Mixed direct-iterative methods for boundary integral formulations of continuum dielectric solvation models

摘要

This paper develops and characterizes mixed direct-iterative methods for boundary integral formulations of continuum dielectric solvation models. We give an example, the Ca(sup ++)(hor ellipsis)Cl(sup (minus)) pair potential of mean force in aqueous solution, for which a direct solution at thermal accuracy is difficult and, thus for which mixed direct-iterative methods seem necessary to obtain the required high resolution. For the simplest such formulations, Gauss-Seidel iteration diverges in rare cases. This difficulty is analyzed by obtaining the eigenvalues and the spectral radius of the non-symmetric iteration matrix. This establishes that those divergences are due to inaccuracies of the asymptotic approximations used in evaluation of the matrix elements corresponding to accidental close encounters of boundary elements on different atomic spheres. The spectral radii are then greater than one for those diverging cases. This problem is cured by checking for boundary element pairs closer than the typical spatial extent of the boundary elements and for those cases performing an ''in-line'' Monte Carlo integration to evaluate the required matrix elements. These difficulties are not expected and have not been observed for the thoroughly coarsened equations obtained when only a direct solution is sought. Finally, we give an example application of hybrid quantum-classical methods to deprotonation of orthosilicic acid in water.