Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii

The volume and surface area of a fused hard-sphere molecule can be calculated by the usual formula relating the union of a number of sets to their intersections. In principle, this formula includes intersections of all orders; however, we show that, by making use of the properties of certain types of double, triple, quadruple and quintuple intersection, all terms of the fifth and higher order and some terms of lower order can be eliminated from the expansion, leaving only spheres and intersections of two, three or four spheres. Analytical expressions are derived for the volumes and surface areas of intersections of two, three or four spheres with unequal radii; these expressions allow the exact calculation of the volume and surface area of any fused hard-sphere molecule.