  Atom-centered point charge model of the molecular electrostatics---a major
workhorse of the atomistic biomolecular simulations---is usually parameterized
by least-squares (LS) fitting of the point charge values to a reference
electrostatic potential, a procedure that suffers from numerical instabilities
due to the ill-conditioned nature of the LS problem. Here, to reveal the
origins of this ill-conditioning, we start with a general treatment of the
point charge fitting problem as an inverse problem, and construct an
analytically soluble model with the point charges spherically arranged
according to Lebedev quadrature naturally suited for the inverse electrostatic
problem. This analytical model is contrasted to the atom-centered point-charge
model that can be viewed as an irregular quadrature poorly suited for the
problem. This analysis shows that the numerical problems of the point charge
fitting are due to the decay of the curvatures corresponding to the
eigenvectors of LS sum Hessian matrix. In part, this ill-conditioning is
intrinsic to the problem and related to decreasing electrostatic contribution
of the higher multipole moments, that are, in the case of Lebedev grid model,
directly associated with the Hessian eigenvectors. For the atom-centered model,
this association breaks down beyond the first few eigenvectors related to the
high-curvature monopole and dipole terms; this leads to even wider spread-out
of the Hessian curvature values. Using these insights, it is possible to
alleviate the ill-conditioning of the LS point-charge fitting without
introducing external restraints and/or constraints. Also, as the analytical
Lebedev grid PC model proposed here can reproduce multipole moments up to a
given rank, it may provide a promising alternative to including explicit
multipole terms in a force field.
