Using an ab initio potential we perform classical and path-integral Mo
nte Carlo simulations together with finite-size scaling to investigate
the head-tail ordering of CO molecules in a full mono-layer on graphi
te. In accordance with experiments, the transition is found to belong
to the 2 D Ising universality class and occurs in the correct temperat
ure range; the quantum shift of the transition amounts to 10%. Beyond
what is known from experiments we predict a particular ferrielectric g
round state, the critical amplitude of the order parameter, and that t
he ordering is caused by molecular shape asymmmetry rather than electr
ostatic dipole forces.