# Classification of rigid rotors

It is possible, and customary in microwave spectroscopy, to **classify rigid rotors** by the relative size of their principal moments of inertia.

## Contents

## Inertia moment

*See the main article: inertia moment.*

The definition of rigid rotor stems from classical mechanics. The concept is applied in molecular physics, especially in microwave spectroscopy. This is the branch of spectroscopy that studies rotational transitions of molecules. In microwave spectroscopy molecules are regarded as rigid rotors (in first approximation).

In classical mechanics, as well as in quantum mechanics, the kinetic energy of rotation of a rigid rotor is linear in a quantity called the inertia tensor. This is a real, symmetric, 3 × 3 tensor which has three real eigenvalues: the *principal moments of inertia*,^{[1]} denoted by *I*_{A}, *I*_{B}, *I*_{C}.
The corresponding eigenvectors (the *principal axes*) are orthogonal and have as common origin the center of mass of the rotor.

## Classification of molecules based on inertia moments

The principal axes are ordered such that associated inertia moments decrease, that is, the *A*-axis has the smallest moment of inertia and other axes are such that *I*_{A} ≤*I*_{B} ≤ *I*_{C}. Depending on the relative size of the inertia moments, rotors can de divided into four classes.

### Linear rotors

For a linear molecule *I*_{A} << *I*_{B} = *I*_{C}. For most purposes *I*_{A} can 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. Since the moment of inertia is quadratic in the bond lengths, the microwave spectrum yields the bond lengths directly, provided the atomic masses are known.

Examples of linear molecules are obviously the diatomics such as oxygen (O=O), carbon monoxide (C≡O), and nitrogen (N≡N). But also many triatoms are linear: carbon dioxide (O=C=O), hydrogen cyanide (HC≡N), and carbonyl sulfide (O=C=S). Examples of larger linear molecules are chloroethyne (HC≡CCl), and acetylene (HC≡CH).

### Symmetric tops

A symmetric top is a rotor in which two moments of inertia are the same. Spectroscopists divide molecules into two classes of symmetric tops, *oblate symmetric tops* (frisbee or disc shaped) with *I*_{A} = *I*_{B} < *I*_{C} and *prolate symmetric tops* (cigar shaped) with *I*_{A} < *I*_{B} = *I*_{C}.
Symmetric tops have a three-fold or higher rotational symmetry axis.

Examples of oblate symmetric tops are: benzene (C_{6}H_{6}), cyclobutadiene (C_{4}H_{4}), and ammonia (NH_{3}). Prolate tops are: chloroform(CHCl_{3}) and methylacetylene (CH_{3}C≡CH).

### Spherical tops

A spherical top molecule is a special case of a symmetric tops with equal moment of inertia about all three axes *I*_{A} = *I*_{B} = *I*_{C}. The spherical top molecules have cubic symmetry.

Examples of spherical tops are: methane (CH_{4}), phosphorus tetramer (P_{4}), carbon tetrachloride (CCl_{4}), ammonium ion (NH_{4}^{+}), and uranium hexafluoride (UF_{6}).

### Asymmetric tops

A rotor is an asymmetric top if all three moments of inertia are different. Most of the larger molecules are asymmetric tops. For such molecules a simple interpretation of the microwave spectrum usually is not possible. Sometimes asymmetric tops have rotational 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. In the determination of the molecular structure of asymmetric tops from microwave spectrum, isotopic substitution is invaluable.

Examples of asymmetric tops: anthracene (C_{14}H_{10}), water (H_{2}O), and nitrogen dioxide (NO_{2}).

## Note

- ↑ Often the adjective
*principal*is omitted, which is somewhat sloppy, because a moment of inertia can be defined with respect to any axis, and a principal moment is defined with respect to a principal axis. Following common usage we will drop the word*principal*, however.