DFT+U: formalism

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Several variants of the DFT+U method exist.[1][2]

Three types of DFT+U approaches are available in VASP. These are the following:

  • LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.[3]
This particular flavour of DFT+U is of the form
and is determined by the PAW on-site occupancies
and the (unscreened) on-site electron-electron interaction
where are real spherical harmonics of angular momentum =LDAUL.
The unscreened electron-electron interaction can be written in terms of the Slater integrals , , , and ( electrons). Using values for the Slater integrals calculated from atomic orbitals, however, would lead to a large overestimation of the true electron-electron interaction, since in solids the Coulomb interaction is screened (especially ).
In practice these integrals are often treated as parameters, i.e., adjusted to reach agreement with experiment for a property like for instance the equilibrium volume, the magnetic moment or the band gap. They are normally specified in terms of the effective on-site Coulomb- and exchange parameters, and (LDAUU and LDAUJ, respectively). and can also be extracted from constrained-DFT calculations.
These translate into values for the Slater integrals in the following way (as implemented in VASP at the moment):
- -
-
The essence of the DFT+U method consists of the assumption that one may now write the total energy as:
where the Hartree-Fock like interaction replaces the semilocal on site due to the fact that one subtracts a double counting energy , which supposedly equals the on-site semilocal contribution to the total energy,
  • LDAUTYPE=2: The simplified (rotationally invariant) approach to the DFT+U, introduced by Dudarev et al.[4]
This flavour of DFT+U is of the following form:
This can be understood as adding a penalty functional to the semilocal total energy expression that forces the on-site occupancy matrix in the direction of idempotency,
.
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 meaningful.
  • LDAUTYPE=4: same as LDAUTYPE=1, but without exchange splitting (i.e., the total spin-up plus spin-down occupancy matrix is used). The double-counting term is given by

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

Note on bandstructure calculation: the CHGCAR file contains only information up to angular momentum quantum number =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 self-consistent run. The deviations are often large for DFT+U calculations. For the calculation of band structures within the DFT+U approach, it is hence strictly required to increase LMAXMIX to 4 ( elements) and 6 ( elements).

Related tags and articles

LDAU, LDAUL, LDAUU, LDAUJ, LDAUPRINT, LMAXMIX

Examples that use this tag

References