Regularized Density-Functional Analysis

The Kohn-Sham approach of density-functional theory (DFT) is the most widely used method in quantum chemistry and its usefulness as a practical tool can hardly be overestimated. The central object is the universal density functional. However, this density functional is nondifferentiable, leaving many practical aspects of the theory unfounded. Nevertheless, extensive work has been done establishing exact conditions for the density functional, constituting one of the cornerstones of functional development. The aim of the proposal is to apply a generalization of the Moreau-Yosida regularization to DFT. This achieves not only differentiability, but also mitigates the problem of potential-representability and provides global solutions of the underlying variational problem. This unconventional approach may have transformative impact on the development of approximate functionals as well as the iterative Kohn-Sham scheme.The first objective is to establish the mathematical foundation of a regularized DFT, akin to the unregularized setting of standard DFT. Close interplay between different theories that use more than just the particle density as variables will be a guide for the regularized theory.Equipped with a regularized formulation, the aim of the second objective is to develop new and understand existing exact constraints for the density functional. This intends to open up a whole new axis of method development for approximate functionals. Since the regularization transformation considered is lossless, REGAL opens up for a new theoretical bridge between formal DFT and density-functional approximations.The third objective is the study of the regularized Kohn-Sham iteration scheme. Here a proof of guaranteed convergence is the ultimate aim. Furthermore, regularization effects to speed up convergence using bounds on the energy curvature will be studied.The feasibility is increased by the PI’s strong background in both applied mathematics and quantum chemistry.