Multidimensional systems classical solution

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

Three general approaches can be used to evaluate the effects of solvent on absorption spectra Continuum theories describe the solute as lying in a cavity in contact with a polarizable continuum. In semicontinuum theories the first few shells of solvent molecules around the solute are treated explicitly, and the remaining solvent molecules are treated as a continuum. Fully discrete theories treat as many solvent molecules as possible in full quantum mechanical detail, on the same footing as the solute molecule. The main problem with this approach is the enormous number of degrees of freedom associated with the multidimensional solute-solvent system. Instead of a fully quantum mechanical treatment, it is imperative to use an approximation that separates the system into classical and quantum mechanical parts. 

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