## 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 endeavours, we pursue a twofold goal: placing the models on a sound mathematical grounding, and improving the numerical approaches.

E. Cancès and N. Mourad have mathematically analyzed the density functional perturbation theory, both in the non-degenerate case (that is, when the Fermi level is not an eigenvalue of the Kohn-Sham hamiltonian) and in the degenerate case. They have in particular proved that Wigner's 2n+1 rule holds in both cases. D. Gontier has obtained a complete, explicit, characterization of the set of spin-polarized densities for finite molecular systems. This problem was left open in the pioneering work of von Barth and Hedin setting up the Kohn-Sham density functional theory for magnetic compounds. He has also extended a previous work by Anantharaman and Cancès, and proved the existence of minimizers for the spin-polarized Kohn-Sham model in the presence of a magnetic field within the local spin density approximation.

E. Cancès has pursued his long-term collaboration with Y. Maday (UPMC)
on the numerical analysis of electronic structure models. With L. He (ENPC) and
R. Chakir (IFSTTAR), they have designed and analyzed a two-grid
methods for nonlinear elliptic eigenvalue problems, which can be
applied, in particular, to the Kohn-Sham model. Some numerical tests
demonstrating the interest of the approach have been performed with the
Abinit software. Together with G. Dusson (UMPC), B. Stamm (UMPC), and
M. Vohralík (Inria), they have designed a new post processing 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).

Implicit solvation models aims 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 (UPMC, 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). The extension of this method to other implicit solvation models is work in progress.

Claude Le Bris, in collaboration with Pierre Rouchon (Ecole des Mines de Paris), 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 specific case under consideration is that of oscillation revivals of a set of atoms interacting resonantly with a slightly damped coherent quantized field of photons. The approach may be employed for other similar equations. Current work is directed towards other numerical challenges for this type of problems.