β coex which gives the curvature of the liquid-vapor coexistence curve in the critical region. All the critical point parameters we calculated are listed in Table I. A detailed description of SCOZA and its application to a Yukawa fluid has been given elsewhere and we refer the interested reader to the previous literature on this subject [1, 23, 27, 28]. Here we recall that, as usual in integral-equation theories, in the SCOZA the Ornstein-Zernike (OZ) equation is closed by an approximate ansatz which involves the direct correlation function c ( r ) and the radial distribution function g ( r ). In the standard formulation of SCOZA, one requires that i) the core condition g ( r ) = 0 be satisfied inside the hard core r < σ ; and ii) the contribution to c ( r ) outside the hard core due to tail of the potential be linear in the potential itself. For a HCAY potential one has then: III. THEORY    g ( r ) = 0 if r < σ , c ( r ) = c HS ( r ) + K $^{exp[}$− κ ( r − σ )] /r if σ ≤ r , (2) where c HS ( r ) is the direct correlation function of the hard-sphere fluid. The functional form of this closure is similar to that of well-established theories such as the mean spherical approximation (MSA), where c HS ( r ) is identically vanishing for r > σ , or the optimized random-phase approximation (ORPA) [40]. However, in these approaches the amplitude K of the direct correlation function coincides with the inverse temperature β = 1 / ( k$_{B}$T ), while in the SCOZA K is an a priori unknown function of the thermodynamic state, to be determined so that consistency between the compressibility and the internal energy route to the thermodynamic is obtained. The consistency requirement is embodied in the following condition on the reduced compressibility χ red and the excess internal energy per unit volume u : where χ red and u are obtained via the compressibility and internal energy route respectively. Eq. (3) together with closure (2) give a closed partial differential equation (PDE) for u , that can be solved numerically, provided the initial condition at β = 0 and the boundary conditions at low and high density are specified. The solution procedure is made easier by the fact that, for a given K , the OZ equation with closure (2) can be solved analytically, 5 ∂ ∂β ( 1 χ red ) = ρ ∂ $^{2}$u ∂ρ 2 , (3)