A Classical and Quantum Trajectory Description of Nonadiabatic Dynamics within Time-Dependent Density Functional Theory

Characterizing and predicting the nuclear dynamics of electronically excited molecules is of paramount importance to the understanding of photochemical and photophysical processes in molecules and to the development of new technologies in domains like solar energy conversion, efficient illumination devices, and medicine. However, the theoretical description of such phenomena remains a challenge for theoretical chemists. The most notable difficulty comes from the breakdown of the widely used Born-Oppenheimer approximation, which considers the time evolution of the nuclei fully decoupled from the one of the electrons. Moving beyond this approximation requires the inclusion of nonadiabatic effects, which induces an entangled electron-nuclear dynamics. Additionally, the convenient approximation that the motion of the nuclear degrees of freedom can be described with classical mechanics is likely to fail in the case of nonadiabatic events. While several theoretical techniques have been proposed to treat both the electronic structure and the nonadiabatic dynamics problems, they all rely on a compromise between accuracy and efficiency. The principal goal of this thesis is to improve the description of molecular nonadiabatic phenomena using classical and quantum trajectories, combined with linear-response time-dependent density functional theory (LR-TDDFT) for the calculation of the electronic structure properties. In the first part of this work, we review several methods used to solve exactly or in an approximate way the nuclear time-dependent Schrödinger equation with quantum or classical trajectories. We then move on to show how the molecular time-dependent Schrödinger equation can be reformulated exactly in terms of quantum trajectories evolving in coupled adiabatic electronic states. This nonadiabatic Bohmian dynamics (NABDY) scheme allows for the description of all nuclear quantum effects like decoherence and tunneling, and is compatible with a nuclear wavepacket dynamics in which the electronic structure information is computed on-the-fly. Furthermore, we discuss how NABDY can be related, through several approximations, to the trajectory surface hopping (TSH) method, which is one of the most commonly applied on-the-fly trajectory-based techniques to describe the dynamics of molecular systems beyond the Born-Oppenheimer approximation in (the unconstrained) configuration space of molecules. The TSH method describes the nuclear wavepacket dynamics with a swarm of uncorrelated classical trajectories, consequently banishing all nuclear quantum effects. Understanding the underlying limitations of TSH is of foremost importance for the improvement of the theory. In this thesis, several one dimensional model systems are used to assess the accuracy of TSH through a comparison with the correlated NABDY dynamics. In the second part, we focus on the electronic structure problem and discuss how LR-TDDFT can be used in the implementation of an efficient on-the-fl