Hindered Internal Motions of Molecules

Hindered Internal Motions of Molecules A. AlicycHic Ring Systems 

The frequency factor and enthalpy of activation for the inversion of the chair of cyclohexane, as below, are important quantities and as a result several laboratories have pursued this problem to arrive, after some controversies, at a solution. 

Cyclohexane exists in two identical chair forms which interconvert. 

If we number hydrogen atoms 1 and 2 at any carbon, their designations axial and equatorial are opposite in the two identical molecules. The inversion process can be labelled by the Larmor frequencies between which hydrogen 1 and 2 oscillate (Lemieux et al., 1958). The rate of inversion of these chair forms comes in the range accessible to N.M.R. exchange broadening studies. Jensen and Berlin (1960) first reported that line broadening occurred when a solution of cyclohexane in carbon 8  

Examination of the symmetry of cyclohexane in the chair form shows that there are two possible chemical shifts and 8 and four types of coupling constant andwith a 12-spin system (Corio, 

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Typical of fast reactions for which rate constants (and activation parameters) have been obtained using NMR relaxation measurements hindered internal motions of molecules exchange... 

An interesting result was also obtained (10) in a study of polyamides with stilbene residues in the chain backbone. In that case,an increasing polymer concentration led to a decreasing fluorescence intensity. This effect can be understood since the trans-cis isomerization and the emission from the excited stilbene moiety compete with each other. Thus, an increasing concentration of chain molecules which hinders the photoisomerization favors the fluorescence. This effect is observed long before the systems becomes glassy and it is, therefore, distinct from the enhancement of fluorescence in rigid media in dyes whose emission is quenched by internal motions of the excited molecules (11). 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

The Bartell mechanical model has also been used to estimate the isotope effect on molar volume due to the over all motion (i.e. hindered translation) of molecules interacting in a Lennard-Jones potential. For C6H6/C6D6 one finds AV/V 4 x 10-5, about two orders of magnitude smaller than the contribution of the internal modes (and experiment). We conclude that for all but very light molecules this contribution can be neglected. 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

The transfer of an H atom from one site to another, as in the HCN — NCH isomerization, can be viewed as a special type of internal rotation. A hindered internal rotor treatment of such motions was found [148, 149] to yield an increase in the reactant state density by a factor of 3 to 4 for both HCN and HCCH at the thresholds for CH bond dissociations. Furthermore, for HCN, where the dissociation energy is well known, the resulting low pressure limit rate coefficients were found to be in much improved agreement with experiment. This study also provided a simple general formula for estimating the effect of such corrections for arbitrary isomerizations (Eq. (2.31) in [149]). Illustrative calculations suggested that such effects may be important even in larger molecules. 

Since detailed investigation of properties of molecules began, the motions of the atomic nuclei within a molecule have been of great interest because they reflect the dynamics of the molecule. Usually the internal motions are adequately described as vibrations, but if there is not a one-to-one correspondence of potential well and nucleus, we speak of inversion or hindered internal rotation. In the latter case, which is the subject of this paper, the change of sites of the nuclei is performed by a rotation of a part of the molecule (top) against the rest of the molecule (frame). Internal rotation and torsion are used synonymously in this paper. 

In molecular solids the molecules cannot move around freely, but they are trapped in relatively deep potential wells, caused by the intermolecular potential. In these wells they can vibrate and since the vibrations of individual molecules are coupled, again by the intermolecular potential, one obtains collective vibrations of all the molecules in the solid, called lattice vibrations or phonons. Phonons associated with the center of mass motions of the molecules are called translational phonons, phonons associated with their hindered rotations or librations are called librons. The degree of hindrance of the rotations may vary. If the molecules have well-defined equilibrium orientations and perform small amplitude librations about these, one speaks about ordered phases. If the molecular rotations are nearly free or if the molecules can oscillate in several orientational pockets and easily jump between these pockets, then the solid is called orientationally disordered or plastic. Several molecular solids may occur in each of these phases, depending on the temperature and pressure they undergo order/disorder phase transitions. Also the intramolecular vibrations are coupled by the intermolecular potential, via its dependence on the internal coordinates. The excitations of the solid associated with such vibrations are called vibrational excitons or vibrons. 

The internal dynamics of the methyl group immensely complicates the spectroscopy of these molecules. Of course, this aspect of the problem also provides much of the spectroscopic interest. When the methyl hydrogens of acetaldehyde oscillate around the CC axis, they experience forces arising from the CHO frame of the molecule which vary sinusoidally. As a result, the potential function for internal rotation can be represented by a cosine function in which the crest to trough distance measures the height of the potential barrier. Since the energy barrier to methyl rotation is low in acetaldehyde, the internal motion is one of hindered internal rotation, rather than torsional oscillation. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

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