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An accurate force calculation with the Poisson-Boltzmann equation ischallenging, as it requires the electric field on the molecular surface. Here,we present a calculation of the electric field on the solute-solvent interfacethat is exact for piece-wise linear variations of the potential and analyzefour different alternatives to compute the force using a boundary elementmethod. We performed a verification exercise for two cases: the isolated andtwo interacting molecules. Our results suggest that the boundary element methodoutperforms the finite difference method, as the latter needs a much finer meshthan in solvation energy calculations to get acceptable accuracy in the force,whereas the same surface mesh than a standard energy calculation is appropriatefor the boundary element method. Among the four evaluated alternatives of forcecalculation, we saw that the most accurate one is based on the Maxwell stresstensor. However, for a realistic application, like the barnase-barstar complex,the approach based on variations of the energy functional, which is lessaccurate, gives equivalent results. This analysis is useful towards using thePoisson-Boltzmann equation for force calculations in applications where highaccuracy is key, for example, to feed molecular dynamics models or to enablethe study of the interaction between large molecular structures, like virusesadsorbed onto substrates.

Accurate boundary-integral formulations for the calculation of electrostatic forces with an implicit-solvent model