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.

[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 |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 [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 e-e interaction, since in solids the Coulomb interaction is screened (especially [math]\displaystyle{ F^0 }[/math]).

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):

[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 L(S)DA+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 L(S)DA 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 L(S)DA 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 LSDA+U, introduced by Dudarev et al.

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

[math]\displaystyle{ E_{\mathrm{LSDA+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 LSDA 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 U and J do not enter seperately, only the difference (U-J) is meaningfull.

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

In the LDA+U case the double counting energy 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]

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT

References

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