# What is the link between the classical and quantum definition of the symmetry number?

From what I understand, the symmetry number for a molecule can be defined in 2 ways:

1. The quantum mechanical symmetry number corrects for overcounting the number of possible rotational states of a rotating molecule. Here's a source using this method.

2. The classical symmetry number corrects for including physical orientations of that molecule that appear the same because of its symmetrical physical structure. Here's a source using this method (a PDF file)

This source claims at the end of page 3, that the classical view is roughly based on the quantum view. I have a very hard time to grasp that.

How does the physical structure of a molecule have anything to do with nuclear spins being integer or half-integers, whether the total wavefunction needs to be symmetrical or antisymmetrical, and thus determining which rotational quantum states are allowed for a molecule?

• Without looking at the actual links, the statement that the "classical view is based on the quantum view" is non-sensical. The classical view for any system is derived from immediate observables. That's why it's classical. The quantum view requires an understanding of quantum mechanics, historically arriving much later. How could the earlier view be based on the latter view? – Zhe Jun 23 '20 at 0:02
• on page 3, $\sigma$ is constant, nothing to do with quantum or classical, but is 2 for homonuclear molecules because it is not possible to tell left from right and so extra counting is accounted for. – porphyrin Jun 23 '20 at 7:46
• @Zhe That is what I am trying to understand. If you look at page 3 of this source, it says the following: The symmetry number is rigorously based on the nuclear spins. Here, the symmetry number corrects for the number of physical orientations of a molecule that are indistinguishable. How is this based on nuclear spins? – Phy Jun 23 '20 at 21:44
• @porphyrin Please also see this end of page 6 of this source. Initially on page 3, it derived the symmetry number $\sigma$ based on nuclear spins and wavefunction symmetry to correct for the number of allowed quantum states with the factor $\frac{1}{\sigma}$ for the partition function. But on page 6 it says $\frac{1}{\sigma}$ is of classical mechanical origin, but must come out of QM – Phy Jun 23 '20 at 21:56
• That appears to be extending classical statistical mechanics so that it's consistent with quantum mechanics. – Zhe Jun 23 '20 at 22:17