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Grid-based Fast Multipole Method : a fully numerical scheme for evaluating two-electron integrals

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Title: Grid-based Fast Multipole Method : a fully numerical scheme for evaluating two-electron integrals
Author(s): Toivanen, Elias Akseli
Contributor: University of Helsinki, Faculty of Science, Department of Chemistry
Discipline: Physical Chemistry
Language: English
Acceptance year: 2014
Two-electron integrals, which arise in the quantum mechanical description of electron-electron repulsion, are needed in electronic structure calculations. In this thesis, a fully numerical scheme for computing them has been developed and implemented. The accuracy and performance of the scheme is also demonstrated with proof-of-concept calculations. The work in this thesis is a part of the ongoing efforts aiming at a fully numerical electronic structure code for massively parallel computer architectures. The power of these emerging computational resources can be seized only if all computational tasks are divided to small and independent parts that are then processed concurrently. Such a divide and conquer approach is indeed the main characteristic of the present integration scheme. The scheme is a variant of the Fast Multipole Method (FMM) that is an algorithm originally designed for rapid evaluation of electrostatic and gravitational potential fields in point particle simulations. Since the two-electron integrals can be formulated as a problem in electrostatics involving electrostatic potentials and continuous charge densities, the FMM algorithm is also applicable for tackling them. The basic idea in the present scheme is to decompose the computational domain to sub-domains in which the electron densities and electrostatic potentials are further decomposed to finite element functions. The two-electron integrals are then computed as a sum of contributions from each sub-domain. As the current scheme performs all integrals on real-space grids, it has been titled as the Grid-based Fast Multipole Method (GB-FMM). Its computational cost scales linearly with respect to the number of sub-domains. The thesis consists of two parts – a literature review discussing the key features of electronic structure calculations at the Hartree-Fock level of theory and a documentation of the GB-FMM. The results of the proof-of-concept calculations are encouraging. The GB-FMM scheme can achieve parts per billion accuracy. In addition, an analysis of its performance in a single-core environment indicates that the computational cost of the GB-FMM scheme has a rather big prefactor but a favorable scaling with respect to system size. However, as the GB-FMM algorithm has been designed with parallel execution in mind, its full power is predicted to become evident only when massively parallel computers will become commonplace.

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