This is a tough question to answer because the intermolecular distances are similar in the solid to liquid transition unlike those in the liquid to gas phase transition.
In the case of the elements there is a correlation between the Debye temperature and the melting temperature. The Debye temperature is that temperature at which the atoms gain their full degrees of freedom, and the correlation, while not prefect provides evidence for a connection between melting point and inter-atomic vibrations.
The idea (originally by Lindemann ) is that when the temperature reaches a certain value, the potential energy of the (harmonic) intermolecular vibration has now increased by a sufficient amount that atoms become independent of one another and the solid melts. Thus the energy profile of a solid is like an egg box with many depressions in which the atoms reside and have some limited freedom, given a bit more energy they become free. An approximate calculation shows that only a 5% increase in intermolecular spacing is enough to cause melting, which to me seems very small. Clearly this is an over simple model but does give some insight into the problem.
In your example it is far more difficult to give any quantitative answer and many examples should be studied to understand the general picture. If we follow this simple model we must assume that the vibrations are of whole CX or water molecules moving in their intermolecular potential. Hopefully someone who is an expert in this area can provide you with a more satisfying answer.