Rotational spectra theory

We make use of the assumption which is conventional in kinetic theory of the harmonic oscillator [193] as well as in energy-corrected IOS [194]. All the transition rates from top to bottom in the rotational spectrum are supposed to remain the same as in EFA. Only transition rates from bottom upwards must be corrected to meet the demands of detailed balance. In the same way the more general requirements expressed in Eq. (5.21) may be met ... 

To compare this result with that obtained within perturbation theory [273, 279], one must additionally assume the perturbation correlation function to be exponential, as in [273, 279, 280]. In this case, the purely rotational spectrum [273, 279] and that obtained with Eq. (7.71) coincide, if the co-dependence of the / operator is neglected and ([Pg.247]

Let us consider the quasi-classical formulation of impact theory. A rotational spectrum of ifth order at every value of co is a sum of spectral densities at a given frequency of all J-components of all branches... 

Directly linked to the geometry and dipole moment of a molecule, the rotational spectrum is an unambiguous fingerprint that has enabled the radioastronomers community to identify more than a hundred species. Optimized geometries of C3H2calculated at increasing levels of theory (from RHF to MP4 [12]) are presented in Table 1. The rotational constants obtained for C3H2 and its deuterated isomers are presented in Table 2. 

As we saw in Chapter 1, the importance of numbers in chemistry derives from the fact that experimental measurement of a particular chemical or physical property will always yield a numerical value to which we attach some significance. This might involve direct measurement of an intrinsic property of an atom or molecule, such as ionization energy or conductivity, but, more frequently, we find it necessary to use theory to relate the measured property to other properties of the system. For example, the rotational constant, B, for the diatomic molecule CO can be obtained directly from a measurement of the separation of adjacent rotational lines in the infrared spectrum. Theory provides the link between the measured rotational constant and the moment of inertia, I, of the molecule by the formula ... 

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... 

In practice values of B are also often quoted in cm-1. For the simple rigid rotor the rotational quantum number J takes integral values, J = 0, 1, 2, etc. The rotational energy levels therefore have energies 0, 2B, 6B, 12B, etc. Elsewhere in this book we will describe the theory of electric dipole transition probabilities and will show that for a diatomic molecule possessing a permanent electric dipole moment, transitions between the rotational levels obey the simple selection rule A J = 1. The rotational spectrum of the simple rigid rotor therefore consists of a series of equidistant absorption lines with frequencies 2B, 4B, 6B, etc. 

In a paper in 1979, Carl Ballhausen [1] expressed the belief that today we realize that the whole of chemistry is one huge manifestation of quantum phenomena, but he was perfectly well aware of the care that had to be taken to express the relevant quantum theory appropriately. So in an earlier review [2] that he had undertaken with Aage Hansen, he scorned the usual habit of chemists in naming an experimental observation as if it was caused by the theory that was used to account for it. Thus in the review they remark that a particular phenomenon observed in molecular vibration spectra is presently refered to as the Duchinsky effect. The effect is, of course, just as fictitious as the Jahn-Teller effect. Their aim in the review was to make a start towards rationalization of the nomenclature and to specify the form of the molecular Hamiltonian implicit in any nomenclature. In an article that Jonathan Tennyson and I published in the festschrift to celebrate his sixtieth birthday in 1987 [3], we tried to present a clear account of a molecular Hamiltonian suitable for treating the vibration rotation spectrum of a triatomic molecule. In an article that I wrote that appeared in 1990 [4], I discussed the difficulty of deciding just how far the basic chemical idea of molecular structure could really be fitted into quantum mechanics. 

Elsewhere, in the mid-IR, photon energy is sufficient to modify the quantized terms vib and iJjo in expression 10.2. This is therefore a vibration-rotation spectrum, that is, several tens of rotational transitions accompany each vibrational transition. For the simplest molecules it is possible to interpret particular aspects of the absorption bands. Experience and theory have enabled rules of the permitted transitions to be drawn up. Small molecules as carbon monoxide and hydrogen chloride (Figure 10.5) have been intensely studied from this point of view. 

In the case of the top we do not denote the quantum number of the resultant angular momentum by j, an in the general theory, but by m, because this letter is used to denote the terms of a molecular rotation spectrum (see Rotator, 12). 

In the far-infrared spectral region water vapour can absorb radiation by transition between different rotational levels without any vibrational or electronic changes. Such a pure rotational spectrum can only be exhibited by a molecule with a permanent dipole moment. The rotational spectrum has been measured for wavelengths up to 2 mm(Herzberg, 1945, p. 58 Furashov, 1966), where this type of absorption ceases and, in good agreement with theory, consists of relatively sharp lines distributed in what looks... 

Literatures HgO, rotation spectrum (91, 105, 170, 172), vibration-rotation spectrum (37, 68, 124, 138, 140), Raman spectrum (98, 142), theory (44) HgS, vibration-rotation spectrum (51, 125, 131, 146), Raman spectrum (41) NO2, vibration-rotation spectrum (27, 73, 74,150,161) SOo, vibration-rotation spectrum (18, 24, 25, 28), theory (90) CIO2, vibration-rotation spectrum (24, 25) ClgO, vibration-rotation spectrum (26) O3, vibration-rotation spectrum (39, 71, 92). 

Literature NHg, rotation spectrum (19,172), vibration-rotation spectrum (15, 20, 33, 35, 57, 117,145,157,163), Raman spectrum (14), theory (147) PH3, vibration-rotation spectrum (69,145, 172), AsHg, vibration-rotation spectrum (145) other pyramidal molecules (3, 9). 

Literature HgCO, vibration-rotation spectrum (130, 133), theory (129)... 

The calculations have also been extended to states with very high total angular momentum (J 30) covering the breakdown of vibration-rotation separation (11,12) and onto the point at which there are no true bound states J 50 (13). Dipole transition moment calculations have been used to synthesise spectra (1A,15). In particular, we have predicted the "forbidden pure rotational spectrum of H J (lA) refining previous estimates which were based on perturbation theory (16). These results are now being used in the search for H in interstellar clouds. 

The investigation of the hyperfine stmcture of the rotational spectrum of H CN by means of the Lamb-dip technique [136] provides a good example of how theory may also supply missing experimental data. In detail, it turned out to be essential to fix in... 

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