View source for DFT+U: formalism

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{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., 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>

== References ==
<references/>
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[[Category:INCAR tag]][[Category:Exchange-correlation functionals]][[Category:DFT+U]]