Pure-rotational transition

It is also possible to measure microwave spectra of some more strongly bound Van der Waals complexes in a gas cell ratlier tlian a molecular beam. Indeed, tire first microwave studies on molecular clusters were of this type, on carboxylic acid dimers [jd]. The resolution tliat can be achieved is not as high as in a molecular beam, but bulk gas studies have tire advantage tliat vibrational satellites, due to pure rotational transitions in complexes witli intennolecular bending and stretching modes excited, can often be identified. The frequencies of tire vibrational satellites contain infonnation on how the vibrationally averaged stmcture changes in tire excited states, while their intensities allow tire vibrational frequencies to be estimated. 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

For purely rotational transitions, the initial and final eleetronie and vibrational states are the same. Moreover, the eleetronie and vibrational states are not summed over in the analog of the above development beeause one is interested in obtaining an expression for a partieular Xiv /ie ==> Xfv Vfe eleetronie-vibrational transition s lineshape. As a result, the... 

This integral would arise in the eleetronie-vibration-rotation ease the other two eases would involve integrals of the same form but with the AEi f/h absent in the vibration-rotation situation and with cofy, iv + AEi f/h missing for pure rotation transitions. All sueh integrals ean be earried out analytieally and yield ... 

Pure rotational transitions, vibrorotational transitions and spontaneous radiative lifetimes have been derived by solving numerically [20] the one-dimensional radial part of the Schrodinger equation for the single X state preceded by construeting an interpolation... 

To the best of our knowledge, pure rotational transition calculations for the PN X state are reported for the first time. 

According to apphcation of Dunham s formalism to analysis of molecular spectra, as for GaH and H2, these radial coefficients of seven types represent many Dunham coefficients Ym and their auxiliary coefficients Zki of various types that collectively allow wave numbers of observed transitions to be reproduced almost within their uncertainty of measurement through formula 54. Mostly because of inconsistency between reported values of frequencies of pure rotational transitions [118,119], the reduced standard deviation of the fit reported in table 3 is 1.25, slightly greater than unity that would be applicable with consistent data for which uncertainty of each measurement were carefully assigned. 

Fig. 3.26. Examples of spectra of some -containing complexes which accompany the So(0) pure rotational transition frequency of H2 after [268].

An analytical expression for the Bxl coefficients is often desirable. For the purely rotational transitions (v = v = 0) one may write... 

The dependence of these coefficients on the rotational quantum numbers, j, f 0, was also investigated. For the purely rotational transitions,... 

We see from (4.104) that, although the vibrational quantum number is not changing, the frequency of a pure-rotational transition depends on the vibrational quantum number of the molecule undergoing the transition. (Recall that vibration changes the effective moment of inertia, and thus affects the rotational energies.) For a collection of diatomic molecules at temperature T, the relative populations of the energy levels are given by the Boltzmann distribution law the ratio of the number of molecules with vibrational quantum number v to the number with vibrational quantum number zero is... 

Table 4.2 lists some of the pure-rotation transitions of CO. 

We previously found the selection rule A7 = 1 for a 2 diatomic-molecule vibration-rotation or pure-rotation transition. The rule (4.138) forbids A/ = 1 for homonuclear diatomics this gives us no new information as far as vibration-rotation spectra are concerned, since the absence of a dipole moment insures the absence of a vibration-rotation or pure-rotation spectrum, anyway. 

The electric-dipole selection rules for pure-rotation transitions will be considered in Section 6.5. Here we will simply give the results. 

First of all, a molecule must possess a permanent electric dipole moment to exhibit electric-dipole pure-rotation transitions. 

Symmetric tops with no dipole moment have no microwave spectrum. For example, planar symmetric-top molecules have a C axis and a ak symmetry plane such molecules cannot have a dipole moment. Thus benzene has no microwave spectrum. For a symmetric top with a permanent electric dipole moment, the selection rules for pure-rotation transitions are... 

Aside from the possession of a permanent dipole moment and sufficient volatility, a molecule must be reasonably small for its microwave spectrum to be profitably studied. Large molecules have many low-frequency vibrational modes these modes will be appreciably populated at room temperature, giving many strong pure-rotation transitions between levels with nonzero vibrational quantum numbers. The microwave spectrum of a large molecule will thus have so many lines that assignment of the lines will be virtually impossible. 

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