## Section: New Results

### Computational quantum chemistry

Participants : Hanen Amor, Arnaud Anantharaman, Eric Cancès, Ismaila Dabo, Virginie Ehrlacher, Yanli Li, Gabriel Stoltz.

In computational quantum chemistry as in most of our scientific endeavours, we pursue a twofold goal: giving the models a sound mathematical grounding, and improving the numerical approaches.

Existence results for the extended Kohn-Sham LDA (local density approximation) model as well as for the two-electron Kohn-Sham GGA (generalized gradient approximation) model, have been obtained by A. Anantharaman and E. Cancès, using the concentration-compactness method [8] .

E. Cancès has addressed issues related to the modeling and simulation of local defects in periodic crystals. 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 on the so-called supercell model. They have also introduced a variational method for computing the perturbation in a basis of precomputed maximally localized Wannier functions of the reference perfect crystal. In [22] , E. Cancès and M. Lewin have pursued the analysis of this model and have used the model to construct a rigorous mathematical derivation of the Adler-Wiser formula for the dielectric permittivity of crystals.

E. Cancès has also initiated a collaboration with
Y. Maday and R. Chakir (University of Paris 6) on the numerical analysis
of variational approximations of nonlinear elliptic eigenvalue
problems. In [20] , they provide *a priori* error estimates
for variational approximations of the ground state energy, eigenvalue
and eigenvector of problems of the form
, . They focus in particular on the Fourier spectral
and pseudospectral approximation (for periodic problems) and on the
and finite-element discretizations. Work in progress is
concerned with the planewave approximation of the Thomas-Fermi-von
Weizsäcker and the Kohn-Sham LDA model.

In collaboration with F. Mauri and N. Mingo, physicists respectively from IMPMC (Paris 6 and 7) and CEA Grenoble, G. Stoltz has continued to study the reduction in thermal conductivity for carbon nanotubes when isotope disorder is present, using methods from quantum statistical physics, see [42] . More precisely, the influence of the disorder structure has been investigated in [43] (alloy material versus homogeneous slices).

In collaboration with C. Brouder (IMPMC, Paris 6 and 7) and G. Panati (University La Sapienza, Roma), G. Stoltz has also proved the Gell-Mann and Low formula for systems with degenerate ground states in [18] . This formula relates an eigenstate of an initial reference Hamiltonian to a perturbed one, using some adiabatic switching. The key point of the work has been to identify the directions within the initial degenerate space in which the switching can be performed. Physical implications of this work in the domain of quantum field theory are discussed in [19] .

E. Cancès and G. Stoltz have studied, in collaboration with several researchers in chemistry, the mathematical foundations of the so-called Optimized Effective Potential approach [23] , in the context of the Kohn-Sham equations in Density Functional theory. This method replaces the (exact) non-local exchange operator by some approximate local operator, optimal in some sense. They made precise the necessary optimality condition, and studied also the existence and uniqueness of the solution to the corresponding nonlinear partial differential equation system in a simplified case (for the so-called Slater potential).

The domain decomposition method proposed by M. Barrault (now at EDF), E. Cancès, W. Hager (University of Florida), and C. Le Bris, originally designed to solve the linear subproblem in electronic structure calculations, has been successfully coupled with the nonlinear loop of the Hartree-Fock problem (self-consistent iterations). This work has been completed by H. Amor, under the supervision of G. Bencteux (EDF) and E. Cancès. Besides, test cases with large basis sets including polarization and diffuse atomic orbitals have confirmed the robustness of this approach to compute the ground state of extended linear molecules (polymers and nanotubes).

The PhD thesis [6] of H. Galicher has been defended.