Schedule for: 19w5035 - Optimal Transport Methods in Density Functional Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will survey main results (both at the rigorous and the nonrigorous level) and open questions on the strongly correlated limit of DFT, including: - the connection between Hohenberg-Kohn-Lieb-Levy constrained-search and minimization of the interaction energy over $|\Psi|^2$ (alias Kantorovich optimal transport) - the SCE (alias Monge) ansatz in the Kantorovich problem: where it works, where it fails - the new quasi-Monge ansatz [1] which - unlike the SCE ansatz - always yields the minimum Kantorovich cost, but whose data complexity scales linearly instead of exponentially with the number of particles/marginals - asymptotic and semi-empirical exchange-correlation functionals related to the strictly correlated limit - representability challenges. [1] G.Friesecke, D.Vögler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM J. Math. Analysis Vol. 50 No. 4, 3996-4019, 2018

The strictly correlated electron approach to density functional theory, first proposed by Seidl and coworkers [1-4], offers a unique perspective on finite-temperature density functional theory and one of its application areas, simulations in the warm dense matter regime. In this region of phase space, many of the assumptions of traditional Kohn-Sham density functional theory no longer hold, requiring exchange-correlation free energy approximations that include explicit temperature dependence and better handling of complicated ionization processes. Formal analysis of the strong interaction limit for thermal ensembles will be demonstrated using the asymmetric Hubbard model, accompanied by discussions of the finite-temperature uniform gas and the competition between strong interaction and temperature in complicated physical systems. [1] M. Seidl, Phys. Rev. A 60, 4387 (1999). [2] M. Seidl, J. P. Perdew, and M. Levy, Phys. Rev. A 59, 51 (1999). [3] M. Seidl, P. Gori-Giorgi, and A. Savin, Phys. Rev. A 75, 042511 (2007). [4] P. Gori-Giorgi, G. Vignale, and M. Seidl, J. Chem. Theory Comput. 5, 743 (2009).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Time-dependent density functional theory relies on a one-to-one correspondence between external potentials in the $N$-particle Schrodinger equation and the one-particle densities they generate. Runge and Gross argued that this correspondence holds (up to a time-dependent constant in the potential) for sufficiently "nice" external potentials. I will discuss a mathematical setting for this argument and explain when it can be made fully rigorous. The interaction potentials play an important role for this and I will discuss the problems that arise in the presence of singular interactions, like the Coulomb potential. This is joint work with S. Fournais, M. Lewin, and T. Ostergaard Sorensen.

The talk will focus on the leading term in the strong-interaction limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory. Such term can be fully determined from a functional of the Hartree-Fock density. We analyse this functional and highlight similarities and differences with the strong-interaction limit of the density-fixed adiabatic connection case of Kohn-Sham density functional theory.

We compute explicitly the functional derivative of the subleading term in the strong coupling limit expansion of the generalized Hohenberg-Kohn functional for the special case of two electrons in one dimension, analyzing its features.

The Hohenberg-Kohn theorem is one of the cornerstones of Density functional theory, and it relies on a unique continuation property, a mathematical tool which is in general used to prove uniqueness of Cauchy problems. We present a recent result on unique continuation which can treat potentials in many-body magnetic Schrödinger operator, and apply it to show the Hohenberg-Kohn theorem in presence of a fixed magnetic field.

We here discuss Hohenberg-Kohn-like theorems for systems with magnetic fields where different current densities are used together with the particle density. The situation is much more complicated than the case of no magnetic field, a fact that we further explore.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This theorem says that the true electron density of a molecule has a convergent Taylor expansion away from the positions of the nuclei. We discuss this, and an open problem about its structure at the nuclei.

In this talk, we discuss the phase diagram of the Hartree-Fock (HF) Homogeneous Electron Gas (HEG), both theoretically and numerically. We first recall the Overhauser spatial symmetry breaking which occurs for $T=0$ at all densities and we present a new lower bound on the energy gain due to this symmetry breaking. At high density we can prove that this gain can only be exponentially small. Then we focus on the fluid phase of the HF HEG and discuss spin symmetry breaking. We find two regions where the gas is either ferromagnetic or paramagnetic. We can partly justify the paramagnetic phase theoretically. This is joint work with Mathieu Lewin and Christian Hainzl.

The Kohn-Sham SCE formulation is a natural way for using the strictly correlated electron (SCE) functional via an optimal transport formulation, especially when the Coulomb interaction dominates over the kinetic part. In this talk I will discuss some numerical issues associated with solving Kohn-Sham SCE using primal and dual formulations, as well as modeling issues for molecular systems regardless of how the optimal transport problem is solved. In the end I will discuss some on-going work for applying the Kohn-Sham SCE formulation to model problems with effective convex relaxation strategies. (Joint work with Yuehaw Khoo, Michael Lindsey and Lexing Ying)

We extend the zero-temperature formalism of strictly correlated electrons in second quantization to the setting of finite temperature, naturally motivating the entropic regularization of a multi-marginal optimal transport problem on the binary hypercube. We introduce a semidefinite relaxation of this problem (mostly analogous to that of the zero-temperature setting) and briefly discuss its connection to the literature of probabilistic graphical models.

We consider an infinite number of (interacting) quantum particles with constant spatial density filling the whole 2-dimensional space. We show that for small enough perturbations of this state at initial time, the system returns to this equilibrium for large times. The dynamics which we consider is of Hartree-type with localized interactions. This is a joint work with Mathieu Lewin.

In this talk, we consider one relativistic nucleon interacting with classical $\sigma$ and $\omega$ mesons fields in an atomic nucleus. I will explain that, in the nonrelativistic limit of nuclear physics, the model converges to a specific nonlinear Schrödinger-type equation with a mass depending on the solution itself. I will also discuss open problems concerning the case of several nucleons. Joint work with Mathieu Lewin.

First I will introduce the strong and the relaxed formulations of the multi-marginal optimal transportation problem and I will compare them with the 2-marginal version. I will then explain which are the key tools which allowed the solution in the 2-marginal case and are missing in the multi-marginal problem. For particular costs advancement have been made and, for sufficiently general costs, partial solutions have been given. This positive results will be discussed in the second part of the expository talk. Finally some attention will be dedicated to costs which appeared in applications related to the theme of the meeting.

I shall present the mathematical analysis of a limit problem in Density Functional Theory as a small parameter in front of the kinetic energy (the Planck constant) tends to zero, and explain how this relates to a bond dissociating model. This recent work shows that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect, and how the theme of fractional number of electrons appears naturally in this context, along with many open questions.

After introducing the general setting of multimarginal optimal transport, we present some operators designed to approximate and regularize the tansport plans by keeping the marginals fixed. As an application to the Density Functional Theory, we show how to use this technique to study the properties of the mapping from the wave-functions to the single particle marginals.

In this short communication, we will introduce the dual formulation of Multi-marginal Optimal Transport focusing on the case of repulsive costs functions (e.g. Coulomb). We will briefly discuss the connection with DFT and point out some recent results that appears in the literature in the last three years.

In several applications where one studies interactions mediated by a long-range kernel, it is useful to decompose this kernel into finite-range contributions, and study them separately, without losing information on the system. In my talk I present a finite-range decomposition method developed in joint work with C. Cotar for the study of multimarginal transport with interaction kernels of the type 1/|x-y|^s. This helped apply to the study of Density Functional asymptotics, decomposition techniques that extend to much higher generality, and which seem to have high potential for algorithmic applications.

In this talk, we introduce methods from convex optimization to solve the multi-marginal transport type problems arise in the context of density functional theory. Convex relaxations are used to provide outer approximation to the set of N-representable 2-marginals and 3-marginals, which in turn provide lower bounds to the energy. We further propose rounding schemes to obtain upper bound to the energy.

We consider two sharp next-order asymptotics problems, namely the asymptotics for the minimum energy for optimal point configurations and the asymptotics for the many-marginals Optimal Transport, in both cases with Coulomb and Riesz costs with inverse power-law long-range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, while the second appears as a semi-classical limit of the Density Functional Theory energy modelling a quantum version of the same system. Recently the second-order term in these expansions was precisely described for inverse power-law interactions with power $\max(0,d-2)\le s (TCPL 201)

When a free positive charge is inserted in a metal, electrons flock towards it. This creates a reaction potential that effectively nullifies the Coulomb potential of the charge at long range. I will explain how this phenomenon occurs in the framework of the linear response of the reduced Hartree-Fock model (a simplified version of DFT) of defects at finite temperature. The analysis also sheds light on the convergence of algorithms to solve the self-consistent equations of DFT, and the phenomenon of "charge sloshing".

In this talk, we introduce the spectral renormalization method to solve the Kohn-Sham equation. The idea is to renormalize the electron density of orbital and recast the Kohn-Sham equation as a fixed point equation in spectral space. The resulting system is then numerically solved by implementing a direct renormalized fixed point iteration scheme which conserves the total number of atoms. One interesting aspect of the proposed method is that the electronic ground state energy and density are computed simultaneously. We implement the spectral renormalization scheme on a benchmark problems that include the strongly correlated electronic system using the KS equation and Hartree potential.

We introduce a relaxation of the OT problem (Moments Constrained Optimal Transport, MCOT) by minimizing the OT cost over the set of measures having only N moments against some given test functions equal to the ones of each marginal law. Using Tchakaloff’s theorem, we show that a finite discrete measure minimizes this problem which charges at most DN+2 points, with D the number of marginal laws. In addition, we prove the convergence of the MCOT problems towards the OT problem as the number of moments imposed goes to infinity, under some appropriate assumptions on the set of test functions. In this context, the convergence rate with respect to the number of moment constraints depends on the choice of test functions and cost considered. We present quantitative estimates for this rate in the particular cases of piecewise constant and affine test functions. Although the resulting MCOT problem is non convex, the linear dependency of the number of charged points of an optimum in the number of marginal laws and test functions is of algorithmic interest for the resolution of multi-marginal optimal transport problems with a very large number of marginal laws. This situation is typically encountered in the context of DFT. We thus propose two algorithms for the resolution of the MCOT problem which exploits this particular structure.

In this talk, I will present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from combinatorial in both $N$ and $\ell$ to $\ell(N +1)$, where $\ell$ is the number of marginal states and $N$ the number of marginals. The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to $\ell(N − 1)$ unknowns, and cures the insufficiency of the Monge ansatz; i.e., it is shown that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions. Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which emerges as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context $N$ corresponds to the number of particles, motivating the interest in large $N$. These results were established in collaboration with Gero Friesecke. The corresponding paper can be found under doi:10.1137/17M1150025.

The two main challenges for Density Functional Theory (DFT) are strong correlation and the inclusion of London dispersion interactions. In this talk I will introduce our new formalism for dispersion interactions with fixed marginals (arXiv:1812.11840). The emphasis will be on the mathematical aspects, starting with the simple system consisting of two hydrogen atoms.

In this talk, we consider disordered quantum crystals in the simplest Kohn-Sham model with no exchange-correlation, that is, the reduced Hartree-Fock (rHF) framework. The nuclei are supposed to be classical particles arranged around a reference periodic configuration. In particular, we consider a family of nuclear distributions $\mu(\omega,\cdot)$, where $\omega$ spans a probability space $\Omega$. Under some assumptions on the nuclear distribution $\mu$, the average energy per unit volume admits a minimizer, which is a solution of the self-consistent rHF equations. We mainly deal with short-range Yukawa interaction and obtain partial results for Coulomb systems. We also study localization properties of the mean-field Hamiltonian numerically. Joint works with Eric Cancès and Mathieu Lewin.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.