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*{{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>l</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> (<math>f</math> electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true electron-electron interaction, since in solids the Coulomb interaction is screened (especially <math>F^0</math>).
:In practice these integrals are often treated as parameters, ''i.e.'', adjusted to reach agreement with experiment for a property like for instance 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.
:These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
::{| cellpadding="5" cellspacing="0" border="1"
| <math>L\;</math> || <math>F^0\;</math> || <math>F^2\;</math> || <math>F^4\;</math> || <math>F^6\;</math>
|-
| <math>1\;</math> || <math>U\;</math> || <math>5J\;</math> || - || -
|-
| <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:
::<math>
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}}
*{{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:
:This flavour of DFT+U is of the following form:


::<math>
*{{TAG|LDAUTYPE}}=4: same as {{TAG|LDAUTYPE}}=1, but without exchange splitting.
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., the total spin-up plus spin-down occupancy matrix is used). The double-counting term 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).
'''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).

Revision as of 12:44, 11 April 2022

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

Description: LDAUTYPE specifies which type of DFT+U approach will be used.

Three types of DFT+U approaches are available in VASP. These are the following:

  • LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.[1]
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.[2]
This flavour of DFT+U is of the following form:

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 and (LDAUU and LDAUJ, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different and/or , or and in case of Dudarev's approach (LDAUTYPE=2).

Note on bandstructure calculation: the CHGCAR file contains only information up to angular momentum quantum number =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 be necessarily not identical to a self-consistent 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 LMAXMIX to 4 ( elements) and 6 ( elements).

Related tags and articles

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX

Examples that use this tag

References

  1. ↑ A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).
  2. ↑ S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
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