Relative translational motion

The tilde over operator r here and below indicates that the operator is calculated in the EFA, as was done in [185, 186], This treatment ignores the influence of rotational transitions, caused by the anisotropic part of the interaction, on relative translational motion of colliding particles. Therefore f (.K , differs slightly from the true operator r(K . What... 

The reaction pair 9.100 and 9.101 can also be considered from the point of view of an association reaction of A and B to form product C. If species A and B collide and form a bond, the resultant species will contain a great deal of vibrational energy in the newly formed bond. Essentially all of the energy of the relative translational motion of... 

The subsequent fate of C is a competition among the other reactions listed in the scheme. The reverse of reaction 9.132 occurs when the highly energetic C decomposes unimolecularly back to the reactant molecules, with rate constant kd, the internal energy in its vibrations is converted to relative translational motion of A and B when C falls apart. 

Generally, T2 relaxation times are very sensitive to slower relative translational motions of the polymer chains and can provide information on intramolecular couplings, such as chemical crosslinks and chain entanglements. Numerous studies on both permanent and temporary networks are presented in a series of papers by Charlesby and co-workers 74,86 94). In the case of extracted polymer networks, T2s relaxation is observed in the crosslinked (gel) fraction, while T2 relaxation occurs in the soluble fraction of the irradiated polymer86 . It is shown that the fraction of more mobile protons, (1-f). has the same general trend with increasing... 

The function tr describes the relative translational motion of the photofragments. This motion can be described in the semi-classical approximation (except in the turning point region) by an oscillating wavefunction for which the number of oscillations increases with an increase of the relative momentum. An increase of the number of oscillations results in a decrease of the FC factor. Hence, the semiclassical behavior of the translational wavefunction makes a transition to a state with large momentum less favorable. Because of conservation of energy, the resulting state is characterized by a small vibrational quantum number. 

Figure 1. The frames of reference S and S in relative translational motion. In the frame of reference S, Lorentz transformation and special relativity principles are valid. In the frame of reference S, superluminal transformation and SLRT principles are valid.

In simple collision theory, for a bimolecular reaction this critical energy, eo, is the kinetic energy of relative translational motion along the line of centres of the colliding molecules, loosely described as the violence of the collision. 

Since the outcome of the collision only depends on the relative motion of the reactant molecules, we begin with an elimination of the center-of-mass motion of the system. From classical mechanics it is known that the relative translational motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass fj, = rri nif)/(m + mB), relative to a fixed center of force. This result can be generalized to molecules by introducing proper relative coordinates, to be described in detail in Section 4.1.4. 

We insert Eq. (5.2) into Eq. (5.1), and note that the factors in front of the integral are related to the partition function associated with the relative translational motion of the reactants, see Eq. (A. 14) ... 

To describe how the required superposition state [Eq. (7.5)] can be constructed in the laboratory requires some introductory remarks. Note first that Eqs. (7.2) to (7.7) and the E, q, m 0) states are understood to be in the center-of-mass coordinate system and describe the relative translational motion as well as the internal state of A and B. In typical A-B scattering, separating out the center-of-mass motion comes about in a straightforward way. That is, let rA and rB denote the laboratory position of A and B and ikA, Mr denote their laboratory momenta. The relative momentum k, relative coordinate r, center-of-mass momentum K and position Rcm are defined as... 

Fig. 2.1. Bead-spring model. Each circle represents frictional point which exerts force on surrounding liquid in relative translational motion. Zigzag line represents spring which exerts force proportional to its length on beads at both ends...

Consider first the case of a polar molecule that is not rotating as it collides with an ion. For this case the angular momentum of the system is all in relative translational motion and depends only on the impact parameter b. If the dipole orientation can adjust sufficiently rapidly as the collision develops, the angle y in the potential of (11) is always zero, and the largest possible cross section results. This maximum cross section is calculated according to the procedure outlined in the introduction with the effective potential... 

If, however, the same amount of energy is present solely as translational kinetic energy, then the system moves along a successful encounter trajectory C and travels smoothly over the saddle point into products. We can therefore conclude that reactions with attractive potential energy surfaces proceed more efficiently if the energy is in relative translational motion. Moreover, the potential surface shows that once past the saddle point the trajectory runs up the steep wall of the product valley, and then rolls from side to side as it falls to the foot of the valley as the products separate. In other words, the products emerge in a vibrationally excited state. 

The time dependence of the interaction can arise from variations in the interspin vector, r, due to the relative translational motions of the / and 5 spins. It can be shown that if the spins I and S are in different molecules and can be treated as diffusing independently, an expression for the reduced spectral density 7)1 =7d,co/- d,o can be obtained. The result is—... 

This may be called a body-frame label set. Alternatively one may use the space-frame labels v, j, and / of the initial collision pair, where / is the orbital angular momentum quantum number of the relative translational motion of FF with respect to H2. The quantum number l is constrained by (137)... 

It is rather fascinating to note that the dynamic properties of the protein hydration layer have been studied by so many different techniques. Initially, there was controversy about the accuracy of the different techniques employed. The situation became cleared when proper care was taken to isolate and interpret the results. For example, DR and SD are mostly sensitive to the rotational motion of the water molecules and the protein side-chain motions, while NOE is sensitive to the relative translational motion between the protein and the water molecules. Naturally they... 

In most of the numerous theoretical treatments of V-T energy transfer in non-chemical collisions, a simplified intermolecular potential is assumed for example, angular anisotropy is usually ignored. Furthermore, the coupling between the relative translational motion of A and BC and the vibration of BC is assumed to be weak. In the theory of Schwartz, Slawsky, and Herzfeld, -" for example, only collinear collisions are treated explicitly and A is assumed to interact only with B, so that the potential is... 

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