Degrees of freedom for vibration

Isotopic molecules with a comparable structure have more degrees of freedom for vibration if the number of atoms in the molecule is higher. Therefore, the reduced partition function is larger in such molecules where the number of atoms is higher. 

As was shown for translational and rotational motions, there are three degrees of freedom for vibrational motion for every center of mass in the molecule. The number six on the right hand side term of equation (2.9) arises from the total number of degrees of freedom for translational and rotational motion, which do not belong to vibrational motion. It should be known that for linear molecules, there are only two degrees of freedom for rotational motion. This is why for this case there is a special equation for the calculation of the degrees of freedom for vibrational motion (2.10). 

In the linear model which has just been discussed the number of modes is one less than the number of masses. In general, three-dimensional modes have to be considered, and here the relation obeyed is that if there are N atoms in the molecule, there are ZN—6 normal modes of vibration. This is easily proved. N isolated masses have ZN translational degrees of freedom and no others. Into whatever system the masses may be combined, they retain these. But since when N atoms constitute a molecule they must preserve certain relations between their coordinates, it becomes convenient to formulate some modes of motion in common namely 3 for the translation of the centre of gravity of the entire system and 3 for the rotation of the whole about three axes. This leaves ZN—6 degrees of freedom for vibration. 

In contrast, three angles abouMhre mutuffly perpenScuIar axes Uffough the center of mass are needed to describe the rotation of a nonlinear molecule, such as H2O or CH4, leaving (3M - 6) degrees of freedom for vibration. 

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

There are 78 vibrational degrees of freedom for TgHg and it has been shown that the molecule has 33 different fundamental modes under Oh symmetry, 6 are IR active, 13 are Raman active, and 14 vibrations are inactive. The experimental fundamental IR active vibrational frequencies have been assigned as follows 2277 (v Si-H), 1141 (vas Si-O-Si), 881 5 O-Si-H), 566 ( s O-Si-O), 465 (v O-Si-O), and 399 cm ( s O-Si-O). These generally agree well with calculated values The IR spectrum recorded in the solid state shows bands at 2300 and 2293 cm ... 

For reactants having complex intramolecular structure, some coordinates Qk describe the intramolecular degrees of freedom. For solutions in which the motion of the molecules is not described by small vibrations, the coordinates Qk describe the effective oscillators corresponding to collective excitations in the medium. Summation rules have been derived which enable us to relate the characteristics of the effective oscillators with the dielectric properties of the fi edium.5... 

Figure 10.2 Translational, rotational, and vibrational degrees of freedom for a simple diatomic molecule.

The transition state theory of reaction rates [21] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom. For an extensive discussion of transition state theory applied to surface reactions we refer to books by Zhdanov [25] and by Van Santen and Niemantsverdriet [27]. The desorption of a molecule M proceeds as follows ... 

Fig. 2 Schematic representation of the so-called semiclassical treatment of kinetic isotope effects for hydrogen transfer. All vibrational motions of the reactant state are quantized and all vibrational motions of the transition state except for the reaction coordinate are quantized the reaction coordinate is taken as classical. In the simplest version, only the zero-point levels are considered as occupied and the isotope effect and temperature dependence shown at the bottom are expected. Because the quantization of all stable degrees of freedom is taken into account (thus the zero-point energies and the isotope effects) but the reaction-coordinate degree of freedom for the transition state is considered as classical (thus omitting tunneling), the model is ealled semielassieal.

For a molecule with N atoms, its 3iV degrees of freedom would be split into three translational degrees of freedom (corresponding to x-, y-, and z-directions), and three rotational degrees of freedom for nonlinear molecules and two for linear ones. Therefore, 3N—6 and 3N— 5 vibrational degrees of freedom exist for nonlinear and linear molecules, respectively. Vibrational frequencies can be obtained from convenient tabulations (see, for example, Shimanouchi, 1972 Chase et al., 1985). 

In its order of magnitude, this value of b0 coincides with the statistical result. (The difference between the two derivations is that in the second one a translatory degree of freedom is replaced by a vibrational degree of freedom for the adsorbed state.)... 

In summary, although the BH model predicts an inverted region for the kinetics of proton in the nonadiabatic regime, the BH model is only in qualitative accord with the data derived from the proton transfer within the benzophenone-N, A -dimethylaniline contact radical ion pairs. The failure of the model lies in its ID nature as it does not take into account the degrees of freedom for the vibrations associated with the proton-transfer mode. By incorporating these vibrations into the BH model, the LH model provides an excellent account of the parameters serving to control the kinetics of nonadiabatic proton transfer. A more rigorous test for the LH model will come when the kinetic deuterium isotope effects for benzophenone-A, A -dimethylaniline contact radical ions are examined as well as the temperature dependence of these processes are measured. 

The 3N degrees of freedom for nuclear motion are divided into 3 translational, 3 (or 2) rotational, and 3N-6 (or 3N-5) vibrational (degrees of freedom. (The translations and rotations are often called nongenuine vibrations.) The 9 irreducible representations in (9.104) include the 3 translations and the 3 rotations. To find the symmetry species of the 3 vibrations, we must find the symmetry species of the translations and rotations. 

When the total energy of a system is the sum of the energies from the different degrees of freedom, for example, translation, rotation, vibration, and electronic, then the partition function for a combination of the energy levels is the product of the partition functions for each type. Thus,... 

In C70, because of its lower DSh symmetry, there are five kinds of non-equivalent atomic sites and eight kinds of non-equivalent bonds. This means that the number of normal vibrations increases for C70 in comparison to C60. Although there are now 204 vibrational degrees of freedom for the 70-atom molecule, the symmetry of C70 gives rise to a number of degenerate modes so that only 122 modes are expected. Of these 31 are infrared-active and 53 are Raman-active. 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

Question. Given that the total number of degrees of freedom for a polyatomic molecule is 3N, calculate the number of vibrational modes open to (a) an atom, (b) a diatomic molecule, (c) a non-linear polyatomic molecule with N atoms and (d) a non-linear activated complex with N atoms. 

---