  It has been extensively demonstrated through first principles quantum
mechanics calculations that water exhibits strong hydrogen bond cooperativity.
Classical molecular simulation and statistical mechanics methods typically
assume pairwise additivity, meaning they cannot account for these 3-body and
higher cooperative effects. In this document, we extend second order
thermodynamic perturbation theory to correct for hydrogen bond cooperativity in
4 site water. We show that the association theory gives substantially different
predictions than the first order result, which does not include hydrogen bond
cooperativity. By comparison to spectroscopy, neutron diffraction and molecular
simulation data, we show that the theory accurately predicts the hydrogen
bonding structure of water as a function of temperature and density.
