where Ω = (ϑ, ϕ) comprises the azimuthal and polar angle in spherical coordinates and γ k is the great circle distance between Ω and Ω k . We have associated with each charge a finite size λ k through a normal distribution on the sphere ; when λ → ∞, this reduces to the Dirac delta function. For simplicity, we will assume that λ k = λ ∀k, and we set λ = 100, since the exact choice of this parameter does not significantly influence the results . In Eq. ( ), we have already made a distinction between the charges based on whether they acquire a positive charge, q + k = eη k , or negative charge, q - k = e(η k -1); η k ∈ [0, 1] determines the protonation/deprotonation state of each charge. The surface charge density can be expanded in terms of spherical harmonics Y lm (Ω), with the expansion coefficients given by σ(lm) = 4π k q k g l (λ k )Y * lm (Ω k ) , where we have defined g l (λ) = λ i l (λ)/sinh λ and i l (x) = π/2x I l+1/2 (x) are the modified spherical Bessel functions of the first kind.