Polyatomic molecules rotational spectra

Figure 5.15 Rotational Raman spectrum of a diatomic or linear polyatomic molecule...

The pure-rotational Raman spectrum of a polyatomic molecule provides information on the moments of inertia, hence allowing a structural determination. For a molecule to exhibit a pure-rotational Raman spectrum, the polarizability must be anisotropic that is, the polarizability ellipsoid must not be a sphere. As noted in Section 5.2, a spherical top has a spherical polarizability ellipsoid, and so gives no pure-rotational Raman spectrum. Symmetric and asymmetric tops have asymmetric polarizabilities. The structures of several nonpolar molecules (which cannot be studied by microwave spectroscopy) have been determined from their pure-rotational Raman spectra these include F2, C2H4, and C6H6. 

Figure 3.25 Outline of the absorption spectrum of a rigid polyatomic molecule. The bands corresponding to electronic transitions are broad as they include vibrational and rotational transitions and they coalesce to form an absorption continuum...

First, we describe briefly the calculation of the absorption spectrum for bound-bound transitions. In order to keep the presentation as clear as possible we consider the simplest polyatomic molecule, a linear triatom ABC as illustrated in Figure 2.1. The motion of the three atoms is confined to a straight line overall rotation and bending vibration are not taken into account. This simple model serves to define the Jacobi coordinates, which we will later use to describe dissociation processes, and to elucidate the differences between bound-bound and bound-free transitions. We consider an electronic transition from the electronic ground state (k = 0) to an excited electronic state (k = 1) whose potential is also binding (see the lower part of Figure 2.2 the case of a repulsive upper state follows in Section 2.5). The superscripts nu and el will be omitted in what follows. Furthermore, the labels k used to distinguish the electronic states are retained only if necessary. 

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

The vibration-rotation hamiltonian of a polyatomic molecule is more complicated than that of a diatomic molecule, both because of the increased number of co-ordinates, and because of the presence of Coriolis terms which are absent from the diatomic hamiltonian. These differences lead to many more terms in the formulae for a and x values obtained from the contact transformation, and they also lead to various kinds of vibrational and rotational resonance situations in which two or more vibrational levels are separated by so small an energy that interaction terms in the hamiltonian between these levels cannot easily be handled by perturbation theory. It is then necessary to obtain an effective hamiltonian over these two or more vibrational levels, and to use special techniques to relate the coefficients in this hamiltonian to the observed spectrum. 

The study of the rotation-vibration spectra of polyatomic molecules in the gas phase can provide extensive information about the molecular structure, the force field and vibration-rotation interaction parameters. Such IR-spectra are sources of rotational information, in particular for molecules with no permanent dipole moment, since for these cases a pure rotational spectrum does not exist. Vibrational frequencies from gas phase spectra are desirable, because the molecular force field is not affected by intermolecular interactions. Besides, valuable support for the assignment of vibrational transitions can be obtained from the rotational fine structure of the vibrational bands. Even spectra recorded with medium resolution can contain a wealth of information hot bands , for instance, provide insight into the anharmonicity of vibrational potentials. Spectral contributions of isotopic molecules, certainly dependent on their abundance, may also be resolved. 

Condition 4. For most polyatomic molecules monochromatic light is impossible to obtain. Monochromatic light for experiments of this type would be defined as one for which the absorption coefficient is constant. To say the least, the incident radiation will cover much unresolved rotational structure within a single band. The more common case found with high-pressure arcs and color filters will be that two or more vibration transitions within a given electronic transition will be involved. Since the absorption coefficient will never be a true constant, the strongly absorbing parts of the spectrum will be important at low pressures and will decrease in relative importance as the pressure increases. 

A linear polyatomic molecule such as HCN, 0=C=0 or HC=CH has a rotational spectrum closely analogous to that of a diatomic molecule, if one takes into account the more complicated form of the moment of inertia. Consider the most general case of a linear triatomic molecule ... 

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. 

The electronic spectrum of a nonlinear polyatomic molecule is very complicated. In addition to three modes of rotation with distinct moments of inertia, there are 3N — 6 modes of vibration. While some of these may be forbidden in the infrared or Raman spectrum on the basis of symmetry, there is no rule to forbid their appearance in the electronic spectrum, which is extraordinarily complex as a consequence. For our purposes here, we mention only a few fundamental points and present one example. 

The importance of the Raman spectrum lies especially in the fact that it also occurs for homonuclear molecules, which, according to sections 22 and 23, have no rotation and vibration-rotation spectra. Hence, it may be used to supplement the evidence derived from electronic bands, regarding the energy of vibrational and rotational levels in the ground state, and for a confirmation of the values of and thus obtained. Researches of this sort have actually been carried out on HCl by Wood and on Hg, Ng, Og, CO by Rasetti (for literature see G) and (lO)) and, more recently, on CO by Amaldi(is). Really essential, however, is the Raman effect in analysing the possible vibrations of polyatomic molecules, as we shall see in the next chapter. For such molecules very rarely have sharply defined electronic bands, while the rotation and vibration-rotation data usually are insufficient to arrive at a unique description of the molecular behaviour. 

The effects of isotopic substitution on the vibrational spectra of small polyatomic molecules have resulted in two recent astrophyslcal observations of considerable interest. The first extra-terrestrial detection of deuterium was reported in 19 by Beer al. (37) > who found 11 lines of the P-branch of the 2200 cm vibration-rotation band of ClfeD in the spectrum of Jupiter. More recently Beer and Taylor (38) have examined the intensities of these lines and concluded that the d/h ratio in the atmosphere of Jupiter is between I/2 and I/6 the terrestrial value. 

The H+ molecular ion is the simplest polyatomic molecule, and was discovered by J.J. Thompson in 1911 (1). Although its chemistry has been studied extensively using mass spectrometric methods, its spectrum has only recently been observed. The first spectroscopic studies were described by Oka (2) for H+, and by Shy, Farley, Lamb and Wing (3) for D+ and H2D+. These studies were confined to the first few vibration-rotation levels of the molecules and confirmed the essential correctness of the theoretical descriptions of the molecule in these low energy states. 

The selective excitation of single upper levels, which is possible in molecular beams with sufficiently good collimation, results even in polyatomic molecules in astonishingly simple fluorescence spectra. This is, for example, demonstrated in Fig. 1.56 for the NO2 molecule, which is excited at a fixed wavelength Xl = 592 nm. The fluorescence spectrum consists of readily assigned vibrational bands that are composed of three rotational lines (strong P and R lines, and weak Q lines). 

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