Incorporating Nuclear Quantum Effects for Investigating Non-Adiabatic Dynamics

Accurately and efficiently simulating open quantum systems, such as electronic states interacting with nuclear vibrations or the photon field are one of the central challenges in theoretical chemistry and condensed matter physics. Directly performing exact quantum dynamics simulations of these systems remain computationally demanding. It is thus ideal to develop trajectory-based approaches, which can accurately describe the non-adiabatic electronic transitions among various electronic states while at the same time captures nuclear quantum effects (NQEs) through classical-like trajectories. Among the trajectory based approaches, recently emerged state-dependent Ring Polymer Molecular Dynamics (RPMD) approaches are promising to provide accurate non-adiabatic electronic dynamics while captures the NQEs with ring polymer quantization. Despite the initial success of state-dependent RPMD formalisms, they either can not be able to capture the electronic coherence effects due to the lack of explicit electronic state description (mean-field RPMD) or can not accurately predict the correct Rabi oscillation (mapping variable RPMD) or can not preserve quantum Boltzmann distribution (ring polymer Ehrenfest) in general. Here, in this thesis, we first developed a novel state-dependent RPMD formalism named Coherent State RPMD (CS-RPMD), which can be viewed as a unified classical theory for electronic states (mapping Hamiltonian) and nuclei (ring polymer). Our preliminary investigations suggest that CS-RPMD holds the promise to be the first trajectory-based approach that simultaneously preserves quantum Boltzmann distribution (in a limiting case) over an ensemble of trajectory and captures electronic Rabi oscillation, which is the first key findings of this thesis. Even though state-dependent RPMD based formalisms were originally developed for investigating quantum dynamics under thermal-equilibrium conditions, we investigated the validity of state-dependent RPMD formalism to study the non-adiabatic dynamics under non-equilibrium initial conditions. Our numerical results suggest that state-dependent RPMD has the potential to accurately describe the non-equilibrium dynamics while accurately captures the NQEs with ring polymer quantization. This is the second key findings of this thesis. Moreover, we apply the ring-polymer (RP) quantization to the cavity photon field in polariton chemistry. To the best of our knowledge, this is the first numerical demonstration of using RP quantization of cavity photon field, which is the third key findings of this thesis. Finally, we derived the state-dependent RPMD formalism from non-adiabatic Matsubara dynamics. This work provides a strong theoretical justification for the dynamics governed by these approaches and not only it explained the numerical success of the state-dependent RPMD approaches for equilibrium and non-equilibrium dynamics, but it also provides a general theoretical framework to understand the limitations of these approaches and further improving them. The rigorous justification of the state-dependent RPMD formalisms is one of the crucial discovery in this thesis