Section: New Results

Electronic structure calculations

Participants : Éric Cancès, Virginie Ehrlacher, Antoine Levitt, Sami Siraj-Dine, 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 by an appropriate mathematical analysis, and improving the numerical approaches by a dedicated numerical analysis. We also insist on rigorously studying current materials of technological interest.

Mathematical analysis

In [42], E. Cancès and N. Mourad performed a detailed study of the extended Kohn-Sham model for atoms subjected to cylindrically-symmetric external potentials. In particular, they computed the occupied and unoccupied energy levels of all the atoms of the first four rows of the periodic table for the reduced Hartree-Fock (rHF) and the extended Kohn-Sham X$\alpha$ models. These results allowed them to test numerically the assumptions on the negative spectra of atomic rHF and Kohn-Sham Hamiltonians used in their previous theoretical works on density functional perturbation theory and pseudopotentials. Interestingly, they observed accidental degeneracies between s and d shells or between p and d shells at the Fermi level of some atoms.

Numerical analysis

E. Cancès has pursued his long-term collaboration with Y. Maday (UPMC) on the numerical analysis of linear and nonlinear eigenvalue problems. Together with G. Dusson (UMPC), B. Stamm (UMPC), and M. Vohralík (Inria SERENA), they have designed a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition [15]. In particular, upper and lower bounds for any simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with the optimal speed to the exact eigenvalue. In [41], this analysis is extended to all standard numerical methods, including nonconforming discontinuous Galerkin, and mixed finite element approximations or arbitrary polynomial degree.

It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. In [14], E. Cancès and G. Dusson initiated the quantitative study of discretization error cancellation. Discretization error is the error component due to the fact that the model used in the calculation (e.g. Kohn−Sham LDA) must be discretized in a finite basis set to be solved by a computer. They first reported comprehensive numerical simulations showing that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cut-off goes to infinity. They then analyzed a simple one-dimensional periodic Schrödinger equation with Dirac potentials, for which analytic solutions are available. This allowed them to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.

E. Cancès, V. Ehrlacher and A. Levitt, together with D. Gontier (Dauphine) and D. Lombardi (Inria REO), have studied the convergence of properties of periodic systems as the size of the computing domain is increased. This convergence is known to be difficult in the case of metals. They have characterized the speed of convergence for a number of schemes in the metallic case, and studied the properties of a widely used numerical method that adds an artificial electronic temperature.

A. Levitt has continued his study of Wannier functions in periodic systems, after the work [16] with E. Cancès, G. Panati (Rome) and G. Stoltz was published. With H. Cornean (Aalborg), D. Gontier (Dauphine) and D. Monaco (Rome), they introduced a mathematical definition of Wannier functions for metals, used routinely in materials science but not studied theoretically until now. They proved that, under generic assumptions, there exists a set of localized Wannier functions that span a given set of bands, even if this set is not isolated from the others [50]. With A. Damle (Cornell) and L. Lin (Berkeley), they proposed an efficient numerical method for the computation of maximally-localized Wannier functions in metals, and showed on the example of the free electron gas that they are not in general exponentially localized. With D. Gontier (Dauphine) and S. Siraj-Dine, they proposed a new method for the computation of Wannier functions which applies to any insulator, and in particular to the difficult case of topological insulators.

New materials

As an external collaborator of the MURI project on 2D materials (PI: M. Luskin), E. Cancès has collaborated with P. Cazeaux (Kansas) and M. Luskin (University of Minnesota) on the computation of the electronic and optical properties of multilayer 2D materials. In particular, they have adapted the C${}^{*}$-algebra framework for aperiodic solids introduced by J. Bellissard and collaborators, to the case of tight-binding models of incommensurate (and possibly disordered) multilayer systems [13].

The optimal design of new crystalline materials to achieve targeted electronic properties is a very important issue, in particular for photovoltaic applications. In the context of a collaboration with IRDEP, A. Bakhta (CERMICS), V. Ehrlacher and D. Gontier (Dauphine) studied the following inverse problem in [37]: given desired functions defined over the Brillouin zone of a crystalline structure, is it possible to compute a periodic potential so that the first bands of the associated periodic Schrödinger operator are as close as possible to these functions? Theoretical results were obtained for the corresponding variational problem in one dimension for the first band, and it appears from the mathematical analysis that the potential has to belong to a Borel measure space. In addition, a numerical method has been developped to solve the resulting optimization problem where the different discretization parameters are adjusted throughout the calculation, which leads to significant computational gains.