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# Rotational spectroscopy

Rotational spectroscopy or microwave spectroscopy studies the absorption and emission electromagnetic radiation (typically in the microwave region of the electromagnetic spectrum) by molecules associated with a corresponding change in the rotational quantum number of the molecule. 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.

Rotational spectrum from a molecule (to first order) requires that the molecule 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). Diatomic molecules such as dioxygen (O2), dihydrogen (H2), etc. do not have a dipole moment and hence no purely rotational spectrum. However, electronic excitations can lead to asymmetric charge distributions and thus provide a net dipole moment to the molecule. Under such circumstances, these molecules will exhibit a rotational spectrum.

Amongst the diatomic molecules, carbon monoxide (CO) has one of the simplest rotational spectra. As for tri-atomic molecules, hydrogen cyanide (HC≡N) has a simple rotational spectrum for a linear molecule and hydrogen isocyanide (HN=C:) for a non-linear molecule. As the number of atoms increases the spectrum becomes more complex as lines due to different transitions start overlapping.

## Understanding the rotational spectrum

In quantum mechanics the free rotation of a molecule is quantized, that is the rotational energy and the angular momentum can only take certain fixed values; what these values are is simply related to the moment of inertia, I, of the molecule. In general for any molecule, there are three moments of inertia: IA, IB and IC about three mutually orthogonal axes A, B, and C with the origin at the center of mass of the system. A linear molecule is a special case in this regard. These molecules are cylindrically symmetric and one of the moment of inertia (IA, which is the moment of inertia for a rotation taking place along the axis of the molecule) is negligible (i.e. IA < < IB = IC).

### Classification of molecules based on rotational behavior

The general convention is to define the axes such that the axis A has the smallest moment of inertia (and hence the highest rotational frequency) and other axes such that IA < = IB < = IC. Sometimes the axis A may be associated with the symmetric axis of the molecule, if any. If such is the case, then IA need not be the smallest moment of inertia. To avoid confusion, we will stick with the former convention for the rest of the article. The particular pattern of energy levels (and hence of transitions in the rotational spectrum) for a molecule is determined by its symmetry. A convenient way to look at the molecules is to divide them into four different classes (based on the symmetry of their structure). These are,

1. Linear molecules (or linear rotors)
2. Symmetric tops (or symmetic rotors)
3. Spherical tops (or spherical rotors) and
4. Asymmetric tops

Dealing with each in turn:

1. Linear molecules:
• As mentioned earlier, for a linear molecule IA < < IB = IC. For most of the purposes, IA is taken to be zero. 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).
• Examples or linear molecules: dioxygen (O=O), carbon monoxide (O≡C*), hydroxy radical (OH), carbon dioxide (O=C=O), hydrogen cyanide (HC≡N), carbonyl sulfide (O=C=S), chloroethyne (HC≡CCl), acetylene (HC≡CH)
2. Symmetric tops:
• A symmetric top is a molecule in which two moments of inertia are the same. As a matter of historical convenience, spectroscopists divide molecules into two classes of symmetric tops, Oblate symmetric tops (saucer or disc shaped) with IA = IB < IC and Prolate symmetric tops (rugby football, or cigar shaped) with IA < IB = IC. The spectra look rather different, and are instantly recognizable. As for linear molecules, the structure of symmetric tops (bond lengths and bond angles) can be deduced from their spectra.
• Examples of symmetric tops:
3. Spherical tops:
4. Asymmetric tops:
• As you would have guessed a molecule is termed an asymmetric top if all three moments of inertia are different. Most of the larger molecules are asymmetric tops, even when they have a high degree of symmetry. Generally for such molecules 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.
• Examples of asymmetric tops: anthracene (C14H10), water (H2O), nitrogen dioxide (NO2)

### Structure of rotational spectrum

• Linear molecules

These molecules have two degenerate modes of rotation (IB = IC, IA = 0). Since we cannot distinguish between the two modes, we need only one rotational quantum number (J) to describe the rotational motion of the molecule.

The rotational energy levels ($F \left( J \right)$) of the molecule based on rigid rotor model can be expressed as,

$F\left( J \right) = \tilde B_{e} J \left( J+1 \right) \qquad J = 0,1,2,...$

where $\tilde B_e$ is the rotational constant of the molecule and is related to the moment of inertia of the molecule IB = IC by,

$\tilde B_e = {h \over{8\pi^2cI_B}}$

Selection rules dictate that during emission or absorption the rotational quantum number has to change by unity i.e. $\Delta J = J^{\prime} - J^{\prime\prime} = \pm 1$. Thus the locations of the lines in a rotational spectrum will be given by,

$\tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = F\left( J^{\prime} \right) - F\left( J^{\prime\prime} \right) = 2 \tilde B_e \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,...$

where $J^{\prime\prime}$ denotes the lower energy level and $J^{\prime}$ denotes higher energy level involved in the transition. The height of the lines is determined by the distribution of the molecules in the different levels and the probability of transition between two energy levels.

We observe that, for a rigid rotor, the transition lines are equally spaced in the wavenumber space. However, this is not always the case, except for the rigid rotor model. For non-rigid rotor model, we need to consider changes in the moment of inertia of the molecule. Two primary reasons for this are,

• Centrifugal distortion:

When a molecule rotates, the centrifugal force pulls the atoms apart. As a result, the moment of inertia of the molecule increases, thus decreasing $\tilde B_e$. To account for this a centrifugal distortion correction term is added to the rotational energy levels of the molecule.

$F\left( J \right) = \tilde B_{e} J \left( J+1 \right) - \tilde D_{e} J^2 \left( J+1 \right)^2 \qquad J = 0,1,2,...$

where $\tilde D_e$ is the centrifugal distortion constant.

Accordingly the line spacing for the rotational mode changes to,

$\tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime}} = 2 \tilde B_e \left( J^{\prime\prime} + 1 \right) - 4\tilde D_e \left( J^{\prime\prime} +1 \right)^3 \qquad J^{\prime\prime} = 0,1,2,...$
• Effect of vibration on rotation:

A molecule is always in vibration. As the molecule vibrates, its moment of inertia changes. Further there is a fictitious force, Coriolis coupling, between the vibrational motion of the nuclei in the rotating (non-inertial) frame. However, as long as the vibrational quantum number does not change (i.e. the molecule is in only one state of vibration), the effect of vibration on rotation are not important, because the time for vibration is much greater than the time required for rotation. The Coriolis coupling is often negligible, too, if one is interested in low vibrational and rotational quantum numbers only.

• Symmetric Top

The rotational motion of a symmetric top molecule can be described by two independent rotational quantum numbers (since two axes have equal moments of inertia, the rotational motion about these axes requires only one rotational quantum number for complete description). Instead of defining the two rotational quantum numbers for two independent axes, we associate one of the quantum number (J) with the total angular momentum of the molecule and the other quantum number (K) with the angular momentum of the axis which has different moment of inertia (i.e. axis C for oblate symmetric top and axis A for prolate symmetric tops). The rotational energy $F\left(J,K\right)$ of such a molecule, based on rigid rotor assumptions can be expressed in terms of the two previously defined rotational quantum numbers as follows,

$F\left( J,K \right) = \tilde B J \left( J+1 \right) + \left( \tilde A - \tilde B \right) K^2 \qquad J = 0,1,2,... \quad \mbox{and}\quad K = -J, -J+1, ...,-1, 0, 1, ..., J-1, J$

where $\tilde B = {h\over{8\pi^2cI_B}}$ and $\tilde A = {h\over{8\pi^2cI_A}}$ for a prolate symmetric top molecule or $\tilde A = {h\over{8\pi^2cI_C}}$ for an oblate molecule.

Selection rule for the these molecules provide the guidelines for possible transitions. Accordingly,

$\Delta J = \pm 1 \quad \mbox{and} \quad \Delta K = 0$.

This is so because K is associated with the axis about which the molecule is symmetric and hence has no net dipole moment in that direction. Thus there is no interaction of this mode with the light particles (photon).

This gives the transition wavenumbers of,

$\tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) = 2 \tilde B \left( J^{\prime\prime} + 1 \right) \qquad J^{\prime\prime} = 0,1,2,...$

which is the same as in the case of a linear molecule.

In case of non-rigid rotors, the first order centrifugal distortion correction is given by,

$F\left( J,K \right) = \tilde B J \left( J+1 \right) + \left( \tilde A - \tilde B \right) K^2 - \tilde D_J J^2\left(J+1\right)^2 - \tilde D_{JK}J\left(J+1\right)K^2 - D_KK^4 \qquad J = 0,1,2,... \quad \mbox{and}\quad K = -J,...,0, ..., J$

The suffixes on the centrifugal distortion constant D indicate the rotational mode involved and are not a function of the rotational quantum number. The location of the transition lines on a spectrum are given by,

$\tilde \nu_{J^{\prime}\leftrightarrow J^{\prime\prime},K} = F\left( J^{\prime},K \right) - F\left( J^{\prime\prime},K \right) = 2 \tilde B \left( J^{\prime\prime} + 1 \right) -4D_J\left(J^{\prime\prime}+1\right)^3 - 2D_{JK}\left(J^{\prime\prime}+1\right)K^2 \qquad J^{\prime\prime} = 0,1,2,...$
• Spherical Tops

Unlike other molecules, spherical top molecules have no net dipole moment, and hence they do not exhibit a pure rotational spectrum.

• Asymmetric Tops

The spectrum for these molecules usually involves many lines due to three different rotational modes and their combinations. There is no general rule for studying the rotational spectrum of these molecules.

Hyperfine interactions:

In addition to the main structure that is observed in microwave spectra 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:

1. electron spin - electron spin interaction (this occurs in molecules with two or more unpaired electrons, and is a magnetic-dipole / magnetic-dipole interaction)
2. electron spin - molecular rotation (the rotation of a molecule corresponds to a magnetic dipole, which interacts with the magnetic dipole moment of the electron)
3. 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)).
4. electric field gradient - nuclear electric quadrupole interaction (the interaction between the electric field gradient of the electron cloud of the molecule and the electric quadrupole moments of nuclei (if present)).
5. 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.

## Experimental determination of the spectrum

Fourier transform infrared (FTIR) spectroscopy can be used to experimentally study rotational spectrum.

## Applications

Microwave spectroscopy is commonly used in physical chemistry to determine the structure of small molecules (such as ozone, methanol, or water) with high precision. Other common techniques for determining molecular structure, such as X-ray crystallography don't work very well for some of these molecules (especially the gases) and are not as precise. However, microwave spectroscopy is not useful for determining the structures of large molecules such as proteins.

## References

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