Infrared rigid rotors

Selection Rules. The harmonic-oscillator, rigid-rotor selection mles are Av = 1 and AJ = 1 that is, infrared emission or absorption can occur only when these allowed transitions take place. For an anharmonic diatomic molecule, the A7 = 1 selection mle is still valid, but weak transitions corresponding to An = 2, 3, etc. (overtones) can now be observed. Since we are interested in the most intense absorption band (the fundamental ), we are concerned with transitions from various J levels of the vibrational ground... 

Principal axes can easily be identified in a molecule which possesses symmetry elements e.g., symmetry axes that coincide with principal ones, and a symmetry plane that is oriented perpendicularly to one of the principal axes. The simplest models discussed here are rigid rotor - harmonic oscillator models, which can be extended on demand to better fit the spectral data. For a more complete coverage, the reader is referred to other text books. As a first introduction to infrared rotation-vibration spectra the author prefers Barrow (1962). The topic is discussed in greater details by publications such as by Allen and Cross (1963), Herzberg (1945, 1950), and Hollas (1982). 

The thermodynamic functions of this table are analogous to those in the JANAF table for H20(g) (j ) both tables are taken from Freidman and Haar (1 ). Friedman and Haar applied their non-rigid-rotor, anharmonic-oscillator treatment (with vibrational-rotational coupling terms and low-temperature rotational corrections) to the infrared-spectra analyses of Benedict et al. (J ), and... 

These operators are the algebraic counterpart of the single-bond electric dipole operator in the rigid-rotor, harmonic approximation, which corresponds only to An = 1 infrared transitions. In this elementary... 

Fi om Table XVI-3, several statements about the rotational fine structure in the Raman effect may be made immediately. The polarized, totally symmetric band of a linear rigid rotor will resemble a perpendicular (.L) infrared band of the same type of rotor, except that the line spacing is twice as great ( AJ1 = 2 instead of jA./j = 1). The degenerate Raman band, on the other hand, will more nearly resemble a parallel (j ) infrared... 

Other Complications. When the fine structure is resolved, a number of refinements of the vibration-rotation theory are usually required. In the first place, since the molecule is not really a rigid rotor, the variation of effective moments of inertia with vibrational state must be considered. This introduces the possibility that the rotational constants a, 6, and c are not the same in upper and lower vibrational states, and would change the simple expression for the fine structure of a parallel infrared band of a linear molecule to (72 branch)... 

The microwave spectrum of 1,2-dimethylenecyclobutane in the 26.5—40 GHz range has been investigated, and rigid-rotor constants have been derived for the ground state. A dipole moment fi = 0.457 + 0.002 D has been found. The infrared and Raman spectra of perfluorocyclobutane in the gas and solid states have been discussed. The spectra are assigned on the basis of 2 symmetry in the gas phase. The barrier to methyl rotation in methylcyclobutane has been found to be 2.47 kcal mol-. ... 

Since the energy changes in cm" for transitions from state (J — 1) to state J are equal to 2BJ for a rigid rotor, the observed rotational absorption lines are predicted to be unifomly separ ed by the spacing 2B (Fig. 3.5). This is borne out to within experimental error in the far-infrared spectrum of HCl, Fig. 3.2. If the nuclear masses (and hence the nuclear reduced mass pj ) are independently... 

Diatomic molecules are relatively unimportant in the present context since there are very few whose vibrational bands are fortuitously located with respect to pump laser frequencies. This does not exclude the possibility of pumping via other mechanisms, such as electrical or chemical excitation, and a number of (generally mid-infrared) emissions have been observed from such molecules as HCl, HF and OH radical [2.7]. None has been observed so far in true cw mode. However the case of the diatomic molecule will be treated as it introduces most of the main concepts. To a first approximation the diatomic molecule is a rigid rotor with a negligible moment of inertia about one axis and a value of... 

Thus, for a rigid rotor, the pure rotation spectmm would be comprised of a series of lines with equal spacing of 2B cm. For most molecules, B is sufficiently small that the pure rotation spectrum is found in the microwave region of the spectrum however, for light molecules such as HCl, H2O, or CO, rotational transitions absorb in the far infrared. 

It is possible to deduce the same kinds of structural information from Raman spectra as from infrared and microwave spectra. From the selection rule for rotation, Eq. (23.7-3a), the Raman shift of the Stokes rotational lines of a diatomic molecule is given in the rigid-rotor approximation by... 

Sketch an infrared spectrum of H F at 300K and at lOOOK using the harmonic oscillator and rigid rotor approximations. 

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