Wannier Functions and Their Role in Improving Density Functional Approximations

Abstract

Density functional theory has proven to be an invaluable tool for modeling matter and chemistry . This can be seen from the fact that density functional theory papers are far and away the most cited theory from the physical sciences. While density functional theory excels at predicting total energies and equilibrium geometries, standard approximate functionals can be inadequate for determining some properties such as dissociation energies, reaction barriers, and band gaps. These deficiencies can be traced to delocalization error in density functional approximations. In finite systems, delocalization error can be attributed to the incorrect treatment of fractional electron charge whereby the total energy deviates from the correct behavior of linear interpolation between integer points. For bulk systems the delocalized nature of the orbitals results in a linear total energy at fractional charges, but the slope is incorrect due to delocalization error. Multiple methods have been proposed to fix this deficiency and produce the correct linearity condition such asthe Fermi-Löwdin orbital self-interaction correction, Koopmans-compliant functionals, the screened range-separated hybrid functional, the generalized transition state method, and the localized orbital scaling correction. All of these methods rely on spatially localized orbitals for their corrections, highlighting the importance of localized orbitals in modern density functional theory. The traditional method of obtaining localized orbitals minimizes the spatial variance, but here we explore an alternative approach that minimizes the combination of spatial and energetic variance. Minimizing the energetic variance allows for the occupied and unoccupied spaces to be considered together, a feature that is not prescribed in other localization schemes. The localization in energy results in localized orbitals that are more correlated with certain energy ranges, thereby making them more chemically relevant for the states that are associated with frontier energies. We show how these localized functions can be used in the localized orbital scaling correction to remedy many of the density functional approximation shortcomings related to delocalization error.