On site Coulomb interaction: L(S)DA+U

`LDAU`= `.TRUE. | .FALSE.`

`LDAUTYPE`= `1 | 2 | 4`

`LDAUL`= [`0 | 1 | 2 | 3 ` array] `LDAUU`= [real array] `LDAUJ`= [real array]

`LDAUPRINT`= `0 | 1 | 2`

Defaults: | |

LDAU |
= .FALSE. |

LDAUTYPE |
= 2 |

LDAUPRINT |
= 0 |

The L(S)DA often fails to describe systems with localized (strongly correlated)
and 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.

By means of the `LDAUTYPE`-tag on specifies which type of L(S)DA+U approach
will be used:

`LDAUTYPE`=1: The rotationally invariant LSDA+U introduced by Liechtenstein*et al.*[90], which is of the formThe unscreened e-e interaction can be written in terms of Slater's integrals , , , and (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 ).

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, and . ( and 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):

- -electrons: ,
- -electrons: , , and
- -electrons: , , , and

The essence of the L(S)DA+U method consists of the assumption that one may now write the total energy as:

(6.55)

`LDAUTYPE`=2 (Default): The simplified (rotationally invariant) approach to the LSDA+U, introduced by Dudarev*et al.*[91]. This flavour of LSDA+U is of the following form: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, i.e., . (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 and do not enter seperately, only the difference 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,(6.56)

`LDAUL`= ... specifies the -quantum number (one number for each species)
for which the on-site interaction is added.

(-1=no on-site terms added, 1= p, 2= d, 3= f, Default: `LDAUL`=2)

`LDAUU`= ... specifies the effective on-site Coulomb interaction parameters.

`LDAUJ`= ... specifies the effective on-site Exchange interaction parameters.

NB: `LDAUL`, `LDAUU`, and `LDAUJ` must be specified for *all*
atomic species!

`LDAUPRINT`= `0 | 1 | 2 ` Controls the verbosity of the L(S)DA+U module.

(0: silent, 1: Write occupancy matrix to OUTCAR, 2: idem 1., plus potential matrix dumped
to stdout, Default: `LDAUPRINT`=0)

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 and . It is therefore not meaningful to compare the total energies resulting from calculations with different and/or (c.q. in case of Dudarev's approach).

Furthermore, since LDA+U usually results in aspherical charge densities at
and atoms we recommend to set `LASPH` = .TRUE. in the INCAR file for
gradient corrected functionals (see Sec. 6.44).
For CeO for instance, identical results to the FLAPW methods
can be only obtained setting `LASPH` = .TRUE.

Note on bandstructure calculation:
The CHGCAR file also contains only information up to `LMAXMIX` (defaulted to 2) 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
can be (or actually *are*) 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).
(see Sec. 6.63).