Collision calculation runs

Qei and Qvtbrot denote electronic and rovibrational partition fimctions, respectively. In general, the contributions of the internal degrees of freedom of A and B cancel in g and gviiroXA)gv iro((B ), such that only contributions Irom the external rotations of A and B and the relative motion, summarized as "transitional modes", need to be considered. Under low temperature quantum conditions, these can be obtained by statistical adiabatic channel (SACM) calculations [9],[10] while classical trajectory (CT) calculations [11]-[14] are the method of choice for higher temperatures. CT calculations are run in the capture mode, i.e. trajectories are followed Irom large separations of A and B to such small distances that subsequent collisions of AB can stabilize the adduct. 

The connection between the classical trajectories and the quantum transition is established by the classical excitation function 2( i), an example of which is depicted in Fig. la. ri2 q ) is the final classical vibrational quantum number (not necessarily an integer) as a function of the initial phase of the oscillator. It is calculated by running - for a particular collision energy - trajectories with different initial phase angles q. The particular trajectories which contribute to the probabilities in Eq. (1) are found using the equation... 

Figures 15(a) and 15(b) show the calculation results of EEPF and VDF of N2 X, respectively. The calculations were run at Tg = 1200 K, Ne = 5.0 x IQU cm-3 and Te = 2.5 - 4.5 eV. These parameters were chosen to correspond to our experimental results at P = 1.0 Torr and 2 = 60 mm obtained in the experimental apvparatus shown in Fig. 3. It should be repeated that we choose a reduced electric field so that the electron mean energy s) equals (3/2)/cTe when we compare the numerical calculation with the number densities obtained experimentally by OES measurement. Obviously, the EEPF is not like Maxwellian. It has a dip in the range from 2 to 3 eV owing to frequent consumption of electrons with this energy range due to inelastic collisions to make vibrationally excited molecules. Meanwhile, Fig. 15(b) shows that the VDF is also quite far from the Maxwellian distribution. The number density of the vibrational levels shows rapid decrease first, then moderate decrease, and rapid decrease again as the vibrational quantum number increases. This behaviour of the VDF of N2 X state has been frequently reported, and consequently, our model is also considered to be appropriate. If we can assume corona equilibrium of some excited states of N2 molecule, for example, N2 C state, we can calculate the number density of the vibrational levels of the excited state that can be experimentally observed. This indicates that we can verify the appropriateness of the calculated VDF of the N2 X state as shown in Fig. 15(b).

To evaluate the overall speed-up obtained as a result of the program restructuring we have performed test runs of the lOS program on the IBM 3090. Calculations have been carried out for the Li- HF reaction at one value of 7 and using two different values of the partial wave set dimension (NV). Detailed run times obtained for two values of the collision energy are reported in Table I. 

In previous simulation studies of collision induce effects in rare gases, y=0 and only the first term in (56b) contributes to AA2 [92]. The first and second rank collision induced correlation functions, (57) and (60), are notoriously difficult to calculate they involve four point correlation functions, (i induced by j correlated with k induced by Z and the simulation runs in [90, 91, 92] are for tens of thousands of timesteps). 

The time of arrival of the next train on an adjacent line ry is calculated based on the timetable. Based on the train type of this train (speed and emergency braking capability) and the time to warn the approaching train, the time required for this train to stop short of the derailed train is estimated, tj. The arrival time of the next train will follow a distribution to reflect the possibility of late (or early) running. It is not expected that this distribution is symmetrical, but for simplicity a symmetrical triangular distribution is assumed, based around t J but with a half-width C". The probability of a secondary collision is... 

Most of the details of the trajectory calculations are the same as described previously. All of the calculations are quasiclassical, i.e,y the vibrational and rotational action variables are restricted to have the correct quantized values at the start of the collision but, except for this and the final-state analysis after the collision, the internuclear motion is purely classical under the influence of the quantum mechanical potential energy surface.The only initial states considered were para-hydrogen states, i.e., states with even rotational quantum numbers. As will be indicated, some runs are for a given initial vibration-rotation state while other runs are for vibration-rotation states selected by Monte Carlo methods from the distribution of quantized states that has the desired temperature. Similarly, the relative translational energy is sometimes fixed for a run while for other runs a random selection is made from a predetermined distribution characteristic of a temperature. In the latter case importance sampling is sometimes used to improve the convergence of the calculated quantities such as the... 

Features of the collision dynamics have just been discussed in terms of diagrams that simply show how potential energy varies along a single progress coordinate. Even in collinear collisions in which the system only just surmounts the barrier, inertial effects on the downhill run ensure that no trajectory exactly follows this path. Consequently, it is important that simplified explanations of collision dynamics are only given after full three-dimensional trajectory calculations. Nevertheless, such simplified descriptions, usually based on a consideration of the dynamics of collinear collisions, are frequently valid and valuable. 

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