Conical Intersections and Avoided Crossings of Electronic Energy Levels

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dc.contributor.authorGamble, Stephanie Nicoleen
dc.contributor.committeechairHagedorn, George A.en
dc.contributor.committeememberFermanian Kammerer, Clotildeen
dc.contributor.committeememberElgart, Alexanderen
dc.contributor.committeememberKlaus, Martinen
dc.contributor.committeememberValeyev, Eduard Faritovichen
dc.contributor.departmentMathematicsen
dc.date.accessioned2021-01-15T09:00:22Zen
dc.date.available2021-01-15T09:00:22Zen
dc.date.issued2021-01-14en
dc.description.abstractWe study the unique phenomena which occur in certain systems characterized by the crossing or avoided crossing of two electronic eigenvalues. First, an example problem will be investigated for a given Hamiltonian resulting in a codimension 1 crossing by implementing results by Hagedorn from 1994. Then we perturb the Hamiltonian to study the system for the corresponding avoided crossing by implementing results by Hagedorn and Joye from 1998. The results from these demonstrate the behavior which occurs at a codimension 1 crossing and avoided crossing and illustrates the differences. These solutions may also be used in further studies with Herman-Kluk propagation and more. Secondly, we study codimension 2 crossings by considering a more general type of wave packet. We focus on the case of Schrödinger equation but our methods are general enough to be adapted to other systems with the geometric conditions therein. The motivation comes from the construction of surface hopping algorithms giving an approximation of the solution of a system of Schrödinger equations coupled by a potential admitting a conical intersection, in the spirit of Herman-Kluk approximation (in close relation with frozen/thawed approximations). Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase and its proof is based on a normal form approach.en
dc.description.abstractgeneralWe study energies of molecular systems in which special circumstances occur. In particular, when these energies intersect, or come close to intersecting. These phenomena give rise to unique physics which allows special reactions to occur and are thus of interest to study. We study one example of a more specific type of energy level crossing and avoided crossing, and then consider another type of crossing in a more general setting. We find solutions for these systems to draw our results from.en
dc.description.degreeDoctor of Philosophyen
dc.format.mediumETDen
dc.identifier.othervt_gsexam:29059en
dc.identifier.urihttp://hdl.handle.net/10919/101899en
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectquantum mechanicsen
dc.subjectconical intersectionsen
dc.subjectenergy level crossingsen
dc.subjectavoided crossingsen
dc.subjectsemiclassical wave packetsen
dc.subjectpropagation of coherent statesen
dc.titleConical Intersections and Avoided Crossings of Electronic Energy Levelsen
dc.typeDissertationen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.leveldoctoralen
thesis.degree.nameDoctor of Philosophyen

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