Understanding Surface Hopping Algorithms and Their Applications in Condensed Phase Systems

While electron transfer plays an important role in a variety of fields, our understanding of electron transfer relies heavily on quantum mechanics. Given the high computational cost of quantum mechanics calculations and the limits of a computer\u27s capability nowadays, the straightforward use of the Schrödinger equation is extremely limited by the dimensionality of the system, which has spurred the advent of many approximate methods. As a mixed quantum-classical approach, fewest-switches surface hopping (FSSH) can treat many nuclei as classical particles while retaining the quantum nature of electrons. However appealing, though, FSSH has some notable drawbacks: FSSH suffers from over-coherence (in addition to its inability to capture presumably rare nuclear quantum effects). Here, in this thesis, we revisit the issue of decoherence from the perspective of entropy, unraveling the nature of the erroneous coherence associated with FSSH trajectories and further justifying the improvements made by the recently proposed augmented-FSSH. Going beyond traditional Tully-style surface hopping technique, we also study new flavors of surface hopping that treat a manifold of electronic states to capture dynamics near metal surfaces. Moreover, we highlight how surface hopping can be used to study electrochemistry and we thoroughly benchmark the surface hopping algorithms against mean-field approaches. This thesis captures 4 years of research which has successfully analyzed the guts of the surface hopping approach for nonadiabatic dynamics both in solution and at a metal surface