Mathematical foundations of density functional theory - the representability problem

• Applicant: Dr. Thiago Carvalho Corso
• Subject Area: Mathematics ; Theoretical Chemistry: Electronic Structure, Dynamics, Simulation
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 572811220

Density functional theory (DFT) is the most widely used method for electronic structure calculations. One of the key reasons for the success of DFT is the Kohn-Sham scheme, which together with current approximate exchange-correlation (xc-)functionals, allows practitioners to compute ground-state properties of electronic systems with a reasonable accuracy and at relatively low computational cost. However, a significant drawback of DFT, as compared to wave-function methods, is that systematic improvements of DFT calculations are in general not possible. One of the reasons is that the approximation properties of the various xc-approximate functionals suggested in the literature are not well understood. In fact, even the existence of an exact exchange-correlation potential, which one is supposed to approximate, is mathematically questionable. In this project, our goal is to investigate, from a rigorous mathematical perspective, the existence, uniqueness, and regularity properties of the exact xc-functional. More precisely, we would like to address the following points. First, we seek to establish a complete solution to the v-representability problem. Roughly speaking, the v-representability problem consists in characterizing the set of all possible ground-state single-particle densities for the family of Schrödinger operators with a given class of external potentials. In particular, the v-representability problem plays a fundamental role in a mathematically sound foundation of the Kohn and Sham scheme. In this project, we propose to address this problem by studying suitable classes of distributional potentials. This will be achieved by combining convex and functional analytic techniques with potential theoretic tools. In the second part of this project, we aim to study asymptotic properties of several functionals in certain regimes of physical relevance. Such studies are paramount to the development of approximate functionals and will contribute to a better understanding of their limitations. At this stage, we plan to benefit from combining the results established in the first part of this project with optimal transportation theory and tools from semi-classical analysis.

• DFG Programme: Research Grants