein ''et al.''{{cite|liechtenstein:prb:95}} | | *{{TAG|LDAUTYPE}}=1: The rotationally invariant DFT+U introduced by Liechtenstein ''et al.''{{cite|liechtenstein:prb:95}} |
| :This particular flavour of DFT+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 <math>|m\rangle</math> are real spherical harmonics of angular momentum <math>\ell</math>={{TAG|LDAUL}}.
| |
|
| |
|
| :The unscreened electron-electron interaction <math>U_{\gamma_{1}\gamma_{3}\gamma_{2}\gamma_{4}}</math> 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 often treated as parameters, ''i.e.'', adjusted to reach agreement with experiment for a property like the equilibrium volume, the magnetic moment or the band gap. They are normally specified in terms of the effective on-site Coulomb- and exchange parameters, <math>U</math> and <math>J</math> ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). <math>U</math> and <math>J</math> can also be extracted from constrained-DFT 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 DFT+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 semilocal on site due to the fact that one subtracts a double counting energy <math>E_{\mathrm{dc}}</math>, which supposedly equals the on-site semilocal 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 DFT+U, introduced by Dudarev ''et al.''{{cite|dudarev:prb:98}}
| |
| :This flavour of DFT+U is of the following form:
| |
| | |
| ::<math>
| |
| E_{\mathrm{DFT+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 semilocal 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 <math>U</math> and <math>J</math> do not enter seperately, only the difference <math>U-J</math> is meaningful.
| |
| | |
| *{{TAG|LDAUTYPE}}=4: same as {{TAG|LDAUTYPE}}=1, but without exchange splitting (i.e., LDA instead of LSDA)
| |
| :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 DFT+U, in general the total energy will depend on the parameters <math>U</math> and <math>J</math> ({{TAG|LDAUU}} and {{TAG|LDAUJ}}, respectively). 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> and 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 <math>\ell</math>={{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 DFT+U calculations. For the calculation of band structures within the DFT+U approach, it is hence strictly required to increase {{TAG|LMAXMIX}} to 4 (<math>d</math> elements) and 6 (<math>f</math> elements).
| |
| | |
| == Related Tags and Sections == | |
| {{TAG|LDAU}}, | | {{TAG|LDAU}}, |
| {{TAG|LDAUL}}, | | {{TAG|LDAUL}}, |
| Line 89: |
Line 27: |
| {{TAG|LDAUJ}}, | | {{TAG|LDAUJ}}, |
| {{TAG|LDAUPRINT}}, | | {{TAG|LDAUPRINT}}, |
| {{TAG|LMAXMIX}} | | {{TAG|LMAXMIX}}, |
| | {{TAG|DFT+U: formalism}} |
|
| |
|
| {{sc|LDAUTYPE|Examples|Examples that use this tag}} | | {{sc|LDAUTYPE|Examples|Examples that use this tag}} |
| Line 95: |
Line 34: |
| == References == | | == References == |
| <references/> | | <references/> |
|
| |
| ---- | | ---- |
| [[The_VASP_Manual|Contents]]
| |
|
| |
|
| [[Category:INCAR]][[Category:Exchange-correlation functionals]][[Category:DFT+U]] | | [[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:DFT+U]] |
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]
| 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.
|
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 ( elements) or 6 ( 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  (LDAUU) and  (LDAUJ). It is, therefore, not meaningful to compare the total energies resulting from calculations with different and/or ; or  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.
Related tags and articles
LDAU,
LDAUL,
LDAUU,
LDAUJ,
LDAUPRINT,
LMAXMIX,
DFT+U: formalism
Examples that use this tag
References
- ↑ A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).
- ↑ S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
- ↑ M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).
