Bugeanu, Monica. The Wavelet Galerkin method for the polarizable continuum model in quantum chemistry. 2017, Doctoral Thesis, University of Basel, Faculty of Science.

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Abstract
The polarizable continuum model (PCM) is a well established method for computing solvation effects. Its attractiveness comes from the fact that the solution is modelled as a continuum instead of modelling each atom individually. This allows for more complex simulations. Nevertheless, there are still a couple of challenges to face. One of them being the computational time required at the limit of large systems, or when high accuracy is needed.
The wavelet boundary element method can be used to overcome these problems, provided a reliable cavity generator is used. The PCM calculates the molecular free energy in solution as the sum of electrostatic, dispersionrepulsion and the cavitation energy by solving the underlying partial differential equations. Those differential equations can be transformed into integral equations, solely defined on the boundary of the molecular cavity, by applying the integral equation formalism (IEFPCM). The integral equations can then be discretized using the wavelet boundary element method. The resulting sparse system of linear equations can be solved reliably by iterative solvers, leading to high accuracy solutions even for large systems.
The challenge of this approach lies in the generation and refinement of the molecular cavities on which the interactions take place. Since wavelets are defined as a tensor product of onedimensional functions, the wavelet boundary element method needs a discretization into quadrangular patches of the molecular cavity. This feature is missing in common commercial mesh generation tools, which focus on generating triangular meshes. Making use of characteristic functions for defining which points are inside the cavity, a model for generating the molecular surface independently of its exact description is achieved. We present here a way for generating triangular meshes which are subsequently merged into quadrangular ones. We apply geometrical quantities that measure the fitness of the shapes involved and improve the position of individual points iteratively. Thereby, we generate quadrangular meshes with an unprecedented quality, which ultimately can be refined in a hierarchical manner for the use of wavelets.
Overall, this method can generate reliable parametrizations on a variety of smooth cavities. With the help of the wavelet boundary element method, the PCM equations can thus be solved with high accuracy also on large systems.
The wavelet boundary element method can be used to overcome these problems, provided a reliable cavity generator is used. The PCM calculates the molecular free energy in solution as the sum of electrostatic, dispersionrepulsion and the cavitation energy by solving the underlying partial differential equations. Those differential equations can be transformed into integral equations, solely defined on the boundary of the molecular cavity, by applying the integral equation formalism (IEFPCM). The integral equations can then be discretized using the wavelet boundary element method. The resulting sparse system of linear equations can be solved reliably by iterative solvers, leading to high accuracy solutions even for large systems.
The challenge of this approach lies in the generation and refinement of the molecular cavities on which the interactions take place. Since wavelets are defined as a tensor product of onedimensional functions, the wavelet boundary element method needs a discretization into quadrangular patches of the molecular cavity. This feature is missing in common commercial mesh generation tools, which focus on generating triangular meshes. Making use of characteristic functions for defining which points are inside the cavity, a model for generating the molecular surface independently of its exact description is achieved. We present here a way for generating triangular meshes which are subsequently merged into quadrangular ones. We apply geometrical quantities that measure the fitness of the shapes involved and improve the position of individual points iteratively. Thereby, we generate quadrangular meshes with an unprecedented quality, which ultimately can be refined in a hierarchical manner for the use of wavelets.
Overall, this method can generate reliable parametrizations on a variety of smooth cavities. With the help of the wavelet boundary element method, the PCM equations can thus be solved with high accuracy also on large systems.
Advisors:  Harbrecht, Helmut and Stamm, Benjamin 

Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht) 
UniBasel Contributors:  Bugeanu, Monica and Harbrecht, Helmut 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  12457 
Thesis status:  Complete 
Number of Pages:  1 OnlineRessource (129 Seiten) 
Language:  English 
Identification Number: 

edoc DOI:  
Last Modified:  05 Apr 2018 17:36 
Deposited On:  23 Feb 2018 13:02 
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