Analytical solutions to quantum mechanical or quantum chemical problems exist only for very simple systems. A phase transition in a liquid is a many-body problem and thus not a simple phenomenon, and we have to resort to approximate numerical methods.

If you want to be as rigorous as possible within the quantum chemical framework, I'd say the biggest difficulty is simulating a proper ensemble of configurations in the liquid phase. Although the relevant (London dispersion) interactions are quite short-ranged, you'd need to calculate the free energy of interaction with the environment, i.e., Boltzmann-average over multiple configurations (possibly generated with a low-level method, using either Monte Carlo or Molecular Dynamics). Then use an accurate QM method that takes into account as much electron correlation as possible, since for this system correlation is basically the only cause of the attractive intermolecular force. To get the temperature dependence right, you'd probably have to decompose the free energy into enthalpic and entropic contributions.

Of course this is the 'ab initio' approach. If you use empirically parametrized methods, it may be possible to analytically calculate the interaction free energy using some implicit solvent-like methods. But it depends if you consider these parametrized theories to be "other mathematical frameworks".