  We analyze the mathematically rigorous BIBEE (boundary-integral based
electrostatics estimation) approximation of the mixed-dielectric continuum
model of molecular electrostatics, using the analytically solvable case of a
spherical solute containing an arbitrary charge distribution. Our analysis,
which builds on Kirkwood's solution using spherical harmonics, clarifies
important aspects of the approximation and its relationship to Generalized Born
models. First, our results suggest a new perspective for analyzing fast
electrostatic models: the separation of variables between material properties
(the dielectric constants) and geometry (the solute dielectric boundary and
charge distribution). Second, we find that the eigenfunctions of the
reaction-potential operator are exactly preserved in the BIBEE model for the
sphere, which supports the use of this approximation for analyzing
charge-charge interactions in molecular binding. Third, a comparison of BIBEE
to the recent GB$\epsilon$ theory suggests a modified BIBEE model capable of
predicting electrostatic solvation free energies to within 4% of a full
numerical Poisson calculation. This modified model leads to a
projection-framework understanding of BIBEE and suggests opportunities for
future improvements.
