Multiple critical points for square-well potential with repulsive shoulder

A global renormalization procedure used recently to calculate thermal volumetric properties near and to far from the critical point for several simple square-well and Lennard-Jones fluids for which comparison could be made with Monte Carlo and molecular dynamic simulations is here applied to a square-well with repulsive shoulder of width and height that gives ratios of first and second critical point temperatures and pressures similar to ones proposed recently as possibilities for water, if indeed water has a second critical point below which, briefly, before ice forms, it has separate low density and high density liquid phases. Thermal behavior at and below the second critical point is found to be quite sensitive to choices of height and width of the shoulder relative to the depth and width of the well, with, for some choices, a subsequent closing of the liquid-liquid coexistence curve as temperature drops, and emergence at a yet lower temperature of what appears to be a third critical point.