Collision Trajectories

The evaluation of the reaction cross sections as a function of the initial state of the reactants and final state of the products has been described by Karplus, Porter and Sharma (1965), and Greene and Kuppermann (1968) using classical equations of motions for interacting species. For a given potential V(rb r2, r3), a set of initial coordinates and momenta for the particles determine uniquely the collision trajectory and the occurrence of reaction. The method is described as follows. 

After defining the initial values of the dynamical variables, we can define the final state of collision trajectory as the state in which system leaves the reaction shell, i.e. by inspecting when the distance between any one of the atoms and center of mass of the other two atoms is equal to r0. This will reflect which atom is bound to which other atom. 

For determination of reaction probability and reaction cross section, a large number of collision trajectories have to be considered and appropriate averages over the initial conditions performed. The reaction probability is calculated for a specified initial relative velocity vR (i.e. initial relative kinetic energy), rotational state /, and impact parameter b. The reaction probability is the ratio of number of reactive trajectories to the total number trajectories, i.e. 

Figure 3.19 The collision trajectories of particles in a shear field, r0 = hQ + 2a...

Bakalov et al. treated the trajectories of the helium atom in collision with pHe+ in a semiclassical way, and calculated the pressure shifts and broadening. They obtained numerical values for a numer of transitions, as presented in Table 2. For the precisely known transitions (39,35) —> (38,34) and (37,34) —> (36,33) their theoretical values with realistic collision trajectories (not the linear approximation) turned out to be in excellent agreement with the experimental values. The theoretical treatment of Bakalov et al. was the first quantum chemistry type calculation on the interaction of antiprotonic helium with other atoms and molecules. 

Consider a setup where the two-I-frame states are sent in a collision trajectory remember the I-frame is a classical physics device. The initial state is a direct product of state functions for each I-frame system at a collision point, they continue to be a simple product and they separate away as a simple product. This product defines a nonentangled state. 

At mtermediate electrolyte concentrations ( 10 mol dm" ) and at low volume fractions of the dispersed phase, the charged particles occupy random positions in the system and undergo continuous Brownian motion with transient repulsive contacts when the particles approach each other. The range of the electrostatic repulsive forces is represented by the dashed circle in Fig. 1. which implies that when a similar circle on another particle overlaps with it on a collision trajectory, a transient electrostatic repulsion occurs and the particles move out of range. With most latices the particles. have a real refractive index and their visual appearance is milky white. 

The theory above does not fully take into consideration the energy states of the reacting molecules and the offset of the colhsion as measured by the impact parameter. To accormt for these parameters, Karplus carried out a number of dynamic simulations to calculate collision trajectories. 

Bemshtein, V. and Oref, I. (1995) Minimal separation distance in energy transferring collisions - trajectory calculations, Chem. Phys. Lett., 233, 173-178. 

Equation (45) shows that all we need for the calculation of the cross-section is E (R) — E (R) along the collision trajectory x. However, the accurate wave-mechanical calculation of E (R) — E (R) as a function of the internuclear distance R can only be made for H [38, 39]. A less accurate wave-mechanical calculation has also been made for He [40]. 

FIGURE 17.28 Schematic of the flow around a falling drop. The dashed lines are the trajectories of small drops considered as mass points. Trajectory a is a grazing trajectory, while b is a collision trajectory. 

Calculations for larger drops are complicated by phenomena such as shape deformation, wake oscillations, and eddy shedding, making theoretical estimates of E difficult. The overall process of rain formation is further complicated by the fact that drops on collision trajectories may not coalesce but bounce off each other. The principal barrier to coalescence is the cushion of air between the two drops that must be drained before they can come into contact. An empirical coalescence efficiency Ec suggested by Whelpdale and List (1971) to address droplet bounce-off is... 

Inclusion of the possibility of interaction of the atom with different normal modes of the molecule in proportion to the squares of the projections of the normal modes along the collision trajectory. 

Fig. 2-4 Collision trajectories, (a) Elastic hard spheres (b) elastic hard spheres with superposed weak central attractions (c) molecules with central finite repulsive and attractive forces.

Fig. 2-7 Calculated trajectories for collisions of H + H2. (a) Typical nonreactive H + H2 collision trajectory, (b) Typical reactive H -I- H2 collision trajectory, j — 5, = 0, i = A x 10 cm/sec. (Adapted from Karplus et al [24].)...

The basic principles of CPS are described in detail in Ref 2. In essence, the method is based on generating collisions between two mierometer-sized particles (droplets) under simple shear flow eonditions and extracting the foree-distance relationships by analyzing the asymmetry of collision trajectories before and after the collision. Figure... 

Microscopic reversibility is a non-thermodynamic argument for the reversibility of elementary chemical reactions. This principle derives from the fact that the equations of motion for atoms and molecules are invariant under time reversal. The momenta of a pair of atoms or molecules on a particular collision trajectory change as a result of the collision. If, in the hnal state after the collision, all of the momenta of the translational, rotational, and internal motions of the collision products are reversed, then the trajectory is exactly retraced and a state similar to the initial state is reached, except that the momenta are reversed. This will be true irrespective of whether classical or quantum mechanics are used to describe the equations of motion, and is a consequence of the fact that the equations of motion have the same form when t is replaced by —t. It is true of both unreactive collisions (energy transfer) and reactive collisions. Thus, we see that reactions are required to be reversible when the time dependent behavior of the energy states is considered. 

Similarly, the rate coefficient for a thermal reaction occurring with the influence of a spherically symmetric potential V(r) can be calculated from equation (63) by relating the cross-section to the potential. A useful relationship from classical scattering dynamics [16] is found in terms of the impact parameter, b. The impact parameter is the distance of closest approach between two particles in the absence of an interparticle force. At large separation, the collision trajectories of two particles will be parallel straight lines, and the impact parameter is the perpendicular distance between the trajectories. The cross-section is given by equation (64),... 

Figure 4.2 The collision trajectory in the c.m. system. The solid curve represents a trajectory with initial velocity v, impact parameter b, and mass fi. The relative separation R(t) is uniquely defined in terms of the distance /land the orientation angle if. The trajectory is symmetric about the apse line, which passes from the origin through Rq where Rq is the distance of closest approach. The final deflection angle is X =jt — 2 0 where ifo is the value of f at the mid-point of the trajectory.

The potential deflects the colliding molecules from their original, pre-collision paths. The deflection caused by the collision is defined as the angle between the final and initial relative velocity vectors. Section 4.1 showed that the initial collision energy and the impact parameter specify a imique collision trajectory. It follows that the deflection is a unique function of the initial collision energy and impact parameter and can be computed. Section 4.2.3, once we have determined the colhsion trajectory R t). 

Figure 4.4 Collision trajectories at different impact parameters for a given collision energy. The ordinate is the reduced impact parameter b —b/Rrn where fim is the equilibrium distance of the well in the potential. The resulting deflection function is shown on the left-hand side. Note the qualitative difference between hard-sphere scattering and scattering by a realistic potential for hard spheres there is no attractive part of the potential so going from the deflection xlWto bis single valued x(b) is defined uniquely by the value of b. But in the presence of a well in the potential there can be more than one value of bthat results in scattering into a given value of x(b).

Figure 5.13 Plot of a collision trajectory for a reactive coiiision of the "direct" type (the time scale is that for the H + H2 reaction). The collision is direct because of the fast switchover between the old and new bonds. Note also the vibration of the reactant BC molecule prior to the encounter with the reactant atom and the low-amplitude oscillation of the newly formed AB bond, indicating only a minimal product vibrational excitation. To interpret the dynamics recall that the slope of a plot of distance vs. time is the velocity. From Newton s first law, a change in the slope in such a plot indicates that a force is acting.

H. A hard-sphere model, (a) Develop the hard-sphere model of Problem D in Chapter 5 for a non-reactive A - - BC collinear colUsion. In this model the interaction time is infinitesimally short so that energy transfer is efficient. Hence the model is used to determine the pre-exponential factor for the efficiency of tile collision, (b) The role of the masses. Is it more efficient for A to come from the direction of tiie B atom or from the direction of the C atom (c) Draw a trajectory for a BC molecule that is initially vibrationally excited. Does the outcome depend on tile phase, cf Section 5.2.2, of the initial vibration (d) Under what conditions will A collide more than once with the B atom (e) If you are geometrically minded, show that you can draw a construction that, for a given mass combination, allows you to determine the collision trajectory as a straight line superimposed on your drawing. There can be more than one line. What is tiie physics of the different lines (f) Hint for part (e). Show that certain mass combinations, with a BC molecule that is initially vibrationally cold, will not result in any energy transfer. 

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