Calculation of solvation free energy utilizing a constrained QM/MM approach combined with a theory of solutions

In an extended QM/MM (quantum mechanical/molecular mechanical) description of a solution, the solvent molecules surrounding the solute are incorporated into the QM region besides the solute. In a recent development, we introduced a simple and efficient method, referred to as boundary constraint with correction (BCC), to prevent the diffusion of the QM solvent into the bulk. The major purpose of the present work is to develop a method to compute the solvation free energy of a QM solute in an extended QM/MM simulation by means of the BCC method. The strategy of our development is to utilize the QM/MM-ER method which combines the QM/MM simulation and the theory of solutions termed energy representation (ER) to expedite the free energy calculation. A theory is, then, formulated to couple QM/MM-ER and the BCC method on the basis of the statistical mechanics. A notable feature of our method is that the effect of the constraint potentials on the free energy completely vanishes when the force field of the QM solvent coincides with that of the MM solvent. The method is applied to the calculations of the solvation free energies of a water molecule and a hydronium ion in water solutions. It turns out that the present method can offer a significant improvement in describing the free energy Δν of the hydronium ion, in particular, as compared with the conventional QM/MM approach. Explicitly, Δν is obtained as -98.0 kcal/mol showing a good agreement with an experimental value of -103.5 kcal/mol, while -86.1 kcal/mol by the conventional method.