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# LDAUTYPE

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

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

The L(S)DA often fails to describe systems with localized (strongly correlated) d and f-electrons (this manifests itself primarily in the form of unrealistic one-electron energies). In some cases this can be remedied by introducing a strong intra-atomic interaction in a (screened) Hartree-Fock like manner, as an on-site replacement of the L(S)DA. This approach is commonly known as the L(S)DA+U method. Setting LDAU=.TRUE. in the INCAR file switches on the L(S)DA+U. The first VASP LDA+U calculations, including some additional technical details on the VASP implementation, can be found in Ref. [1] (the original implementation was done by Olivie Bengone [2] and Georg Kresse).

• LDAUTYPE=1: The rotationally invariant LSDA+U introduced by Liechtenstein et al.[3]
This particular flavour of LSDA+U is of the form
${\displaystyle E_{\rm {HF}}={\frac {1}{2}}\sum _{\{\gamma \}}(U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}-U_{\gamma _{1}\gamma _{3}\gamma _{4}\gamma _{2}}){\hat {n}}_{\gamma _{1}\gamma _{2}}{\hat {n}}_{\gamma _{3}\gamma _{4}}}$
and is determined by the PAW on-site occupancies
${\displaystyle {\hat {n}}_{\gamma _{1}\gamma _{2}}=\langle \Psi ^{s_{2}}\mid m_{2}\rangle \langle m_{1}\mid \Psi ^{s_{1}}\rangle }$
and the (unscreened) on-site electron-electron interaction
${\displaystyle U_{\gamma _{1}\gamma _{3}\gamma _{2}\gamma _{4}}=\langle m_{1}m_{3}\mid {\frac {1}{|{\mathbf {r} }-{\mathbf {r} }^{\prime }|}}\mid m_{2}m_{4}\rangle \delta _{s_{1}s_{2}}\delta _{s_{3}s_{4}}}$
where |m〉 are real spherical harmonics of angular momentum L=LDAUL.
The unscreened e-e interaction Uγ1γ3γ2γ4 can be written in terms of the Slater integrals ${\displaystyle F^{0}}$, ${\displaystyle F^{2}}$, ${\displaystyle F^{4}}$, and ${\displaystyle F^{6}}$ (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 ${\displaystyle F^{0}}$).
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):
 ${\displaystyle L\;}$ ${\displaystyle F^{0}\;}$ ${\displaystyle F^{2}\;}$ ${\displaystyle F^{4}\;}$ ${\displaystyle F^{6}\;}$ ${\displaystyle 1\;}$ ${\displaystyle U\;}$ ${\displaystyle 5J\;}$ - - ${\displaystyle 2\;}$ ${\displaystyle U\;}$ ${\displaystyle {\frac {14}{1+0.625}}J}$ ${\displaystyle 0.625F^{2}\;}$ - ${\displaystyle 3\;}$ ${\displaystyle U\;}$ ${\displaystyle {\frac {6435}{286+195\cdot 0.668+250\cdot 0.494}}J}$ ${\displaystyle 0.668F^{2}\;}$ ${\displaystyle 0.494F^{2}\;}$
The essence of the LSDA+U method consists of the assumption that one may now write the total energy as:
${\displaystyle E_{\mathrm {tot} }(n,{\hat {n}})=E_{\mathrm {DFT} }(n)+E_{\mathrm {HF} }({\hat {n}})-E_{\mathrm {dc} }({\hat {n}})}$
where the Hartree-Fock like interaction replaces the LSDA on site due to the fact that one subtracts a double counting energy ${\displaystyle E_{\mathrm {dc} }}$, which supposedly equals the on-site LSDA contribution to the total energy,
${\displaystyle E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).}$
• LDAUTYPE=2: The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev et al.[4]
This flavour of LSDA+U is of the following form:
${\displaystyle E_{\mathrm {LSDA+U} }=E_{\mathrm {LSDA} }+{\frac {(U-J)}{2}}\sum _{\sigma }\left[\left(\sum _{m_{1}}n_{m_{1},m_{1}}^{\sigma }\right)-\left(\sum _{m_{1},m_{2}}{\hat {n}}_{m_{1},m_{2}}^{\sigma }{\hat {n}}_{m_{2},m_{1}}^{\sigma }\right)\right].}$
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,
${\displaystyle {\hat {n}}^{\sigma }={\hat {n}}^{\sigma }{\hat {n}}^{\sigma }}$.
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 meaningful.
• 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,
${\displaystyle E_{\mathrm {dc} }({\hat {n}})={\frac {U}{2}}{\hat {n}}_{\mathrm {tot} }({\hat {n}}_{\mathrm {tot} }-1)-{\frac {J}{2}}\sum _{\sigma }{\hat {n}}_{\mathrm {tot} }^{\sigma }({\hat {n}}_{\mathrm {tot} }^{\sigma }-1).}$

Warning: it is important to be aware of the fact that when using the L(S)DA+U, in general the total energy will depend on the parameters U and J (LDAUU and LDAUJ, respectively). It is therefore not meaningful to compare the total energies resulting from calculations with different U and/or J, or U-J in case of Dudarev's approach (LDAUTYPE=2).

Note on bandstructure calculation: the CHGCAR file contains only information up to angular momentum quantum number L=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 L(S)DA+U calculations. For the calculation of band structures within the L(S)DA+U approach, it is hence strictly required to increase LMAXMIX to 4 (d elements) and 6 (f elements).