Rotational spectroscopy

Rotational spectroscopy is concerned with the measurement of the energies of transitions between quantized rotational states of molecules in the gas phase. The rotational spectrum of polar molecules can be measured in absorption or emission by microwave spectroscopy or by far infrared spectroscopy. The rotational spectra of non-polar molecules cannot be observed by those methods, but can be observed and measured by Raman spectroscopy. Rotational spectroscopy is sometimes referred to as pure rotational spectroscopy to distinguish it from rotational-vibrational spectroscopy where changes in rotational energy occur together with changes in vibrational energy, and also from ro-vibronic spectroscopy where rotational, vibrational and electronic energy changes occur simultaneously.
For rotational spectroscopy, molecules are classified according to symmetry into spherical tops, linear molecules, and symmetric tops; analytical expressions can be derived for the rotational energy terms of these molecules. Analytical expressions can be derived for the fourth category, asymmetric top, for rotational levels up to J=3, but higher energy levels need to be determined using numerical methods. The rotational energies are derived theoretically by considering the molecules to be rigid rotors and then applying extra terms to account for centrifugal distortion, fine structure, hyperfine structure and Coriolis coupling. Fitting the spectra to the theoretical expressions gives numerical values of the angular moments of inertia from which very precise values of molecular bond lengths and angles can be derived in favorable cases. In the presence of an electrostatic field there is Stark splitting which allows molecular electric dipole moments to be determined.
An important application of rotational spectroscopy is in exploration of the chemical composition of the interstellar medium using radio telescopes.

Applications

Rotational spectroscopy has primarily been used to investigate fundamental aspects of molecular physics. It is a uniquely precise tool for the determination of molecular structure in gas-phase molecules. It can be used to establish barriers to internal rotation such as that associated with the rotation of the group relative to the group in chlorotoluene. When fine or hyperfine structure can be observed, the technique also provides information on the electronic structures of molecules. Much of current understanding of the nature of weak molecular interactions such as van der Waals, hydrogen and halogen bonds has been established through rotational spectroscopy. In connection with radio astronomy, the technique has a key role in exploration of the chemical composition of the interstellar medium. Microwave transitions are measured in the laboratory and matched
to emissions from the interstellar medium using a radio telescope. was the first stable polyatomic molecule to be identified in the interstellar medium. The measurement of chlorine monoxide is important for atmospheric chemistry. Current projects in astrochemistry involve both laboratory microwave spectroscopy and observations made using modern radiotelescopes such as the Atacama Large Millimeter/submillimeter Array.

Overview

A molecule in the gas phase is free to rotate relative to a set of mutually orthogonal axes of fixed orientation in space, centered on the center of mass of the molecule. Free rotation is not possible for molecules in liquid or solid phases due to the presence of intermolecular forces. Rotation about each unique axis is associated with a set of quantized energy levels dependent on the moment of inertia about that axis and a quantum number. Thus, for linear molecules the energy levels are described by a single moment of inertia and a single quantum number,, which defines the magnitude of the rotational angular momentum.
For nonlinear molecules which are symmetric rotors, there are two moments of inertia and the energy also depends on a second rotational quantum number,, which defines the vector component of rotational angular momentum along the principal symmetry axis. Analysis of spectroscopic data with the expressions detailed below results in quantitative determination of the value of the moment of inertia. From these precise values of the molecular structure and dimensions may be obtained.
For a linear molecule, analysis of the rotational spectrum provides values for the rotational constant and the moment of inertia of the molecule, and, knowing the atomic masses, can be used to determine the bond length directly. For diatomic molecules this process is straightforward. For linear molecules with more than two atoms it is necessary to measure the spectra of two or more isotopologues, such as ¹⁶O¹²C³²S and ¹⁶O¹²C³⁴S. This allows a set of simultaneous equations to be set up and solved for the bond lengths). A bond length obtained in this way is slightly different from the equilibrium bond length. This is because there is zero-point energy in the vibrational ground state, to which the rotational states refer, whereas the equilibrium bond length is at the minimum in the potential energy curve. The relation between the rotational constants is given by
where v is a vibrational quantum number and α is a vibration-rotation interaction constant which can be calculated if the B values for two different vibrational states can be found.
For other molecules, if the spectra can be resolved and individual transitions assigned both bond lengths and bond angles can be deduced. When this is not possible, as with most asymmetric tops, all that can be done is to fit the spectra to three moments of inertia calculated from an assumed molecular structure. By varying the molecular structure the fit can be improved, giving a qualitative estimate of the structure. Isotopic substitution is invaluable when using this approach to the determination of molecular structure.

Classification of molecular rotors

In quantum mechanics the free rotation of a molecule is quantized, so that the rotational energy and the angular momentum can take only certain fixed values, which are related simply to the moment of inertia,, of the molecule. For any molecule, there are three moments of inertia:, and about three mutually orthogonal axes A, B, and C with the origin at the center of mass of the system. The general convention, used in this article, is to define the axes such that, with axis corresponding to the smallest moment of inertia. Some authors, however, define the axis as the molecular rotation axis of highest order.
The particular pattern of energy levels 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

Selection rules

Microwave and far-infrared spectra

Transitions between rotational states can be observed in molecules with a permanent electric dipole moment. A consequence of this rule is that no microwave spectrum can be observed for centrosymmetric linear molecules such as or HCCH, which are non-polar. Tetrahedral molecules such as , which have both a zero dipole moment and isotropic polarizability, would not have a pure rotation spectrum but for the effect of centrifugal distortion; when the molecule rotates about a 3-fold symmetry axis a small dipole moment is created, allowing a weak rotation spectrum to be observed by microwave spectroscopy.
With symmetric tops, the selection rule for electric-dipole-allowed pure rotation transitions is,. Since these transitions are due to absorption of a single photon with a spin of one, conservation of angular momentum implies that the molecular angular momentum can change by at most one unit. Moreover, the quantum number K is limited to have values between and including +J to -J.

Raman spectra

For Raman spectra the molecules undergo transitions in which an incident photon is absorbed and another scattered photon is emitted. The general selection rule for such a transition to be allowed is that the molecular polarizability must be anisotropic, which means that it is not the same in all directions. Polarizability is a 3-dimensional tensor that can be represented as an ellipsoid. The polarizability ellipsoid of spherical top molecules is in fact spherical so those molecules show no rotational Raman spectrum. For all other molecules both Stokes and anti-Stokes lines can be observed and they have similar intensities due to the fact that many rotational states are thermally populated. The selection rule for linear molecules is ΔJ = 0, ±2. The reason for the values ±2 is that the polarizability returns to the same value twice during a rotation. The value ΔJ = 0 does not correspond to a molecular transition but rather to Rayleigh scattering in which the incident photon merely changes direction.
The selection rule for symmetric top molecules is
Transitions with ΔJ = +1 are said to belong to the R series, whereas transitions with belong to an S series. Since Raman transitions involve two photons, it is possible for the molecular angular momentum to change by two units.

Units

The units used for rotational constants depend on the type of measurement. With infrared spectra in the wavenumber scale, the unit is usually the inverse centimeter, written as cm⁻¹, which is literally the number of waves in one centimeter, or the reciprocal of the wavelength in centimeters. On the other hand, for microwave spectra in the frequency scale, the unit is usually the gigahertz. The relationship between these two units is derived from the expression
where ν is a frequency, λ is a wavelength and c is the velocity of light. It follows that
As 1 GHz = 10⁹ Hz, the numerical conversion can be expressed as