Optimal-transport formulation of electronic density-functional theory

Abstract

The most challenging scenario for Kohn-Sham density-functional theory, that is, when the electrons move relatively slowly trying to avoid each other as much as possible because of their repulsion (strong-interaction limit), is reformulated here as an optimal transport (or mass transportation theory) problem, a well-established field of mathematics and economics. In practice, we show that to solve the problem of finding the minimum possible internal repulsion energy for N electrons in a given density ρ(r) is equivalent to find the optimal way of transporting N − 1 times the density ρ into itself, with the cost function given by the Coulomb repulsion. We use this link to set the strong-interaction limit of density-functional theory on firm ground and to discuss the potential practical aspects of this reformulation.

Exchange–correlation functionals from the strong interaction limit of DFT: applications to model chemical systems

“…(6) for a given density ρ(x), 11,18,33 choosing boundary conditions that make the density between two adjacent strictlycorrelated positions always integrate to 1 (total suppression of fluctuations), 11 in Fig. 3,…”

Kantorovich dual solution for strictly correlated electrons in atoms and molecules

“…Recently, the behavior of the exchange-correlation functional has been revealed in the limit of strictly correlated electrons (SCE) [12][13][14][15] . The many-body Coulomb repulsive energy of SCE determines the exact exchangecorrelation functional in the strong interaction limit, without artificially breaking any symmetry of the system or introducing any tunable parameters.…”

“…These co-motion functions characterize the relative positions of all the electrons with respect to one given electron in the SCE limit. To the extent of our knowledge, in practice the co-motion functions can only be determined for one dimensional systems 14 and spherically symmetric atoms 12,13,15 , with the help of semianalytic methods. Little is known about the shape or even the existence of the co-motion functions for general systems including small molecules.…”

“…For given electron density profile, the SCE limit is described by minimizing the many-body Coulomb interaction energy with respect to all wavefunctions in a 3N dimensional space, where N is the number of electrons in the system, under the additional constraint that the wavefunction is consistent with the electron density 12 . Mathematically, this daunting minimization task is an optimal transport problem with Coulomb cost function 13,15,16 . The optimal transport theory finds the optimal way for transferring masses from one position to another.…”

“…Little is known about the shape or even the existence of the co-motion functions for general systems including small molecules. On the other hand, the optimal transport problem with Coulomb cost function can be equivalently solved by its dual formulation, called the Kantorovich dual problem 13,[17][18][19][20] . The main advantage of the Kantorovich dual problem is its potential applicability to general systems, ranging from atoms and molecules to condensed matter systems.…”