Local density approximation Hubbard

Similarly to the OP scheme, the LDA + U method (Anisimov et al., 1991) adds a quadratic term to the LSD Hamiltonian to improve the description of the correlated f-manifold, namely 

In its simplest form (Anisimov et al., 1991), the correlation energy term includes a Hubbard repulsion term (Hubbard, 1963) and the so-called double-counting term 

indices k and / rrm over the 14 f orbitals on the site considered, and U is the f-f Coulomb interaction. The double-cormting term Edc is needed to correct for the fact that already includes some Coulomb correlation contribution. 

The LDA -I- U has been extensively used in studies of lanthanides, but a comprehensive review will not be given here. Some significant applications and reviews are reported in Antonov et al. (1998), Gotsis and Mazin (2003), Duan et al. (2007), Larson et al. (2007), and Torumba et al. (2006). The method is almost as fast as a conventional band structure method, and when comparisons to experimental photoemission experiments are made, the LDA + U method provides a much improved energy position of localized bands over the LDA/LSD. In addition, often, the precise position of occupied f-states is not essential to describe bonding properties, rather the crucial effect is that the f-states are moved away from the Fermi level. 

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In the calculation of electronic structures, the presence of correlations thus always represents a difficulty. Perturbation expansions can account for the two extreme cases the delocalized limit in which the effective repulsion U is low compared to the band width, and the quasi-atomic limit where the electron delocalization modifies only slightly the correlated ground state (Anderson, 1959). Some variational techniques (Hubbard, 1964 Gutzwiller, 1965) allow a treatment of systems with U of the order of jS, but they are difficult to use. New methods have recently been developed for adding a part of the Hubbard Hamiltonian to the LDA (local density approximation) ground state (Czyzyk and Sawatzky, 1994). 

First-principles calculations based on the density functional theory (DFT) within the local density approximation plus the Hubbard model (LDA + U) can predict not only the probability of the defect formation but also the energy levels of defects in... 

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