Selective initial conditions

We have investigated another procedure for reducing the computational expense of the AIMS method, which capitalizes on the temporal nonlocality of the Schrodinger equation and the deterministic aspect of the AIMS method. Recall that apart from the Monte Carlo procedure that we employ for selecting initial conditions, the prescription for basis set propagation and expansion is deterministic. We emphasize the deterministic aspect because the time-displaced procedure relies on this property. 

Going back to Miller s synthesis in the flask, one question is why a-amino acids have been obtained and not, for example, p-amino acids, cyclic diketopiperazines, or some other isomers. The answer is important a-amino acids form because they are the most stable products under the selected initial conditions. In other words the formation of those a-amino acids is under thermodynamic control. The same can be said for Oro s synthesis of adenine and other prebiotically low-molecular-weight substances formed in hypothermal vents, or found in space certain molecules and not others form because they are thermodynamically more stable. 

Continuity of classical dynamics with respect to initial conditions suggests that the search for quaslperlodlc trajectories in 3D should begin by selecting initial conditions close to the known colllnear RPOs, but rotated slightly out of the colllnear plane. To a... 

Finally, in selecting initial conditions for bimolecular reactions, it is important that the coordinates and momenta for the atoms have random classical phases as given, for example, by Eqs. (2.10) and (3.20). If this is done, the trajectory results will be independent of the initial separation between the reactants, if they no longer interact [46]. 

Guide for selecting initial conditions for capillary-electromigration separation techniques... 

The laws of physics needed to answer the question are well known. Since the potential energy surface is given, one knows the masses of the colliders and so one only needs to solve the SchrUdinger equation. The problem of course is that the number of coupled equations that need to be solved is enormous and not yet within reach of present day computers. Necessarily then the theorist is restricted to studying model systems and construction of approximations. One type of approximation is to solve the exact classical mechanical equations of motion. One selects initial conditions which correspond to the experimental initial state, integrates the equations of motion forward in time till the process is over and then obtains cross sections, product distributions etc. In essence, Hamilton s equations of motion serve as a black box , whose structure is determined by the masses and the potential energy surface. This black box provides the necessary transformation from initial conditions to final conditions. 

The choice of initial conditions will depend on the safety aspects and acceptance criteria considered for analyses. Table 11 may serve as a guide to selecting initial conditions for analyses of accidents associated with degradation of heat removal from the core and of accidents leading to a pressure rise in the coolant circuit. 

In some molecular dynamics work, the process of equilibration has accounted for a substantial fraction of the total computer time devoted to the calculation. If the intermolecular potential has a sufficiently short range, however, it is possible to use only a fraction of the molecular dynamics system for equilibration. One can set aside a subsystem containing, say, an eighth of the total number of molecules, select initial conditions, and allow the... 

Fig. 6. The time-dependent energies of the first three vibrational levels of the coupling oscillator on the two diabatic surfaces according to the MTDM (see text). The numbering in the two panels refers to the initial conditions of Fig. 1(c). As explained in the text, the plotted energies are obtained considering that the motion along the tuning oscillator is a classical trajectory on the initial diabatic surface. As in Fig. 2, the points are placed at the crossing which are relevant for the selected initial condition, while the arrows indicate possible paths.

This example shows what one might have expected. The freedom in selecting initial conditions is limited by the constraint. Clearly, the initial conditions must meet the constraints. 

Using the thin layer initial condition the lower layer increases in height due to mass transfer from the upper to lower layer. Thus the initial mass of solids in the upper layer must be sufficient to form the lower layer. In order to achieve this the volume fraction of the gas in the upper layer must be reduced, otherwise the model will fail when the gas volume fraction in the upper layer reaches unity. Using the thick layer option there is no need to do this since the mass transfer is in the opposite direction. As a result selecting initial conditions using the thick layer option is more straightforward. This corresponds with the almost foil-bore flow of densely packed material may be observed at the discharge from a blow tank (pressure vessel) feeder. 

The above presentation describes how the collision impact parameter is sampled to calculate reaction cros.s-sections. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is de.scribed for atom -I- polyatomic collisions. Initial energies are chosen for the reactants, which correspond to quantum mechanical vibrational-rotational energy levels. This is the quasiclassical model. " Extensive reviews have been given of the procedure for selecting initial conditions for atom + diatom collisions. 

Since the methods for selecting initial conditions are thoroughly described elsewhere, we will give only a brief overview of them, mentioning some recent advances, and refer the interested reader to the previous, more complete presentations of the methods. 

Various forms can be used for S(r). The period over which the coupling is turned on must be long compared to the periods of the vibrations. An interesting extension of adiabatic switching to select initial conditions for a polyatomic molecule (CD4) has recently been reported. ... 

Recently, an extensive series of calculations for state-selected initial conditions for... 

For regions of intense seismic activity, the crustal stmcture is frequently defined in terms of one-dimensional velocity models. Detailed three-dimensional crustal models have also become available for specific regions in the benefit of three-dimensional wave propagation codes that may effectively take into account basin effects and complex fault geometries at the cost of increased computational demands. The characterization of the seismic source is a more complicated issue. For kinematic descriptions of the earthquake source, source parameters such as slip, rise time, rupture velocity, and slip function should properly be quantified and a priori defined. On the other hand, for dynamic descriptions of the earthquake source, the source parameters may vary as long as the elastodynamics equation with a prescribed fracture criterion on a predetermined fault plane is satisfied. The selected initial conditions and failure criterion determine the time and space evolution of the fault rupture in a dynamic source model. 

Procedures for selecting initial values of coordinates and momenta for an ensemble of trajectories has been described in detail in recent chapters entitled Monte Carlo Sampling for Classical Trajectory Simulations and Classical Trajectory Simulations Initial Conditions. In this section a brief review is given of methods for selecting initial conditions for trajectory simulations of unimolecular and bimolecular reactions and gas-surface collisions. 

Here, the nonrandom excitation of C2H4F is described by the dynamics of the F - - C2H4 bimolecular reaction. To simulate chemical activation, proper initial conditions must be chosen for the reactants and for their relative properties. The procedure for choosing initial conditions for the reactant s relative properties is given below in the discussion of bimolecular reactions. The quasi-classical method may be used to select initial conditions for molecular reactants. The energy for a symmetric-top polyatomic molecule in a specific vibrational-rotational state may be approximated by the harmonic oscilla-tor/rigid rotor model... 

Fig. 14. Rate coefficients for the decay of clusters as a function of energy. The points represented by triangles are the trajectory results for the decay of clusters formed in A + A3 collisions. The circles are trajectory results for decay of clusters with Monte Carlo selected initial conditions. The solid curve is the best fit of the RRK equation to the results for the noncollisionally formed clusters. Reproduced from reference 3 with permission of copyright holder.

---