DFT+U: formalism

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DFT+U is a method that was proposed to improve the description of systems with strongly correlated or electrons, like antiferromagnetic NiO for instance, that are usually inaccurately described with the standard LDA and GGA functionals[1]. Several variants of the DFT+U method exist (see Refs. [2][3] for reviews) that differ for instance in the way the double counting term is calculated. Three of them are implemented in VASP, whose formalism is briefly summarized below.

  • LDAUTYPE=1: The rotationally invariant DFT+U introduced by Liechtenstein et al.[4]
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.[5]
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

References