LDA+U Method

Anisimov V I, Aryasetiawan F and Liechtenstein A I 1997 First-principles calculations of the electronic structure and spectra of strongly correlated systems The LDA+U method J. Phys. Condens Matters 767... 

Shiok A B, Lieohtenstein A i and Piokett W E 1999 impiementation of the LDA+U method using the fuii-potentiai iinearized augmented piane-wave basis Phys. Rev. B 60 10 763... 

The intraatomic d-d electron-electron interaction includes Coulomb and exchange interactions, and it is responsible for orbital and spin polarization. To account for orbital polarization, the idea of the LDA + U method was followed.70 A generalized Hartree-Fock approximation including all possible pairings was then used to write... 

In this paper we begin to address the correlation question using the correlated band theory LDA+U method. We focus on the x 1/3 regime where superconductivity emerges. We find that U > Uc = 3 eV leads to charge ordering... 

Figure 1. Effect of the intraatomic repulsion U on the magnetic moment of the Col and Co2 ions for ferromagnetic order. The LDA+U method in the FPLO code was used. Disproportionation to formal charge states Co3+ and Co4+ states occurs above Uc = 3 eV.

As has been alluded to already (see Section 2.12.1), some insulating transition-metal oxides (also called Mott insulators) are falsely predicted as being metals when their electronic structures are calculated on the basis of the LDA because the full amount of the local Coulomb repulsion experienced by the electrons within the d orbitals is underestimated, and the inclusion of some extra repulsion U is needed to theoretically change them into insulators. This is how the LDA+U method is typically justified [174]. The method works well for transition-metal oxides and may also be used for "correlated" metals, in particular, the / elements. One may well ask whether a method that goes by the noble name of first principles or ab initio should be augmented by the introduction of a somewhat arbitrary U energy parameter, but that is matter of taste. The calculation of U requires additional approximations (the so-called constrained LDA method may deal with the problem [175]) and the spatial extent of the basis orbitals effectively determines the size of U, as expected by any quantum chemist. 

The LDA+U method (Liechtenstein et al. 1995) is one of the most frequently used corrections for orbital energy gaps in band calculations. In the LDA+U method, the difference of the Coulomb-exchange interactions from their averaged value is added... 

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... 

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. 

Current approximations to density functional theory are not equally successful for all materials. While its formulation is general, there are some materials for which the EDA and GGA do not seem to be adequate. Examples include the transition metal oxides, and presumably transition metal bearing silicates as well. The problem is that the strongly localized Coulomb repulsion between d electrons does not seem to be adequately represented. As a consequence, FeO wustite is predicted to be a metal in LDA and GGA, whereas experimental observations find an insulator. Despite this failure, it is interesting to note that the structural and elastic properties of FeO are well reproduced by LDA (Isaak et al. 1993). In any case, the complete understanding of Mott insulators will require new advances in theory. These will need to go beyond such developments as the LDA+U method which has yielded considerable insight but adds the local Coulomb repulsion (U parameter) in an ad hoc manner (Mazin and Anisimov 1997). 

In the LDA+U method electrons are separated into two subsystems localized d- or /-electrons for which Coulomb d-d (/-/) interaction is taken into account by a term if/ riirij ni are d- or /-orbital occupancies) as in a mean-field (HF) approximation... 

In the LDA+U method the total Coulomb-interaction energy expression E = UN N — 1)/2 N = is substracted from the LDA total energy and the orbital-... 

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