∈ Ω, the model defined by eqs. 1 and 2 can be mapped onto an equivalent boundary-element or apparent surface charge (ASC) problem, defined at the cavity surface Γ that is indicated in Fig. . That remapping defines the PCM. The most fundamental version of this remapping has been called the integral equation formulation (IEF). For classical solutes where there is no charge leakage into the continuum, IEF-PCM is an exact reformulation of isotropic Poisson boundary conditions, as can be demonstrated numerically. For QM charge densities, the tails of ρ sol (r) do penetrate into the medium, to the tune of ∼ 0.1-0.2 electrons for small molecules. However, an alternative derivation of the IEF-PCM equation demonstrates that this approach implicitly (albeit approximately) accounts for the volume polarization due to this escaped charge. Therefore, IEF-PCM is an accurate reformulation of Poisson boundary conditions even in the case of a QM solute. In matrix form, the IEF-PCM equation is 3