Selection rules pure rotational transitions

Pure rotational spectra can be observed in the gas phase however the selection rule for rotational transitions requires the molecule to have a permanent electric dipole. Homonuclear diatomics are, again, an important group of molecules which do not show microwave absorption because of this selection rule. 

The selection rules for rotational transitions in linear polyatomic molecules are also the same as for diatomic molecules. The transition AJ is equal to 1 in infrared spectroscopy and -nl in purely rotational spectroscopy (i.e. microwave spectroscopy) but only if the molecule has a non-zero dipole moment (see Section 6.7). A rotational transition for H-C=C-C1 will be observed whereas for H-C=C-H no rotational transition will be observed due to its zero dipole moment. 

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. 

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. 

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... 

Selection Rules for Vibration-Rotation and Pure-Rotation Transitions... 

The selection rules for pure-rotation transitions are found by evaluation of the nine integrals IXOa,/ZOc these involve the nine direction cosines6 cos(XOa),..., cos(ZOc). The volume element in Eulerian angles can be shown to be... 

The component Mz belongs to the species 4" in the Dah group because fiz is not changed by pure permutations and it changes sign by permutation—inversion operations (Section 4.1). The overall symmetry selection rule therefore allows transitions only between vibration—inversion-rotation states with opposite parity with respect to the operation of inversion (cf. Fig. 6). 

Forbidden pure rotational transitions of H3, following the selection rules Ak = +3, occur in the wide region from millimetre wave to mid-infrared.These transitions are caused by centrifugal distortions of the symmetric structure. No laboratory observation of them has been reported so far. These transitions are much weaker than the usual dipole-allowed rotational transitions in polar molecules, and their spontaneous emission rates range from ca. 10" s" to ca. 10" s". Nevertheless, such weak transitions may be observable in low-density regions just like the Hj quadrupole transitions. Also, the spontaneous emission lifetimes are short compared with the collisional time in low-density areas, making the forbidden rotational transitions important processes for cooling the rotational temperature of Hj. ... 

Rotational features of almost aU H-bonded complexes in the gaseous phase appear in the microwave region, with wavenumbers less than 10 cm They correspond to transitions between pure rotational levels, pure meaning that vibrations remain unchanged, or no vibrational transition accompanies such rotational transitions. Rotational features, however, also appear in the IR spectra of these H-bonded complexes. IR bands correspond to transitions between various vibrational levels of a molecule. When this molecule is isolated, as in the gas phase, these transitions are always accompanied by transitions between rotational levels that obey the same selection rules as pure rotational transitions detected in microwave spectroscopy. The information conveyed by these rotational features in IR spectra are therefore most similar to those conveyed by microwave spectra, even if the mechanism at the origin of their appearance is different. Their interests lie in the use of an IR spectrometer, a common instrument in many laboratories, instead of a microwave spectrometer, which is a much more specialized instrament. However, the resolution of usual IR spectrometers are lower than that of microwave spectrometers that use Fabry-Perot cavities. This IR technique has been used in the case of simple H-bonded dimers with relatively small moments of inertia, such as, for instance, F-H- -N C-H (3). Such complexes are far from simple to manipulate, but provide particularly simple IR spectra with a limited number of bands that do not show any overlap. 

Pure rotational transitions of symmetrical diatomic molecules like dihydrogen are forbidden in infrared spectroscopy by the dipole selection rule but are active in Raman spectroscopy because they are anisotropically polarisable. They are in principle observable in INS although the scattering is weak except for dihydrogen. These rotational transitions offer the prospect of probing the local environment of the dihydrogen molecule, as we shall see in this chapter. 

The propensity rules for collision-induced transitions between electronic states and among the fine-structure components of non-1E+ states depend on the identity of the leading term in the multipole expansion of the molecule/collision-partner interaction potential. Alexander (1982a) has considered the dipole-dipole term, which included both permanent and transition dipole contributions. In the limit that first-order perturbation theory applies (not the usual circumstance for thermal molecular collisions), the following collisional propensity rules for the permanent dipole term can be enumerated from the selection rules for both perturbations and pure rotational transitions... 

The molecule PH3 (C3V symmetry) is an oblate symmetric top (Crotational constants C (refers to rotation around the C3 axis) and B (perpendicular to C3). Since the permanent electric dipole moment is pointed parallel to the C3 axis, only pure rotational transitions with the selection rule AK=0 are allowed (K is the quantum number of the component about the C3 axis of the total angular momentum J). Their analysis leads to the parameters B, Dj, Djk, and Hjk. From the perturbation-allowed transitions AK= 3n (n=1,2,...), which become weakly allowed by centrifugal distortion effects (inducing a small dipole moment of about 8x10 D perpendicular to the C3 axis [1, 2, 3]), the K-related constants (C, Dk, Hk) were obtained see, e.g. [1, 3, 4]. 

The rotational levels of a diatomic molecule can be well approximated by the two-particle rigid-rotor energies (6.52). It is found (Levine, Molecular Spectroscopy, Section 4.4) that when a diatomic molecule absorbs or emits radiation, the allowed pure-rotational transitions are given by the selection rule... 

Thus, either < JKM 9 J K M y vanishes, or J = J l (the quantity in the second set of square brackets cannot vanish when J J and J, J 0). Hence we find the selection rule AJ = 1 for pure rotational transitions in a symmetric top. The symmetric top El selection rules can therefore be summarized SiS AJ = +1, AK = 0. In the presence of strong external electric fields, the rotational energy levels depend on M as well as on J and K (via the Stark affect) the selection rules AM = 0 and AM = 1 become... 

Pure rotational transitions in H2, following electric quadrupole selection rules, are now often observed in emission from interstellar gas, and the signals are one of many indicators that astronomers use to assess the energy distribution in those clouds. Those signals are too weak for routine observation in the laboratory. 

When the transition is allowed, the electric dipole selection rules are A/ = 1 for pure rotational transitions, obeying our general rule that the photon behaves like a particle with one unit of angular momentum. For a transition / / + 1, the photons absorbed or emitted have energy (neglecting distortion terms)... 

The selection rules governing allowed changes in the rotational quantum number J (and K in the case of the symmetric top) depend on whether changes are taking place in other quantized molecular properties at the same time. They are therefore different for a pure rotational transition, for a vibrational transition with associated rotational changes, or for an electronic transition with associated vibrational and rotational changes. These selection rules are all based on symmetry, but here we simply present the results, rather than attempt to use symmetry to derive them. 

The selection rules for pure rotational transitions of symmetric tops are A7 = 1, AAi = 0 for direct absorption or emission, and A/= 1 or 2, AAi = 0 for the Raman effect. We obtain simple spectra in both cases, with a single series of lines (A/= +1) in absorption and two series (A7 = +1 and +2) in the Raman effect. Neglecting centrifugal distortion, these series have constant spacings of 2B or AB, and lines for all values of K coincide. If there is centrifugal distortion, separate lines can be observed for the different K values that are possible for each value of J, with frequencies (for A7 = 1) given by... 

Having identified a molecule, the next objective is to estimate its (relative) abundance - generally expressed relative to that of H2. The first stage in this procedure is to measure the absolute strength of the transitions that are observed. Quantum mechanics demonstrates (a) that molecules without an electric dipole moment (like H2) do not undergo pure rotational transitions, and (b) in molecules that have a dipole moment the transitions are limited by selection rules for example, in linear molecules, N can only change by one that is, AM = 1. 

Linear polyatomic molecules, like diatomic molecules, have only one important moment of inertia and the rotational energies have the same form as (3). The selection rule for pure rotation transitions likewise has the same form as rule (4) above. 

The o e, )/e, and /3e are the rotation-vibration interaction constants representing corrections for the effect of vibration. The selection rules for pure rotational transitions are J J + I,v v, and the rotational frequencies are easily shown to be... 

Applying the selection rule A/ = 1 for purely rotational transitions to the rotational energy [Eq. (3.3.19)], the allowed transition wavenumbers are... 

The methyl iodide molecule is studied using microwave (pure rotational) spectroscopy. The following integral governs the rotational selection rules for transitions labeled J, M, K... 

Here a third selection rule applies for linear molecules, transitions corresponding to vibrations along the main axis are allowed if Aj = 1. The A/=0 transition is only allowed for vibrations perpendicular to the main axis. Note that because of this selection rule the purely vibrational transition (called Q branch) appears in the gas phase spectrum of C(X but is absent in that of CO. In both cases, two branches of rotational side bands appear (called P and R branch) (see Fig. 8.3 for gas phase CO). 

Upon absorption of light of an appropriate wavelength, a diatomic molecule can undergo an electronic transition, along with simultaneous vibrational and rotational transitions. In this case, there is no restriction on Au. That is, the selection rule Av = +1 valid for purely vibrational and vibrational-rotational transitions no longer applies thus numerous vibrational transitions can occur. If the molecule is at room temperature, it will normally be in its lower state, v" = 0 hence transitions corresponding to v" = 0 to v = 0,... 

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