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The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated) <math>d</math> or <math>f</math> electrons (this manifests itself primarily in the form of unrealistic one-electron energies or too small magnetic moments). In some cases this can be remedied by introducing on the <math>d</math> or <math>f</math> atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner (<math>E_{\text{HF}}(\hat{n})</math>), as an on-site replacement of the LDA/GGA functional:
The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated) <math>d</math> or <math>f</math> electrons (this manifests itself primarily in the form of unrealistic one-electron energies or too small magnetic moments). In some cases this can be remedied by introducing on the <math>d</math> or <math>f</math> atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner (<math>E^{\text{HF}}[\hat{n}]</math>), as an on-site replacement of the LDA/GGA functional:
:<math>E_{\text{xc}}^{\text{LDA/GGA}+U}(n,\hat{n}) = E_{\text{xc}}^{\text{LDA/GGA}}(n) + E_{\text{HF}}(\hat{n}) - E_{\text{dc}}(\hat{n})</math>
:<math>E_{\text{xc}}^{\text{DFT}+U}[n,\hat{n}] = E_{\text{xc}}^{\text{DFT}}[n] + E^{\text{HF}}[\hat{n}] - E_{\text{dc}}[\hat{n}]</math>
where <math>E_{\text{dc}}(\hat{n})</math> is the double-counting term and <math>\hat{n}</math> is the on-site occupancy matrix of the <math>d</math> or <math>f</math> electrons. This approach is known as the DFT+U method (traditionally called LSDA+U{{cite|anisimov:prb:91}}).
where <math>E_{\text{dc}}[\hat{n}]</math> is the double-counting term and <math>\hat{n}</math> is the on-site occupancy matrix of the <math>d</math> or <math>f</math> electrons. This approach is known as the DFT+U method (traditionally called LSDA+U{{cite|anisimov:prb:91}}).


The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. {{cite|rohrbach:jcp:03}} (the original implementation was done by Olivier Bengone {{cite|Bengone:prb:00}} and Georg Kresse).
The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. {{cite|rohrbach:jcp:03}} (the original implementation was done by Olivier Bengone {{cite|Bengone:prb:00}} and Georg Kresse).


For more details, read the article about the [[DFT+U: formalism|formalism of the DFT+U method]].
More detail about the formalism is provided below.
 
==Theory==
DFT+U is a method that was proposed to improve the description of systems with strongly correlated <math>d</math> or <math>f</math> electrons, like antiferromagnetic NiO for instance, that are usually inaccurately described with the standard LDA and GGA functionals{{cite|anisimov:prb:91}}. Several variants of the DFT+U method exist (see Refs. {{cite|Ylvisaker:prb:2009}}{{cite|Himmetoglu:ijqc:2014}} for reviews) that differ for instance in the way the double counting term <math>E_{\text{dc}}[\hat{n}]</math> is calculated. Three variants of them are implemented in VASP, whose formalism is briefly summarized below.
 
*{{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}[{\hat n}]=\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> represents a 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{{cite|vaugier:prb:2012}}{{cite|kaltak:thesis2015}}.
 
: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{DFT}+U}[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{DFT}}+\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}}=3: This option is for the calculation of the parameter <math>U</math> using the linear response approach from Ref. {{cite|cococcioni:2005}}. The steps to use this method are shown for [[Calculate U for LSDA+U|the example of NiO]].
 
*{{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>


==How to==
==How to==
DFT+U can be switched on with the {{TAG|LDAU}} tag, while the {{TAG|LDAUTYPE}} tag determines the DFT+U flavor that is used. {{TAG|LDAUL}} specifies the <math>l</math>-quantum number for which the on-site interaction is added, and the effective on-site Coulomb and exchange interactions are set (in eV) with the {{TAG|LDAUU}} and  {{TAG|LDAUJ}} tags, respectively. Note that it is recommended to increase {{TAG|LMAXMIX}} to 4 for ''d''-electrons or 6 for ''f''-elements.
DFT+U can be switched on with the {{TAG|LDAU}} tag, while the {{TAG|LDAUTYPE}} tag determines the DFT+U flavor that is used. {{TAG|LDAUL}} specifies the <math>l</math>-quantum number for which the on-site interaction is added, and the effective on-site Coulomb and exchange interactions are set (in eV) with the {{TAG|LDAUU}} and  {{TAG|LDAUJ}} tags, respectively. Note that it is recommended to increase {{TAG|LMAXMIX}} to 4 for ''d''-electrons or 6 for ''f''-elements.
==Tutorials==
*Lecture on [https://youtu.be/6F_WNIh6V7I the optical gap], introduces DFT+U towards the end of the lecture.


== References ==
== References ==

Latest revision as of 10:39, 22 September 2025

The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated) [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] electrons (this manifests itself primarily in the form of unrealistic one-electron energies or too small magnetic moments). In some cases this can be remedied by introducing on the [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner ([math]\displaystyle{ E^{\text{HF}}[\hat{n}] }[/math]), as an on-site replacement of the LDA/GGA functional:

[math]\displaystyle{ E_{\text{xc}}^{\text{DFT}+U}[n,\hat{n}] = E_{\text{xc}}^{\text{DFT}}[n] + E^{\text{HF}}[\hat{n}] - E_{\text{dc}}[\hat{n}] }[/math]

where [math]\displaystyle{ E_{\text{dc}}[\hat{n}] }[/math] is the double-counting term and [math]\displaystyle{ \hat{n} }[/math] is the on-site occupancy matrix of the [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] electrons. This approach is known as the DFT+U method (traditionally called LSDA+U[1]).

The first VASP DFT+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. [2] (the original implementation was done by Olivier Bengone [3] and Georg Kresse).

More detail about the formalism is provided below.

Theory

DFT+U is a method that was proposed to improve the description of systems with strongly correlated [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] electrons, like antiferromagnetic NiO for instance, that are usually inaccurately described with the standard LDA and GGA functionals[1]. Several variants of the DFT+U method exist (see Refs. [4][5] for reviews) that differ for instance in the way the double counting term [math]\displaystyle{ E_{\text{dc}}[\hat{n}] }[/math] is calculated. Three variants of them are implemented in VASP, whose formalism is briefly summarized below.

  • LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.[6]
This particular flavour of DFT+U is of the form
[math]\displaystyle{ E^{\rm HF}[{\hat n}]=\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 PAW on-site occupancies
[math]\displaystyle{ {\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle \langle m_1 \mid \Psi^{s_1} \rangle }[/math]
and the (unscreened) on-site electron-electron interaction
[math]\displaystyle{ 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]\displaystyle{ |m\rangle }[/math] represents a real spherical harmonics of angular momentum [math]\displaystyle{ l }[/math]=LDAUL.
The unscreened electron-electron interaction [math]\displaystyle{ U_{\gamma_{1}\gamma_{3}\gamma_{2}\gamma_{4}} }[/math] can be written in terms of the Slater integrals [math]\displaystyle{ F^0 }[/math], [math]\displaystyle{ F^2 }[/math], [math]\displaystyle{ F^4 }[/math], and [math]\displaystyle{ F^6 }[/math] ([math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ U }[/math] and [math]\displaystyle{ J }[/math] (LDAUU and LDAUJ, respectively). [math]\displaystyle{ U }[/math] and [math]\displaystyle{ J }[/math] can also be extracted from constrained-DFT calculations[7][8].
These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
[math]\displaystyle{ L\; }[/math] [math]\displaystyle{ F^0\; }[/math] [math]\displaystyle{ F^2\; }[/math] [math]\displaystyle{ F^4\; }[/math] [math]\displaystyle{ F^6\; }[/math]
[math]\displaystyle{ 1\; }[/math] [math]\displaystyle{ U\; }[/math] [math]\displaystyle{ 5J\; }[/math] - -
[math]\displaystyle{ 2\; }[/math] [math]\displaystyle{ U\; }[/math] [math]\displaystyle{ \frac{14}{1+0.625}J }[/math] [math]\displaystyle{ 0.625 F^2\; }[/math] -
[math]\displaystyle{ 3\; }[/math] [math]\displaystyle{ U\; }[/math] [math]\displaystyle{ \frac{6435}{286+195 \cdot 0.668+250 \cdot 0.494}J }[/math] [math]\displaystyle{ 0.668 F^2\; }[/math] [math]\displaystyle{ 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]\displaystyle{ E^{\mathrm{DFT}+U}[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]\displaystyle{ E_{\mathrm{dc}} }[/math], which supposedly equals the on-site semilocal contribution to the total energy,
[math]\displaystyle{ 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]
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.[9]
This flavour of DFT+U is of the following form:
[math]\displaystyle{ E^{\mathrm{DFT+U}}=E^{\mathrm{DFT}}+\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 on-site occupancy matrix in the direction of idempotency,
[math]\displaystyle{ \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]\displaystyle{ U }[/math] and [math]\displaystyle{ J }[/math] do not enter seperately, only the difference [math]\displaystyle{ U-J }[/math] is meaningful.
  • LDAUTYPE=3: This option is for the calculation of the parameter [math]\displaystyle{ U }[/math] using the linear response approach from Ref. [10]. The steps to use this method are shown for the example of NiO.
  • LDAUTYPE=4: same as 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]\displaystyle{ 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]

How to

DFT+U can be switched on with the LDAU tag, while the LDAUTYPE tag determines the DFT+U flavor that is used. LDAUL specifies the [math]\displaystyle{ l }[/math]-quantum number for which the on-site interaction is added, and the effective on-site Coulomb and exchange interactions are set (in eV) with the LDAUU and LDAUJ tags, respectively. Note that it is recommended to increase LMAXMIX to 4 for d-electrons or 6 for f-elements.

Tutorials

  • Lecture on the optical gap, introduces DFT+U towards the end of the lecture.

References

  1. ↑ a b V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
  2. ↑ A. Rohrbach, J. Hafner, and G. Kresse J. Phys.: Condens. Matter 15, 979 (2003).
  3. ↑ O. Bengone, M. Alouani, P. Blöchl, and J. Hugel, Phys. Rev. B 62, 16392 (2000).
  4. ↑ E. R. Ylvisaker and W. E. Pickett, Phys. Rev. B 79, 035103 (2009).
  5. ↑ B. Himmetoglu, A. Floris, S. de Gironcoli, and M. Cococcioni, Int. J. Quantum Chem. 114. 14 (2014).
  6. ↑ A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (1995).
  7. ↑ L. Vaugier, H. Jiang, and S. Biermann, Phys. Rev. B 86, 165105 (2012).
  8. ↑ M. Kaltak, Thesis: Merging GW with DMFT (2015).
  9. ↑ S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
  10. ↑ M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).

Pages in category "DFT+U"

The following 6 pages are in this category, out of 6 total.

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