Section: New Results

Electronic structure calculations

Participants : Éric Cancès, Virginie Ehrlacher, Claude Le Bris, Antoine Levitt, 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.

Molecular systems

The work of the project-team on molecular systems has focused on advanced approaches for the computation of the electronic state of molecular systems, including the effects of electronic correlation and of the environment.

In [12], E. Cancès, D. Gontier (former PhD student of the project-team, now at Université Paris Dauphine) and G. Stoltz have analyzed the GW method for finite electronic systems. This method enables the computation of 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.

Implicit solvation models aim at computing the properties of a molecule in solution (most chemical reactions indeed take place in the liquid phase) by replacing all the solvent molecules except the 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). 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 has pursued his collaboration with Pierre Rouchon (Ecole des Mines de Paris) on the study of high dimensional Lindblad type equations at play in the modelling of open quantum systems. In order to complement and better understand the numerical approaches developed in the past couple of years, some theoretical aspects are now under study, in particular regarding the well-posedness of the equations and their convergence in the long time limit.

Crystals and solids

Periodic systems are mathematically treated using Bloch theory, raising specific theoretical and numerical issues.

A. Bakhta (CERMICS) and V. Ehrlacher are 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.

As an external collaborator of the MURI project on 2D materials (PI: M. Luskin), E. Cancès has started a collaboration with P. Cazeaux and M. Luskin (University of Minnesota) on the computation of the electronic and optical properties of multilayer 2D materials. Together with E. Kaxiras (Harvard) and members of his group, they have developped a perturbation method for computing the Kohn-Sham density of states of incommensurate bilayer systems. They have also 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 [36].

É. Cancès, A. Levitt and G. Stoltz, in collaboration with G. Panati (Rome) have proposed a new method for the computation of Wannier functions, a standard post-processing of density functional theory computations [38]. Compared to previous approaches, it does not require an initial guess for the shape of the Wannier functions, and is therefore more robust.

Numerical analysis

Members of the project-team have worked on the numerical analysis of partial differential equations arising from electronic structure theory.

E. Cancès and N. Mourad (CERMICS) 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.

E. Cancès has pursued his long-term collaboration with Y. Maday (UPMC) on the numerical analysis of electronic structure models. 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). 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 homogeneous Dirichlet boundary conditions [37]. In particular, upper and lower bounds for any simple eigenvalue are established. These bounds are guaranteed, fully computable, and converge with the optimal rate to the exact eigenvalue.

A. Levitt, in collaboration with X. Antoine and Q. Tang (Nancy), has proposed a new numerical method to compute the ground state of rotating Bose-Einstein condensates [31]. This method combines a nonlinear conjugate gradient method with efficient preconditionners. Compared to the state of the art (implicit timestepping on the imaginary-time equation), gains of one to two orders of magnitude are achieved.