A theoretical model for molecules interacting with intense laser pulses: The Floquet–based quantum–classical Liouville equation

The Floquet–based quantum–classical Liouville equation (F–QCLE) is presented as a novel theoretical model for the interaction of molecules with intense laser pulses. This equation efficiently combines the following two approaches: First, a small but spectroscopically relevant part of the molecule is treated quantum–mechanically while the remaining degrees of freedom are modelled by means of classical molecular dynamics. The corresponding non–adiabatic dynamics is given by the quantum–classical Liouville equation which is a first–order approximation to the partial Wigner transform of full quantum dynamics. Second, the dynamics of the quantum subsystem is described in terms of instantaneous Floquet states thus eliminating highly oscillatory terms from the equations of motion. The resulting F–QCLE is shown to have a well defined adiabatic limit: For infinitely heavy classical particles and for infinitely slow modulation the dynamics adiabatically follows the Floquet quasi–energy surfaces for a strictly time– periodic field. Otherwise, non–adiabtic effects arise both from the motion of the classical particles and from the modulation of the field which is assumed to be much slower than the carrier frequency. A numerical scheme to solve the F–QCLE is based on a Trotter splitting of the time evolution. The simplest implementation can be realized by an ensemble of trajectories stochastically hopping between different Floquet surfaces. As a first application we demonstrate the excellent agreement of quantum–classical and fully quantum–mechanical dynamics for a two–state model of photodissociation of molecular fluorine. In summary, due to the favorable scaling of the numerical effort the F–QCLE provides an efficient tool for the simulation of medium to large molecules interacting with intense fields beyond the perturbative regime.