Determination of approximate quantum labels based on projections of the total angular momentum on the molecule-fixed axis

Molecular line lists, particularly those computed for high temperature applications, often have very few states assigned local quantum numbers. These are often important components for accurately determining line shape parameters required for radiative transfer simulations. The projection of the total angular momentum onto the molecule fixed axis ($k$) is investigated in the Radau internal coordinate system to determine when it can be considered a good quantum number. In such a coordinate system, when the square of the $k^{th}$ component of the wavefunction is greater than one half, then we can classify $k$ as a good quantum number in accordance with the theorem of Hose and Taylor. Furthermore, it is demonstrated that when this holds true, oblate and prolate quantum labels $K_{a}$ and $K_{c}$ can reliably be predicted. This is demonstrated for the water and ozone molecules.