The Rotation of Polyatomic Molecules

The straightforward way to treat the rotational and vibrational motion of a polyatomic molecule would be to set up the wave equation for (Eq. 34-4), introducing for [/ ( ) an expres- 

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Microwave spectroscopy often gives more definite and precise information on the structure of polyatomic molecules than vibration-rotation and electronic spectra. For example, consider the simplest oxime formald-oxime, CH2=NOH. There are two likely structural configurations for this... 

As might be expected, the rotational spectra of polyatomic molecules are more complex than those for diatomic molecules. For example, depending on their structures polyatomic molecules can have as many as three different moments of inertia. Although more complicated, the analysis of the structures of polyatomic molecules using rotational spectra follows the same principles as we have discussed for diatomic molecules. 

In the case of polyatomic molecules, such as CO2, H2O, O3, etc., the principles discussed above still apply, but the spectra become more complex. Polyatomic molecules do not rotate only about one single axis, but about three mutually perpendicular axes. In addition, the number of vibrational degrees of freedom is also increased. 

In addition to the bands centered on the fundamental frequencies, other bands appear in the spectra of polyatomic molecules. We have mentioned overtone bands in the spectrum of diatomic molecules due to violation of the selection rule, Ap = +1, that is permitted because of anharmonicity. But in polyatomic molecules, combination bands also appear. For example, in the case of water if the absorbed quantum splits to raise from 0 to 1 and V2 from 0- 1, there will be a vibration-rotation band centered on the combination frequency, + V2 This process is relatively less probable than the absorbtion of a single quantum at either fundamental frequency, so the intensity of the band is relatively weak. Nonetheless, combination bands appear with sufficient intensity to be an important feature of the infrared spectra of polyatomic molecules. Even in the case of a simple molecule like water, there are a large number of prominent bands, several of which are listed in Table 25.2. 

State geometries involves analysis of the rotational fine structure of the electronic bands. Since the spectra of polyatomic molecules are complex, most analyses of this sort have been carried out on the triatomic or tetra-atomic molecules. For molecules whose spectra contain unresolved fine structure, estimates of the excited state geometry can be obtained by vibrational analyses. 

A second rotational effect comes into play when rotations are strongly coupled to the vibrations, via, for instance, coriolis interactions. In that case, the projection of the principle rotational quantum number, the K quantum number in symmetric top molecules, is no longer conserved. The energy associated with this quantum number then gets mixed in with the molecule s vibrational energy, thereby increasing the density and sums of states. When this happens we say that the A -rotor is active. If the T-rotor does not couple with the vibrations, it is inactive. We first discuss what happens when a diatom dissociates and follow that with the dissociation of polyatomic molecules. 

Ab initio studies of the PESs of triatomic molecules [17,59,60,88] have shown the importance of appending so-called small corrections to standard non-relativistic valence-only ab initio predictions. So far these have not been considered for the DMSs of polyatomic molecules. It is up to future high-accuracy computation of DMSs and the utilization of new measurements to decide whether such corrections have a significant effect on computed rotational-vibrational intensities making their computation worth pursuing. 

Naturally, this strongly affects the values of AH calculated by expression (O 53.2). In the case of polyatomic molecules, when one has to take into account the entropy variation due to variation of internal degrees of freedom, rotation of the molecule, and probably chemical bond of molecules with the surface, a theoretical calculation of AS becomes much more complicated. Thus, the uncertainty in evaluation of AS results in approximate values of AH only. Several attempts have been made to calculate empirically AS and AH for various compounds, e.g., chlorides and oxides (B. Eichler 1976), and to determine experimentally these two main parameters of the adsorption interaction, but their values obtained for a majority of adsorbate-adsorbent pairs are still estimates. Therefore, it is desirable to cite these values together, no matter how they were obtained. 

In Section 22.3, we found that only half of the values of the rotational quantum number J occurred for a homonuclear diatomic molecule because of the indistinguishability of the nuclei. In the case of polyatomic molecules the effect of the indistinguishability of identical nuclei is more complicated. We assert without proof that the fraction of the conceivable rotational states that can occur is 1 /a, where a is called the symmetry number of the molecule. The symmetry number is defined as the number of equivalent orientations of the molecule in its equilibrium conformation, which means the number of orientations in which the molecule can be placed and have each nuclear location occupied by a nucleus of the same kind as in the first orientation. 

The states of polyatomic molecules are governed by the same Boltzmann probability distribution as those of atoms and diatomic molecules. The rotational levels of polyatomic molecules are generally large enough that many rotational states are occupied. The rotation of a linear polyatomic molecule such as acetylene or cyanogen is just... 

The spectra of polyatomic molecules are more complicated than those of atoms or diatomic molecules. As with diatomic molecules, rotational transitions can occur without vibrational or electronic transitions, vibrational transitions can occur without electronic transitions but are generally accompanied by rotational transitions, and electronic transitions are accompanied by both vibrational and rotational transitions. 

The basic theory of vibration-rotation of polyatomic molecules has been worked out for many years, including the form of the vibration-rotation Hamiltonian, the quantum mechanical solution of certain anharmonic oscillators," and the relationship, in terms of perturbation theory,between experimental spectroscopic data and higher than quadratic terms in the potential function. Still, prediction of harmonic... 

It is quite straightforward to perform quasiclassical trajectory computations (QCT) on the reactions of polyatomic molecules providing a smooth global potential energy surface is available from which derivatives can be obtained with respect to the atomic coordinates. This method is described in detail in Classical Trajectory Simulations Final Conditions. Hamilton s equations are solved to follow the motion of the individual atoms as a function of time and the reactant and product vibrational and rotational states can be set or boxed to quantum mechanical energies. The method does not treat purely quantum mechanical effects such as tunneling, resonances. or interference but it can treat the full state-to-state, eneigy-resolved dynamics of a reaction and also produces rate constants. Numerous applications to polyatomic reactions have been reported. ... 

As stated earlier, in considering rotation of polyatomic molecules we can follow the approximation used for diatomic molecules, that the rotation can be treated as independent of vibration. Much of the stmcture of rotational spectra of polyatomic molecules can be understood by using as a model an aggregate of nuclear masses connected rigidly at their equilibrium positions. In contrast to diatomic molecules where only one axis of rotation is required, in polyatomic molecules we must consider rotation about any axis. 

In the previous chapter, vibrational/rotational (i.e. infrared) spectroscopy of diatomic molecules was analyzed. The same analysis is now applied to polyatomic molecules. Polyatomic molecules have more than one bond resulting in additional vibrational degrees of freedom. Rotation of linear polyatomic molecules is mechanically equivalent to that of diatomic molecules however, the rotation of non-linear polyatomic molecules results in more than one degree of rotational freedom. The result of the additional vibrational and rotational degrees of freedom for polyatomic molecules is to complicate the vibrational/rotational spectra of polyatomic molecules relative to spectra of diatomic molecules. Though the spectra of polyatomic molecules are more complicated, many of the same features exist as in the spectra of diatomic molecules. As a result, a similar approach wiU be used in this chapter. The mechanics of a model system will be solved, determine the selection rules, and the features of a spectrum will be predicted. 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

This Schrodinger equation relates to the rotation of diatomic and linear polyatomic molecules. It also arises when treating the angular motions of electrons in any spherically symmetric potential... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

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