Discrepancy in the near-solute electric dipole moment calculated from the electric field

The electric dipole moment p(r) was computed as the integral of the permanent dipole moment of the solvent molecule μ(r) weighted by the orientational probability distribution Ω(r;O) over all orientations, where O is the orientation of the solvent molecule at r. The relationship between Ω(r;O) and the potential of the mean torque was derived; p(r) is proportional to the electric field E(r) under the following assumptions: (1) the van der Waals (vdW) interaction is independent of the orientation of the solvent molecule at r; (2) the solvent molecule and its electrical effect are modeled as a point dipole moment; (3) the solvent molecule at r is in a region far from the solute; and (4) μE(r) ≪ k(B) T, where k(B) is Boltzmann's constant and T is absolute temperature. The errors caused by calculating near-solute Ω(r) and p(r) from E(r) are unclear. The results show that Ω(r) is inconsistent with the value calculated from E(r) for water molecules in the first and second shells of solute with charge state Q = ±1 e, and a large variation in solvent molecular polarizability γ(mol) (r), which appeared in the first valley of 4πr(2) E(r) for |Q| < 1 e. Nonetheless, p(r) is consistent with the values calculated from E(r) for |Q| ≤ 1 e. The implication is that the assumptions for calculating p(r) can be ignored in the calculation of the solvation free energy of biomolecules, as they pertain to protein folding and protein-protein/ligand interactions.