Noura Dawass

3.3. F INITE - SIZE EFFECTS OF RADIAL DISTRIBUTION FUNCTIONS 3 43 work. To study the RDF correction methods, MD simulations of WCA molecules are performed. The system conditions and the simulation details are provided in this section. 3.3.1. V AN DER V EGT CORRECTION When computing KBIs using finite systems, Ganguly and van der Vegt [113] ob- serve a drifting asymptote due to the asymptotic behavior of RDFs in finite sys- tems. Specifically, this asymptotic behavior of the RDF does not converge to one. These authors proposed that the RDF could be corrected using a correlation that takes into account the excess, or depletion of the bulk density of molecule type β around a central molecule of type α at a distance r due to the finite-size of the system (simulation box). The bulk density of molecules β is compensated by computing the excess or depletion of number of molecules of species β inside the considered subvolume, V . The subvolume is formed by taking a distance r from the central molecule of type α . The correlation takes into account the de- pletion of molecule type β around a central molecule of type α , ∆ N αβ ( r ), and provides the corrected RDF, g vdV αβ ( r ): g vdV αβ ( r ) = g αβ ( r ) N β ³ 1 − V L 3 box ´ N β ³ 1 − V L 3 box ´ − ∆ N αβ ( r ) − δ αβ (3.3) where g αβ ( r ) is the RDF obtained from a MC/MD simulation of a closed system (Eq. (1.2) ). For infinitely large and open systems, g αβ ( r ) and g vdV αβ ( r ) are equal. N β is the number of molecules of type β , L 3 is the volume of the closed simulation box (which is assumed to be cubic), and δ αβ is the Kronecker delta. V is the subvolume that surrounds a molecule of type α , with radius r . This volume is calculated according to whether r extends to outside the simulation box or not. When r L box < 1 2 , V is simply the volume of the sphere, 4 3 π r 3 . However, when 1 2 < r L box < p 2 2 , V corresponds to the volume in Eq. (3.2) . The excess or depletion of molecule type β around a molecule of type α , ∆ N αβ ( r ), can be calculated from, depending on whether r extends to outside the simulation box or not [82] : ∆ N αβ ( r ) =   R r 0 d r 0 4 π r 0 2 ρ β £ g αβ ( r 0 ) − 1 ¤ , r L box < 1 2 R L box /2 0 d r 0 4 π r 0 2 ρ β £ g αβ ( r 0 ) − 1 ¤ + R r L / box 2 d r 0 2 π r 0 (3 L box − 4 r 0 ) ρ β £ g αβ ( r 0 ) − 1 ¤ , 1 2 < r L box < p 2 2 (3.4)