Internal vibrations

A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Once prepared in S q witli well defined energy E, donor molecules will begin to collide witli batli molecules B at a rate detennined by tire batli-gas pressure. A typical process of tliis type is tire collision between a CgFg molecule witli approximately 5 eV (40 000 cm or 460 kJ mor ) of internal vibrational energy and a CO2 molecule in its ground vibrationless state 00 0 to produce CO2 in tire first asymmetric stretch vibrational level 00 1 [11,12 and 13]. This collision results in tire loss of approximately AE= 2349 cnA of internal energy from tire CgFg,... 

Figure C3.3.4. A schematic diagram of an apparatus described in the text for studying vibrational energy transfer to small bath molecules from donor molecules having chemically significant amounts of internal vibrational energy.

In many cases the dynamical system consists of fast degrees of freedom, labeled x, and slow degrees of freedom, labeled y. An example is that of a fluid containing polyatomic molecules. The internal vibrations of the molecules are often very fast compared to their translational and orientational motions. Although this and other systems, like proteins, have already been treated using RESPA,[17, 34, 22, 23, 24, 25, 26] another example, and the one we focus on here, is that of a system of very light particles (of mass m) dissolved in a bath of very heavy particles (mass M).[14] The positions of the heavy particles are denoted y and the positions of the light particles rire denoted by X. In this case the total Liouvillian of the system is ... 

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... 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

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. 

The equipartition principle is a classic result which implies continuous energy states. Internal vibrations and to a lesser extent molecular rotations can only be understood in terms of quantized energy states. For the present discussion, this complication can be overlooked, since the sort of vibration a molecule experiences in a cage of other molecules is a sufficiently loose one (compared to internal vibrations) to be adequately approximated by the classic result. 

An eight-membered ring molecule has 3x8-6=18 intramolecular vibrations [77]. Since the molecular symmetry of Ss belongs to the point group 04a the representation of the internal vibrations is given by... 

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

Figures 4 and 5 show the Raman and IR spectra of ce-Ss in the range up to about 100 cm A comparison of these spectra with those presented in Figs. 2 and 3 reveals that the linewidths are much smaller at low temperatures (ca. 0.02-0.2 cm ). The wavenumbers and assignments of the external and torsional modes as reported by Gautier and Debeau [106] and Becucci et al. [107] are listed in Table 3. The spectra in Figs. 4 and 5 clearly demonstrate that there is no gap between the external vibrations and the crystal components of the lowest internal vibration Vg. Moreover, at about 76 cm an IR active lattice mode appears between two components of the fundamental Vg at 74 cm and 79 cm respectively.

Fig. 29 Raman spectrum of p-S at high pressure and room temperature [109]. The wavenumbers indicated are given for the actual pressure. No signals of other allotropes have been detected. The line at 48 cm (ca. 25 cm atp 0 GPa) may arise from lattice vibrations, while the other lines resemble the typical pattern of internal vibrations of sulfur molecules...

Experimental studies of liquid crystals have been used for many years to probe the dynamics of these complex molecules [12]. These experiments are usually divided into high and low-frequency spectral regions [80]. This distinction is very important in the study of liquid crystalline phases because, in principle, it can discriminate between inter- and intramolecular dynamics. For many organic materials vibrations above about 150 cm are traditionally assigned to internal vibrations and those below this value to so-called lattice modes . However, the distinction is not absolute and coupling between inter- and intramolecular modes is possible. 

The variation in the lattice vibration of the solid products was examined by utilizing the FT-IR technique at successive DGC process times and the results are presented in Fig. 5. The absorption bands at 550 cm and 450 cm" are assigned to the vibration of the MFI-type zeolite and the internal vibration of tetrahedral inorganic atoms. The band 960 cm" has been assigned to the 0-Si stretching vibration associated with the incorporation of titanium species into silica lattice [4], This indicates that the amorphous wall of Ti-MCM-41 was transformed into the TS-1 structure. 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

---