DFT+U: formalism

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

• LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.

This particular flavour of DFT+U is of the form

$E_{\rm HF}=\frac{1}{2} \sum_{\{\gamma\}} (U_{\gamma_1\gamma_3\gamma_2\gamma_4} - U_{\gamma_1\gamma_3\gamma_4\gamma_2}){ \hat n}_{\gamma_1\gamma_2}{\hat n}_{\gamma_3\gamma_4}$

and is determined by the PAW on-site occupancies

${\hat n}_{\gamma_1\gamma_2} = \langle \Psi^{s_2} \mid m_2 \rangle \langle m_1 \mid \Psi^{s_1} \rangle$

and the (unscreened) on-site electron-electron interaction

$U_{\gamma_1\gamma_3\gamma_2\gamma_4}= \langle m_1 m_3 \mid \frac{1}{|\mathbf{r}-\mathbf{r}^\prime|} \mid m_2 m_4 \rangle \delta_{s_1 s_2} \delta_{s_3 s_4}$

where $|m\rangle$ are real spherical harmonics of angular momentum $\ell$=LDAUL.

The unscreened electron-electron interaction $U_{\gamma_{1}\gamma_{3}\gamma_{2}\gamma_{4}}$ can be written in terms of the Slater integrals $F^0$, $F^2$, $F^4$, and $F^6$ ($f$ 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 $F^0$).

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, $U$ and $J$ (LDAUU and LDAUJ, respectively). $U$ and $J$ 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):

$L\;$	$F^0\;$	$F^2\;$	$F^4\;$	$F^6\;$
$1\;$	$U\;$	$5J\;$	-	-
$2\;$	$U\;$	$\frac{14}{1+0.625}J$	$0.625 F^2\;$	-
$3\;$	$U\;$	$\frac{6435}{286+195 \cdot 0.668+250 \cdot 0.494}J$	$0.668 F^2\;$	$0.494 F^2\;$

The essence of the DFT+U method consists of the assumption that one may now write the total energy as:

$E_{\mathrm{tot}}(n,\hat n)=E_{\mathrm{DFT}}(n)+E_{\mathrm{HF}}(\hat n)-E_{\mathrm{dc}}(\hat n)$

where the Hartree-Fock like interaction replaces the semilocal on site due to the fact that one subtracts a double counting energy $E_{\mathrm{dc}}$, which supposedly equals the on-site semilocal contribution to the total energy,

$E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) - \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).$

• LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.

This flavour of DFT+U is of the following form:

$E_{\mathrm{DFT+U}}=E_{\mathrm{LSDA}}+\frac{(U-J)}{2}\sum_\sigma \left[ \left(\sum_{m_1} n_{m_1,m_1}^{\sigma}\right) - \left(\sum_{m_1,m_2} \hat n_{m_1,m_2}^{\sigma} \hat n_{m_2,m_1}^{\sigma} \right) \right].$

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,

$\hat n^{\sigma} = \hat n^{\sigma} \hat n^{\sigma}$.

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 $U$ and $J$ do not enter seperately, only the difference $U-J$ is meaningful.

• 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

$E_{\mathrm{dc}}(\hat n) = \frac{U}{2} {\hat n}_{\mathrm{tot}}({\hat n}_{\mathrm{tot}}-1) - \frac{J}{2} \sum_\sigma {\hat n}^\sigma_{\mathrm{tot}}({\hat n}^\sigma_{\mathrm{tot}}-1).$

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