Rotation, internal polyatomic molecule

Another important mode of rotation in polyatomic molecules is internal modes of rotation. As an example, consider the rotation of a methyl group about the C-C bond axis in ethane. The rotation of the methyl group can be approximated as a free rotor about the < i angle as in the Particle-on-a-Ring model problem (see Section 3.1). From the moment of inertia of the methyl group, the energy of the internal rotational states can be obtained from Equation 3-6. 

Treating the full internal nuclear-motion dynamics of a polyatomic molecule is complicated. It is conventional to examine the rotational movement of a hypothetical "rigid" molecule as well as the vibrational motion of a non-rotating molecule, and to then treat the rotation-vibration couplings using perturbation theory. 

When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 

Clodius, W. B., and Quade, C. R. (1985), Internal Coordinate Formulation for the Vibration-Rotation Energies of Polyatomic Molecules. III. Tetrahedral and Octahedral Spherical Top Molecules, /. Chem. Phys. 82, 2365. 

Equations (6)-(7) apply to the rigid rotation of a molecule as a whole. Most polyatomic molecules possess one or more modes of internal rotation. If the internal rotation is unhindered, the partition function is... 

As shown in Chapter 17, the rotations and vibrations of diatomic or polyatomic molecules make additional contributions to the energy. In a monatomic gas, these other contributions are not present thus, changes in the total internal energy AU measured in thermodynamics can be equated to changes in the translational kinetic energy of the atoms. If n moles of a monatomic gas is taken from a temperature Ti to a temperature Tx, the internal energy change is... 

The complete partition function of a polyatomic molecule may now be represented by the product Qt X Q., where Qt is the tran.slational, including the electronic, factor, as derived above, and Q is the combined rotational and vibrational, i.e., internal, factor. Since In Q is then equal to In Qt + In Qiy equation (24.12) may be written in the form... 

Even for molecules in the ground electronic state, our knowledge about cross sections is largely limited to the room-temperature condition, in which vibrational and rotational states are populated in a thermal distribution. Then, for a diatomic molecule, the ground vibrational state is predominantly populated. However, for a polyatomic molecule, normal modes with small quanta must be appreciably excited. For the full understanding of kinetics in plasma chemistry, it is important to assess the role of the internal energy of reactant molecules. 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

Polyatomic molecules differ from diatoms in that the departing fragments can themselves rotate so that the angular momentum can be conserved in many different ways. Secondly, the dissociation of polyatomic species may involve a complex series of rearrangements in which the transition state may have a structure that is very different from the molecule so that its rotational constants may differ as well. If the transition state has a real barrier and is described solely in terms of vibrational oscillators (plus perhaps one or two internal rotors), that is, a vibrator transition state, angular momentum conservation results in a much larger rotational barrier than the previously discussed centrifugal barrier in the diatom dissociation. 

Here the reaction process is one of spontaneous transformation, which requires sufficient energy to be present in the ether molecule to permit the rupture of a carbon-carbon bond. This energy is obviously internal to the molecule and cannot be represented as a translational-energy term. The pertinent question to ask is how a molecule acquires the required energy for the transformation to occur, and the answer lies in a consideration of the energy exchange from external (kinetic) to internal (rotational and vibrational) modes in polyatomic molecules. 

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