# Do we have a fundamental Hamiltonian for the system of H$_2$O molecules?

From the quantum mechanics(QM) viewpoint, does there exist a Hamiltonian $H$ for the system of H$_2$O molecules? Assume that the number of H$_2$O molecules is fixed.

Imagine that by calculating the free energy $F(T)$ from the Hamiltonian $H$, one can reproduce the three common phases ice, water, and steam at different temperatures $T$(Here do we need order parameters to characterize these phases?). On the other hand, the Hamiltonian $H$ may depend on some parameters, i.e., $H=H(\lambda )$, and at $T=273K$, varying the parameter $\lambda$ causes the phase transition between ice and water without temperature changing.

So can we write a microscopic Hamiltonian $H$ of this kind in terms of the QM language?

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 $N$.