Degrees of freedom, vibrational and

Both heavy (atoms, ions, molecules) and light (electrons, photons) particles can be involved in collisions. Polyatomic molecules have internal degrees of freedom (vibrational and rotational motion of atoms) and, in this sense, they have an internal structure. 

Systems with two or more degrees of freedom vibrate in a complex manner where frequency and amplitude have no definite relationship. Among the multitudes of unorderly motion, there are some very special types of orderly motion called principal modes of vibration. 

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

Different kinds of molecules have different degrees of translational, vibrational, and rotational freedom and, hence, different average degrees of molecular disorder or randomness. Now, if for a chemical reaction the degree of molecular disorder is different for the products than for the reactants, there will be a change in entropy and AS0 A 0. 

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

Let X = q,p) denote the one-degree-of-freedom reaction coordinate. For M-degrees-of-freedom vibrational modes, 7 e R" and 0 G T" denote their action and angle variables, respectively, where T = [0,27t]. These action and angle variables would be obtained by the Lie transformation, as we have discussed in Section IV. In reaction dynamics, the variables (/, 0) describe the degrees of freedom of the intramolecular and possibly the intermolecular vibrational modes that couple with the reaction coordinate. In the conventional reaction rate theory, these vibrational modes are supposed to play the role of a heat bath for the reaction coordinate x. 

Due to the mutual exclusion rule, g modes (except Ajg) are Raman active, while u modes (except Ai ) are infrared active. We thus expect the internal modes to give rise to three bands in the Raman spectrum and to three bands in the infrared. In order to determine the lattice modes, we have to consider the carbonate anions with 6 degrees of freedom each and two calcium cations with three degrees of freedom. One obtains 2x6-1-2x3 3=15 lattice vibrations and 3 acoustic modes. These can be classified... 

The remarks in the previous paragraph apply, of course, only to the case of electronically adiabatic molecular collisions for which all degrees of freedom refer to the motion of nuclei (i.e. translation, rotation and vibration) if transitions between different electronic states are also involved, then there is no way to avoid dealing with an explicit mixture of a quantum description of some degrees of freedom (electronic) and a classical description of the others.9 The description of such non-adiabatic electronic transitions within the framework of classical S-matrix theory has been discussed at length in the earlier review9 and is not included here. 

It is possible to define a classical path theory by introducing a single configuration type of approximation. Consider, e.g., a system with two degrees of freedom r and R where the first is the vibrational and the second the translational degree of freedom. Thus we... 

Molecules exhibit a normal mode of vibration for each vibrational degree of freedom. Infrared and Raman spectroscopy are useful tools to measure normal modes of vibration. 

This kinetic approximation assumes a single vibrational temperature 77 for CO2 molecules and, therefore, is sometimes referred to as quasi equilibrium of vibrational modes. As one can see from (5-20), most of the vibrationally excited molecules can be considered as being in quasi continuum in this case. Vibrational kinetics of polyatomic molecules in quasi continuum was discussed in Chapter 3. The CO2 dissociation rate is limited not by elementary dissociation itself, but via energy transfer from a low to high vibrational excitation level of the molecule in the W-relaxation processes. Such a kinetic situation was referred to in Chapter 3 as the fast reaction limit. The population of highly excited states with vibrational energy E depends in this case on the number of vibrational degrees of freedom 5 and is proportional to the density of the vibrational states p E) a. The... 

In this equation, (qmt)i is the partition function associated with the internal degrees of freedom (rotations and vibrations) of species i. In addition, the ( inOi can be written as products of vibrational ( vib) and rotational ( roOi partition functions. It is also instructive to write (qvib)R Rf as the product of terms associated with the conserved modes, which correlate with vibrational modes in the reactants, and the transitional modes, which correlate with relative translational and rotational motions in the reactants. Then ... 

In order to illustrate the approximations involved when trying to mix quantum and classical mechanics we consider a simple system with just two degrees of freedom r and R. The R coordinate is the candidate for a classical treatment - it is, e.g., the translational motion of an atom relative to the center of mass of a diatomic or polyatomic molecule. This motion is slow compared to the vibrational motion of the diatom - here described by the r coordinate. Thus if we treat the latter quantum mechanically we could introduce a wavefunction t) and expand this in eigenstates of the molecule, i.e., eigenstates to the hamiltonian operator Hq for the isolated molecule. Thus we have... 

Stochastic Structural Identification from Vibrational and Environmental Data, Fig. 3 Three degree of freedom system and its properties... 

We now provide an example in wdiich eigenvalue analysis is of direct interest to a problem from chemical engineering practice. Let us say that we have some sfructme (it could be a molecule or some sohd object) whose state is described by the positional degrees of freedom q and the corresponding velocities q. We have some model for the total potential energy of the system U q) and some model of the total kinetic energy K q, q). We wish to compute the vibrational frequencies of the structure. Such a normal mode analysis problem arises when we wish to compute the IR spectra of a molecule (Allen Beers, 2005). 

Table 1. Loss of rotational degrees of freedom (n) and gain of vibrational degrees of freedom (m) in various types of bimolecular reactions.

Although a diatomic molecule can produce only one vibration, this number increases with the number of atoms making up the molecule. For a molecule of N atoms, 3N-6 vibrations are possible. That corresponds to 3N degrees of freedom from which are subtracted 3 translational movements and 3 rotational movements for the overall molecule for which the energy is not quantified and corresponds to thermal energy. In reality, this number is most often reduced because of symmetry. Additionally, for a vibration to be active in the infrared, it must be accompanied by a variation in the molecule s dipole moment. 

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