# An efficient numerical solver for the Poisson equation in heterogeneous polarizable media

# An efficient numerical solver for the Poisson equation in heterogeneous polarizable media

Promotor(en):**17MODEV05**/ Model and software development**T. Verstraelen**/Molecular dynamics simulations of realistic models of supramolecular systems have had an enormous impact on various scientific disciplines, far beyond the scope of molecular physics. For example, in biology, the molecular simulation of proteins, cell membranes, DNA helices, ... shaped our understanding of biological processes at the nanoscale. Molecular dynamics simulations require a model that accurately and efficiently computes the forces acting on all atoms at every time step in the simulation. Even though many contributions to these forces can be approximated quite well with pairwise interatomic potentials, a part of the potential energy of the molecular system has an inherent many-body character. For such many-body interactions, good models and efficient implementations are currently still lacking.

The largest many-body interaction in molecular systems is the so-called electronic induction interaction: when two molecular fragments (A+B) approach, A exerts an electrostatic field that induces a change in the electron density in B and vice versa. This rearrangement of density affects the electrostatic interaction in a non-additive way: the induction interaction of a molecular trimer (A+B+C) is not simply to the sum of interaction energies of the pairs (A+B, B+C, C+A). The macroscopic manifestation of the induction interaction is the dielectric constant of a material. However, at the scale of atoms, the introduction of a dielectric constant in the coulomb law only leads to a very poor additive model of the induction interaction.**Goal**At the Center for Molecular Modeling, a new model for induction interactions was recently proposed: Atom-Condensed Kohn-Sham approximated to second order (ACKS2). This is a linear-response model that can describe induction interactions in molecular systems. However, before it can be used in molecular dynamics simulations, an efficient implementation is required. The goal of this thesis is to develop and implement efficient algorithms to solve the ACKS2 equations and compute the forces, due to induction interaction, acting on atoms in a molecular system.

An essential ingredient in this work is the correct treatment of long-range electrostatics in periodic boundary conditions. This is in general a non-trivial problem for which many different Poisson solvers can be found in the literature. However, none of these make use of the fact that electronic induction weakens the strength of long-range electrostatics, such that they can be computed more easily. In this thesis, a combined solver for the Poisson and ACKS2 equations can exploit this advantage.

During this thesis, the student will learn how to transform fundamental theories into practical algorithms and their implementation on high-performance computers. Depending on the interests of the student, one may also explore more fundamental theoretical questions, e.g. how to define the polarization of a periodic system in the context of the ACKS2 model.**Mobility:**Mobility is not required for this thesis. However, the topic of the thesis fits well in ongoing collaborations, such that there will be interactions with international partners.**Motivation Appl. Phys.:**The physics aspect is a linear response theory for media with nanoscale heterogeneity. The engineering aspect is the development and implementation of algorithms to solve the governing equations in molecular simulation software.

- Study programmeMaster of Science in Engineering Physics [EMPHYS], Master of Science in Physics and Astronomy [CMFYST]ClustersFor Engineering Physics students, this thesis is closely related to the cluster(s) NANO, MODELLINGKeywordsnumerical algorithms, Computational physics, linear response theory, long-range interactions