A physically meaningful method for the comparison of potential energy functions

In the study of the conformational behavior of complex systems, such as proteins, several related statistical measures are commonly used to compare two different potential energy functions. Among them, the Pearson's correlation coefficient r has no units and allows only semiquantitative statements to be made. Those that do have units of energy and whose value may be compared to a physically relevant scale, such as the root-mean-square deviation (RMSD), the mean error of the energies (ER), the standard deviation of the error (SDER) or the mean absolute error (AER), overestimate the distance between potentials. Moreover, their precise statistical meaning is far from clear. In this article, a new measure of the distance between potential energy functions is defined that overcomes the aforementioned difficulties. In addition, its precise physical meaning is discussed, the important issue of its additivity is investigated, and some possible applications are proposed. Finally, two of these applications are illustrated with practical examples: the study of the van der Waals energy, as implemented in CHARMM, in the Trp-Cage protein (PDB code 1L2Y) and the comparison of different levels of the theory in the ab initio study of the Ramachandran map of the model peptide HCO-L-Ala-NH2.