Selection Rules for Pure Rotational Transitions

We begin this section by deriving the El selection rules for rotational transitions in symmetric tops, since spherical and linear molecules may be considered special cases of symmetric tops. Foi El-allowed transitions from state JKM to state J K M y, we require a nonzero electric dipole transition moment 

Here ft is the permanent molecular electric dipole moment. An intuitive argument will suffice for the selection rule on AK. Since ft must be parallel to the figure axis by symmetry in any prolate or oblate top, and since K controls the velocity of rotation about the figure axis, changing K has no effect on the motion of the molecule s permanent dipole moment. Accordingly, the presence of an oscillating external electric field cannot influence K, and we have the selection rule AK = 0. 

To obtain the selection rules on AJ and AM, we exploit the properties of vector operators. All quantities that transform like vectors under three-dimensional rotations have operators exhibiting commutation rules that are identical to those shown by the space-fixed angular momentum operators f,c-  

Such operators, which we will denote V = (P, Vy, Pg), exhibit the commutation rules 

These operators are termed vector operators with respect to J. The space-fixed angular momentum components fx,Jy, Jz are obviously vector operators with respect to themselves (cf. Eq. 5.11). The position and linear momentum r = (x, y, z) and p = (p c, p, Pz) are also vector operators, as is the electric dipole moment operator i. Since we have from Eq. 5.28 that for any vector operator V 

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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... 

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 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... 

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... 

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. 

In a pure rotational spectrum, the upper and lower states of the transition belong to the same vibrational level. If the vibrational state is different in the lower and upper states, then there is a rotational broadening of the vibrational level in the IR region. The selection rule for the rotational transition is the same as in a pure rotational spectrum. 

The pure rotation spectrum of an asymmetric top is very complex, and cannot be reduced to a formula giving line positions. Instead, it has to be dealt with by calculation of the appropriate upper and lower state energies (Section 7.2.2). The basic selection rule, A7 = 0, 1, applies to absorption/emission spectra, and there are other selection rules. These depend on the symmetry of the inertial ellipsoid, which is always Dan, but the orientations of the dipole moment components depend on the symmetry of the molecule itself. For the rotational Raman effect A7= 2 transitions are allowed as well. The selection rules for pure rotational spectra are described in more detail in the on-line supplement for Chapter 7. 

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). 

SELECTION RULES FOR VIBRATION-ROTATION AND PURE-ROTATTON TRANSITIONS... 

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

As in the case of absorption and fluorescence emission spectroscopy, selection rules apply for the Raman transitions between rotational energy levels. However, since two photons are involved in the process, each of angular momentum Lphoton = 1. angular momentum conservation requires that the difference between the initial and final rotational levels must be two. As the selection rule for pure Raman spectra, one finds... 

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... 

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)... 

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

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. 

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... 

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. 

Consideration of the matrix elements m a n of the polarizability shows that the selection rule for a pure-rotational Raman transition of a l2 diatomic molecule is (see Wilson, Decius, and Cross)... 

Stoicheff investigated the pure rotational Raman spectrum of CS2. The first few lines could not be observed because of the width of the exciting line. The average values of the Stokes and anti-Stokes shifts for the first few observable lines (accurate to 0.02 cm-1) are Ap = 4.96, 5.87, 6.76, 7.64, and 8.50 cm-1, (a) Calculate the C=S bond length in carbon disulfide. (Assume centrifugal distortion is negligible. The rotational Raman selection rule for linear molecules in 2 electronic states is AJ = 0, 2.) (b) Is this an R0 or Re value (c) Predict the shift for the 7 = 0—>2 transition. 

These rules show that the G<- G transition, in contrast with the others, is purely rotational. In the coordinate system shown in Figure 8.20, the transition states for the cis and trans paths of interconversion have symmetry axes and C2y and relate to the symmetry groups and C2h, respectively. The different symmetries of the transition states results from the fact that the same permutation relates to different symmetry operations in C2v and C2h. For example, (ab)(14)(28)(36) is equivalent to inversion in C2h, while in it corresponds to the reflection in the axy plane. The symmetry of the reaction path does not affect the symmetry of states with even Ka (and Ka = 0). However, the selection rules for transitions Ka = 1 0 are different for cis and trans paths. The classifica-... 

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. ... 

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... 

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. 

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. 

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