Noura Dawass

1 14 I NTRODUCTION of these molecular pairs to the KBIs, the function Q αβ is defined by splitting the the integral domain in Eq. (1.18) over the surrounding R V R L 3 and R V R L 3 − V (this is possible as RDFs have a finite range), Q αβ ≡ Z V d r 1 Z L 3 − V d r 2 ( g αβ ( r 12 ) − 1) = Z V d r 1 Z L 3 d r 2 ( g αβ ( r 12 ) − 1) − Z V d r 1 Z V d r 2 ( g αβ ( r 12 ) − 1) (1.26) where non-zero contributions to Q αβ originate from molecule pairs where one molecule is inside V and the other one outside V . Assuming a finite correlation length ζ for a layer surrounding the subvolume, we have [ g αβ ( r 12 ) − 1] ≈ 0 for r 12 > ζ . The volume of this layer, and thus Q αβ , increases linearly with the surface area A of the spherical subvolume (for a radius of the sphere much larger than ζ ). In the case of an infinitely large system, the homogeneous conditions allow for the substitution r = r 1 − r 2 in the integral over L 3 , resulting in: Q αβ = Z V d r 1 Z L 3 d r ( g αβ ( r 12 ) − 1) − Z V d r 1 Z V d r 2 ( g αβ ( r 12 ) − 1) ≈ VG ∞ αβ − VG V αβ (1.27) where G ∞ αβ is the KB integral for an infinite volume. As Q αβ scales linearly with the surface area A , the difference ( G ∞ αβ − G V αβ ) scales as A / V , i.e. inversely with the linear dimension of the subvolume ( A / V ∼ 1/ L ), G V αβ ( L ) = G ∞ αβ + F ∞ αβ L (1.28) where F ∞ αβ is a scaling constant proportional to the function Q αβ defined ear- lier. From extrapolating G V αβ to 1/ L → 0, KBIs in the thermodynamic limit are obtained. O THER METHODS FOR COMPUTING K IRKWOOD –B UFF I NTEGRALS Similar to the method of Krüger and co-workers [74] , the approach of Cortes- Huerto et al. [83] uses small subvolumes to estimate KBIs in the thermodynamic limit. In their approach, Cortes-Huerto et al. [83] apply a correction for RDFs that is independent of the interparticle distance. The methods of Krüger and co-workers [74, 80, 81, 84] , and Cortes-Huerto et al. [83] provide practical ap- proaches to computing KBIs for any isotropic fluid, while addressing system size effects and RDF–related finite–size effects. Other available methods for comput- ing KBIs are more complicated, and found to be difficult to extend to systems