Energy Levels and Transition Frequencies

The rotational energy levels of a molecule result from the quantization of its rotational kinetic energy \begin{equation}E=\frac{1}{2}\left(\frac{P_x^2}{I_x}+\frac{P_y^2}{I_y}+\frac{P_z^2}{I_z}\right)\end{equation} where the $P_j$ are the components of the molecular angular momentum and the $I_j$ are the components of the principal moments of inertia. All real molecules have additional effects (e. g. centrifugal distortion, perturbations, and internal rotations) which significantly complicate the spectroscopic problem [1]. but which have minimal impact on the overall character of the spectra.

The fundamental underpinnings of the high specificity of THz spectroscopy are (1) that the rotational degree of freedom is unique in that many levels are thermally populated and (2) that the strong fundamental rotational transitions which arise from these levels are not associated with functional groups which may be constituents of many similar molecules. Rather they depend upon the global moment of inertia tensor $I$ of the molecule. Since THz spectroscopy is sensitive to changes in each of the $I_j$ of $< 10^{-7}$, each molecule has an unique signature if even a few of the lines of its rotational structure can be detected and measured.