  In the mean-field theory, the stabilization of polar chiral molecules is
understood as a quantum phase transition where the mean-field ground state of
molecules changes from the achiral eigenstate of the molecular Hamiltonian to
one of the degenerated chiral states as the increase of the intermolecular
interaction. Starting from the many-body Hamiltonian of the molecular gases
with electric dipole-dipole interactions, we give the static non-linear
Schr\"{o}dinger equations without free parameters to explore the achiral-chiral
transitions of polar chiral molecules. We find that the polar chiral molecules
of different species can be classified into two categories: At the critical
point for the achiral-chiral transition, the mean-field ground state changes
continuously in one category, and changes discontinuously in the other
category. We further give the mean-field phase diagram of the achiral-chiral
transitions for both two categories.
