Evaluation of molecular radial distribution function and solvent-excluded volume with the numerical integration of the union of spheres

The atoms within a composite solute are modeled by spheres, which overlap to construct a molecular domain. The volume of the union of spheres was calculated with the numerical integration such that it is the total volume of a large number of tiny tetrahedra, filled a whole domain. These molecular volumes are utilized to calculate the volumes of solvation shells for the evaluation of an averaged number of solvents, which is used to compare with the actual counting number of solvents within the same interval, giving rise to the corrected molecular radial distribution function (RDF). The values of corrected RDFs especially in the bulk were converged to unity, according to the definition. This method was also applied to evaluate the solvent-excluded volume by the supplement of inaccessible radii, obtained from the atomic RDF of constituent atoms.