The failure of U i to converge in real space can be demonstrated with a lattice model of an uncorrelated Gaussian density field. Specifically, consider the Madelung potential, U 0 , of a central site within a three-dimensional cubic lattice. Each lattice site is assigned a random charge q i drawn from a Gaussian distribution, P (q i ) ∝ exp(-q 2 i ). The value of ũ0 (r) for several random charge configurations is plotted in Fig. . As this model illustrates, ũ0 (r) contributes random noise with constant amplitude at all radial distances. Thus, the value of U i for a given charge configuration is determined primarily by long-ranged uncorrelated noise, and not (as one might incorrectly intuit) the fluctuations of the local electrostatic environment. Therefore, Madelung potential quantities such as P (U α ) and ⟨(δU α ) 2 ⟩ are not reliable reporters of the local electrostatic environment. Notably, ⟨U i ⟩ reports on the structure of the local electrostatic environment because the uncorrelated random noise at long range averages to zero.