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