Implementing and evaluating a fictitious electron dynamics method for the calculation of electronic structure

Abstract

English: Quantum chemical calculations are an invaluable tool in the determination of electronic structure. However, the size of systems studied using these calculations are severely limited due to the highly unfavourable scaling of computational time - typically to the third or fourth order of the number of atoms in the system. Molecular dynamics calculations, on the other hand, can model systems consisting of thousands of atoms. They are, however, insufficient in describing chemical events - quantum chemical calculations are necessary for this. The quest for O(N) electronic structure calculation methods led to the investigation of a recently proposed fictitious electron dynamics method for calculating electronic structure which uses the idempotency of the density matrix to develop first and second order equations of motion. These equations are implemented in a semi-empirical environment, supplied by the MOPAC software package. The velocity Verlet scheme is used to integrate these equations and the enforcement of constraints is accomplished through McWeeny purification and the RATTLE algorithm. The essential role that parameters play in the effectiveness of the equations of motion is investigated and suggestions are made for these parameters. The requirements of energy conservation of the equations of motion, as well as the stability of the velocity Verlet integrator are addressed and the parameters are revised in order to comply to these requirements. The importance of the Si(100)2xl:H system as a test system is emphasized and the barriers for hydrogen atom diffusion are calculated using an existing parameter set in MOPAC. This system is used to determine the efficiency of the fictitious electron dynamics method to keep the calculated minimum energy close to the Born-Oppenheimer energy surface. Issues of relevance for further development of this method are discussed. These include the potential combination of this method with atomic dynamics to successfully describe chemical reactions on crystal surfaces as well as making use of the principle of nearsightedness of the density matrix to achieve linear scaling.Afrikaans: Kwantum chemiese berekeninge is 'n onmisbare hulpmiddel in die berekening van elektroniese struktuur. Tog, die grootte van sisteme wat bestudeer kan word is grootliks beperk vanweë die hoogs ongewenste skaling van berekeningstyd - tipies die aantal atome in die sisteem tot die derde of vierde orde. Molekulêre dinamika berekeninge, in teenstelling, kan sisteme wat duisende atome bevat modelleer. Ongelukkig is molekulêre dinamika nie in staat om chemiese reaksies te beskryf nie - hiervoor is kwantum chemiese berekeninge nodig. Die strewe na O(N) metodes om elektroniese struktuur te bereken het gelei tot die ondersoek van 'n onlangs voorgestelde fiktiewe elektron dinamika metode vir die berekening van elektroniese struktuur wat die idempotensie van die digtheidsmatriks gebruik om eerste en tweede orde bewegingsvergelykings te ontwikkel. Hierdie vergelykings is geïmplementeer in 'n semi-empiriese omgewing, soos verskaf deur die MOPAC sagteware pakket. Die snelheids Verlet skema is gebruik om hierdie vergelyking te integreer en die afdwing van beperkings is bewerkstellig deur McWeeny suiwering en die RATTLE algoritme. Die essensiële rol wat parameters in die effektiwiteit van die bewegingsvergelykings speel, is ook ondersoek en voorstelle is gemaak vir hierdie parameters. Die vereiste van energiebehoud in die bewegingsvergelykings, tesame met die stabiliteit van die snelheids Verlet integreerder, is aangespreek en die waardes wat parameters moet hê sodat daar aan hierdie vereistes voldoen word, is hersien. Die belangrikheid van die Si(100)2xl:H sisteem as 'n toetssisteem is benadruk en die versperrings vir waterstof atoom diffusie is bereken deur gebruik te maak van bestaande parameters verskaf deur MOPAC. Dié sisteem is dan gebruik om die effektiwiteit van die fiktiewe elektron dinamika metode te bepaal om die berekende minimum energie naby die Bom-Oppenheimer energievlak te hou. Aangeleenthede van belang vir verdere ontwikkeling van hierdie metode is bespreek. Dit sluit in die potensiële kombinasie van hierdie metode met atoomdinamika om chemiese reaksies op kristaloppervlaktes suksesvol te beskryf asook om gebruik te maak van die bysiendheidsbeginsel van die digtheidsmatriks om lineêre skaling met die aantal atome in die sisteem te bereik.