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## Mathematical and numerical analysis of electronic structure models

#### Station biologique de Roscoff (UMPC/CNRS)

#### Organisers

The workshop will focus on the following mathematical and numerical issues, for which promising results have recently been obtained and pave the way to further developments and interdisciplinary collaborations:

• Error estimators for quantifying model, discretization, and algorithmic errors (1 day). Assessing the accuracy of the computed quantities of interest (equilibrium geometries of molecules, vibration spectra, activation energies of chemical reactions, band gaps of semiconductors, …) is a major challenge in first-principle molecular simulation. New approaches based on a posteriori error estimators have recently been proposed to quantify the discretization (e.g. energy cut-off in planewave calculations) and algorithmic (e.g. stopping criteria in SCF algorithms) errors for Kohn-Sham calculations. Works aiming at quantifying the model error in wavefunctions methods are also in progress. This emerging topic will be discussed at large during the workshop.

• Low-rank tensor methods (1 day). Several methods used in electronic structure calculations (MCSCF, coupled-cluster, DMRG, …) fall into the scope of low-rank tensor methods. Such objects are encountered in many different fields including uncertainly quantification (UQ), mathematical finance, or machine learning. A unified mathematical framework and new algorithms to manipulate high-dimensional low-rank tensor methods have been recently introduced. Applications of these techniques to electronic structure calculation will be discussed.

• Multiscale models (1 day). Coupling electronic structure models with atomistic (QM/MM) or macroscopic (implicit solvent, quasi-continuum) models gives rise to very interesting mathematical and numerical problems. Very recent results have been obtained by a few groups of mathematicians who will participate in the workshop and present their works.

• Mathematical analysis of models for ground and excited states (1 day). While the mathematical nature of some electronic structure models (Hartree-Fock, Kohn-Sham LDA, CI, MCSCF, CC, …) is well-understood, the mathematical frameworks underlying, in particular, the TDDFT, GW and Bethe-Salpeter models are still unclear. The recent advances and the remaining challenges will be discussed.

• Topology and geometry in electronic structure calculation (1/2 day). Overviews of the applications of topology and geometry in electronic structure calculations will be presented in two keynote lectures. The lectures will focus on 1) the localization of electrons in molecules by means of topological methods, and 2) the physical interpretation of geometric phases and topological invariants in electronic structure models. Some numerical issues related to theses topics, which should be of high interest to the applied mathematics community, will also be presented.

## References

**China**

Aihui Zhou (Chinese Academy of Sciences) - Organiser

**France**

Eric Cancès (CERMICS - ENPC) - Organiser

Yvon Maday (UPMC and IUF) - Organiser

**Germany**

Gero Friesecke (TU Munchen) - Organiser

Reinhold Schneider (Technical University Berlin) - Organiser

Harry Yserentant (Technische Universität Berlin) - Organiser