Moments of inertia of benzene and ammonia as symmetrial tops

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?

• The three I axes are always orthogonal and along x, y, z, but the I values along these directions can differ depending on geometry. You can choose any set of xyz orientation wrt to the molecule but its easier to use the molecules symmetry axes if possible. The molecule rotates about the total angular momentum which is the vector sum the 3 components. See chapter 7 of applying-maths-book.com for a discussion and examples of the 3x3 matrix you may need to diagonalise in the general case if you choose arbitrary xyz. Diagonalising rotates the axes to be along molecules own MOI direction. Oct 20 at 9:56
• There is non fundamental need for the axes to be orthogonal, merely non-coplanar is enough for a complete description of the object. This is so the unit vectors in the directions of the axes form a complete basis for the three dimensional space. But certainly if you can choose your axes an orthogonal choice will almost always make your life easier. Oct 20 at 11:15
• So @IanBush both options I mentioned for the two inertia axis (a,b) for benzene that I mentioned can be considered correct? Oct 20 at 12:39
• All the above are valid choices. They just may not have the same values. See @porphyrin comment on diagonalisation about relating them. Oct 20 at 13:01
• @IanBush But we can only measure two rotational constants for the molecule can’t we? 🤔 So there should be only two or am I wrong? And about porphyrins comment, I am not that deep into the topic as my expertise is in bioorganic chemistry so its a bit difficult to understand 🥲 Oct 20 at 15:25