Density Functional Theory and Optimal Transportation with Coulomb Cost

By Codina Cotar, G. Friesecke, Claudia Klüppelberg. 2011

180 citations recorded by the source corpus.

We present here novel insight into exchange‐correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange‐correlation functional reduces to a very interesting functional that depends on an optimal transport map T associated with a given density ρ. The limit problem has been suggested, on grounds of formal arguments, in the physics literature, but it appears that it has not hitherto been interpreted as an optimal transport problem. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each ρ and determine the map explicitly in the case when ρ is radially symmetric. © 2012 Wiley Periodicals, Inc.

Subfields: Atomic and Molecular Physics, and Optics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fefc8c45194f8346a64aca9e15f1659b3f0ed3b5

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In Atomic and Molecular Physics, and Optics

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Citations in the corpus

Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “The minimization of the above energy becomes a multi marginal transport problem as soon as one takes the semi-classical limit h̄→ 0 (see [129, 130], the rigorous limit for N > 2 is an ongoing work).”
Iterative Bregman Projections for Regularized Transportation Problems (2014, background): “Multi-marginal Optimal Transport, [48, 47], is a natural extension of Optimal Transport with many potential ﬁelds of applications : Economics [22] [19], Density Functionnal Theory in Quantum Chemistry [24].”
Computational Optimal Transport (2018, methodology): “A typical application of multi-marginal OT is to compute approximation of solutions to quantum chemistry problems, and in particular, for the so-called Density Functional Theory [Cotar et al., 2013].”
Multi-marginal optimal transport: theory and applications (2014, methodology): “The following theorem from [9, 21] is similar in spirit to Theorem 2.”
Multimarginal Optimal Transport Maps for One–dimensional Repulsive Costs (2015, background): “At the same time, in [6], Cotar, Friesecke, and Klüppelberg showed that an analogous optimal transportation problem describes the semiclassical limit of DFT in the case of two electrons and provides estimates from below in the general case.”