if c ( r ) for r > σ is given by a superposition of Yukawa tails [41]. This is indeed the case, provided one adopts the Waisman parametrization for the hard-sphere direct correlation function c HS ( r ) [42]. According to this, c HS ( r ) for r > σ is also given by a Yukawa tail with known, density-dependent amplitude and range, so that c ( r ) for r > σ consists of the superposition of two Yukawas. On the other hand, for non-Yukawa intermolecular potentials the SCOZA c ( r ) for σ ≤ r is not of Yukawa form, irrespective of the parametrization adopted for c HS ( r ), and in order to implement SCOZA a fully numerical solution of the OZ equation is usually required [43, 44]. In conventional SCOZA, the virial route is not included in the consistency condition. In order to enforce consistency among all the three routes, one has to endow Eq. (2) with one more degree of freedom. To this purpose, the following form for c ( r ) was proposed [29]: where the amplitude of the (attractive) Yukawa potential has now been set to β as in the MSA, while both the amplitude H and the inverse range z of the extra Yukawa tail must be determined by imposing consistency among virial, compressibility, and internal energy. This approach is very similar to that originally proposed in Ref. [45], which is obtained from Eq. (4) by having the last Yukawa to account also for the hard-sphere contribution to c ( r ) outside the repulsive core, thereby setting c HS ( r ) = 0 in Eq. (4). A fully numerical implementation of that closure was considered in Ref. [23], but the numerical algorithm failed to converge in the critical region. In a recent work [29], a different strategy was adopted: on the one hand, the analytical solution of the OZ equation for a c ( r ) given by a superposition of Yukawas (three when also c HS ( r ) is taken into account) for σ ≤ r was again exploited. On the other hand, consistency with the virial route was required to hold only at the critical point, and z was fixed at the value at which this condition is satisfied, thereby making the theory considerably easier to implement. This is the approach used here together with the standard SCOZA of Eqs. (2), (3). In the following the standard and modified versions of SCOZA will be referred to as S-SCOZA and M-SCOZA respectively. Before concluding this Section, it is worthwhile adding some remarks about the relevance of the high-density boundary condition for the SCOZA PDE. While the boundary condition at low density is trivial, since one must have χ red ( ρ =0 , β ) = 1, u ( ρ =0 , β ) = 0 for every β , 6 c ( r ) = c HS ( r ) + β $^{exp[}$− κ ( r − σ )] /r + H $^{exp[}$− z ( r − σ )] /r if σ ≤ r , (4)