In the use of dielectric continuum theory to model bulk solvation effects on the electronic structure and properties of a solute, volume polarization contributions due to quantum mechanical penetration of the solute charge density outside the cavity nominally enclosing it are known to be significant. This work provides a new formulation and implementation of methods for solution of the requisite Poisson equation. In previous formulations the determination of the surface polarization contribution required evaluation of the difficult to calculate electric field generated by the volume polarization. It is shown that this problematic quantity can be eliminated in favor of other more easily evaluated quantities. That formal advance also opens the way for a more efficient apparatus to be implemented for calculation of the direct contribution of volume polarization to the solvation energy. The new formulation and its practical implementation are described, and illustrative numerical results are given for several neutral and ionic solutes to study the convergence and precision in practice.