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Extra resources for Advances in Chemical Physics (Volume 139)
Duh and Haymet Division Scheme (DHHDS) To address the problem of uniqueness of the closure, these authors  proposed a method for partitioning the potential energy, by defining the perturbation part to be density dependent (see Fig. 4), so that uLR ðrÞ ¼ À4E s6 r exp À1 s6rÃ rÃ r ð70Þ The physical picture behind this choice is that (1) at high densities, the system is dominated by the repulsive part of the potential, while (2) at low densities, the attractive part begins to play a more important role.
Then (82) can be rewritten as: Z bmex ¼ r Z Z 1 1 2 hðrÞcðrÞ qBðr; lÞ h ðrÞ À cðrÞ þ BðrÞ À dl l dr þ r dr hðrÞ 2 2 ql 0 ð88Þ recent advances in the field of integral equation theories 45 The second assumption is the unique functionality  of the bridge function, meaning that B ¼ B½g is a simple function of the indirect correlation function, as expressed in the majority of the approximations for the bridge function. Changing the integration variable from l to g, and integrating by parts yields Z r Z1 dr hðrÞ l 0 qBðr; lÞ dl ¼ r ql Z dr½hðrÞBðrÞ À SðrÞ ð89Þ where hðrÞ SðrÞ ¼ gðrÞ gÃZ ðl¼1Þ Bðg0Þdg0 ð90Þ gÃ ðl¼0Þ Again, a homotopy has to be constructed (the parameter l varies from 0 to 1), so that gÃ ðr; lÞ ¼ g0 ðrÞ þ lg1 ðrÞ ð91Þ Then, the calculation of the chemical R potential requires the knowledge of the Star function  defined by SÃ ¼ r drSðrÞ.
After a carefull analysis of diagrams expansion, it has been concluded that it was more efficient to expand Eq. (29) in terms of ½oðrÞ À buLR ðrÞ rather than in terms of oðrÞ . Instead of BðrÞ ¼ À1=2o2 ðrÞ, the bridge function is approximated 26 jean-marc bomont by BðrÞ ¼ À1=2½oðrÞ À buLR ðrÞ2, providing an extension in terms of gÃ ðrÞ, so that BðrÞ ¼ ½1 þ 2gÃ 1=2 À gÃ ðrÞ À 1 ð59Þ In this framework, inspired by the BPGG approach, a parameter a, whose role is to fulfil the thermodynamic consistency condition, has been introduced by Charpentier and Jakse .