An Algorithm for the Calculation of Volume and Surface of Unions of Spheres. Application for Solvation Shells

A simple algorithm for the calculation of the volume and surface area of a union of spheres of different radii is presented. It is based on the ideas published in S. Sastry et al, Phys. Rev. E, v.56, pp.5524-5532, 1997 [1], where they computed volume and surface of interatomic voids in simple liquids. They proposed to work with the intersection of Delaunay simplexes and the corresponding Voronoi polyhedra. Analytical formulas for volume and surface area were derived for the atoms occupying this region. This could be achieved without explicit calculation of multiple intersections of the overlapping atoms. We have implemented such ideas for the calculation of the occupied volume and its surface inside the polyhedra defined by power Voronoi diagram. This allows calculating the required values for spheres with different radii. Simple analytical formulas are also valid in this case. We applied our algorithm to the calculation of the solvation shell volume for complex solute molecules in molecular dynamics models of solutions. A comparison of our program with the available implementation of the certified algorithm for unions of spheres by F. Cazals et al. (ACM Trans. Math. Soft. 38 (1), 2011) [2] shows coincidence of the results.