Determining the optimal U correction – Self-consistent Hubbard-U corrections for transition metal oxides

As outlined in the theory section, a choice for the U correction strength can be made such that the spurious curvature in DFT energy as a function of occupation can be corrected. This can be done on a site by site basis for each hubbard atom in the system (transition metals) with DFPT. Once the U parameters are obtained for a system using DFPT, the DFT + U calculations can be performed with at least partially corrected SIE. This can result in a correction of the common errors due to SIE such as the underpredicted band gap, incorrect lattice parameters, and overly-delocalized charges. However, once the U corrections are applied and the ionic positions are relaxed, the Hubbard manifold may change as the projections of the Kohn-Sham states on the local basis change. As a result, the U correction itself may change.

In order to address the structure-dependent nature of the U parameter, it is best to determine the U correction self-consistently. Simply put, the U parameter depends on the structure (e.g. the lattice constant) and the structure on the U parameter. The following workflow outlines a self-consistent procedure to determine the correction.

Fig.1) Self-consistent workflow for determining optimal effective U parameters

First an initial structure is used to get the initial estimate of the $U^i$ per site. Using GGA or experimental lattice constants is usually suitable. After DFPT is used to calculate the parameters, the structure should be relaxed. A comparison should be made between the $U^i_n$ for the current iteration, $n$ , and the previous iteration, $U^i_{n-1}$ . Though a set tolerance has not been established for the convergence of the $U^i$ per site, a convergence tolerance of $U^i = 10 meV/site$ is recommended. Other properties can be checked for convergence at this stage such as the lattice constant, $a_n$ . If the $U^i$ are converged as well as other desired quantities, a final SCF calculation can be performed to obtain system energies. If not, the $U^i$ should be recalculated with the adjusted lattice parameters using DFPT and the process repeated. This should be done until the desired tolerance(s) is/are achieved.