Category:DFT+U

The LDA and semilocal GGA functionals often fail to describe systems with localized (strongly correlated) ${\displaystyle d}$ or ${\displaystyle f}$ 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 ${\displaystyle d}$ or ${\displaystyle f}$ atom a strong intra-atomic interaction in a simplified (screened) Hartree-Fock like manner (${\displaystyle E^{\text{HF}}[\hat{n}]}$), as an on-site replacement of the LDA/GGA functional:

${\displaystyle E_{\text{xc}}^{\text{DFT}+U}[n,\hat{n}] = E_{\text{xc}}^{\text{LDA/GGA}}[n] + E_{\text{x}}^{\text{HF}}[\hat{n}] - E_{\text{dc}}[\hat{n}]}$

where ${\displaystyle E_{\text{dc}}[\hat{n}]}$ is the double-counting term and ${\displaystyle \hat{n}}$ is the on-site occupancy matrix of the ${\displaystyle d}$ or ${\displaystyle f}$ 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 ${\displaystyle d}$ or ${\displaystyle f}$ 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 ${\displaystyle E_{\text{dc}}[\hat{n}]}$ 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

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

and is determined by the PAW on-site occupancies

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

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

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

where ${\displaystyle |m\rangle}$ represents a real spherical harmonics of angular momentum ${\displaystyle l}$=LDAUL.

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

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, ${\displaystyle U}$ and ${\displaystyle J}$ (LDAUU and LDAUJ, respectively). ${\displaystyle U}$ and ${\displaystyle J}$ 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):

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

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

${\displaystyle E^{\mathrm{DFT}+U}[n,\hat n]=E^{\mathrm{DFT}}[n]+E_{\mathrm{x}}^{\mathrm{HF}}[\hat n]-E_{\mathrm{dc}}[\hat n] }$

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

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

• 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:

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

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,

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

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

• LDAUTYPE=3: This option is for the calculation of the parameter ${\displaystyle U}$ 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

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

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 ${\displaystyle l}$-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.

References