Quantum Dynamics of Interacting Electrons and Phonons

In this thesis I explore the dynamical behavior of electrons and excitons interacting with quantized nuclear vibrations. In the first chapter I begin by introducing the notion of vibronic models and discussing their utility for modeling dynamical phenomena in the condensed phase. In the second chapter, I continue to detail a collaborative effort for modeling photophysics and transport dynamics in aggregates of the organic dye molecule perylene diimide (PDI). There I discuss how the vibronic signatures in steady-state photoluminescence spectra may be used to decode the microscopic couplings that determine the hybrid H and J aggregate behavior in PDI crystals. I then show how interference between these couplings has a substantial effect on controlling ballistic and diffusive transport dynamics. In the third chapter I continue to address the challenge of describing finite temperature dynamics in the Holstein model in the thermodynamic limit. Toward this end, I present approximate solutions via the cumulant expansion and discuss in detail the successes and limitations of this method. Finally, in the interest of providing fully quantum mechanical solutions for vibronic models in the nonperturbative intermediate coupling regime, in the fourth chapter I discuss the application of the numerically exact reduced hierarchical equations of motion (HEOM) method. I expose how for models such as the Holstein model that incorporate a finite bath of undamped harmonic oscillators, temperature-dependent instabilities arise in HEOM which corrupt the long-time dynamics. Through a projection-based approach, I demonstrate how these instabilities may be removed, obviating the need for a costly and poorly-behaved convergence procedure with respect to the hierarchy depth. I also present a numerical iterative approach for accomplishing this projection, intended for use in cases where a diagonalization-based projection proves too costly. Overall, this thesis delves into applications as well as approximate and numerically exact solutions of vibronic models.