Recent theoretical developments together with the increase in computational power hasmade the study of the nonadiabatic quantum dynamics of complex systems accessible tocomputational chemistry and physics. These simulations can provide important new insightsinto a wide variety of photochemical and photophysical problems such as photovoltaic solarenergy conversion, design of data storage devices, and the study of radiation induced cellulardamage mechanisms.Nonadiabatic dynamics implies the breakdown of the Born-Oppenheimer approximation,which makes the implementation of the dynamics more involved because it requires theaccurate description of the nuclear quantum effects together with the calculation of thenonadiabatic couplings between the electronic states. A wide variety of methods have beendeveloped to simulate the quantum dynamics of molecular systems, which differ in thetreatment of the nuclear dynamics. Among these, two main classes have emerged. In thefirst case, the nuclei are represented by a wavepacket and the dynamics typically follows pre-computed potential energy surfaces. In the second class, the nuclear dynamics is describedby a collection of classical trajectories following PESs commonly computed on-the-fly usingelectronic structure methods such as DFT/TDDFT, CASSCF, CC2 and others.The main goal of this tutorial is to introduce the students to the theoretical and practicalconcepts related to these two approches for nonadiabatic dynamics. To this end, we will usethe full nuclear quantum dynamics package MCTDH  and the on-the-fly TDDFT-basedtrajectory surface hopping approach within the CPMD  code. This tutorial will fill a voidin the training on nonadiabatic quantum dynamic, giving the opportunity to learn aboutthe fundamental theory of these processes and (especially) the available methodological andapplicative tools.
In systems which contain multiple pathways and nuclear quantum effects, an explicit quantum treatment of the nuclear degrees of freedom is required. Multi Configuration Time Dependent Hartree, MCTDH  is currently the method of choice, and can treat systems up to about 1000 degrees of freedom. However, the computational cost of such simulations and the requirement of an a priori knowledge of the potential energy surfaces (PESs) is often a limiting factor. On-the-fly calculation of the PESs with MCTDH formulation is now possible but is still in the early stages of development .
On-the-fly nonadiabatic dynamics with TDDFT potential energy surfaces
In many systems of interest quantum nuclear effects are small and therefore one can use the mixed quantum/classical approximation where electrons are treated at quantum mechanical level (using DFT and TDDFT), while the nuclei follow mainly classical trajectories on a surface (or average surface) determined by the nonadiabatic couplings. The most widely used approach for excited state dynamics using trajectories is Tully’s surface hopping  (TSH). This scheme relies on the idea that a complete swarm of independent trajectories can reproduce the dynamics of the density probability of a nuclear wavepacket. Following Tully’s algorithm, trajectories are allowed to ”hop” between electronic surfaces reproducing therefore the splitting of the nuclear wavefunction. Many different implementations of trajectory surface hopping have been proposed, which differ mainly in the way PESs, forces, nonadiabatic couplings, etc. are computed. Within post Hartree Fock methods, multireference configuration interaction singles doubles (MR-CISD) has been successfully coupled to TSH [5, 6]. From the density functional theory side, linear response time-dependent density functional theory (LR-TDDFT) has become the method of choice to perform nonadiabatic dynamics. This method can be applied to the study of rather large system (up to several hundreds of atoms) at the cost of a (sometimes) less accurate description of the potential energy surfaces (see for example [7, 8]). An alternative approach to nonadiabatic dynamics is Ehrenfest dynamics. In this formulation, all quantities required for the dynamics are calculated as mean expectation values giving rise to a so called mean field type of dynamics. When coupled to TDDFT, Ehren- fest dynamics can be performed on-the-fly, the electrons being propagated according to the time-dependent Kohn-Sham equations and the nuclei moving on the corresponding time-dependent mean field electronic potential .