## Section: New Results

### Computational quantum chemistry

Participants : Eric Cancès, Ismaila Dabo, Virginie Ehrlacher, Salma Lahbabi, Francis Nier, Gabriel Stoltz.

In computational quantum chemistry 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, V. Ehrlacher, S. Lahbabi and G. Stoltz have addressed issues related to the modeling and simulation of local defects in periodic crystals (see [61] for a pedagogical introduction).

Computing the energies of local defects in crystals is a major issue in quantum chemistry, materials science and nano-electronics. In collaboration with M. Lewin (CNRS, Cergy), E. Cancès and A. Deleurence have proposed in 2008 a new model for describing the electronic structure of a crystal in the presence of a local defect. This model is based on formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics (QED) in the presence of an external electrostatic field. The justification of this model is obtained using a thermodynamic limit of Kohn-Sham type models. In [29] , E. Cancès and G. Stoltz studied the time evolution of defects within this model, in the context of linear response, which allowed them to give a rigorous meaning to the Adler-Wiser formula for the frequency-dependent dielectric permittivity of crystals. In [27] E. Cancès et V. Ehrlacher have proved that local defects are always neutral in the Thomas-Fermi-von Weisäcker (TFW) theory. In this respect, all TFW crystals behave like metals. In collaboration with M. Lewin, E. Cancès and S. Lahbabi are working on the extension of Kohn-Sham like models to disordered systems.

On the numerical side, E. Cancès has worked with Y. Maday and R. Chakir (University Paris 6) on the numerical analysis of the electronic structure models. In [26] , they have obtained optimal *a priori* error bounds for the the planewave approximation of the Thomas-Fermi-von Weizsäcker and the Kohn-Sham LDA models. Together with Y. Maday, E. Cancès and V. Ehrlacher have analyzed the computation of eigenvalues in spectral gaps of locally perturbed periodic Schrödinger operators [28] .

Photovoltaic cells based upon organic photovoltaic (OPV) semiconductors are cost-effective, light-weight alternatives to conventional silicon and thin-film technologies. This year, in collaboration with A. Ferretti, N. Poilvert, N. Marzari (MIT and University of Oxford), M. Cococcioni (University of Minnesota), and Y. L. Li (Xiamen University), I. Dabo has worked on the prediction of the electronic spectra and electrical response of molecular and polymer compounds for OPV applications. Specifically, Y. L. Li and I. Dabo have implemented an efficient computational method for studying the electrical response of semiconducting polymers, in close agreement with more expensive local basis-set algorithms [52] . In parallel, A. Ferretti, I. Dabo, M. Cococcioni, and N. Marzari have applied recently developed electronic-structure theories, namely, orbital-dependent density-functional theories (OD-DFTs), to describe donor and acceptor levels in semiconducting organic materials, demonstrating that OD-DFT is apt at describing donor and acceptor levels within 0.1-0.4 and 0.2-0.6 eV of experiment [41] . Future work includes the prediction of the band structure of semiconductor alloys and the simulation of molecular heterojunctions.

The current works by F. Nier related to electronic structure calculations concern the mean field limit of the quantum dynamics in the bosonic setting; and the non linear modelling of Bose-Einstein condensates. A few years ago, Z. Ammari and F. Nier initiated a program about the Hamilton mean field dynamics of a large numbers of bosons. Their approach reconsiders the old idea that the mean field limit in the bosonic setting is actually a semiclassical limit in infinite dimension. In previous works, they proved results which are general with respect to the initial data but hold for bounded interactions. They also proved that the dynamics of the BBGKY hierarchy, often used within the mean field theory, is actually a projected picture of the dynamics of Wigner measures in the infinite dimensional phase-space. Recently in [10] , they obtained such results for singular pair interaction potentials, which include the Coulombic (attractive or repulsive) case in dimension $d=3$. Regarding the modelling of Bose-Einstein condensates, F. Nier and collaborators have developed in [9] complete analysis of the nonlinear adiabatic ansatz proposed by physicists for the simulation of rotating Bose-Einstein condensates.