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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
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| ::<math>
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| E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}}
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| (U_{\gamma_1\gamma_3\gamma_2\gamma_4} -
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| U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat
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| n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4}
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| </math>
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| :and is determined by the <span id="occmat">PAW on-site occupancies
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| ::<math>
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| {\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle
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| \langle m_1 \mid \Psi^{s_1} \rangle
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| </math></span>
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| :and the (unscreened) on-site electron-electron interaction
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| ::<math>
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| U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid
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| \frac{1}{|\mathbf{r}-\mathbf{r}^\prime|} \mid m_2 m_4 \rangle
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| \delta_{s_1 s_2} \delta_{s_3 s_4}
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| </math>
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| :where <math>|m\rangle</math> are real spherical harmonics of angular momentum <math>l</math>={{TAG|LDAUL}}.
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|
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| :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>).
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| :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.
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|
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| :These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
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|
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| ::{| cellpadding="5" cellspacing="0" border="1"
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| | <math>L\;</math> || <math>F^0\;</math> || <math>F^2\;</math> || <math>F^4\;</math> || <math>F^6\;</math>
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| |-
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| | <math>1\;</math> || <math>U\;</math> || <math>5J\;</math> || - || -
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| |-
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| | <math>2\;</math> || <math>U\;</math> || <math>\frac{14}{1+0.625}J</math> || <math>0.625 F^2\;</math> || -
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| |-
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| | <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>
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| |}
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|
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| :The essence of the DFT+U method consists of the assumption that one may now write the total energy as:
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|
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| ::<math>
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| E_{\mathrm{tot}}(n,\hat n)=E_{\mathrm{DFT}}(n)+E_{\mathrm{HF}}(\hat n)-E_{\mathrm{dc}}(\hat n)
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| </math>
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|
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| :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,
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|
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| ::<math>
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| E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
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| \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
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| </math>
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|
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| *{{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: |
|
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|
| ::<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[
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| \left(\sum_{m_1} n_{m_1,m_1}^{\sigma}\right) - \left(\sum_{m_1,m_2}
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| \hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right].
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| </math>
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| | |
| :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,
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| ::<math>\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}</math>.
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| | |
| :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.
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| | |
| :'''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.
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| *{{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 | |
|
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| ::<math>
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| E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) -
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| \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
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| </math>
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| ---- | | ---- |
| '''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). |
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
- ↑ 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).
