Variationally localized search direction method for constrained optimization of non-orthogonal, localized orbitals in electronic structure calculations.

A new method for the constrained optimization of non-orthogonal, spatially localized orbitals using direct energy minimization techniques, in the context of electronic structure calculations, is presented. The variationally localized search direction (VLSD) method, as it was named, ensures that strict localization constraints are imposed upon the search direction vectors exactly, analytically and in a fully variational fashion. In contrast, the truncated search direction (TSD) method, of standard use in many electronic structure approaches with localization constraints, relies on the approximation that the truncated search direction vectors of the unconstrained problem resemble the exact search direction vectors of the constrained problem. With the TSD method, in order to maintain the localization constraints, a part of the pre-calculated information that is stored in the search direction vectors has to be deleted via an ad hoc, non-variational truncation step. The results on an extensive set of test molecules show that, in general, calculations with the VLSD method require less iterations to converge than with the TSD method for any size of the localization region. It was found that in calculations on certain systems where the TSD method is forced to delete a very large amount of information, the VLSD method is capable of achieving convergence in up to three times less iterations. Validation tests show that structural and electronic properties calculated with either method are accurate and in agreement with other electronic structure approaches.