Section: New Results

Electronic structure calculations

Participants : Eric Cancès, Virginie Ehrlacher, David Gontier, Claude Le Bris, Gabriel Stoltz.

In electronic structure calculation as in most of our scientific endeavors, we pursue a twofold goal: placing the models on a sound mathematical grounding, and improving the numerical approaches.

E. Cancès and N. Mourad have clarified the mathematical framework underlying the construction of norm-conserving semilocal pseudopotentials for Kohn-Sham models, and have proved the existence of optimal pseudopotentials for a family of optimality criteria [34] .

E. Cancès and R. Scott (University of Chicago) have examined a technique of Slater and Kirkwood which provides an exact resolution of the asymptotic behavior of the van der Waals attraction between two hydrogen atoms. They have modified their technique to make the problem more tractable analytically and more easily solvable by numerical methods [35] .

In [33] , E. Cancès, D. Gontier and G. Stoltz analyze the GW method for finite electronic systems. This method allows to compute excited states. To understand it, a first step is to provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. It is then possible to study the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. With this framework, results can be established for the well-posedness of the GW${}_0$ equations, a specific instance of the GW model. In particular, the existence of a unique solution to these equations is proved in a perturbative regime.

D. Gontier extended his last-year result on N-representability by including the characterization of representable paramagnetic currents [42] . Together with Salma Lahbabi (former student of E. Cancès, University Hassan II Casablanca, ENSEM), he proved the exponential convergence rates of the uniform sampling of the Brillouin zone for the calculation of crystalline structure properties, in linear and nonlinear settings [43] .

A. Bakhta, E. Cancès and V. Ehrlacher have recently been working on the design of an efficient numerical method to solve the inverse band structure problem. The aim of this work is the following: given a set of electronic bands partially characterizing the electronic structure of a crystal, is it possible to recover the structure of a material which could achieve similar electronic properties? The main difficulty in this problem relies in the practical resolution of an associated optimization problem with numerous local optima.

E. Cancès has pursued his long-term collaboration with Y. Maday (Paris 6) on the numerical analysis of electronic structure models. Together with G. Dusson (Paris 6), B. Stamm (Paris 6), and M. Vohralík (Inria), they have designed a new postprocessing method for planewave discretizations of nonlinear Schrödinger equations, and used it to compute sharp a posteriori error estimators for both the discretization error and the algorithmic error (convergence threshold in the iterations on the nonlinearity). They have then extended this approach to the Kohn-Sham model. In parallel, they have derived a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition [32] . In particular, upper and lower bounds for the first eigenvalue are given. These bounds are guaranteed, fully computable, and converge with the optimal speed to the exact eigenvalue.

Implicit solvation models aim at computing the properties of a molecule in solution (most chemical reactions take place in the liquid phase) by replacing all the solvent molecules but the few ones strongly interacting with the solute, by an effective continuous medium accounting for long-range electrostatics. E. Cancès, Y. Maday (Paris 6), and B. Stamm (Paris 6) have recently introduced a very efficient domain decomposition method for the simulation of large molecules in the framework of the so-called COSMO implicit solvation models. In collaboration with F. Lipparini (Paris 6), B. Mennucci (Department of Chemistry, University of Pisa) and J.-P. Picquemal (Paris 6), they have implemented this algorithm in widely used computational software products (Gaussian and Tinker). E. Cancès, Y. Maday, F. Lipparini and B. Stamm have also extended this approach to the more complex polarizable continuum model (PCM).

C. Le Bris, in collaboration with P. Rouchon (École des Mines de Paris) and with J. Roussel, in the context of an internship at École des Ponts, has pursued the study of a new efficient numerical approach, based on a model reduction technique, to simulate high dimensional Lindblad type equations at play in the modelling of open quantum systems. The added value of the most recent contribution with respect to the previous studies lies in two different aspects. First, the rank of the reduced model used as surrogate for the full model can now be dynamically adjusted, in an adaptive strategy. Second, a variance reduction approach based on the technique of control variate has been developed. The noise intrinsically present in the Monte-Carlo simulation of the underlying stochastic dynamics may indeed be reduced by using the deterministic reduced model as control variate. A publication collects these two aspects and reports on the results achieved [19] .