Supported only by VASP.4.6 and on. THIS FEATURE IS IN LATE BETA STAGE (BUGS ARE POSSIBLE).
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.
VASP allows one to choose between two different approaches to L(S)DA+U:
The 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 wave functions, 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):
The essence of the L(S)DA+U method consists of the assumption that
one may now write the total energy as:
Currently VASP allows for the choice between two different definitions for the double counting energy:
LSDA+U | |
LDA+U |
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.
The L(S)DA+U in VASP is switched on by means of the following tags
NB: LDAUL, LDAUU, and LDAUJ must be specified for all
atomic species!
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].
Note on bandstructure calculation: The CHGCAR file also contains only information up to 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 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. 6.56).