Gaussian fluctuation formula for electrostatic free-energy changes in solution

Linear-response theory is used to derive a microscopic formula for the free-energy change of a solutesolvent system in response to a change in the charge distribution of the solutes. The formula expresses the change in the solvent polarization energy as a quadratic function of the changes in the partial charges at the atomic centers of the solute atoms. The average electrostatic potential at the sites of the solute charges and the second moment of the fluctuations in the electrostatic potential at these sites enter as parameters in the formula. These parameters can be obtained from computer simulations of a reference system with fixed solute charges and the results then compared with explicit free-energy simulations of the corresponding processes or with experiment. The formula provides a microscopic definition of the dielectric response function for the combined solute plus solvent system which can be related to standard formulas for the dielectric response. A simple numerical example involving a simulation of the charging free energy of two ions in aqueous solution is discussed. The change in the solvent polarization with solute charge predicted using the molecular dielectric response function agrees remarkably well with the results of corresponding free-energy simulations for large changes in the solvent polarization energy. Some applications of the molecular dielectric response formula are discussed.