Research

Electronic structure calculations are paramount in theoretical chemistry, physics, and material science. Indeed, among the top ten most cited scientific articles, two papers are related to this topic. Despite the fact that this field deals with the discretization of eigenvalue problems, which is a well-established subject in numerical analysis, the expertise of applied mathematics is little involved.Within the proposed project, the aim is to develop novel domain decomposition (DD) algorithms for eigenvalue problems which arise in electronic structure calculation like the Kohn-Sham DFT (Density Functional Theory) equations. The approach within this project is based on the idea that domain decomposition for eigenvalue problems is not fundamentally different than for source problems. Despite this fact, DD-methods for eigenvalue problems are less popular than for source problems. Additionally, recent work in the context of implicit solvation models and the analysis of those methods show that the DD-method is scalable for domains of an increasing number of fixed-size sub-domains, like e.g. for chain-like molecules or proteins, even without a so-called coarse-correction.The problems to be embraced within this project are manifold and contain: i) non-linear eigenvalue problems, ii) a large number of eigenvalues to be determined, iii) eigenvalue problems on unbounded domains and iv) dealing with potentials that contain Coulomb-like singularities.The project is a first step within a broader long-term plan to derive efficient local basis functions based on local reduced order modeling as an alternative of the widely-used but empirical contracted Gaussian basis functions. In fact, the domain decomposition strategy allows to localise the equations and opens the door to the application of reduced order modeling with certified a posteriori error estimates in a second step.

Continuum solvation models (CSM) are widely used in Quantum Mechanical (QM) calculations to take environ- mental effects, caused by polarizable solvents, into account. CSMs are invented for small systems where the associated computational effort is negligible compared to the QM-computations. With many recent advances, such as improved hardware, linear scaling methods or efficient semi-empirical models, the computational part devoted to the CSM has become a bottleneck. A domain-decomposition strategy for the COSMO solvation model (ddCOSMO) proposed by the PI is linearly scaling with respect to the size of the solute molecule and is overall very efficient allowing to consider medium to large-sized solutes. But recent applications revealed that the assumption of the cavity being modelled as a union of scaled van der Waals spheres yields nonphysical results suggesting the Solvent Excluded Surface as a physically more meaningful definition of the solute’s cavity. Recently, the PI has analyzed the SES and proposed a proof of concept of a new domain decomposition-strategy for a SES-based solvation model having the potential to become an accurate and efficient solvation model for medium to large molecules. Goal of this project is to develop this proof of concept to a fully functional and mature tool in theoretical chemistry, including the derivation of an efficient solution strategy and the calculation from the solvent to first and second derivatives.

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