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We investigate two approaches to derive the proper Floquet-basedquantum-classical Liouville equation (F-QCLE) for laser-driven electron-nucleardynamics. The first approach projects the operator form of the standard QCLEonto the diabatic Floquet basis, then transforms to the adiabaticrepresentation. The second approach directly projects the QCLE onto the Floquetadiabatic basis. Both approaches yield a form which is similar to the usualQCLE with two modifications: 1. The electronic degrees of freedom are expandedto infinite dimension. 2. The nuclear motion follows Floquet quasi-energysurfaces. However, the second approach includes an additional cross derivativeforce due to the dual dependence on time and nuclear motion of the Floquetadiabatic states. Our analysis and numerical tests indicate that this crossderivative force is a factitious artifact, suggesting that one cannot safelyexchange the order of Floquet state projection with adiabatic transformation.Our results are in accord with similar findings by Izmaylov et al., who foundthat transforming to the adiabatic representation must always be the lastoperation applied, though now we have extended this result to a time-dependentHamiltonian. This paper and the proper derivation of the F-QCLE should lay thebasis for further improvements of Floquet surface hopping.

journal of chemical physics online/the journal of chemical physics/journal of chemical physics

On the Proper Derivation of the Floquet-based Quantum Classical  Liouville Equation and Surface Hopping Describing a Molecule or Material  Subject to an External Field