Statistical convergence

Iterate from step (1) and run as many trajectories as needed to reach statistical convergence. 

In order to study the origin of the deviations observed, we first consider the statistical convergence of the QCL data. As a representative example. Fig. 14 shows the absolute error of the adiabatic population as a function of the number of iterations N—that is, the number of initially starting random walkers. The data clearly reveal the well-known 1/Vn convergence expected for Monte Carlo sampling. We also note the occurrence of the sign problem mentioned above. It manifests itself in the fact that the number of iterations increases almost exponentially with propagation time While at time t = 10 fs only 200 iterations are sufficient to obtain an accuracy of 2%, one needs N = 10 000 at t = 50 fs. 

The surface hopping study was rather expensive in terms of CPU time, and consequently large numbers of trajectories could not be run. This is important to obtain statistically converged dynamical properties. The main goal of the surface hopping study was thus not to obtain such information but to provide mechanistic insight into the photodissociation and subsequent relaxation processes. The semi-classical work in the full space of nuclear coordinates provides the important vibrational degrees of freedom that one needs to include in any quantum model of the nuclear motion. This will now be described. 

First, we know that the denominator of the F statistic converges to a2. Therefore, the limiting distribution of the F statistic is the same as the limiting distribution of the statistic which results when the denominator is replaced by a2. It is useful to write this modified statistic as W = (l/o2)(Rp - q) [R(X X )"1R ]"1(RP - q)//. 

A comparison between these three theoretical models will be made here. One aspect that should be emphasized is that the results reported are statistically converged, and therefore, statistical discrepancies are not included. All the QM calculations were made with the Gaussian98 program [28]. 

Having obtained statistically converged results for the dipole polarizability of liquid argon, we now consider the resulting values for the dielectric constant. Again, we use the three theoretical models and simply obtain the dielectric constant e, from the dipole polarizability, using the Clausius-Mossotti equation [34] ... 

Appendix 6. Brillouin-Wigner Perturbation Theory of the Quasi-species. Appendix 7. Renormalization of the Perturbation Theory Appendix 8. Statistical Convergence of Perturbation Theory Appendix 9. Variables, Mean Rate Constants, and Mean Selective Values for the Relaxed Error Threshold... 

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