Sampling of initial conditions

The sampling of initial conditions for the purpose of executing an average over them is often done by a so-called Monte Carlo (i.e., a random selection) procedure. Thus the computation of tiie observable dynamical quantities the... 

The Monte Carlo sampling of initial conditions in the method of classical trajectories is illustrated here for the particular problem of computing the reaction cross-section ctr given in terms of the opacity function P b) by, Eq. (3.14),... 

J. Monte Carlo sampling of initial conditions, (a) Discuss why the reaction cross-section computed using Eq. (A.5.7) need not change in value if the value, B, of the maximal impact parameter for reaction is taken larger than it has to be. (b) The reaction rate constant for thermal reactants is obtained. Section 3.1, as an average over the reaction cross-section. Determine a sampling of the relative... 

K. Limitations on the Monte Carlo sampling of initial conditions. Say you want to compute the reaction rate constant when there is a rather small steric factor and/or a rather high activation energy, (a) Will you use a Monte Carlo sampling of initial conditions Discuss in detail why we think that getting an accurate result will require a considerable consumption of computer time. 

Since the early 1960s classical trajectory simulations, with Monte Carlo sampling of initial conditions, have been widely used to study the uni-molecular and intramolecular dynamics of molecules and clusters reactive and nonreactive collisions between atoms, molecules, and clusters and the collisions of these species with surfaces. " For a classical trajectory study of a system, the motions of the atoms for the system under study are determined by numerically integrating the system s classical equations of motion. These equations are usually expressed in either Hamilton s form ... 

Nonrandom sampling of initial conditions for So, following Si So internal conversion, has been performed for chloroacetylene (H—C=C—Cl). The rejection method may be used to sample the different Aij transitions according to their relative probabilities. 

Zero( ) age deposits and waters. Perhaps the most sensitive test of initial conditions for speleothems and analogous carbonate deposits is to undertake U-series measurements of the most recent calcite in actively growing samples such as straw stalactites and the waters that feed them. Surprisingly, few such measurements have been published. 

The answer to the problem lies, I believe, in that a single classical trajectory is never the answer to a dynamical question. There are two reasons. One is a reason of principle. To mimic quantal initial conditions it is necessary to generate an entire ensemble of trajectories, each with somewhat different initial conditions. Then one needs to average over these initial conditions. Now comes a wonderftd technical point (and a tribute to Stan Ulam [27] who, as far as I know, was the first to realize it. The number of initial conditions that one needs to sample is independent of the... 

In principle there are best sets of initial conditions for sampling for a given process, and these can be found by inspecting the matrix of partial... 

The major problem in implementing classical trajectories is that the typical experiment samples a wide distribution of initial conditions. Since each trajectory is run for a specified set of initial coordinates and momenta, a large number of trajectories must be run with a wide range of initial conditions in order to simulate a particular experimental situation. An efficient choice of initial conditions is essential to getting useful information from trajectory calculations. 

An ensemble of 16 nonadiabatic surface hopping trajectories have been calculated sampling different initial conditions from a 300 K ground state simulation. A monoexponential fit to the Sj population gives a lifetime of 1.0 ps. Estimating the lifetime using the relation (10-13) yields the interval [0.9...1.6. ..6.0] ps, which indicates that the result from the exponential fit probably underestimates the lifetime [41,42], The 7H-keto tautomer is thus considerably longer lived than the 9H-keto form. 

While the outcome of an individual trajectory depends in a sensitive way on the details at the surface, the ensemble average is robust. In other words, if an ensemble of initial conditions of the cluster is reused, the yield of dissociation is essentially unchanged even though the new run samples a somewhat different set of conditions at the surface. Similarly, the ensemble-averaged yield is unchanged if a new set of cluster initial conditions is used. We will return to this point later but it is an important practical consideration since it means that a finite number of trajectory computations suffices. 

As has been mentioned above, a new method for the treatment of the dynamics of mixed classical quantum system has been recently suggested by Jung-wirth and Gerber [50,51]. The method uses the classically based separable potential (CSP) approximation, in which classically molecular dynamics simulations are used to determine an effective time-dependent separable potential for each mode, then followed by quantum wave packet calculations using these potentials. The CSP scheme starts with "sampling" the initial quantum state of the system by a set of classical coordinates and momenta which serve as initial values for MD simulations. For each set j (j=l,2,...,n) of initial conditions a classical trajectory [q (t), q 2(t),..., q N(t)] is generated, and a separable time-dependent effective potential V (qj, t) is then constructed for each mode i (i=l,2,...,N) in the following way ... 

Acidity modification of the external surface was characterized by differential scanning calorimetry (DSC) after pre-adsorption of 4-methyl quinoline which cannot enter the pore system. For comparison, sample OW-ZSM-5 and sample lOW-ZSM-5 were subjected to DSC measurements without amine pre-adsorption. In each case, after completion of the first DSC run and re-establishment of initial conditions a second run was performed, and the resulting profile was substracted from that of the first run. In this way, device-dependent factors are eliminated. Measurements were performed by means of a SETARAM DSC-92 equipment under 3 1/h air and nitrogen, respectively, and a heating rate of 5 K/min. 

The general transition path sampling formalism can be applied to various ensembles of pathways differing both in the distributions of initial conditions as well as in the particular transitions probabilities. The specific form of the path probability depends on the process one wants to study and is not imposed by the transition path sampling technique itself. In the following we discuss several path probabilities that frequently occur in the study of condensed matter systems. 

Our trajectories are sampled with the help of a simulated annealing protocol. But how can we test that the sampling is appropriate One measure that can help us to assess the quality of the simulation is the distribution of the orientation of the initial momentum vector. If we sample effectively the space of initial conditions (by sampling complete trajectories), then the momentum vectors should cover all of the orientation space. Alternatively, the vectors of the initial direction of the momentum behave as random vectors (with norm of one). In... 

During initial method development, a set of initial conditions (detector, column, mobile phase) is selected to obtain the first scouting chromatograms of the sample. In most cases, these are based on reversed-phase separations on a Cl 8 column with UV detection. A decision on developing either an iso-cratic or a gradient method should be made at this point. 

Here the q(k (t) are a set of single-mode trajectories for the k mode. The index v refers to the different initial conditions at some time t = 0 satisfied by each of these trajectories. The sampling over initial conditions in each mode (indexed by k) must be large enough to yield a converged average in Eq. (64). To obtain the trajectories, one solves the single-mode Hamilton s equations ... 

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