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In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 300 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 776 entries, the paper represents a broad snapshot of DFT, anno 2022.

Consider the Coulomb potential $-\mu\ast|x|^{-1}$ generated by a non-negative finite measure $\mu$. It is well known that the lowest eigenvalue of the corresponding Schr\"odinger operator $-\Delta/2-\mu\ast|x|^{-1}$ is minimized, at fixed mass $\mu(\mathbb{R}^3)=\nu$, when $\mu$ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator $-i\alpha\cdot\nabla+\beta-\mu\ast|x|^{-1}$. In a previous work on the subject we proved that this operator is self-adjoint when $\mu$ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass $\nu_1$, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all $\mu\geq0$ with $\mu(\mathbb{R}^3), Comment: Final version to appear in Proc. London Math. Soc

The Lieb-Oxford inequality provides a lower bound on the Coulomb energy of a classical system of $N$ identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree-Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy., Final version to appear in Lett. Math. Phys. This article belongs to the themed collection: "Mathematical Physics and Numerical Simulation of Many-Particle Systems"; V. Bach and L. Delle Site (eds.)

We review what is known, unknown and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in $\mathbb{R}^d$ interacting with the Riesz potential $\pm |x|^{-s}$ (resp. $-\log|x|$ for $s=0$). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g. in random matrix theory). The main question addressed in the article is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case $sd$. In the last part we discuss phase transitions and mention what is expected on physical grounds., Comment: Final version to appear in J. Math. Phys. in the special collection of papers honoring Freeman Dyson

We provide upper and lower bounds on the lowest free energy of a classical system at given one-particle density $\rho(x)$. We study both the canonical and grand-canonical cases, assuming the particles interact with a pair potential which decays fast enough at infinity., Comment: Final version to appear in the Journal of Statistical Physics

In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb-Thirring constant when the eigenvalues of a Schr\"odinger operator $-\Delta+V(x)$ are raised to the power $\kappa$ is never given by the one-bound state case when $\kappa>\max(0,2-d/2)$ in space dimension $d\geq1$. When in addition $\kappa\geq1$ we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo-Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb-Thirring inequality, in the same spirit as in Part I of this work (D. Gontier, M. Lewin & F.Q. Nazar, arXiv:2002.04963). In a different but related direction, we also show that the cubic nonlinear Schr\"odinger equation admits no orthonormal ground state in 1D, for more than one function., Comment: Includes some new properties of the one-bound state (Gagliardo-Nirenberg) constant

We study the nonlinear Schr\"odinger equation for systems of $N$ orthonormal functions. We prove the existence of ground states for all $N$ when the exponent $p$ of the non linearity is not too large, and for an infinite sequence $N_j$ tending to infinity in the whole range of possible $p$'s, in dimensions $d\geq1$. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn-Sham model with a large Dirac exchange constant., Comment: Final version, to appear in Arch. Rat. Mech. Anal

We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schr{\"o}dinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose-Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances., Comment: This revised version (to appear in Inventiones Mathematicae) covers both the 2D and 3D cases. It replaces an older 2018 work that was limited to 2D. The older, non-refereed, version is accessible as v1-v2 of the preprint

In quantum field theory, the vacuum is a fluctuating medium which behaves as a nonlinear polarizable material. In this article, we perform the first rigorous derivation of the magnetic Euler-Heisenberg effective energy, a nonlinear functional that describes the effective fluctuations of the quantum vacuum in a classical magnetic field. We start from a classical magnetic field in interaction with a quan-tized Dirac field in its ground state, and we study a limit in which the classical magnetic field is slowly varying. After a change of scales, this is equivalent to a semi-classical limit $\hbar\to0$, with a strong magnetic field of order $1/\hbar$. In this regime, we prove that the energy of Dirac's polarized vacuum converges to the Euler-Heisenberg functional. The model has ultraviolet divergences, which we regularize using the Pauli-Villars method. We also discuss how to remove the regularization of the Euler-Heisenberg effective Lagrangian, using charge renormaliza-tion, perturbatively to any order of the coupling constant., Final version to appear in J. Math. Pure Appl

International audience; We give the first derivation of the Scott correction in the large-$Z$ expansion of the energy of an atom in Dirac-Fock theory without projections.

We study isomerizations in quantum mechanics. We consider a neutral molecule composed of N quantum electrons and M classical nuclei and assume that the first eigenvalue of the corresponding N-particle Schr\"odinger operator possesses two local minima with respect to the locations of the nuclei. An isomerization is a mountain pass problem between these two local configurations, where one minimizes over all possible paths the highest value of the energy along the paths. Here we state a conjecture about the compactness of min-maxing sequences of such paths, which we then partly solve in the particular case of a molecule composed of two rigid sub-molecules that can move freely in space. More precisely, under appropriate assumptions on the multipoles of the two molecules, we are able to prove that the distance between them stays bounded during the whole chemical reaction. We obtain a critical point at the mountain pass level, which is called a transition state in chemistry. Our method requires to study the critical points and the Morse indices of the classical multipole interactions, as well as to improve existing results about the van der Waals force. This paper generalizes previous works by the second author in several directions., Comment: Final version to appear in Arch. Rat. Mech. Anal

International audience; We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $$\Delta u+g(u)=0$$ . Our result applies in particular to the double power non-linearity where $$g(u)=u^q-u^p-\mu u$$ for $$p>q>1$$ and $$\mu >0$$ , which we discuss with more details. In this case, the non-degeneracy of the unique solution $$u_\mu $$ allows us to derive its behavior in the two limits $$\mu \rightarrow 0$$ and $$\mu \rightarrow \mu _*$$ where $$\mu _*$$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $$L^2$$ mass of $$u_\mu $$ in terms of $$\mu $$ , which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $$u_\mu $$ .

We study a system of $N$ interacting fermions at positive temperature in a confining potential. In the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension, we prove the convergence to the corresponding Thomas-Fermi model at positive temperature., Convergence of states rewritten. Some references added

We review our recent result on the rigorous derivation of the renormalized Gibbs measure from the many-body Gibbs state in 1D and 2D. The many-body renormalization is accomplished by simply tuning the chemical potential in the grand-canonical ensemble, which is analogous to the Wick ordering in the classical field theory., Contribution to Proceedings of the International Congress of Mathematical Physics, Montreal, Canada, July 23-28, 2018

Cet expose presente des resultats recents quant a deux modeles effectifs pour les effets de la polarisation du vide quantique sur les champs electromagnetiques. Le modele de Pauli-Villars les decrit de maniere rigoureuse lorsque les champs electromagnetiques sont supposes classiques. A partir de ce premier modele est ensuite proposee une derivation du modele classique d'Euler-Heisenberg dans un regime de champs purement magnetiques et faiblement variables.

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas., Extended version. Includes a new estimate on the critical temperature

We consider a Dirac operator in three space dimensions, with an electrostatic (i.e. real-valued) potential $V(x)$, having a strong Coulomb-type singularity at the origin. This operator is not always essentially self-adjoint but admits a distinguished self-adjoint extension $D\_V$. In a first part we obtain new results on the domain of this extension, complementing previous works of Esteban and Loss. Then we prove the validity of min-max formulas for the eigenvalues in the spectral gap of $D\_V$, in a range of simple function spaces independent of $V$. Our results include the critical case $\liminf\_{x \to 0} |x| V(x)= -1$, with units such that $\hbar=mc^2=1$, and they are the first ones in this situation. We also give the corresponding results in two dimensions., Comment: Final version to appear in Rev. Mat. Iberoam

We give the first mathematically rigorous justification of the Local Density Approximation in Density Functional Theory. We provide a quantitative estimate on the difference between the grand-canonical Levy-Lieb energy of a given density (the lowest possible energy of all quantum states having this density) and the integral over the Uniform Electron Gas energy of this density. The error involves gradient terms and justifies the use of the Local Density Approximation in the situation where the density is very flat on sufficiently large regions in space., Comment: Final version to appear in Pure and Applied Analysis

We modify the "floating crystal" trial state for the classical Homogeneous Electron Gas (also known as Jellium), in order to suppress the boundary charge fluctuations that are known to lead to a macroscopic increase of the energy. The argument is to melt a thin layer of the crystal close to the boundary and consequently replace it by an incompressible fluid. With the aid of this trial state we show that three different definitions of the ground state energy of Jellium coincide. In the first point of view the electrons are placed in a neutralizing uniform background. In the second definition there is no background but the electrons are submitted to the constraint that their density is constant, as is appropriate in Density Functional Theory. Finally, in the third system each electron interacts with a periodic image of itself, that is, periodic boundary conditions are imposed on the interaction potential., Includes some final corrections. Version to appear in Phys. Rev. B

We study a system of $N$ fermions in the regime where the intensity of the interaction scales as $1/N$ and with an effective semi-classical parameter $\hbar=N^{-1/d}$ where $d$ is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas-Fermi minimizers in the limit $N\to\infty$. The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti-Hewitt-Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity., Final version published in Calculus of Variations and Partial Differential Equations (2018)

National audience; Cette étude tente de dresser une cartographie thématique des mathématiques universitaires en France, définies ici comme l'ensemble des électeurs des deux sections CNU 25 et 26. Basée sur les publications référencées dans MathSciNet et la classification MSC, elle met en évidence la répartition des forces entre les différents domaines de recherche, et compare ces résultats avec le découpage en deux sections CNU. L'approche retenue permet aussi de déterminer la part des chercheurs et enseignants-chercheurs travaillant sur les "applications des mathématiques" et de réaliser une cartographie similaire pour les recrutements des CR au CNRS pendant la période 2005-2016.

In this paper we define and study the classical Uniform Electron Gas (UEG), a system of infinitely many electrons whose density is constant everywhere in space. The UEG is defined differently from Jellium, which has a positive constant background but no constraint on the density. We prove that the UEG arises in Density Functional Theory in the limit of a slowly varying density, minimizing the indirect Coulomb energy. We also construct the quantum UEG and compare it to the classical UEG at low density., Comment: Final version to appear in J. Ec. polytech. Math

Conference in honor of Herbert Spohn 70th birthday, Munich, Germany, March 20 – April 1, 2017; International audience; We consider the Gross-Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation $\Omega$. First we study the behavior of the ground state when the coupling constant approaches $a_*$ , the critical strength of the cubic nonlinearity for the focusing nonlinear Schrödinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo-Nirenberg solution. In particular, the blow-up scenario is independent of $\Omega$, to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141–156) in the non-rotating case. In a second part we consider the many-particle Hamiltonian for $N$ bosons, interacting with a potential rescaled in the mean-field manner $−a_N N^{2\beta−1} w(N^{\beta} x), with $w$ a positive function such that $\int_{\mathbb{R}^2} w(x) dx = 1$. Assuming that $\beta < 1/2$ and that $a_N \to a_*$ sufficiently slowly, we prove that the many-body system is fully condensed on the Gross-Pitaevskii ground state in the limit $N \to \infty$.

In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on $\mathbb{R}^{3N}$ where $N$ is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied., Final version to appear in EMS Surv. Math. Sci

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., Final version to appear in Comptes rendus de l'Acad{\'e}mie des Sciences, Math{\'e}matiques

We show local and global well-posedness results for the Hartree equation $$i\partial_t\gamma=[-\Delta+w*\rho_\gamma,\gamma],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2(\R^d)$, $\rho_\gamma(x)=\gamma(x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma(0)$ is assumed to be a perturbation of a translation-invariant state $\gamma_f=f(-\Delta)$ which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state $\gamma(t)$, counted relatively to the stationary state $\gamma_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\Delta)$. Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article., Comment: to appear in Comm. Math. Phys

International audience; We study the long-time behavior of the 3-dimensional repulsive nonlinear Hartree equation with an external attractive Coulomb potential $-Z/|x|$, which is a nonlinear model for the quantum dynamics of an atom. We show that, after a sufficiently long time, the average number of electrons in any finite ball is always smaller than 4Z (respectively 2Z in the radial case). This is a time-dependent generalization of a celebrated result by E.H. Lieb on the maximum negative ionization of atoms in the stationary case. Our proof involves a novel positive commutator argument (based on the cubic weight $|x|^3$) and our findings are reminiscent of the RAGE theorem. In addition, we prove a similar universal bound on the local kinetic energy. In particular, our main result means that, in a weak sense, any solution is attracted to a bounded set in the energy space, whatever the size of the initial datum. Moreover, we extend our main result to Hartree--Fock theory and to the linear many-body Schrödinger equation for atoms.

In this article, we set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered crystals. The electrons are quantum particles and the nuclei are classical point-like articles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density of such disordered crystals, for the reduced Hartree-Fock model (rHF). We consider both (short-range) Yukawa and (long-range) Coulomb interactions. In the former case, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model for disordered crystals is the thermodynamic limit of the supercell model.

We study vacuum polarisation effects of a Dirac field coupled to an external scalar field and derive a semi-classical expansion of the regu-larised vacuum energy. The leading order of this expansion is given by a classical formula due to Chin, Lee-Wick and Walecka, for which our result provides the first rigorous proof. We then discuss applications to the non-relativistic large-coupling limit of an interacting system, and to the stability of homogeneous systems., Revised version to appear in AHP (DOI: 10.1007/s00023-016-0472-y)

International audience; We investigate when Taylor expansions can be used to prove the Runge-Gross Theorem, which is at the foundation of Time-Dependent Density Functional Theory (TDDFT). We start with a general analysis of the conditions for the Runge-Gross argument, especially the time-differentiability of the density. The latter should be questioned in the presence of singular (e.g. Coulomb) potentials. Then, we show that a singular potential in a one-body operator considerably decreases the class of time-dependent external potentials to which the original argument can be applied. A two-body singularity has an even stronger impact and an external potential is essentially incompatible with it. For the Coulomb interaction and all reasonable initial many-body states, the Taylor expansion only exists to a finite order, except for constant external potentials. Therefore, high-order Taylor expansions are not the right tool to study atoms and molecules in TDDFT.

In a recent paper published in Nonlinear Analysis: Theory, Methods & Applications, Argaez and Melgaard have studied excited states for pseudo-relativistic multi-configuration methods. Their paper follows a previous work of mine in the non-relativistic case [M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry, Arch. Ration. Mech. Anal. 171 (1) (2004) 83–114. doi:10.1007/s00205003-0281-6]. The main results of the paper of Argaez and Melgaard are correct, but the proofs are both wrong and incomplete.

We prove that Gibbs measures based on 1D defocusing nonlinear Schr{\"o}dinger functionals with sub-harmonic trapping can be obtained as the mean-field/large temperature limit of the corresponding grand-canonical ensemble for many bosons. The limit measure is supported on Sobolev spaces of negative regularity and the corresponding density matrices are not trace-class. The general proof strategy is that of a previous paper of ours, but we have to complement it with Hilbert-Schmidt estimates on reduced density matrices., Comment: Minor changes and precisions