Since the Coulomb matrix, A, is a symmetric positive-definite matrix, 23 its inverse 𝐀 )𝟏 is also symmetric positive-definite, and the product 𝐱 𝑻 𝐀 )𝟏 𝐱 for an arbitrary vector 𝐱 ≠ 𝟎 is always positive. Hence, the term inside the parathesis of Eq (4), 𝐶 = : 7 (𝐕 𝟎 ) ( 𝐀 )𝟏 𝐕 𝟎 , is positive. For a fixed direction of the dipole moment, 𝐶 is a positive constant, and Δ𝐺 %)&%' is a negative quadratic function of the magnitude of the molecule dipole moment, 𝜇. Therefore, a neutral molecule with a larger dipole moment benefits more from CPCM stabilization, meaning that the formation of localized solute electron density with charge separation is energetically favored in CPCM calculation. The same conclusion can be reached when the derivation is done for the continuous form of the CPCM cavity surface charge distribution 𝜎(𝑟 ⃗) (see Supplementary Material Text S1).