The crucial part of a canonical titration MC is the calculation of pH after the system has equilibrated. The standard Widom particle insertion method is usually not very practical, since to calculate the pH we need the whole electrochemical potential inside the simulation cell, and not just its excess part. At moderate to high pH, the interior of the simulation cell might not have any free hydronium ions at all, preventing us from accurately calculating the electrochemical potential and the activity of hydronium ions. On the other hand, since the condensed protons are in equilibrium with the hydronium ions in the bulk and, therefore, have exactly the same electrochemical potential, we can use them to accurate calculate the pH inside the system. There is, however, an additional complication when working with infinitely replicated Coulomb systems. The Ewald summation, effectively leads to a macroscopic crystal composed of replicated microscopic simulation cells. In general, a simulation cell will have a net electric dipole moment M M M = i q i r r r i and a finite second moment of the charge density tensor. From the electrostatics it is well known that such uniform polarization is analogous to the surface charge density M M M •n n n/V , where n n n represents the unit normal to the boundary of the macroscopic spherical crystal, see Fig. . This effective surface charge, will lead to electric field in the interior of the crystal. Similarly the fact that in general the simulation cell has a non-zero moment of the charge density tensor, results in a dipolar layer at the surface of the macroscopic crystal -so that interior of the crystal has different mean electrostatic potential compared to the exterior . This potential difference is known as the Bethe potential and for a charge neutral systems is given by: