Noura Dawass

2 28 S HAPE E FFECTS 7. Repeat steps 1 to 4 while updating W ( i ) until a satisfactory sampling of p ( i ) is reached. For 10 11 cycles, running the algorithm takes approximately 150 minutes on a modern computer. 2.2.2. C OMPUTING w ( x ) The function w ( i ) is proportional to the distribution function p ( i ) divided by the volume of the bin in a hypersphere with dimension D : w ( i ) = p ( i ) i D − ( i − 1) D (2.3) In Eq. (2.3) , the prefactors for the bin volumes are not included yet since in the next step w ( i ) is normalized using the known value w (0) = 1. Since we do not obtain statistics exactly at r = 0, we interpolate to w (0) using the first two bins w (1) and w (2), w ( i ) → w ( i ) w (1) − ( w (2) − w (1)) 2 (2.4) Similarly, the distances are normalized to find the relative distance x , x ( i ) = ( i − 1/2) ∆ r L max (2.5) Interpolation can be used to find w ( x ) for any value of x . 2.3. S HAPE EFFECTS OF K IRKWOOD –B UFF I NTEGRALS 2.3.1. T HE FUNCTIONS w ( x ) FOR A CUBE , CUBOIDS , AND SPHEROIDS In this section, we present the function w ( x ), computed numerically for differ- ent shapes. To validate our numerical method (section 2.2) , we compute the function w ( x ) for subvolumes where the analytic expressions are known (line, circle, and sphere, see Table 1.1) and make a comparison. In Figure 2.1 (a), the comparison between analytic and numerical functions w ( x ) is shown for a line, circle, and sphere. For these shapes, the numerical results reproduce the theo- retical solution very well. The average absolute difference between analytic and numerical values are 9x10 − 3 , 5x10 − 3 , and 2x10 − 4 for a sphere, circle, and line, respectively. Therefore, the algorithm of section 2.2 can be used to numerically compute the function w ( x ) for any convex subvolume in 1 D , 2 D , or 3 D . We compute the function w ( x ) numerically for subvolumes where analytic expressions for w ( x ) are not available. In Figure 2.1 (b), we show the function w ( x ) computed numerically for a cube and sphere, which are the most com- monly used shapes for the subvolume when computing KBIs . Figure 2.2 shows