LDAUTYPE

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

Description: LDAUTYPE specifies which type of L(S)DA+U approach will be used.

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

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

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⟩ are real spherical harmonics of angular momentum L=LDAUL.

The unscreened e-e interaction U_γ1γ3γ2γ4 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 e-e interaction, since in solids the Coulomb interaction is screened (especially $F^0$).

In practice these integrals are therefore often treated as parameters, i.e., adjusted to reach agreement with experiment in some sense: equilibrium volume, magnetic moment, band gap, structure. 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 are sometimes extracted from constrained-LSDA 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 L(S)DA+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 L(S)DA on site due to the fact that one subtracts a double counting energy $E_{\mathrm{dc}}$, which supposedly equals the on site L(S)DA 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 LSDA+U, introduced by Dudarev et al.

• LDAUTYPE=4: same as LDAUTYPE=1, but LDA+U instead of LSDA+U (i.e. no LSDA exchange splitting).

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT

References

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