Binning trajectory results

Figure 2. Diabatic (left) and adiabatic (right) population probabiUties of the C (fuU line), B (dotted line), and X (dashed line) electronic states as obtained for Model II, which represents a three-state five-mode model of the benzene cation. Shown are (A) exact quantum calculations of Ref. 180, as well as mean-field-trajectory results [(B), (E)] and surface-hopping results [(C),(D),(F),(G)]. The latter are obtained either directly from the electronic coefficients [(C),(F)] or from binned coefficients [(D),(G)].

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. 

Figure 11. Time-dependent population probability of the upper (a) adiabatic and (b) diabatic electronic state of Model 1. The quantum-mechanical results (thick lines) are compared to SH results obtained directly from the electronic coefficients (dashed lines) and to SH results obtained from binned coefficients (thin solid lines), reflecting the percentage N2(t) of trajectories propagating on the upper adiabatic surface. Panel (c) shows the absolute number of successful (thick hue) and rejected (thin line) surface hops occurring in the SH calculation.

To analyze the results in these figures, we note that a purely statistical distribution would lead (in the limit of an infinite number of trajectories) to flat k and m distributions. Recalling the definitions of < k > and < m > from Section II in terms of an average of J, and using bins that are two units of in width, we calculate the statistical value of as follows ... 

---