Question

Hello I was reading about symmetrical tops and how their properties relate to their rotational spectra. According to the 8th edition of Atkins, those molecules have one unique principal axis which includes a moment of inertia (denoted I∥). They also have two orthogonal moments of inertia, which are equal in magnitude (denoted I⊥).

Unfortunately, the book doesn't give any pictural representation, so my question remains: Do all moments of inertia need to be orthogonal to one another or just to the principal axis? On the internet I find conflicting information for the moments of inertia for benzene, as seen here. Here, you can see that the two I⊥ (a,b) are not orthogonal to each other, but in this illustration they are (A, B):
So, which is it? And would this representation of ammonia be correct for the three moments of inertia?

Answer 1

All moments of inertia in symmetrical top molecules are orthogonal to each other, including the two I⊥. This orthogonality is essential for accurately predicting the rotational energy levels of the molecule.

Step-by-step explanation:

In molecules like benzene or ammonia, the moments of inertia play a critical role in understanding their rotational spectra. For symmetrical tops, there is one unique principal axis, often denoted as I∥, and two equal magnitudes of moments of inertia perpendicular to this principal axis, denoted as I⊥.

To clarify, all moments of inertia must indeed be orthogonal to each other. In a symmetrical top molecule, there's a unique axis of rotation about which the molecule has rotational symmetry, giving us the I∥ value. The two I⊥ values correspond to the axes perpendicular to this principal axis and to each other. This orthogonality ensures that the molecule's rotational energy levels can be properly quantized and predicted accurately.

For a molecule like benzene, which is highly symmetrical, the three-dimensional structure dictates the moments of inertia such that the two perpendicular axes in the plane of the molecule (denoted as A and B) are indeed orthogonal to each other and also to the principal axis (often referred to as the C axis).

In the case of ammonia (NH3), a trigonal pyramidal molecule, the moments of inertia are also orthogonal. The axis of the unique moment of inertia passes through the nitrogen atom and is perpendicular to the plane formed by the three hydrogen atoms, while the other two I-values lie in the plane of the hydrogen atoms.