In a recent work, Bindini and De Pascale have introduced a regularization of
$N$-particle symmetric probabilities which preserves their one-particle
marginals. In this short note, we extend their construction to mixed quantum
fermionic states. This enables us to prove the convergence of the Levy-Lieb
functional in Density Functional Theory , to the corresponding multi-marginal
optimal transport in the semi-classical limit. Our result holds for mixed
states of any particle number $N$, with or without spin.Comment: Final version to appear in Comptes rendus de l'Acad{\'e}mie des
  Sciences, Math{\'e}matique

Lewin, Mathieu

Comptes Rendus Mathematique

English

arXiv

Mathieu Lewin

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical physics/Calculus of variationsSemi-classical limit of the Levy–Lieb functional in Density Functional TheoryLimite semi-classique de la fonctionnelle de Levy–Lieb en théorie de la fonctionnelle de densitéMathieu LewinCNRS & CEREMADE, Université Paris-Dauphine, PSL Research University, 75016 Paris, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 15 June 2017Accepted 5 March 2018Available online 12 March 2018Presented by the Editorial BoardIn a recent work, Bindini and De Pascale have introduced a regularization of N-particle symmetric probabilities that preserves their one-particle marginals. In this short note, we extend their construction to mixed quantum fermionic states. This enables us to prove the convergence of the Levy–Lieb functional in Density Functional Theory, to the corresponding multi-marginal optimal transport in the semi-classical limit. Our result holds for mixed states of any particle number N , with or without spin.© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).r é s u m éDans un travail récent, Bindini et de Pascale ont introduit une régularisation des probabilités symétriques décrivant N particules indiscernables, qui préserve la densité à une particule. Nous étendons ici leur construction aux états quantiques mixtes de fermions. Ceci nous permet de démontrer la convergence de la fonctionnelle de Levy–Lieb, objet central de la théorie de la fonctionnelle de densité (DFT), vers le transport optimal multi-marges associé, à la limite semi-classique. Notre résultat est valable pour les états mixtes de n’importe quel nombre de particules N , avec ou sans spin.© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).1. Extending the Bindini–De Pascale constructionLet P be a symmetric N-particle probability measure over (Rd)N and letρP(x1) =ˆ(Rd)N−1dP(x1, x2, ..., xN )E-mail address: mathieu .lewin @math .cnrs .fr.https://doi.org/10.1016/j.crma.2018.03.0021631-073X/© 2018 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).450 M. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455be its one-particle marginal. We assume that √ρP ∈ H1(Rd), as is appropriate in Density Functional Theory (DFT) [11,14]. An interesting question, important for applications in DFT, is to approximate P by a regular N-particle probability density Pε with the same density ρPε = ρP . A more challenging problem, considered first in [6,1] for N = 2, 3 and solved for all N  2 in this note, is to find a fermionic quantum state ε with the same density ρε = ρP and a controlled kinetic energy.Consider a radial function χ ∈ C∞c (Rd, R) with support in the unit ball of Rd , such that ´Rdχ2 = 1, and denote χε(x) =ε−d/2χ(x/ε). In [1], Bindini and De Pascale have introduced the following elegant regularizationPε(x1, ..., xN ) =¨R2dNN∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)dP(y1, ..., yN) dz1 · · ·dzN . (1.1)We assume in the following that P has its support inDcα :={X = (x1, ..., xN ) ∈ (Rd)N : |xi − x j|  α, ∀i = j},for some α > 0, a condition which is satisfied for minimizers of the Coulomb N-particle energy [2]. Since χ is supported in the unit ball, Pε is then supported on the set Dcα−4ε , where all the particles are at a distance α − 4ε to each other. In the following, we always assume that ε < α/4. The purpose of the integration over the zk ’s in (1.1) is to regularize the probability P, since the convolutionQε(y1, ..., yN) =ˆ(Rd)NN∏k=1χε(zk − yk)2dP(y1, ..., yN)is now C∞ . However, its density is ρQε = ρP ∗ χ2, and the purpose of the integration over the yk ’s is to map back the density to ρP . Indeed, integrating (1.1) over x2, ..., xN , we get ∏Nk=2 ρP ∗ χ2ε (yk) in the numerator, which cancels with the denominator. The corresponding integrals over z2, ..., zN give (´χ2ε )N−1 = 1 and we end up withρPε (x1) = ρP(x1)ˆRdˆRdχε(x1 − z1)2χε(z1 − y1)2ρP ∗ χ2ε (z1)ρP(y1)dy1 dz1 = ρP(x1).A somewhat different method was introduced in [6] by Cotar, Friesecke and Kluppelberg.In [1], Bindini and De Pascale prove thatˆRdN|∇√Pε|2  N⎛⎜⎝ˆRd|∇√ρP|2 + 1ε2ˆRd|∇χ |2⎞⎟⎠ (1.2)and use this to get some information on the semi-classical limit of the Levy–Lieb functional (to be discussed later in Section 2). Unfortunately, the probability (1.1) is really a classical object. Although for bosons one can use the symmetric wavefunction ε = √Pε , for fermions the wavefunction must be anti-symmetric. In [6,1], the authors use the spin variable to antisymmetrize √Pε but, in dimension d = 3, this has so far limited the results to N = 2 and N = 3. Note that in dimension d = 1 and d = 2, one can use the multiplication by∏1 j<kNx j − xk|x j − xk|which maps bosons onto fermions and conversely, whatever the value of N (if d = 2 we identify R2 with C). By the Borsuk–Ulam theorem, there is no such simple map in three space dimensions.In this short note, we propose the following simple quantum extension of (1.1), for fermions:ε =¨RdN×RdN√ρP⊗N |χε,z1 ∧ · · · ∧ χε,zN 〉〈χε,z1 ∧ · · · ∧ χε,zN |√ρP⊗N×N∏k=1χε(zk − yk)2ρP ∗ χ2ε (zk)dP(y1, ..., yN) dz1 · · · dzN . (1.3)Here ε is a non-negative self-adjoint operator that acts on the N-particle fermionic space L2a ((Rd)N ) =∧N1 L2(Rd). For simplicity, we do not consider spin here. It is easy to extend the trial state (1.3) to the case of particles with q spin states, for instance by putting all the particles in the same spin state. One can also construct any desired eigenvector of the total spin operator by adding an appropriate spin state to each function χε,z in (1.3).kM. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455 451In (1.3), we use the notation χε,z(x) = χε(x − z) and recall that ε < α/4 such that the functions χε,z1 , ..., χε,zN have disjoint supports. We use the ket-bra notation |〉〈| for the orthogonal projection on , defined by |〉〈|ϕ = 〈,ϕ〉. We callϕ1 ∧ · · · ∧ ϕN(x1, ..., xN ) = 1√N!∑σ∈SNε(σ )ϕσ(1)(x1) · · ·ϕσ(N)(xN )the Slater determinant. Finally, the N-fold tensor product is defined by√ρP⊗N(x1, ..., xN ) =N∏j=1√ρP(x j).In (1.3), √ρP⊗N is understood as a multiplication operator on L2a ((Rd)N ). In particular, we have√ρP⊗N |χε,z1 ∧ · · · ∧ χε,zN 〉〈χε,z1 ∧ · · · ∧ χε,zN |√ρP⊗N= |√ρPχε,z1 ∧ · · · ∧√ρPχε,zN 〉〈√ρPχε,z1 ∧ · · · ∧√ρPχε,zN |.The integral kernel of ε is given byε(x1, ..., xN ; x′1, ..., x′N ) =1N!¨RdN×RdNdet(χε,zi (x j))det(χε,zi (x′j))×N∏k=1√ρP(xk)√ρP(x′k)χ(zk − yk)2ρP ∗ χ2(zk) dP(y1, ..., yN) dz1 · · · dzN . (1.4)Since the χε,zk have disjoint supports, we have(det(χε,zi (x j)))2 = ∑σ ,σ ′∈SNε(σ )ε(σ ′)N∏k=1χε,zσ (k) (xk)χε,zσ ′(k) (xk) =∑σ∈SNN∏k=1χε(xk − zσ (k))2. (1.5)Using the symmetry of P, we haveˆRdNN∏k=1χε(zk − yk)2dP(y1, ..., yN) =ˆRdNN∏k=1χε(zσ (k) − yk)2dP(y1, ..., yN)for every σ ∈ SN . Therefore, in (1.5) the terms in the sum over the permutations σ ∈ SN all contribute the same amount. We thus find that the diagonal of ε coincides with the Bindini–De Pascale probability density:ε(x1, ..., xN ; x1, ..., xN ) = Pε(x1, ..., xN ).From this, we conclude thatTr (ε) =ˆ(Rd)Nε(x1, ..., xN ; x1, ..., xN )dx1 · · ·dxN =ˆRdNdPε(x1, ..., xN ) = 1,and ε is a proper fermionic (mixed) state. We recall that the one-particle density of ε is defined by duality, requiring that Tr (ϕ(x1)ε) =´Rdϕ ρε for every ϕ ∈ L∞(Rd). For a continuous kernel such as ε , we haveρε (x1) =ˆRdNε(x1, ..., xN ; x1, ..., xN )dx2 · · ·dxN = ρPε (x1) = ρP(x1).From the Cauchy–Schwarz inequality, we haveˆ(Rd)N|∇√Pε|2  Tr (−)ε,which is in the wrong direction to conclude anything about the kinetic energy of ε . But we can prove the following theorem, which implies the Bindini–De Pascale estimate (1.2).452 M. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455Theorem 1 (Estimates on ε). Let P be a symmetric N-particle density with support in Dcα for some α > 0 and such that √ρP ∈H1(Rd). Let ε be defined by (1.3). Then, for ε < α/4, we haveTr (−)ε = N⎛⎜⎝ˆRd|∇√ρP|2 + 1ε2ˆRd|∇χ |2⎞⎟⎠ . (1.6)In addition, for every symmetric function  ∈ C2(Dcα−4ε), we have∣∣∣∣∣∣∣Tr (ε) −ˆRdNdP∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣ˆRdNdPε −ˆRdNdP∣∣∣∣∣∣∣ ε2{N∑j=1∣∣∣∣∇ j∣∣∣∣L∞(Dcα−4ε)⎛⎜⎝ˆRd|∇ρP|⎞⎟⎠⎛⎜⎝ˆRd|u|2χ(u)2 du⎞⎟⎠ + 2 N∑j,k=1∣∣∣∣∇ j∇k∣∣∣∣L∞(Dcα−4ε)}. (1.7)In particular, Pε ⇀P, as was proved in [1].Proof. We haveTr (−)ε =N∑=1¨RdN×RdN⎛⎜⎝ ˆRdN∣∣∣∇x (√ρP⊗Nχε,z1 ∧ · · · ∧ χε,zN )∣∣∣2⎞⎟⎠ N∏k=1χε(zk − yk)2ρP ∗ χ2ε (zk)dP(y1, ..., yN) dz1 · · ·dzN .Similarly as in (1.5),∣∣∇x (√ρPχε,z1 ∧ · · · ∧ √ρPχε,zN )∣∣2 = 1N!∑σ∈SN∣∣∇(√ρPχε,zσ ())(x)∣∣2 ∏k=1,...,Nk =ρP(xk)χε,zσ (k) (xk)2.Integrating over x1, ..., xN , we obtainN∑=1ˆRdN∣∣∣∇x (√ρP⊗Nχε,z1 ∧ · · · ∧ χε,zN )∣∣∣2 =N∑j=1⎛⎜⎝ˆRd∣∣∇(√ρPχε,z j )(x)∣∣2 dx⎞⎟⎠ ∏k=1,...,Nk = jρP ∗ χ2ε (z j).Finally, integrating over the yk ’s and zk ’s, we conclude thatTr (−)ε = NˆRdˆRd∣∣∇(√ρPχε,z)(x)∣∣2 dx dz = N⎛⎜⎝ˆRd|∇√ρP(x)|2 dx + 1ε2ˆRd|∇χ(x)|2 dx⎞⎟⎠ .To prove (1.7), we remark that¨R2dNN∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)dx1 · · ·dxN dz1 · · ·dzN = 1.Hence, using the shorter notation X = (x1, ..., xN ), Y = (y1, ..., yN) and Z = (z1, ..., zN ), we haveˆRdNdPε −ˆRdNdP =¨R3dN((X) − (Y )) N∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)dP(Y ) dX dZ .Since |x j − y j |  2ε due to the support of χ , we conclude from the fundamental theorem of calculus that∣∣∣∣ˆRdNdPε −ˆRdNdP−¨R3dNdP(Y )dX dZ(N∑=1∇(Y ) · (x − y))××N∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)∣∣∣∣  2ε2N∑,m=1∣∣∣∣∇x∇xm∣∣∣∣L∞(Dcα−4ε) .M. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455 453It remains to estimate the term involving ∇. By symmetry, it is sufficient to look at the case  = 1, which can be expressed in the form¨R3dN∇1(Y ) · (x1 − y1)N∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)dP(Y )dX dZ=¨RdN+2d∇1(Y ) · (x1 − y1)ρP(x1)χε(x1 − z1)2χε(z1 − y1)2ρP ∗ χ2ε (z1)dP(Y )dx1 dz1.Here we can replace x1 − y1 by x1 − z1 since ´Rd(z1 − y1)χε(z1 − y1)2 dz1 = 0, due to the symmetry of χ . We then estimate∣∣∣∣¨R3dN∇1(Y ) · (x1 − z1)ρP(x1)χε(x1 − z1)2χε(z1 − y1)2ρP ∗ χ2ε (z1)dP(Y )dx1 dz1∣∣∣∣ ||∇1||L∞(Dcα−4ε)¨R2d∣∣∣∣∣∣∣ˆRd(x1 − z1)ρP(x1)χε(x1 − z1)2 dx1∣∣∣∣∣∣∣χε(z1 − y1)2ρP ∗ χ2ε (z1)ρP(y1)dz1 dy1= ||∇1||L∞(Dcα−4ε)ˆRd∣∣∣∣∣∣∣ˆRd(x − z)ρP(x)χε(x − z)2 dx∣∣∣∣∣∣∣ dz.By the symmetry of χ and the fundamental theorem of calculus, we have∣∣∣∣∣∣∣ˆRd(x − z)ρP(x)χε(x − z)2 dx∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣ˆRd(x − z)(ρP(x) − ρP(z))χε(x − z)2 dx∣∣∣∣∣∣∣ 1ˆ0ˆRd|∇ρP(z + tu)| |u|2χε(u)2 du.Integrating over z, we find the claimed estimate∣∣∣∣¨R3dN∇1(Y ) · (x1 − y1)N∏k=1ρP(xk)χε(xk − zk)2χε(zk − yk)2ρP ∗ χ2ε (zk)dP(Y )dX dZ∣∣∣∣ ε2 ||∇1||L∞(Dcα−4ε)⎛⎜⎝ˆRd|∇ρP|⎞⎟⎠⎛⎜⎝ˆRd|u|2χ(u)2 du⎞⎟⎠ . 2. Semiclassical limit of the Levy–Lieb functionalHere we restrict ourselves for simplicity to the physical space R3 and the Coulomb potential. Density Functional Theory is based on the following functional [13,14,4]E(ρ) = min=∗0Tr ()=1ρ=ρTr⎛⎝− N∑j=1x j +∑1 j<kN1|x j − xk|⎞⎠, (2.1)of the density ρ , a given non-negative function such that ´R3ρ = N and √ρ ∈ H1(R3). In the minimum  is an operator acting on the fermionic space ∧N1 L2(R3). Motivated by arguments of Hohenberg and Kohn [12], Levy introduced in [13]a functional similar to (2.1), but with the additional constraint that  = |〉〈| is a rank-one orthogonal projection (pure state). The latter was rigorously studied by Lieb in [14], who proposed to extend the definition to mixed states, as in (2.1). The minimum over mixed states (2.1) has better mathematical properties than with pure states [14]. For instance, E is convex and, by the linearity in , we have the dual formulationE(ρ) = sup{ˆ3ρ(x) V (x)dx : V ∈ L3/2(R3,R) + L∞(R3,R),−N∑j=1x j −N∑j=1V (x j)) +∑1 j<kN1|x j − xk|  0},R454 M. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455which is the quantum equivalent of the Kantorovich duality used in optimal transport [16]. The last inequality is in the sense of self-adjoint operators.It is possible to introduce an effective semi-classical parameter η = h̄2 by scaling the density ρ . Namely, for ρη(x) =η3ρ(ηx), we haveE(ρη)η= min=∗0Tr ()=1ρ=ρTr⎛⎝−η N∑j=1x j +∑1 j<kN1|x j − xk|⎞⎠. (2.2)In the limit η → 0, we prove the convergence to the Coulomb multi-marginal optimal transport problemEOT(ρ) = minP symmetricprobability on (R3)NρP=ρˆ(R3)N∑1 j<kN1|x j − xk|dP(x1, ..., xN ), (2.3)which has recently received a lot of attention [3,6,7,5,8,15]Theorem 2 (Semi-classical limit). Let N  2 and let ρ  0 be such that ́R3ρ = N and √ρ ∈ H1(R3). Then we have for a constant C(depending on N and ρ)EOT(ρ) E(η3ρ(η·))η EOT(ρ) + C(√η + η). (2.4)In particular,limη→0E(η3ρ(η·))η= EOT(ρ).This theorem generalizes the results in [6,1] for N = 2, 3 in the pure state case. Results for N  3 have been announced in [9, Ref. 7], but they were not yet available at the time this note was written. It would be interesting to extend our findings to pure states.Semi-classical analysis suggests that the behavior in √η is optimal for small η. The next order (in √η = h̄) in the expansion of E(ρη)/η was predicted in [10].Proof. Let P be an optimizer for EOT(ρ). It has been shown in [2] that P has its support on Dcα for some α > 0. Taking then our ε as a trial state, we find, by Theorem 1E(ρη)η ηN⎛⎜⎝ˆRd|∇√ρ|2 + 1ε2ˆRd|∇χ |2⎞⎟⎠ + EOT(ρ) + Cε2⎛⎜⎝ N3(α − 4ε)2ˆRd|∇ρP|ˆRd|u|2χ(u)2 du + N4(α − 4ε)3⎞⎟⎠ .We have used here that (X) = ∑1 j<kN |x j − xk|−1 is C∞ on Dcα−4ε . Optimizing in ε gives the result. Remark 1. The convergence of states in the limit η → 0 is proved as in [1].AcknowledgementsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement MDFT No. 725528).References[1] U. Bindini, L. De Pascale, Optimal transport with Coulomb cost and the semiclassical limit of Density Functional Theory, ArXiv e-prints, 2017.[2] G. Buttazzo, T. Champion, L. De Pascale, Continuity and estimates for multimarginal optimal transportation problems with singular costs, Appl. Math. Optim. (2017), https://doi .org /10 .1007 /s00245 -017 -9403 -7.[3] G. Buttazzo, L. De Pascale, P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory, Phys. Rev. A 85 (2012) 062502.[4] É. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris, Y. Maday, Computational Quantum Chemistry: A Primer, Handbook of Numerical Analysis, vol. X, North-Holland, Amsterdam, 2003, pp. 3–270.[5] M. Colombo, S. Di Marino, Equality between Monge and Kantorovich multimarginal problems with Coulomb cost, Ann. Mat. Pura Appl. (4) 194 (2015) 307–320.[6] C. Cotar, G. Friesecke, C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Commun. Pure Appl. Math. 66 (2013) 548–599.M. Lewin / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 449–455 455[7] C. Cotar, G. Friesecke, B. Pass, Infinite-body optimal transport with Coulomb cost, Calc. Var. Partial Differ. Equ. 54 (2015) 717–742.[8] S. Di Marino, A. Gerolin, L. Nenna, Optimal Transportation Theory with Repulsive Costs, ArXiv e-prints, 2015.[9] G. Friesecke, C.B. Mendl, B. Pass, C. Cotar, C. Klüppelberg, N-density representability and the optimal transport limit of the Hohenberg–Kohn functional, J. Chem. Phys. 139 (2013) 164109.[10] P. Gori-Giorgi, G. Vignale, M. Seidl, Electronic zero-point oscillations in the strong-interaction limit of density functional theory, J. Chem. Theory Comput. 5 (2009) 743–753.[11] J.E. Harriman, Orthonormal orbitals for the representation of an arbitrary density, Phys. Rev. A 24 (1981) 680–682.[12] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964) B864–B871.[13] M. Levy, Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem, Proc. Natl. Acad. Sci. USA 76 (1979) 6062–6065.[14] E.H. Lieb, Density functionals for Coulomb systems, Int. J. Quant. Chem. 24 (1983) 243–277.[15] M. Seidl, S. Di Marino, A. Gerolin, L. Nenna, K.J.H. Giesbertz, P. Gori-Giorgi, The strictly-correlated electron functional for spherically symmetric systems revisited, ArXiv e-prints, 2017.[16] C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 338, Springer-Verlag, Berlin, 2009.

Semi-classical limit of the Levy–Lieb functional in Density Functional Theory

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Semi-classical limit of the Levy-Lieb functional in Density Functional Theory

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