## Section: New Results

### Electronic structure calculations and related problems

Participants : Robert Benda, Éric Cancès, Virginie Ehrlacher, Luca Gorini, Gaspard Kemlin, Claude Le Bris, Antoine Levitt, Sami Siraj-Dine, Gabriel Stoltz.

#### Mathematical analysis

The members of the team have continued their systematic study of the properties of materials in the reduced Hartree-Fock (rHF) approximation, a model striking a good balance between mathematical tractability and the ability to reproduce qualitatively complex effects.

In collaboration with L. Cao, E. Cancès and G. Stoltz have studied the nuclear dynamics of infinite crystals with local defects within the Born-Oppenheimer approximation, using the reduced Hartree-Fock model to compute the electronic ground state. In this model, nuclei obey an autonomous classical Hamiltonian dynamics on a potential energy surface obtained by rHF electronic ground-state calculations. One of the main motivations for this work is to study the *nonlinear* collective excitations of nuclei in a crystal, in order to go beyond the simple harmonic approximation of non-interacting phonons. rHF ground states associated with generic nuclear displacements with respect to the periodic configuration are not mathematically well-defined at the time of writing. However, by relying on results by Cancès, Deleurence and Lewin for the rHF ground states of crystals with local defects, it is possible to study the fully nonlinear rHF Born-Oppenheimer dynamics of nuclei in the neighborhood of an equilibrium periodic configuration of a crystal. A Hilbert space of admissible nuclear displacements, and an infinite-dimensional Hamiltonian describing the dynamics of nuclei can then be defined. For small initial data, it is proved that the Cauchy problem associated with this Hamiltonian dynamics is well posed for short times (see the PhD thesis of Lingling Cao). The existence and uniqueness for arbitrary initial data, and/or long times requires a perturbation analysis of the rHF model when the Fermi level is occupied, which is work in progress.

#### Numerical analysis

E. Cancès has pursued his long-term collaboration with Y. Maday
(Sorbonne Université) on the numerical analysis of linear and
nonlinear eigenvalue problems. Together with G. Dusson (Besançon), B. Stamm (Aachen,
Germany), and M. Vohralìk (Inria SERENA),
they have designed *a posteriori* error estimates for conforming
numerical approximation of eigenvalue clusters of second-order
self-adjoint operators on bounded domains [44]. Given a cluster of
eigenvalues, they have estimated the error in the sum of the eigenvalues, as
well as the error in the eigenvectors represented through the density
matrix, i.e. the orthogonal projector on the associated eigenspace.
This allows them to deal with degenerate (multiple) eigenvalues within
this framework. The bounds are guaranteed and converge at the same rate as the exact error. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is notably the case (i) for the Laplace eigenvalue problem discretized with conforming finite elements, and (ii) for a Schrödinger operator with periodic boundary conditions discretized with plane waves.

R. Benda, E. Cancès and B. Lebental (Ecole Polytechnique) have initiated the design and analysis of multiscale models for the electrical conductivity of networks of functionalized carbon nanotubes. Such devices are used as nanosensors, for instance to monitor the quality of water. In [11], they study by means of Monte-Carlo numerical simulations the resistance of two-dimensional random percolating networks of stick, widthless nanowires. They use the multi-nodal representation (MNR) to model a nanowire network as a graph. They derive numerically from this model the expression of the total resistance as a function of all meaningful parameters, geometrical and physical, over a wide range of variation for each. They justify their choice of non-dimensional variables applying Buckingham π−theorem. The effective resistance of 2D random percolating networks of nanowires is found to have a nice expression in terms of the geometrical parameters (number of wires, aspect ratio of electrode separation over wire length) and the physical parameters (nanowire linear resistance per unit length, nanowire/nanowire contact resistance, metallic electrode/nanowire contact resistance). The dependence of the resistance on the geometry of the network, on the one hand, and on the physical parameters (values of the resistances), on the other hand, is thus clearly separated thanks to this expression, much simpler than the previously reported analytical expressions. In parallel, atomic scale models based on electronic structure theory are being developed to parameterize these mesoscale models (PhD thesis of R. Benda).

C. Le Bris has pursued his long term collaboration with Pierre Rouchon (Ecole des Mines de Paris and Inria QUANTIC) on the study of high dimensional Lindblad type equations at play in the modelling of open quantum systems. They have co-supervised the M2 internship of Luca Gorini, that was focused on the simulation of some simple quantum gates, and has investigated several discretization strategies based upon the choice of suitable basis sets.

V. Ehrlacher, L. Grigori (Inria ALPINES), D. Lombardi (Inria COMMEDIA) and H. Song (Inria ALPINES) have designed a new numerical method for the compression of high-order tensors [49]. The principle of the algorithm consists in constructing an optimal partition of the set of indices of the tensor, and construct an approximation of the tensor on each indices subdomain by means of an adapted High-Order Singular Value Decomposition. This method was used, among other examples, for the reduction of the solution of the Vlasov-Poisson system, and enabled to reach very significant compression factors. They also obtained very encouraging results on the compression of the Coulomb potential, which could be very interesting with a view to the resolution of the time-dependent Schrödinger equation in high dimension, which is currently work in progress.

A. Alfonsi, R. Coyaud (Ecole des Ponts), V. Ehrlacher and D. Lombardi (Inria COMMEDIA) studied a different approach for discretizing optimal transport problems, which relies in relaxing the marginal constraints in a finite number of marginal moment constraints, while keeping an infinite state space [40]. The advantage of such an approach is that the approximate solution of the multi-marginal optimal transport problem with Coulomb cost, which is the semi-classical limit of the so-called Lévy-Lieb functional, can be represented as a discrete measure charging a low number of points, thus avoiding the curse of dimensionality when the number of electrons is large.

M. Herbst and his collaborators have developed the `adcc`
Python/C++ software package for performing excited state
calculations based on algebraic-diagrammatic construction methods.
It connects to four SCF packages (pyscf, psifour, molsturm and
veloxchem), allows the inclusion of environmental effects through
implicit or explicit solvent models, and implements methods up to
third order in perturbation theory. Its features are summarized in
[54].