Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Sparsity results on a moment-constrained approximation of classical and quantum optimal transport Aur´ elien Alfonsi1,2, Rafa¨ el Coyaud1,2, Virginie Ehrlacher1,2, Damiano Lombardi2, Luca Nenna3 1Ecole des Ponts ParisTech 2INRIA 3Laboratoire de Math´ ematiques d’Orsay Workshop on Optimal Transport, Berlin, 12th March 2024 1 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Outline of the talk Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 2 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Outline of the talk Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 3 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Motivation: electronic structure calculation for molecules Computation of the ground state of electrons in a molecule: electric, optical, magnetic properties, prediction of chemical reactions, computation of inter-atomic or inter-molecular potentials for molecular dynamics simulations... 4 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Many-body Schr¨ odinger model For the sake of simplicity, atomic units will be used and the influence of spin will be neglected. Born-Oppenheimer approximation: Let us consider a physical system composed of: • M nuclei, that are assumed to be (fixed) classical point charges, whose positions and electric charges are denoted by R1 , . . . , RM ∈ R3 and Z1 , . . . , ZM ∈ N∗ respectively; • N electrons, considered as quantum particles, and represented by a complex-valued function ψ : R3N → C. The function ψ is called the wavefunction of the system of electrons. 5 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Physical interpretation of the wavefunction For x1 , . . . , xN ∈ R3, the quantity |ψ(x1 , . . . , xN )|2 represents the probability density of the positions x1 , . . . , xN of the N electrons. For B ⊂ R3N , B |ψ|2: probability that the positions of the electrons (x1 , · · · , xN ) ∈ B. Consequence: R3N |ψ|2 = ψ 2 L2 = 1 6 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Admissible set of wavefunctions (with finite kinetic energy) The wavefunction ψ of a set of N electrons (fermionic particles) with finite kinetic energy has to belong to the set: AN := ψ ∈ L2(R3N ; C), ∇ψ ∈ L2(R3N ; C), ψ antisymmetric, ψ L2 = 1 . A function ψ ∈ AN is antisymmetric in the following sense: For all σ ∈ SN , the set of permutations of {1, · · · , N}, ψ(xσ(1) , · · · , xσ(N) ) = (σ)ψ(x1 , · · · , xN ), ∀(x1 , · · · , xN ) ∈ R3N , where (σ) is the signature of σ. 7 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Many-body Schr¨ odinger electronic problem The ground state energy Eg[v] of a system of N electrons in the presence of an external potential v : R3 → R is given by: Eg[v] = inf ψ∈AN T[ψ] + C[ψ] + Wv ext [ψ], where • kinetic energy: T[ψ] := 1 2 R3N N i=1 |∇xi ψ|2 • electron-electron interaction energy: Cee[ψ] := R3N c|ψ|2 with c : R3N → R+ ∪ {+∞} is the Coulomb potential ∀x1 , . . . , xN ∈ R3, c(x1 , · · · , xN ) := 1≤i=j≤N 1 |xi − xj | • external potential energy: Wv ext [ψ] := R3N N i=1 v(xi ) |ψ(x1 , · · · , xN )|2 dx1 · · · dxN . 8 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Example of external potential Example: External potential v generated by M nuclei at positions R1 , . . . , RM ∈ R3 and charges Z1 , . . . , ZM > 0 in the Born-Oppenheimer approximation: v(x) := − M k=1 Zk |Rk − x| 9 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem High-dimensional linear eigenvalue problem The ground state wavefunction is then solution to Hv N ψg = Eg[v]ψg (1) with Hv N the Hamiltonian operator Hv N = − 1 2 N i=1 ∆xi + c(x1 , . . . , xN ) + N i=1 v(xi ) = H0 N + N i=1 v(xi ) Traditional methods: curse of dimensionality 10 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Bra-ket notation For ϕ, ψ ∈ L2(R3N ; C) and H a self-adjoint operator acting on L2(R3N ; C). • φ|ψ = φ, ψ L2 = R3N φψ • φ|H|ψ = φ, Hψ L2 = R3N φ(Hψ) • |ψ ψ| is the operator on L2(R3N ; C) defined by ∀ϕ ∈ L2(R3N ; C), |ψ ψ|ϕ = ψ ψ, ϕ L2 In particular, if ψ L2 = 1, |ψ ψ| is the orthogonal projector in L2(R3N ; C) onto Span{ψ} Example: ψ|Hv N |ψ = ψ, Hv N ψ L2 = R3N ψ(x1 , . . . , xN ) − 1 2 N i=1 ∆xi + c(x1 , . . . , xN ) + N i=1 v(xi ) ψ(x1 , . . . , xN ) = T[ψ] + C[ψ] + Wv ext [ψ] 11 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Many-body Schr¨ odinger electronic problem The ground state energy Eg[v] of a system of N electrons in the presence of an external potential v is given by: Eg[v] = inf ψ∈AN ψ|Hv N |ψ where Hv N = H0 N + N i=1 v(xi ) with H0 N = − 1 2 N i=1 ∆xi + 1≤i

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Outline of the talk Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 13 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Hohenberg-Kohn theorem and Density Functional Theory For all ψ ∈ AN , the electronic density ρψ associated to ψ is a real-valued function defined on R3 by ∀x ∈ R3, ρψ (x) := N R3(N−1) |ψ(x, x2 , · · · , xN )|2 dx2 · · · dxN . [Hohenberg,Kohn,1964], [L´ evy,1979], [Lieb,1983] It holds that IN := {ρψ , ψ ∈ AN } = ρ ≥ 0, R3 ρ = N, R3 |∇ √ ρ|2 < +∞ The Hohenberg-Kohn theorem states that Eg[v] = inf FHK [ρ] + R3 vρ, ρ ∈ IN (2) where the Hohenberg-Kohn functional FHK (ρ) is defined by FHK [ρ] := inf ψ|H0 N |ψ , ψ ∈ AN , ρψ = ρ . 14 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Density Functional Theory: approximations of FHK [ρ] Unfortunately, the functional FHK [ρ] is not known explicitly. All DFT models rely on approximations of this functional in order to obtain computable models (Kohn-Sham LDA, GGA, hybrid functionals, machine-learnt exchange-correlation ...) Besides, the functional FHK [ρ] is not convex! As a consequence, even if FHK was known, the computation of Eg[v] out of (2) might be a complicated task. 15 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Lieb functional To alleviate the second point, Lieb [Lieb, 1983] introduced the so-called Lieb functional FL [ρ], which is actually a convexification of the Hohenberg-Kohn functional FHK [ρ]. FL [ρ] = inf αi ≥ 0, ρi ∈ IN , i ∈ N∗ i∈N∗ αi ρi = ρ i∈N∗ αi FHK [ρi ] In particular, it still holds that Eg[v] = inf FL [ρ] + R3 vρ, ρ ∈ IN (3) 16 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Mixed states The Lieb functional can also be seen as a minimization problem defined over the set of mixed states (instead of pure states like in the Hohenberg-Kohn functional). • HN 0 := ψ ∈ L2(R3N ; C), ψ antisymmetric • S+ 1 (HN 0 ): Set of non-negative trace-class operators on HN 0 , that is the set of operators Γ of the form: Γ = +∞ i=1 αi |ψi ψi | for some • αi ≥ 0 s.t. +∞ i=1 αi < +∞, • (ψi )i orthonormal basis of HN 0 Associated electronic density: ρΓ (x) = +∞ i=1 αi ρψi (x) 17 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Lieb functional: a quantum optimal transport problem FL [ρ] = inf Γ ∈ S+ 1 (HN 0 ) ρΓ = ρ Tr H0 N Γ Tr H0 N Γ = +∞ i=1 αi ψi |H0 N |ψi 18 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Strictly Correlated Electrons (SCE) limit of the Hohenberg-Kohn functional The SCE limit of the HK functional was first considered in the series of work: [Seidl,1999], [Seidl,Gori-Giorgi,Savin,2007] H0 N = T + c where T = − 1 2 N i=1 ∆xi and c(x1 , . . . , xN ) = 1≤i 0 and consider Fh HK [ρ] := inf {h ψ|T|ψ + ψ|c|ψ , ψ ∈ AN , ρψ = ρ} . SCE limit of the Hohenberg-Kohn functional: FSCE [ρ] = lim h→0 Fh HK (ρ) 19 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem SCE functional: a classical optimal transport problem If ψ ∈ AN , γψ := |ψ|2 is a symmetric probability measure on R3N , and ψ|c|ψ = R3N cγψ For all γ ∈ Psym (R3N ) symmetric probability measure on R3N , let ργ be its marginal ργ (x) = N (R3)N−1 γ(x, x2 , . . . , xN ) dx2 . . . dxN (ργψ = ρψ ) [Cotar,Friesecke, Kl¨ uppelberg, 2011], [Lewin, 2017], [Cotar,Friesecke, Kl¨ uppelberg, 2018] FSCE [ρ] = inf γ ∈ Psym (R3N ) ργ = ρ R3N cγ 20 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Recap’ Lieb functional FL [ρ] = inf Γ ∈ S+ 1 (HN 0 ) ρΓ = ρ Tr H0 N Γ • Γ mixed state (trace-class non-negative s.a. operator) • ρΓ = ρ: partial trace constraint • Cost functional: Tr H0 N Γ Quantum optimal transport SCE functional FSCE [ρ] = inf γ ∈ Psym (R3N ) ργ = ρ R3N cγ • γ symmetric probability measure on R3N • ργ = ρ: marginal constraint • Cost functional: R3N cγ Classical optimal transport Several recent efforts on the design of numerical schemes for the computation of the SCE functional. Much less (at least up to my knowledge) for the computation of the Lieb functional. 21 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Classical discretization: discrete state space Let y1, · · · , yM ∈ R3 and Y = {y1, · · · , yM } be a discretization grid of R3. Classical discretization (for the SCE problem) approaches consist in approximating the solution γ as a discrete measure defined on YN : γ ≈ 1≤i1,...,iN ≤M γi1,...,iN δ (yi1 ,...,yiN ) Linear problem of size MN ! Curse of dimensionality Complexity is even worse for classical discretizations of the Lieb problem 22 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Numerical methods for the SCE problem • [Benamou,Carlier,Cuturi,Nenna,Peyr´ e,2015], [Nenna,2016] : use of an entropic regularization (using the Kullback-Leibler entropy), together with an iterative algorithm called Sinkhorn algorithm. • [Mendl,Lin,2013] : dual formulation of the Kantorovich problem: needs appropriate treatment of the (infinite-dimensional) inequality constraint. • [V¨ ogler,2019],[Friesecke, Schulz, V¨ ogler,2021] : The Genetic column generation algorithm builds on the sparsity structure of minimizers of classical discretizations of the SCE problem • [Alfonsi, Coyaud, VE, Lombardi,2021], [Alfonsi, Coyaud, VE,2022]: Moment constraints discretization also leads to sparse minimizers [VE, Nenna,2023] Moment constraints discretization also leads to sparse minimizers for the Lieb functional 23 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Outline of the talk Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 24 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem (Simplifying) Assumptions • Assumption on ρ (which can be relaxed): there exists a compact subset Q ⊂ R3 such that Suppρ ⊂ Q. • Assumption on moment functions: (ϕm)m∈N∗ ⊂ C(R3) s.t. • ϕ1(x) = 1; • for all f ∈ C(Q), inf ξM ∈Span{ϕ1,...,ϕM } f − ξM L∞(Q) −→ M→+∞ 0. ∀m ∈ N∗, ρm := R3 ϕmρ the moment of ρ associated with the moment function ϕm 25 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Alternative discretization: moment constraints Let M ∈ N∗ be a discretization parameter. The marginal/partial trace constraint ργ = ρ (SCE case) ρΓ = ρ (Lieb case) (4) is replaced by the M moment constraints: for all 1 ≤ m ≤ M, R3 ϕmργ = ρm (SCE case) R3 ϕmρΓ = ρm (Lieb case) (5) FM SCE [ρ] = inf γ ∈ Psym (R3N ) ργ satisfies (5) R3N cγ FM L [ρ] = inf Γ ∈ S+ 1 (HN 0 ) ρΓ satisfies (5) Tr H0 N Γ 26 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Convergence and existence of minimizers [Alfonsi, Coyaud, VE, Lombardi, 2021], [VE, Nenna, 2023] Theorem For all M ∈ N∗, there exists at least one minimizer to both moment constraint optimal transport problems (SCE and Lieb case). In addition, it holds that FM SCE [ρ] −→ M→+∞ FSCE [ρ] and FM L [ρ] −→ M→+∞ FL [ρ] Links with: • Garrigue, 2022: particular set of moment constraints for the Lieb functional • Lelotte, 2022: dual charge approach: particular set of moment constraints for the SCE functional 27 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Tchakaloff’s theorem The following theorem is the backbone of our analysis to prove the existence of sparse minimizers. [Bayer,Teichmann,2006] Theorem Let µ be a non-negative Borel measure on a Hilbert space H concentrated on a Borel set B, i.e. µ(H \ B) = 0. Let M0 ∈ N∗ and Λ : H → RM0 be a continuous map. Assume that the first moments of Λ#µ exist i.e. H Λ(z) dµ(z) < +∞. Then, there exists an integer 1 ≤ K ≤ M0 , points z1, · · · , zK ∈ B and weights w1 , · · · , wK > 0 such that ∀1 ≤ m ≤ M0 , H Λm(z) dµ(z) = K k=1 wk Λm(zk ), where Λm denotes the mth component of Λ. 28 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Sparse structure of minimizers: SCE case Using the Tchakaloff’s theorem, [Bayer,Teichmann,2006] Theorem ([Alfonsi, Coyaud, VE, Lombardi, 2021]) There exists an integer 1 ≤ K ≤ M + 1, and for all 1 ≤ k ≤ K, points Xk ∈ (R3)N and weights wk > 0 such that the symmetrized measure associated to γ = K k=1 wk δXk (6) is a minimizer to FM SCE [ρ]. Complexity of this sparse representation: O (MN) 29 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Sparse structure of minimizers: Lieb case Using the Tchakaloff’s theorem, [Bayer,Teichmann,2006] Theorem ([VE, Nenna, 2023]) There exists an integer 1 ≤ K ≤ M + 1, and for all 1 ≤ k ≤ K, functions ψk ∈ AN and weights ωk > 0 such that Γ = K k=1 ωk |ψk ψk | (7) is a minimizer to FM L [ρ]. There exists at least a minimizer to FM L [ρ] which has rank at most M + 1. 30 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Back to the SCE case: particle and weight optimization problem Natural idea for a numerical method: Restrict the minimization set of problem FM SCE [ρ] to measures γ that can be written under the form (7) for some weights wk and points Xk . FM,K SCE [ρ] = inf Y := (Xk )1≤k≤K ⊂ (R3)N , W := (wk )1≤k≤K ⊂ R+, K k=1 wk = 1, ∀1 ≤ m ≤ M, K k=1 wk Φm(Xk ) = ρm K k=1 wk c(Xk ). (8) where ∀X = (x1 , · · · , xN ) ∈ (R3)N , Φm(X) := 1 N N i=1 ϕm(xi ). Non convex optimization problem under non convex constraints! ⇒ Stochastic gradient algorithm with constrained overdamped Langevin dynamics 31 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem SCE case: particle and weight optimization problem Minimization set: PK :=    (W, Y) ∈ RK + × ((R3)N )K , W := (wk )1≤k≤K , Y := (Xk )1≤k≤K , K k=1 wk = 1, ∀1 ≤ m ≤ M, K k=1 wk Φm(Xk ) = ρm    FM,K SCE [ρ] = inf (W,Y)∈PK J (W, Y), (9) with J (W, Y) := K k=1 wk c(Xk ). Theorem ([Alfonsi, Coyaud, VE, 2022]) If K ≥ 2M + 6, for any (W0 , Y0 ), (W1 , Y1 ) ∈ PK , there exists a continuous path ζ : [0, 1] → PK such that • ζ(0) = (W0 , Y0 ); • ζ(1) = (W1 , Y1 ); • [0, 1] t → J (ζ(t)) is monotonous. 32 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 1D Numerical results ρ1 ρ2 ρ3 1D numerical tests presented with N = 5. X = [−1, 1] and Legendre polynomial test functions. 33 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 1D Numerical results • blue curve: M = 10 • green curve: M = 20 • red curve: M = 40 34 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 1D Numerical results ρ1 M = 10 M = 20 M = 40 35 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 1D Numerical results ρ2 M = 20 M = 40 36 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 1D Numerical results ρ3 M = 20 M = 40 37 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem 3D numerical results N = 100, M = 52 (polynomial test functions), ρ (normalized) sum of six gaussian functions defined on R3 5 0 5 10 15 20 25 6 4 2 0 2 4 6 0.00 0.02 0.04 0.06 Density 0.00 0.05 0.10 0.15 0.20 Density 38 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Conclusion and perspectives Conclusions: • Alternative way of discretizing optimal transport problems from DFT with moment constraints: sparse minimizers • Numerical particle scheme for the approximation of the SCE functional: encouraging numerical results, but no mathematical analysis Perspectives: • Analysis of constrained stochastic gradient schemes (with high-dim point particles (SDEs) or functions (SPDEs)) • Numerical scheme for the approximation of the Lieb functional (work in progress with Luca Nenna) • Numerical scheme which allows for more moment functions • Choice of the moment functions for best rates of convergence PhD and postdoc positions available! ERC HighLEAP project High-dimensional mathematical methods for LargE Agent and Particle systems 39 / 40

Electronic structure calculations of molecules Density functional theory and optimal transport Moment constrained optimal transport problem Bibliography • M. Seidl. Strong-interaction limit of density-functional theory, Phys. Rev. A, 60, 4387-4395 (1999) • M. Seidl, P. Gori-Giorgi, A. Savin. Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities, Phys. Rev. A 75, 042511 1-12 (2007) • C. Cotar, G. Friesecke, C. Kl¨ uppelberg. Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math. 66, 548-599 (2013). • C. Codina, G. Friesecke, C. Kl¨ uppelberg. Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional, Archive for Rational Mechanics and Analysis (2018): 1-32. • E.H. Lieb. Density functionals for Coulomb systems, International Journal of Quantum Chemistry 24, 243-277 (1983) • M. Lewin. Semi-classical limit of the Levy-Lieb Functional in Density Functional Theory, arXiv 1706.02199 (2017) • C. Bayer, J. Teichmann. The proof of Tchakaloff’s theorem, Proceedings of the American mathematical society, 134(10):3035- 3040, (2006) • F. Santambrogio. Optimal transport for applied mathematicians, Birk¨ auser, NY, pages 99-102, (2015). • M. Cuturi and G. Peyr´ e. Computational Optimal Transport, https://arxiv.org/abs/1803.00567, (2019). • C. Mendl and L. Lin, Kantorovich dual solution for strictly correlated electrons in atoms and molecules, Phys. Rev. B 87, 125106, (2013). • Luca Nenna. Numerical methods for multi-marginal optimal transportation. PhD thesis, PSL Research University, 2016. • J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyr´ e. Iterative bregman projections for regularized transportation problems, SIAM Journal on Scientific Computing,37(2):A1111-A1138, (2015). • M. Beiglb¨ ock and M. Nutz, Martingale inequalities and deterministic counterparts, Electron. J. Probab., 19(95), (2014). • G. Friesecke, D. V¨ ogler, Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spaces, SIAM Journal on Mathematical Analysis 50.4 (2018): 3996-4019. • G. Friesecke, A.S. Schulz, D. V¨ ogler, Genetic column generation: Fast computation of high-dimensional multi-marginal optimal transport problems, arXiv 2021 • A. Alfonsi, R. Coyaud, V. Ehrlacher, D. Lombardi. Approximation of Optimal Transport problems with marginal moments contraints, Maths of Comp., (2019). • A. Alfonsi, R. Coyaud, V. Ehrlacher. Constrained overdamped Langevin dynamics for symmetric multimarginal optimal transportation, M3AS, (2022). 40 / 40