Proximal distributions from angular correlations: a measure of the onset of coarse-graining

In this work we examine and extend the theory of proximal radial distribution functions for molecules in solution. We point out two formal extensions, the first of which generalizes the proximal distribution function hierarchy approach to the complete, angularly dependent molecular pair distribution function. Second, we generalize from the traditional right-handed solute-solvent proximal distribution functions to the left-handed distributions. The resulting neighbor hierarchy convergence is shown to provide a measure of the coarse-graining of the internal solute sites with respect to the solvent. Simulation of the test case of a deca-alanine peptide shows that this coarse-graining measure converges at a length scale of approximately 5 amino acids for the system considered.