# Rotational spectroscopy

Rotational spectroscopy studies the absorption of electromagnetic radiation (typically in the microwave region of the spectrum) by molecules. The use of microwaves in spectroscopy essentially became possible due to the development of microwave technology for RADAR during World War II. Rotational spectroscopy is only really practical in the gas phase where the rotational motion is quantized. In solids or liquids the rotational motion is usually quenched due to collisions.

For a molecule to have a rotational spectrum it is necessary (to first order) that it have a dipole moment, that is a difference between the center of charge and the center of mass, or equivalently a separation between two unlike charges. It is this dipole moment that enables the electric field of the light (microwave) to exert a torque on the molecule causing it to rotate more quickly (in excitation) or slowly (in de-excitation).

The simplest rotational spectra belong to diatomic molecules such as CO (carbon monoxide).

The next simplest spectra belong to linear triatomic molecule, such as HCN (hydrogen cyanide).

The next simplest spectra belong to non-linear triatomic molecules, such as NCH.

What information can be extracted from microwave spectroscopy? In quantum mechanics the free rotation of a molecule is quantized, that is the rotational energy can only take certain fixed values; what these values are is simply related to the moment of inertia, I, of the molecule. For a linear molecule, there is only one moment of inertia ([itex]I_A[itex]) which is the moment of inertia for a rotation taking place perpendicular to the symmetry axis. For a non-linear molecule, there are three moments of inertia ([itex]I_A[itex], [itex]I_B[itex] and [itex]I_C[itex]). The particular pattern of energy levels (and hence of transitions in the rotational spectrum) for a molecule is determined by its symmetry. For convenience molecules are divided into four classes: (1) Linear molecules (2) Symmetric Tops. (3) Asymmetric Tops.

dealing with each in turn:

(1) Linear molecules: For a linear molecule, the separation of lines in the rotational spectrum can be related directly to the moment of inertia of the molecule, and for a molecule of known atomic masses, can be used to determine the bond lengths (structure) directly. For diatomic molecules this process is trivial, and can be made from a single measurement of the rotational spectrum. For linear molecules with more atoms, rather more work is required, and it is necessary to measure molecules in which more than one isotope of each atom have been substituted (effectively this gives rise to a set of simultaneous equations which can be solved for the bond lengths).

(2) For symmetric tops: A symmetric top is a molecule in which two moments of inertia are the same, and the third is different. As a matter of historical convenience, spectroscopists divide molecules into two classes of symmetric tops, Oblate symmetric tops (saucer shaped) and Prolate symmetric tops (rugby football, or cigar shaped). The spectra look rather different, and are instantly recognizable. As for linear molecules, structure (bond lengths and bond angles) can be deduced from the spectra.

(3) For Asymmetric tops all three moments of inertia are different and a simple interpretation of the spectrum is not normally possible. Sometimes asymmetric tops have spectra that are similar to those of a linear molecule or a symmetric top, in which case the molecular structure must also be similar to that of a linear molecule or a symmetric top. For the most general case however, all that can be done is to fit the spectra to three different moments of inertia. If the molecular formula is known, then educated guesses can be made of the possible structure, and from this guessed structure, the moments of inertia can be calculated. If the calculated moments of inertia agree well with the measured moments of inertia, then the structure can be said to have been determined. For this approach to determining molecular structure, isotopic substitution is invaluable.

Hyperfine interactions In addition to the main structure that is observed in microwave spectra that is due to the rotational motion of the molecules, a whole host of further interactions are responsible for small details in the spectra, and the study of these details provides a very deep understanding of molecular quantum mechanics. The main interactions responsible for small changes in the spectra (additional splittings and shifts of lines) are due to magnetic and electrostatic interactions in the molecule. The particular strength of such interactions differs in different molecules, but in general, the order of these effects (in decreasing significance) is: (a) electron spin - electron spin interaction (this occurs in molecules with two or more unpaired electrons, and is a magnetic-dipole / magnetic-dipole interaction) (b) electron spin - molecular rotation (the rotation of a molecule corresponds to a magnetic dipole, which interacts with the magnetic dipole moment of the electron) (c) electron spin - nuclear spin interaction (the interaction between the magnetic dipole moment of the electron and the magnetic dipole moment of the nuclei (if present)). (d) electron charge - nuclear electric quadrupole interaction (the interaction between the electron and the electric quadrupole moments of nuclei (if present)). (e) nuclear spin - nuclear spin interaction (nuclear magnetic moments interacting with one another).

These interactions give rise to the characteristic energy levels that are probed in "magnetic resonance" spectroscopy such as nmr and esr, where they represent the "zero field splittings" which are always present.

## Rotational-Vibrational Spectroscopy

Vibrational transitions occur in conjunction with rotational transitions. Consequently, it is possible to observe both rotational and vibrational transitions in the vibrational spectrum. Although many methods are available for observing vibrational spectra, the two most common methods are infrared spectroscopy and Raman spectroscopy.

The energy of rotational transitions is on the order of [itex]10^{-21}[itex] J/mol whereas vibrational transitions have energies on the order of [itex]10^{-20}[itex] J/mol. Therefore, highly-resolved vibrational spectra will contain fine structure corresponding to the rotational transitions that occur at the same time as a vibrational transition. Although molecular vibrations and rotations do have some effect on one other, this interaction is usually small. Consequently, the rotational and vibrational contributions to the energy of the molecule can be considered independently:

[itex] E_{vib,rot} = E_{vib} + E_{rot} = \left ( n + {1 \over 2} \right )h\nu_0 + hc\bar B J \left ( J + 1 \right ) [itex]

where [itex] n [itex] is the vibrational quantum number, [itex] J [itex] is the rotational quantum number, h is Planck's constant, [itex]\nu_0[itex] is the frequency of the vibration, [itex]c[itex] is the speed of light, and [itex]\bar B[itex] is the rotational constant.

### Evaluating Spectra

Missing image
DCl-Fourier_Transform_Infrared_Spectrum.JPG
DCl (deuterium chloride) IR Spectrum. The P- and R-branches compose a single vibrational line.

The image to the right shows the infrared spectrum of DCl. All of the peaks in this spectrum correspond to a single vibrational transition. The reason that multiple peaks are observed is because of the various rotational transitions that can occur at the same time as the vibrational transition. The peaks on the left are known as the R-branch and corresponds to transitions for which the rotational energy of the system increases ([itex] \Delta J = 1 [itex]). The peaks on the right are called the P-branch and represents transitions for which the rotational energy of the system decreases ([itex] \Delta J = -1 [itex]).

The gap between the R- and P-branches is known as the Q-branch. A peak would appear here for a vibrational transition in which the rotational energy did not change ([itex] \Delta J = 0 [itex]). However, according to the quantum mechanical rigid rotor model upon which rotational spectroscopy is based, there is a spectroscopic selection rule that requires that [itex] \Delta J = \pm 1 [itex]. This selection rule explains why the P- and R-branches are observed, but not the Q-branch (as well as branches for which [itex] \Delta J = \pm 2 [itex], [itex] \Delta J = \pm 3 [itex], etc.).

The positions of the peaks in the spectrum can be predicted using the rigid rotor model. One prediction of the rigid rotor model is that the space between each of the peaks should be [itex] 2\bar B [itex] where [itex] \bar B [itex] is the rotational constant for a given molecule. Experimentally, it is observed that the spacing between the R-branch peaks decreases as the frequency increases. Similarly, the spacing between the P-branch peaks increases as the frequency decreases. This variation in the spacing results from the bonds between the atoms in a molecule not being entirely rigid.

This variation can be mostly accounted for using a slightly more complex model that takes into account the variation in the rotational constant as the vibrational energy changes. Using this model, the positions of the R-branch peaks are predicted to be at:

[itex] \bar \nu_R = \nu_0 + 2\bar B_1 + \left ( 3\bar B_1 - \bar B_0 \right )J + \left ( \bar B_1 -\bar B_0 \right )J^2 \qquad J = 0,1,2... [itex]

where [itex] \bar B_0 [itex] is the rotational constant for the [itex] n = 0 [itex] vibrational level and [itex] \bar B_1 [itex] is the rotational constant for the [itex] n = 1 [itex] vibrational level. Likewise, the P-branch peaks are predicted to be at:

[itex] \bar \nu_P = \bar \nu_0 - \left ( \bar B_1 + \bar B_0 \right )J + \left ( \bar B_1 - \bar B_0 \right )J^2 \qquad J = 1,2,3... [itex]

Rotational-vibrational spectra will also show some fine structure due to the presence of different isotopes in the spectrum. In the spectrum shown above, all of the rotational peaks are slightly split into two peaks. One peak corresponds to 35Cl and the other to 37Cl. The ratio of the peak intensities corresponds to the natural abundance of these two isotopes.

Resources: Microwave Spectroscopy, Townes and Schawlow, Dover; Rotational Spectroscopy, Harry Kroto, Dover; Rotational Spectroscopy of Diatomic molecules, Brown and Carrington; Quantum Mechanics, Mcquarrie, Donald A.

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