Internal vibrational motion

For the model Hamiltonian used in this study it was assumed that bond stretching satisfactorily describes all internal vibrational motions for a system of linear molecules and the split parts of the Hamiltonian were of the form... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

Both the Raman and the infrared spectrum yield a partial description of the internal vibrational motion of the molecule in terms of the normal vibrations of the constituent atoms. Neither type of spectrum alone gives a complete description of the pattern of molecular vibration, and, by analysis of the difference between the Raman and the infrared spectrum, additional information about the molecular structure can sometimes be inferred. Physical chemists have made extremely effective use of such comparisons in the elucidation of the finer structural details of small symmetrical molecules, such as methane and benzene. But the mathematical techniques of vibrational analysis are. not yet sufficiently developed to permit the extension of these differential studies to the Raman and infrared spectra of the more complex molecules that constitute the main body of both organic and inorganic chemistry. 

In the preceding two sections we considered resonances induced by temporary excitation of the mode that finally becomes the vibrational mode of the diatomic fragment. In this section we feature two other types of resonant excitation of internal vibrational motion. 

What kind of internal vibrational motion causes the superimposed structures in the spectrum and how can we characterize and assign the structures ... 

In solution, molecules constantly encounter each other through collisions. At close distances, repulsions from the nuclei dominate. The balance between repulsive and attractive forces often lead to an equilibrium state such that two molecules will remain associated in a complex. In this state, dispersion or London forces dominate. The motion experienced by the molecules also plays a crucial role in this process. A molecule contains translational, rotational, as well as internal vibrational motion. When two molecules associate, conservation of energy dictates... 

When this stable configuration has been formed, each of the acid molecules is restricted in its rotational degrees of freedom, and moreover it is found spectroscopically that some of the internal vibrational motions are modified. The interaction between these molecules has therefore changed the rotational and vibrational states of each. In general, we can classify the molecules in a solution into two groups ... 

The dHH can be corrected for thermal motion,120 but difficulties in accounting for the internal vibrational motion of the H atoms can lead to an overestimation of the correction factor and large uncertainties in Jhh- For example, the corrected c/im for Mo(CO)(dppe)2(H2) is estimated to be 0.85-0.88 A. A summary of neutron diffraction data for dihydrogen complexes, including some that have been corrected for thermal motion, is provided by Koetzle.121... 

Clearly, the assets of a useful, in itself noncontradictory, and physically based CNM analysis are the internal vibrational motions and their properties as well as the amplitudes that relate internal modes to normal modes. As shown in the previous section, the adiabatic internal modes an are the appropriate candidates for internal modes. Adiabatic modes are based on a dynamic principle, they are calculated by solving the Euler-Lagrange equations, they are independent of the composition of the set of internal coordinates to describe a molecule, and they are unique in so far as they provide a strict separation of electronic and mass effects [18,19]. Therefore, they fulfil the first requirement for a physically based CNM analysis. 

Finally, when studying dilute solutions of high-molecular-weight flexible coil macromolecules, an additional contribution to the spectral dispersion of the scattered light can arise from the dynamic behavior of the low frequency, long wavelength internal vibrational motions of the chains (70). Additional Lorentzian spectral components (or exponential components of the correlation function) arise through this mechanism characterized by halfwidth (time constant) ... 

Chemistry gives reason to suppose that molecules, being groups of atoms, should possess shapes, and therefore be capable of rotations. Since, moreover, the union of atoms implies some kind of force to hold them together, there is the further possibility of internal vibrational motions. The existence of liquids and solids, as well as the departure of real gases from Boyle s law, shows that there are forces between molecules themselves, so that there must be potential energy stored up in any collection of them. 

In fact, as the temperature increases specific intermolecular interactions should slacken and weaken probably because of the increased internal vibrational motion... 

A molecular structure, at a minimum, consists of the molecular formula of the molecule, together with a list of the coordinates of each atom in the molecule. The structure for a given molecule can be determined by a variety of ways, some of them experimental and some of them computational. But it is important to note at the outset that the structure that is determined for a particular molecule depends to a considerable extent on just what method is used to determine it. The reason for this is that the molecules undergo an internal vibrational motion. What that means is that the atoms do not occupy fixed points, as in the mechanical model, but they are moving about some... 

The dynamical computation for the ASVRT model given in eqn (14.59) is essentially identical to that for the basic SVRT model of eqn (14.17). Thus the TD wavepacket treatment described in the previous section for the SVRT model can be applied directly to the ASVRT model with minor modifications. The ASVRT reaction model allows the internal vibrational motions of molecules to adjust adiabatically to the change of the special coordinate. s and thus can describe the change of internal structure of both the target and reactant molecules. The simplest choice of. s would be the reactive coordinate r. However, it is better to choose something like. s = rjR which is closer to the reaction coordinate. 

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