Algorithms for the Computation of Solutions and Bifurcations of the Ornstein–Zernike relations A. T. Peplow Department of Mathematics, Imperial College, 180 Queen’s Gate, South Kensington, London, SW7 2AZ.; a.peplow@ic.ac.uk. R. E. Beardmore Department of Mathematics, Imperial College, 180 Queen’s Gate, South Kensington, London, SW7 2AZ; r.beardmore@ic.ac.uk F. Bresme Department of Chemistry, Imperial College, South Kensington, London, SW7 2AZ; f.bresme@ic.ac.uk (Dated: May 26, 2006) Abstract Direct and indirect correlation functions of simple liquids associated through the Ornstein-Zernike (OZ) relation are calculated with the hyper-netted chain (HNC). Gas-liquid transition regions for Lennard-Jones and other similar systems are studied to a high degree of numerical accuracy. These systems exhibit a vapour–liquid phase separation. We are particularly concerned with the behaviour of the HNC solution near the two–phase region in the phase diagram. It is believed that systems with HNC closure do not have solutions inside a certain region whose boundary line is not a spinodal line. We are able to identify precisely the values of temperature and density along the envelope of a so-called no-solution region in parameter space that has been reported to exist in the literature. The computa- tional analysis shows a rather complicated solution structure, with the existence of real-valued solutions inside the no-solution region and the existence of hysteresis or multiple fold bifurcations near the region of zero inverse isothermal compressibility. PACS numbers: 1