Browsing by Author "Gamble, Stephanie Nicole"

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    Conical Intersections and Avoided Crossings of Electronic Energy Levels
    Gamble, Stephanie Nicole (Virginia Tech, 2021-01-14)
    We 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.
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    Vibrational Energies of the Hydrogen Bonds of H₃O₂⁻ and H₅O₂⁺
    Gamble, Stephanie Nicole (Virginia Tech, 2016-06-24)
    We approximate the vibrational energies of the symmetric and asymmetric stretches of the hydrogen bonds of the molecules H_3O_2^- and H_5O_2^+ by applying an improvement to the standard time-independent Born-Oppenheimer approximation. These two molecules are symmetric around a central hydrogen which participates in hydrogen bonding. Unlike the standard Born-Oppenheimer approximation, this approximation appropriately scales the hydrogen nuclei differently than the heavier oxygen nuclei. This results in significantly more accurate approximations for the stretching vibrational energies, which we compare to experimental measurements.