An overview of nonadiabatic dynamics simulations methods, with focus on the direct approach versus the fitting of potential energy surfaces

We review state-of-the-art nonadiabatic molecular dynamics methods, with focus on the comparison of two general strategies: the "direct" one, in which the potential energy surfaces (PES) and the couplings between electronic states are computed during the integration of the dynamics equations; and the "PES-fitting" one, whereby the PES and couplings are preliminarily computed and represented as functions of the nuclear coordinates. Both quantum wavepacket dynamics (QWD) and classical trajectory approaches are considered, but we concentrate on methods for which the direct strategy is viable: among the QWD ones, we focus on those based on traveling basis functions. We present several topics in which recent progress has been made: quantum decoherence corrections in trajectory methods, the use of quasi-diabatic representations, the sampling of initial conditions and the inclusion of field-molecule interactions and of spin-orbit couplings in the dynamics. Concerning the electronic structure calculations, we discuss the use of ab initio, density functional and semiempirical methods, and their combination with molecular mechanics (QM/MM approaches). Within the semiempirical framework, we provide a concise but updated description of our own method, based on configuration interaction with floating occupation molecular orbitals. We discuss the ability of different approaches to provide observables directly comparable with experimental results and to simulate a variety of photochemical and photophysical processes. In the concluding remarks, we stress how the border between direct and PES-fitting methods is not so sharp, and we briefly discuss recent trends that go beyond this traditional distinction.