OTmeetsDFT · Multi-marginal Optimal Transport and Density Functional Theory: a mathematical setting for physical ideas

Multi-marginal Optimal Transport and Density Functional Theory: a mathematical setting for physical ideas

Being able, by sole means of computer simulations, to make a preselection of pharmaceutical drugs that deserve to be tested experimentally, to predict the mechanisms of DNA damage by specific compounds, or to design materials with specific properties, is of unquestionable interest for the whole society, in terms of both technological progress, health and economic and energetic saving. In some aspects, this is already a reality: computer simulations of many physical, chemical, biochemical and biomedical processes are successful, and of great help in understanding and guiding experiments. Despite all these advances, there are still basic unsolved problems that hamper a complete reliability of the results and conclusions of such calculations. Density Functional Theory (DFT) is the standard approach to quantum chemistry in simulations with more than a dozen electrons. The classical way of breaking the curse of dimensionality in DFT is through the Kohn-Sham (KS) formalism, which has been extremely successful in predicting properties, for instance, in materials science and chemistry. Unfortunately, KS DFT relies on hand-crafted, highly problem-dependent (semi-empirical) approximations (LDA, B3LYP, PBE, etc), which can only be validated a posteriori through experiments or extremely costly calculations. In particular, KS DFT approximations fail in accurately predicting the physics of systems in which electronic correlation plays a prominent role, e.g. transition metals, which are the workhorse of catalysis. In response to this, Gori-Giorgi, Friesecke, and others developed a modification of the KS setup, by considering also the the Coulomb interaction term and developing the so-called Strictly-Correlated Electron (SCE) formalism in Density Functional Theory. Such an approach has shown to be very promising specially to describe strong-correlation effects in atoms and molecules and in describing dissociation energies at long range. The SCE approach has been developed mainly in physics and chemistry literature and still lacks a rigorous mathematical and computational grounds. The objective of the MSCA-IF "OTmeetsDFT" is to develop a mathematical formalism towards a rigorous SCE DFT theory, by developing rigorous analytical and computational algorithms of a new instance of optimal transport problem with finitely many marginals and Coulomb cost.

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