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.

Inertia moment

See the main article: inertia moment.

The idea of a rigid rotor originates in classical mechanics, and finds a most important application 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 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, 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 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 $I_{A}<=I_{B}<=I_{C}$. Sometimes the axis $A$ may be associated with the symmetric axis of the molecule, if any. If such is the case, then $I_{A}$ 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. 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 of the purposes, $I_{A}$ 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: Oxygen (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)

Symmetric tops

A symmetric top is a rotor 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 $I_{A}=I_{B}<I_{C}$ and Prolate symmetric tops (rugby football, or cigar shaped) with $I_{A}<I_{B}=I_{C}$. 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:

• Oblate: Benzene (C₆H₆), Cyclobutadiene (C₄H₄), Ammonia (NH₃)
• Prolate: Chloroform (CHCl₃), Propyne (CH₃C≡CH)

Spherical tops

A spherical top molecule, can be considered as a special case of symmetric tops with equal moment of inertia about all three axes ($I_{A}=I_{B}=I_{C}$).

Examples of spherical tops: Phosphorus tetramer (P₄), Carbon tetrachloride (CCl₄), Nitrogen tetrahydride (NH₄), Ammonium ion (NH₄⁺), Sulfur hexafluoride (SF₆)

Asymmetric tops

A rotos is 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)