LDAUTYPE

LDAUTYPE = 1 | 2 | 4
Default: LDAUTYPE = 2 

Description: LDAUTYPE specifies which type of DFT+U approach will be used.

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Three types of DFT+U approaches are available in VASP. These are the following:

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

This particular flavour of DFT+U is of the form

[math]\displaystyle{ 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} }[/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] are real spherical harmonics of angular momentum [math]\displaystyle{ \ell }[/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] (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 [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.

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{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]\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.

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

[math]\displaystyle{ 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 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=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]

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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]\displaystyle{ U }[/math] and [math]\displaystyle{ J }[/math] (LDAUU and LDAUJ, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different [math]\displaystyle{ U }[/math] and/or [math]\displaystyle{ J }[/math], or [math]\displaystyle{ U-J }[/math] and in case of Dudarev's approach (LDAUTYPE=2).

Note on bandstructure calculation: the CHGCAR file contains only information up to angular momentum quantum number [math]\displaystyle{ \ell }[/math]=LMAXMIX for the on-site PAW occupancy matrices. When the CHGCAR file is read and kept fixed in the course of the calculations (ICHARG=11), the results will be necessarily not identical to a selfconsistent 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 LMAXMIX to 4 ([math]\displaystyle{ d }[/math] elements) and 6 ([math]\displaystyle{ f }[/math] elements).

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX

Examples that use this tag

References

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