DFT+U: formalism: Difference between revisions

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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 of them are implemented in VASP, whose formalism is briefly summarized below.  
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}}
*{{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
:This particular flavour of DFT+U is of the form
::<math>
::<math>
E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}}
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_2\gamma_4} -
U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat
U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat
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\delta_{s_1 s_2} \delta_{s_3 s_4}
\delta_{s_1 s_2} \delta_{s_3 s_4}
</math>
</math>
:where <math>|m\rangle</math> are real spherical harmonics of angular momentum <math>\ell</math>={{TAG|LDAUL}}.
: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>).
: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.
: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):
:These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
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</math>
</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,
: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>
::<math>
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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.
:'''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
*{{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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\frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
\frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).
</math>
</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 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 {{TAG|LMAXMIX}} to 4 (<math>d</math> elements) and 6 (<math>f</math> elements).


== Related tags and articles ==
== Related Tags and Sections ==
{{TAG|LDAU}},
{{TAG|LDAU}},
{{TAG|LDAUTYPE}},
{{TAG|LDAUL}},
{{TAG|LDAUL}},
{{TAG|LDAUU}},
{{TAG|LDAUU}},
{{TAG|LDAUJ}},
{{TAG|LDAUJ}},
{{TAG|LDAUPRINT}},
{{TAG|LDAUPRINT}},
{{TAG|LMAXMIX}}
{{TAG|LMAXMIX}},
 
{{sc|LDAUTYPE|Examples|Examples that use this tag}}


== References ==
== References ==
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----
----


[[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:DFT+U]]
[[Category:Exchange-correlation functionals]][[Category:DFT+U]]

Latest revision as of 15:35, 9 May 2023

DFT+U is a method that was proposed to improve the description of systems with strongly correlated or 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. [2][3] for reviews) that differ for instance in the way the double counting term 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.[4]
This particular flavour of DFT+U is of the form
and is determined by the PAW on-site occupancies
and the (unscreened) on-site electron-electron interaction
where represents a real spherical harmonics of angular momentum =LDAUL.
The unscreened electron-electron interaction can be written in terms of the Slater integrals , , , and ( 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 ).
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, and (LDAUU and LDAUJ, respectively). and can also be extracted from constrained-DFT calculations[5][6].
These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
- -
-
The essence of the DFT+U method consists of the assumption that one may now write the total energy as:
where the Hartree-Fock-like interaction replaces the semilocal on-site due to the fact that one subtracts a double-counting energy , which supposedly equals the on-site semilocal contribution to the total energy,
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.[7]
This flavour of DFT+U is of the following form:
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,
.
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 and do not enter seperately, only the difference is meaningful.
  • LDAUTYPE=3: This option is for the calculation of the parameter using the linear response approach from Ref. [8]. 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

Related Tags and Sections

LDAU, LDAUTYPE, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX,

References