Quantum and Semiclassical Dynamics

We introduce quantum, semiclassical and mixed quantum-classical approaches for the study of the dynamics of chemical phenomena with a specic emphasis on photoinduced molecular processes occurring on coupled electronic states. We start from basic concepts presenting the equation of motions for pure and mixed states and for the Wigner representation of operators, as well as the hydrodynamic formulation of quantum dynamics. Afterward we introduce the Born-Oppenheimer approximation and a suitable form of the molecular Hamiltonian to study nonadiabatic dynamical processes and we brie y discuss the representation of the evolving wavepacket on basis sets and on grids of points. In the second part of this contribution we illustrate a number of popular approaches for the numerical propagation of quantum states, like second order dierencing scheme, split-operator method, short iterative Lanczos and the multicongurational time dependent Hartree method. We then introduce semiclassical methods, discussing the initial value representation and the Hermann-Kluk propagator and mixed quantum-classical methods like surface hopping, full-multiple spawning, quantum-classical Liouville approaches, and methods based on quantum trajectories. We conclude brie y comparing the presented approaches, highlighting their strengths and limitations.