Rotational energy polyatomic

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

In the experimental and theoretical study of energy transfer processes which involve some of the above mechanisms, one should distingiush processes in atoms and small molecules and in large polyatomic molecules. For small molecules a frill theoretical quantum treatment is possible and even computer program packages are available [, and ], with full state to state characterization. A good example are rotational energy transfer theory and experiments on Fie + CO [M] ... 

Table B2.5.3. Product energy distribution for some IR laser chemical reactions. (E ) is the average relative translational energy of fragments, is the average vibrational and rotational energy of polyatomic fragments, and/ is the fraction of the total product energy appearing as translational energy [109],...

As for diatomic molecules, there are stacks of rotational energy levels associated with all vibrational levels of a polyatomic molecule. The resulting term values S are given by the sum of the rotational and vibrational term values... 

Equation (4.18) applies only to a diatomic or linear polyatomic molecule. Similar kinds of rotational energy levels are present in more complicated molecules. We will describe the various kinds in more detail in Chapter 10. 

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. 

Harter, W. G and Patterson, Ch. W. (1984), Rotational Energy Surfaces and High-7 Eigenvalue Structure of Polyatomic Molecules, 7. Chem. Phys. 80, 4241. 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

Fig. 14.4 Vibrational and rotational energy levels of a polyatomic molecule.

Most solution-phase spectra of organic compounds show broad absorption bands, as in Figure 1.7, unlike atomic spectra, which consist of sharp lines. The main reason for this is that there are a large number of vibrational and rotational energy levels associated with polyatomic molecules, and absorption of a photon can result in conversion of a portion of its energy into vibrational or rotational... 

For electronic states of polyatomic molecules for which the net electronic spin S is nonzero, we get an interaction between rotational and spin angular momenta. We shall not consider this case, but will restrict ourselves to rotational energies of singlet electronic states. Polyatomic molecules with nonsinglet ground electronic states are rare. 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

To get the total energy (excluding translational energy) of a polyatomic molecule in a given state of nuclear and electronic motion, we add the equilibrium electronic energy Ue and the rotational energy to (6.56) ... 

I 2.1 Rotational Energy Levels of Diatomic Molecules, K I 2.2 Vibrational Energy Levels of Diatomic Molecules, 10 I 2.3 Electronic Stales of Diatomic Molecules, 11 I 2.4 Coupling of Rotation and Electronic Motion in Diatomic Molecules Hund s Coupling Cases, 12 1-3 Quantum States of Polyatomic Molecules, 14... 

The photofragmentation that occurs as a consequence of absorption of a photon is frequently viewed as a "half-collision" process (16)- The photon absorption prepares the molecule in assorted rovibrational states of an excited electronic pes and is followed by the half-collision event in which translational, vibrational, and rotational energy transfer may occur. It is the prediction of the corresponding product energy distributions and their correlation to features of the excited pes that is a major goal of theoretical efforts. In this section we summarize some of the quantum dynamical approaches that have been developed for polyatomic photodissociation. For ease of presentation we limit consideration to triatomic molecules and, further, follow in part the presentation of Heather and Light (17). 

C. Rotational Structure of Electronic Bands The rotational energy of a polyatomic molecule is expressed in terms of three moments of inertia Ia, Ib, and Ic about the principal axes... 

For non-linear polyatomic molecules, Eq. (4) can be replaced by Eq. (5) if the temperature is so high that k x T is large compared to the separations of the rotational energy levels, as it is in most cases, and if the vibrational energy levels are sufficiently close to harmonic ... 

Further Applications.—It is obvious that further equilibria, in particular those of a chemical nature, may now be calculated with the aid of the chemical constants which have been determined theoretically in what seems to be a perfectly reliable manner. In practice, however, we meet the difficulty that it is only in the case of hydrogen that the loss in rotational energy of a polyatomic molecule is at present known, so that all chemical processes are excluded for the time being other than the reaction,... 

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