Subspace representations inab initiomethods for strongly correlated systems

Abstract

We present a generalized definition of subspace occupancy matrices in ab initio methods for strongly correlated materials, such as DFT+U (density functional theory + Hubbard U ) and DFT+DMFT (dynamical mean-field theory), which is appropriate to the case of nonorthogonal projector functions. By enforcing the tensorial consistency of all matrix operations, we are led to a subspace-projection operator for which the occupancy matrix is tensorial and accumulates only contributions which are local to the correlated subspace at hand. For DFT+U , in particular, the resulting contributions to the potential and ionic forces are automatically Hermitian, without resort to symmetrization, and localized to their corresponding correlated subspace. The tensorial invariance of the occupancies, energies, and ionic forces is preserved. We illustrate the effect of this formalism in a DFT+U study using self-consistently determined projectors.

Maximally localized Wannier functions: Theory and applications

“…The isosurface is set to half of the maximum for the s and p-like NGWFs and 10 À3 times the maximum for the d-like NGWFs. Adapted from O'Regan,Payne, and Mostofi, 2011. …”

Linear-scaling DFT + Uwith full local orbital optimization

“…Ground state calculations employing our method have previously been demonstrated on both bulk and molecular strongly interacting systems 22,23 , with further examples on large-scale systems such as dilute magnetic semiconductor (Ga,Mn)As 57 and disordered VO 2 58 , using an extension of the method to DFT+DMFT, forthcoming. Further examples of candidate systems include organometallic molecules, such as metalloproteins and molecular magnets, where the method is particularly efficient for a low density of strongly interacting subspaces, and solids such as magnetic heterostructures, defective and doped oxides or catalytic interfaces with oxide surfaces.…”

“…Due to the subspace-localized nature of the DFT+U correction in the tensorial representation 23 , only those local orbitals |φ δ in Eqn. 29 which explicitly overlap with the Hubbard projectors |ϕ m ′ contribute and thus require summation over.…”

Theory of projections with nonorthogonal basis sets: Partitioning techniques and effective Hamiltonians

“…Some of the mathematical and conceptual issues discussed here have been previously addressed in the literature [27,[32][33][34][35][36][37][38][39], but we believe that our approach to the problem may shed a new light on some of the unresolved or controversial points. We begin by revisiting some basic operator operations in a nonorthogonal metric in Sec.…”