Trial configuration

The step size, Ar, is the maximum allowed atomic displacement used in the generation of trial configurations. The default value of r in HyperChem is 0.05 Angstroms. For most organic molecules, this will result in an acceptance ratio of about 0.5, which means that about 50% of all moves are accepted. 

In the Metropolis algorithm the components of the displacement vector 8y are obtained by uniformly sampling from the domain D, centered in the coordinates of the molecule y in the i configuration, and defined by the maximum allowed displacement max an(j maximum allowed rotation 8(>MAX parameters (convergence celerity greatly depends on the values used for these two parameters). That is, all the positions inside domain D have the same probability to be chosen as new trial configurations. Thus ... 

The constraints changed from one trial configuration of the reaction system to the next, but typically included things like the minimum coolant temperature to permit efficient utilization of the heat of reaction as process steam, the maximum allowable aldehyde concentration in the condensed crude product to avoid refining and product specification problems, and a prescribed reactor pressure drop to insure adequate flow distribution among the reactor tubes at a minimum energy cost. All of these are implicit constraints — they establish the maximum or minimum levels for certain response variables. Explicit constraints comprise the ranges for search variables. 

Having generated a trial configuration, the total energy of the new configuration, new is determined. Next, calculate the quantity... 

SA(GSA) employs a three step mechanism to build a new trial configuration [7] ... 

In our application, r is therefore an estimate of the statistical inefficiency in the measurements that has routine use in numerical analysis. Later it will be shown that r has a dramatic effect on the results found u.sing Eqs. 11-16, and hence the choice of estimators is an important consideration. To estimate r,-, two different techniques have been employed. A crude estimator can be constructed from the acceptance ratio with number of trial configurations number of accepted configurations... 

The matrix elements are the product of two factors, namely an a priori probability aij of generating a trial configuration Sj from a configuration si and an acceptance probability Pij of accepting the trial configuration as the new state. The detailed balance condition can thus be written as... 

It follows from these considerations that it is not always desirable to sample the property in (12) in each successive configuration, in particular not if the calculation of is computationally expensive. However, it is crucial that the sampling takes place at a regular interval. A typical mistake in Monte Carlo calculations is that a sample only is taken after a fixed number of trial configurations have been accepted. This is wrong, as can also easily be seen from the following example At low temperatures a configuration with a low... 

Several variants of the basic SA algorithm have been developed, which differ in the choice of the CF, in the design of the annealing schedule or in the procedure for the generation of the trial configurations. For example, Andreev et al. reduce T at a preset rate and, for each T, perform several moves which increase as the acceptance ratio decreases David et introduced the following relevant novelties, included in the computer program DASH ... 

Let us go back to generalities. We want to generate a Markov chain of trial configurations. A Markov chain is a sequence of trials in which the outcome of each trial is one of a set of states, the state space or configurational space, and depends only on the state that immediately precedes it... 

To emphasize how the Metropolis scheme works in practice, let us return once more to the hard-disk example. Here the matrix am represents the probability of moving a particle to a new trial positions, which is achieved by the generation of two random numbers for displacements in the x and directions. To make the underlying matrix symmetric, the displacements are calculated from dmlJ2 — 1), where [Pg.6]

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