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.

This particular flavour of LSDA+U is of the form

$E_{\rm {HF}}={\frac {1}{2}}\sum _{\{\gamma \}}(U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}-U_{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}){\hat {n}}_{\gamma _{1}\gamma _{2}}{\hat {n}}_{\gamma _{3}\gamma _{4}}$

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):

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

The essence of the LSDA+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 LSDA on site due to the fact that one subtracts a double counting energy $E_{\mathrm {dc} }$, which supposedly equals the on-site LSDA contribution to the total energy,

$E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).$

• 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:

$E_{\mathrm {LSDA+U} }=E_{\mathrm {LSDA} }+{\frac {(U-J)}{2}}\sum _{\sigma }\left[\left(\sum _{m_{1}}n_{m_{1},m_{1}}^{\sigma }\right)-\left(\sum _{m_{1},m_{2}}{\hat {n}}_{m_{1},m_{2}}^{\sigma }{\hat {n}}_{m_{2},m_{1}}^{\sigma }\right)\right].$

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,

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

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 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,

$E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).$

Related Tags and Sections

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT

References

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