Section: New Results

Electronic structure calculations

Participants : Robert Benda, 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.

Mathematical analysis

The members of the team have continued their systematic study of the properties of materials in the reduced Hartree-Fock approximation, a model striking a good balance between mathematical tractability and the ability to reproduce qualitatively complex effects.

E. Cancès and G. Stoltz have studied with L. Cao models for certain extended defects in materials [37]. These extended defects typically correspond to taking out a slab of finite width in the three-dimensional homogeneous electron gas. The work is performed in the framework of the reduced Hartree-Fock model with either Yukawa or Coulomb interactions, using techniques previously developed to study local perturbations of the free-electron gas. It is shown that the model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. These minimizers are unique for Yukawa interactions, and are characterized by a self-consistent equation. Numerical simulations show evidence of Friedel oscillations in the total electronic density.

A. Levitt has examined the phenomenon of screening in materials. In [54] he has studied the effect of adding a small charge to a periodic system modeled by the reduced Hartree-Fock at finite temperature. He has showed that the reaction potential created by the rearrangement of the electrons counteracts exactly the free charge, so that the effective interaction in such systems is short-range. The proof proceeds by studying the properties of the linear response operator, which also sheds some light on the charge-sloshing instability seen in numerical methods to solve the self-consistent equations.

Numerical analysis

E. Cancès has pursued his long-term collaboration with Y. Maday (Sorbonne Université) on the numerical analysis of linear and nonlinear eigenvalue problems. Together with G. Dusson (Warwick, United Kingdom), B. Stamm (Aachen, Germany), and M. Vohralík (Inria SERENA), they have designed a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with homogeneous Dirichlet boundary conditions. In [38], they prove a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works. They consider a Schrödinger operator $H=-\frac{1}{2}\Delta +V$ on $L^2\left(\Omega \right)$, where $\Omega$ is a cubic box with periodic boundary conditions. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest $N$ eigenvalues of $H$ , and, on the other hand, the ground-state density matrix, that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn-Sham density functional theory (DFT). Interpreting the exact eigenpairs of H as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e. Fourier) basis, they compute first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This allows them to increase the accuracy of both the ground-state energy and the ground-state density matrix at a low computational extra-cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger plane-wave basis and $2N$ Fast Fourier Transforms.

Implicit solvation models aim 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 (Sorbonne Université), and B. Stamm (Aachen, Germany) have introduced a few years ago 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 and B. Mennucci (Chemistry, Pisa, Italy) and J.-P. Piquemal (Sorbonne Université), they have implemented this algorithm in widely used computational software products (Gaussian and Tinker). Together with L. Lagardère (Sorbonne Université) and G. Scalmani (Gaussian Inc., USA), they illustrate in [29] the domain decomposition COSMO (ddCOSMO) implementation and how to couple it with an existing classical or quantum mechanical (QM) codes. They review in detail what input needs to be provided to ddCOSMO and how to assemble it, describe how the ddCOSMO equations are solved and how to process the results in order to assemble the required quantities, such as Fock matrix contributions for the QM case, or forces for the classical one. Throughout the paper, they make explicit references to the ddCOSMO module, which is an open source, Fortran 90 implementation of ddCOSMO that can be downloaded and distributed under the LGPL license.

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 in [39] the speed of convergence for a number of schemes in the metallic case, and have 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. With A. Damle (Cornell, USA) and L. Lin (Berkeley, USA), they have proposed an efficient numerical method for the computation of maximally-localized Wannier functions in metals, and have showed on the example of the free electron gas that they are not in general exponentially localized [42]. 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 [45].