Reduction of electronic structure problems of density functional theory
We investigate a model reduction technique for the electronic structure problem of the Density Functional Theory (DFT). This investigation is motivated by the need to extend the applicability of DFT methods to crystalline systems with numbers of atoms far beyond what can be treated using current DFT techniques. The model reduction technique presented here is based on direct minimization of the energy functional of the electronic system with respect to either the electron density in the case of Orbital-Free DFT method, or with respect to the one-electron wave functions in case of Kohn-Sham DFT approach. This solution of the minimization problem is aided by a two-level interpolation of the unknown functions using finite element (FEM) discretization of space over which the solution is sought. Two FEM grids are used. The first is a subatomic grid spanning the space over the individual unit cells, wherein the local spatial behavior of the unknown electronic functions is represented. The second is a lattice-level grid, over which the profile of the unknown functions over the unit cells interior to lattice elements is interpolated in terms of its counterparts at the unit cells located at the lattice grid points. We present the formalism of the proposed reduction technique for Orbital-Free DFT methods, and give numerical examples showing the effectiveness of the reduction technique.