Noura Dawass

1.3. K IRKWOOD –B UFF I NTEGRALS FROM MOLECULAR SIMULATIONS 1 13 Table 1.1 Exact expressions of the geometrical function w ( x ) for hyperspheres in 1-3 dimensions (i.e. line, circle, and sphere) [74] . Here, x = r / L max (0 < x < 1), L max is the maximum distance between two points, and c ( r )d r is the hyperspherical volume element with thickness d r . For x ≥ 1, w ( x ) = 0. Dimension c ( r ) w ( x ) 1 D 2 1 − x 2 D 2 π r 2/ π (arccos( x ) − x p 1 − x 2 ) 3 D 4 π r 2 1 − 3 x /2 + x 3 /2 R V d r 1 R V 0 d r 0 2 δ ( r 0 2 − r 1 ), where V 0 is the subvolume V shifted by r (see Figure 1.3) . The function w ( r ) is obtained from τ ( r ) by integrating over 4 π solid angle ( Ω ) and dividing by V . We have w ( r ) = 1 V Z d r 0 τ ( r 0 ) δ ( r − | r 0 | ) = r D − 1 V Z d Ω τ ( r ) (1.23) where D is the dimensionality of space. In the following, we consider for V hy- perspheres of radius R , where by symmetry, the overlap volume does not depend on Ω , so τ ( r ) = τ ( r ). The volume of a hypersphere is V = R D R d Ω / D which, to- gether with Eq. (1.23) , yields w ( r ) = τ ( r ) D r D − 1 R D (1.24) The overlap volumes τ ( r ) of hyperspheres in D =1–3 dimensions ( i.e. seg- ment, circle and sphere) are well known [82] . From these, the corresponding functions w ( r ) are obtained using Eq. (1.24) . The functions w ( r ) are computed up to the maximum distance between two points in a subvolume L max . It is con- venient to define the dimensionless distance x = r / L max . The corresponding functions w ( x ) are listed in Table 1.1. Using w ( x ) and the L.H.S of Eq. (1.18) , we arrive at the final expression for KBIs for finite subvolumes, G V αβ = Z L max 0 £ g αβ ( r ) − 1 ¤ c ( r ) w ( x )d r (1.25) where we have used the fact that w ( x ) = 0 for x ≥ 1. KBIs computed from small subvolumes scale with the inverse size of the sub- volumes. This scaling law can be explained by the concept of thermodynamics of small systems as mentioned earlier. Alternatively, Krüger et al. [74] showed that finite-size effects of the subvolume emerge from pairs of molecules α − β , where particle α is inside subvolume V , and particle β is outside V (the simu- lation box which contains V is denoted by L 3 ). To account for the contribution