Breaking degeneracy in phosphorus pentachloride Recently, in a class discussion, the following point was brought up: 


  Although the axial and equatorial bond lengths in $\ce {PCl5}$ are inequivalent, when we consider the time-averaged $\ce {P-Cl}$ bond length, there should only be one value sincethe molecule is symmetric and thus, all the chlorine atoms interact with the phosphorus atom to the same extent. 


The idea of "time-averaged" means that we are considering the $\ce {P-Cl}$ bond over a long duration. This bond length varies over time since the atoms are constantly in motion. When this bond length is "time-averaged", it means that we are taking the average of the bond lengths measured at infinitely many instants in time. 

The argument is that although at any particular instant, the molecule would adopt a configuration with axial bonds being longer than equatorial bonds, when the bonds are time-averaged, they should all be the same length. 

This has caused much dispute among members of the class and it has also led me to consult a professor who is well-versed with quantum chemistry. The professor mentioned that the argument put forward is not valid because in $\ce {PCl5}$, there is a breaking of degeneracy due to perturbation. He cited the Jahn-Teller distortion as a situation similar to that in $\ce {PCl5}$, where degeneracy is broken. 

I am very confused by this. What exactly is this perturbation? Why does it exist for $\ce {PCl5}$ and not for similar molecules, such as $\ce {SF6}$ (as seen in the comments for this question)?
I would guess he means a spontaneous symmetry breaking, in which case the "perturbation" is any old force from the environment.  This can happen in physical systems: it's possible for a system of high symmetry to spontaneously fall to a lower-energy state of lower symmetry.  Only the tiniest of perturbations is required, if you wait long enough.  A familiar classical example is crystallization, in which below a certain temperature a system of atoms spontaneously loses complete translational symmetry even in the absence of an external potential.  A quantum example is magnetization when a ferromagnet cools below the Curie temperature -- the spins lose rotational symmetry and pick some essentially random preferred direction, in response to the tiniest of perturbations from the environment.  In the case of PCl5, one would assume, very roughly (and it would take detailed calculation to verify) that if 3 of the Cls pull in closer to the nucleus and 2 move further away, you get a better balance of nuclear-electron attraction and nuclear-nuclear and electron-electron repulsion than if all 5 remain at the same distance.  So the system spontaneously breaks rotational symmetry, in response to any old tiny perturbation in the environment.

That said, in the absence of an external potential, the rotational quantum state of the molecule will be entirely symmetric, and you will have an equal chance of finding the long Cl-P-Cl axis pointing in any direction.