Mean-field trajectory method discussion

Let us first consider the relation to the mean-field trajectory method discussed in Section III. To make contact to the classical limit of the mapping formalism, we express the complex electronic variables imaginary parts, that is, mean-field Hamiltonian function which may be defined as... 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

Finally, we consider Model V by describing two examples of outer-sphere electron-transfer in solution. Figures 7 and 8 display results for the diabatic electronic population for Models Va and Vb, respectively. Similar to the mean-field trajectory calculations, for Model Va the SH results are in excellent agreement with the quantum calculations, while for Model Vb the SH method only is able to describe the short-time dynamics. As for the three-mode Model IVb discussed above, the SH calculations in particular predict an incorrect long-time limit for the diabatic population. The origin of this problem will be discussed in more detail in Section VI in the context of the mapping formulation. 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

---