Category:DFT+U

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