Internal motional freedom

Let us now suppose that all n atoms move simultaneously by the same amount in the x direction. This will displace the center of mass of the entire molecule in the x direction without causing any alteration of the internal dimensions of the molecule. Thesame may of course be said of similar motions in the y and z directions. Thus, of the 3n degrees of freedom of the molecule, three are not genuine vibrations but only translations. Similarly, concerted motions of all atoms in circular paths about the jt, y, and z axes do not constitute vibrations either but are instead, molecular rotations. Thus, of the 3/i degrees of motional freedom, only 3n — 6 remain to be combined into genuine vibratory motions. 

Al3+-exchanged synthetic hectorite is a good catalyst for these conversions, and the 13C NMR spectrum obtained in the interlamellar, proton-catalyzed addition of water to 2-methylpropene is indistinguishable (Fig. 79) from that of f-butanol. Doubtless studies of this kind, where natural-abundance, 3C NMR signals are used to probe the chemical identity and motional freedom of reactant and product species situated in the interlamellar spaces of clays or pillared clays (see below), will become increasingly popular. Using l3C NMR linewidths and spin-lattice relaxation studies, Matsumoto et al. (466) have succeeded in discriminating between the internal and external surfaces of pillared montmorillonites. 

In order to obtain molecular systems in which the internal motion is easier to study, it is customary to introduce halogen atoms in the molecules because of the enhanced scattering power of these atoms. On the other hand, the larger halogen atoms restrict the internal motion more than is the case in unsubstituted molecules. Halogen substitution thus leads to systems with less torsional freedom than the parent hydrocarbons. 

Humphreys and Hammett have estimated that in solution the entropy of acetic acid or its derivative is about 4-6 e.u. greater than the entropy of formic acid or its corresponding derivative due to the internal freedom of the methyl group. On this basis the authors concluded that the entropy of the acetate ion must be about the same as that of the formate ion, meaning that the internal motion of the methyl group is frozen out in the ionic species. It would appear from the data, however, that the entropy of the activated complex for acetate hydrolysis is more negative than that for formate hydrolysis by another 5 e.u. A possible explanation is that the charge becomes more concentrated in the acetate complex with a resultant increase in solvent electrostriction. 

In the gas phase it was shown that as the complexity of the reacting molecules increased, A S decreased see Sections 4.3.5 and 4.4.2. This is a consequence of including the effect of the internal motions of rotation and vibration in reactants and activated complex. The change in the number of degrees of freedom is a major contribution to the entropy of activation see Problems 4.12-4.15. 

Table 7.2 shows these predictions to be home out by a large number of reactions, but there are exceptions where the A factors are much lower than expected. These can be explained if the internal degrees of freedom are considered. Effects of electrostatic interactions and of solvation are small, and could be in either direction, but the dominant effect is probably due to internal motions. 

Within the adiabatic approximation, we consider the translational motion to be slow and the internal motion to be fast. The most familiar example is the Born-Oppenheimer approximation employed in Section 2.3 to decouple the fast electronic motion from the slow motion of the heavy nuclei. In the same spirit, the adiabatic approximation may be utilized to decouple two nuclear degrees of freedom within the same electronic state. To be specific, we discuss the photodissociation of the linear triatom ABC as defined in Figure 2.1. The translational motion associated with R is... 

Hamilton s equations form a set of coupled first-order differential equar tions which under normal conditions can be numerically integrated without any problems. The forces —dVi/dR and —dVi/dr and the torque —dVj/d7, which reflect the coordinate dependence of the interaction potential, control the coupling between the translational (R,P), the vibrational (r,p), and the rotational (7,j) degrees of freedom. Due to this coupling energy can flow between the various modes. The translational mode becomes decoupled from the internal motion of the diatomic fragment (i.e., dP/dt = 0 and dR/dt =constant) when the interaction potential diminishes in the limit R — 00. As a consequence, the translational energy... 

The existence of chiral pathways in this molecule is made possible by the existence of the two independent degrees of freedom that govern internal motion, rotation, and inversion. As molecular complexity increases, the number of degrees of freedom also increases and, unless an achiral pathway is energetically much preferred, it becomes more and more likely that enantiomerization proceeds by a chiral pathway. For example, it is extremely improbable that reversal of helicity in a polymeric chain involves an achiral intermediate or transition state. There is a strong resemblance here to the stochastic achirality of ensembles of achiral molecules discussed previously. 

As a consequence of the collective motion of the neutral system across the homogeneous magnetic field, a motional Stark term with a constant electric field arises. This Stark term inherently couples the center of mass and internal degrees of freedom and hence any change of the internal dynamics leaves its fingerprints on the dynamics of the center of mass. In particular the transition from regularity to chaos in the classical dynamics of the internal motion is accompanied in the center of mass motion by a transition from bounded oscillations to an unbounded diffusional motion. Since these observations are based on classical dynamics, it is a priori not clear whether the observed classical diffusion will survive quantization. From both the theoretical as well as experimental point of view a challenging question is therefore whether quantum interference effects will lead to a suppression of the diffusional motion, i.e. to quantum localization, or not. 

In the association process some degrees of freedom of the reacting system change their nature (from translation and rotation to internal motions). Statistical thermodynamics suggests us the procedures to be used in gas phase calculations application to processes in solution requires a careful analysis. The additional internal motions are in general quite floppy, and their separation from rotational motions of the whole C is a delicate task. 

It is worthwhile to reiterate that the quantum problem still involves both the translations and internal motion of the gas molecule, or 6 degrees of freedom. Solution of the six-dimensional TDSE is still a very difficult matter (unless the TDQMC approaches become practical). There are a multitude of further approximations that can be made if the scattering does not involve breaking the molecular bond, which are equivalent to the plethora of methods developed for gas-phase inelastic scattering. We will not consider these further here, but will simply refer the interested reader to the excellent review by Gerber (1987). 

Stochastic dynamics has been found to be particularly useful for introducing simplified descriptions of the internal motions of complex systems. When applied to small systems (e.g., a peptide or an amino acid sidechain) it is possible to do simulations that extend into the microsecond range, where many important phenomena occur. Simulation studies using this method have been carried out, for example, to explore solvent effects on the dynamics of internal soft degrees of freedom in small biopolymers, e.g., the dynamics of dihedral angle rotations in the alanine dipeptide (see Chapt. IX.B.l). 

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