Noura Dawass

3 48 F INITE -S IZE E FFECTS Table 3.1: Differences (%), calculated using Eq. (3.12) , between KBIs obtained form direct numer- ical integration of Eq. (1.25) , G ∞ ,num / σ 3 , and integrals computed using MC simulations of vari- ous simulation box sizes, G ∞ / σ 3 . The system used is the fluid described by the analytic RDF of Eq. (2.6) . The values of the KBIs from numerical integration, G ∞ ,num / σ 3 , are − 1.785, − 2.041, and − 2.172 for χ = 1, χ = 2, and χ = 4, respectively. The values of G V / σ 3 obtained fromMC simulations of finite simulation boxes are extrapolated to the thermodynamic limit to obtain G ∞ / σ 3 . Box length ( L box ) χ = 1 χ = 2 χ = 4 7.5 1.979 2.262 2.408 10 1.052 1.042 1.000 15 0.323 0.228 0.140 20 0.154 0.105 0.069 40 0.005 0.036 0.007 50 0.003 0.017 0.055 and the subvolume cannot be considered as grand-canonical. Thus, extending the computations of KBIs beyond L box /2 is not necessary and does not improve the accuracy of the computed KBIs. In fact, when extrapolating G ∞ this range should be avoided and only the linear part of the function G V should be used. In section 3.4.2, the best range used for extrapolating the scaling of 1/ R and G V , to properly obtain KBIs, is further discussed. Finally, we examine the scaling of the function Q (Eq. (1.27) ) with the surface area of the spherical subvolume. Fig- ure 3.3 shows how the function Q scales linearly with the surface area. The in- tegrals over L 3 box in Eq. (1.27) become larger as the number of molecules around surface area of the subvolume increase. For all simulation box sizes, the values of the function Q decrease when R is larger than L box /2 due to the decrease in the number of molecules at the boundary of the sphere V . In any case, regardless of the shape of the subvolume, whether a fully embedded sphere or a truncated sphere, the function Q scales linearly with the surface area.