Independent trajectories

The key practical question is How can the sample size be quantified Initial approaches to answering this question were provided by Grossfield et al. [1] and by Lyman and Zuckerman [10]. Grossfield et al. employed a bootstrap analysis to a set of 26 independent trajectories for rhodopsin, extending the previous... 

Figure 1 Schematic illustration of the parallel-replica method. The four steps, described in the text, are (A) replication of the system into M copies, (B) dephasing of the replicas, (C) propagation of independent trajectories until a transition is detected in any of the replicas, and (D) brief continuation of the transitioning trajectory to allow for correlated events such as recrossings or follow-on transitions to other states. The resulting configuration is then replicated, beginning the process again. Reprinted, with permission, from ref. 6. Copyright 2002 by Annual Reviews, www.annualreviews.org...

The Tar—Tar sequence as well as the A-form RNA and B-form DNA equivalents of Tar—Tar (see Fig. 20.3) do not contain palindromes. However, we have data from multiple independent simulations and the analysis of Ponomarev et al. can be repeated using data from pairs of simulations. The equivalent test is to compare the residue-resolved ion-contact distributions between replicate trajectories. Such comparisons are a stringent test if each simulation was separately initialized with randomized ion starting positions and velocities. The PCCs for the A-form RNA simulations are shown in Fig. 20.4. Error bars denote the standard errors across the six possible pairwise comparisons across four independent trajectories, each of... 

Preliminary MD runs were done, in the absence of the solvent, on a system composed by three PGA chains each counting eight imits. Three independent trajectories each of the duration of 50 ps were generated after... 

F (s) — Fa (s,t)) computed over 1000 independent trajectories is represented as a dashed line, with the error bar given by (10). The units of the error is the same as the units of the free energy. After [56]... 

The main practical problem in the implementation of the mixed quantum-classical dynamics method described in Section 4.2.4 is the nonlocal nature of the force in the equation of motion for the stationary-phase trajectories (Equation 4.29). Surface hopping methods provide an approximate, intuitive, stochastic alternative approach that uses the average dynamics of swarm of trajectories over the coupled surfaces to approximate the behavior of the nonlocal stationary-phase trajectory. The siu--face hopping method of Tully and Preston and Tully describes nonadiabatic dynamics even for systems with many particles. Commonly, the nuclei are treated classically, but it is important to consider a large niunber of trajectories in order to sample the quantum probability distribution in the phase space and, if necessary, a statistical distribution over states. In each of the many independent trajectories, the system evolves from the initial configuration for the time necessary for the description of the event of interest. The integration of a trajec-... 

Equations [18] and [20] are stochastic differential equations because the force iR is taken from a random distribution. The stochastic nature means that one must produce many independent trajectories that are averaged together, producing the time evolution of an ensemble-averaged property. 

Not will, but may. Trajectories that originate from the reactants and cross the barrier an odd number of times will form products. The theory counts each such crossing as an independent trajectory and also for this reason it overestimates the rate, i.e., it provides an upper bound. 

Therefore, we present here our semiclassical Field-Induced Surface Hopping (FISH) method [59] for the simulation and control of the laser-driven coupled electron-nuclear dynamics in complex molecular systems including all degrees of freedom. It is based on the combination of quantum electronic state population dynamics with classical nuclear dynamics carried out on the fly . The idea of the method is to propagate independent trajectories in the manifold of adiabatic electronic states and allow them to switch between the states under the influence of the laser field. The switching probabilities are calculated fully quantum mechanically. The application of our FISH method will be illustrated in Sect. 17.6 on the example of optimal dynamic discrimination (ODD) of two almost identical flavin molecules. 

In order to connect Eqs. 17.14-17.15 with classical molecular dynamics on the fly the diagonal density matrix elements pu q, p, t) which are functions of the coordinates q and momenta p can be represented by independent trajectories propagated in the ground and excited electronic states, respectively. Thus, employing a number of Ntraj trajectories, puiq,p,t) can be represented by a swarm of time-dependent 5 functions... 

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