From the quantum mechanics(QM) viewpoint, does there exist a Hamiltonian H for the system of H2O molecules? Assume that the number of H2O molecules is fixed.

Yes, the multi-particle Hamiltonian with the Coulomb potential energy. But calculating the free energy $F(T,V,N)$ from such Hamiltonian, although straightforward in principle, would be mathematically $very$ difficult.

How much gaseous, liquid and solid phase is present at the triple point is not determined by any parameter $\lambda$ in the microscopic Hamiltonian; it is rather part of the macroscopic description, similarly to volume $V$ and molar number $N$.

I do not know of approach that would describe water by free energy function(al) that uses order parameter (similar to Landau theory of magnetism). The Landau theory was meant for second-order phase transitions ; it does not seem to make sense for first-order transition that water may undergo at the triple point.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user Ján Lalinský