It is demonstrated that Coulombic infinities of both nuclear-electron and electron-electron type can be eliminated through a transformation of the Hamiltonian. The transformed Hamiltonian is no longer self-adjoint and will contain three-particle interactions. The new pair interaction, depending only on the interparticle distance, can be chosen almost at will, however. If the new pair-potential terms are chosen to not contain an infinity, the corresponding right-hand wave function no longer contains a cusp, and we expect this feature to help improve the convergence of ab initio quantum chemical calculations with respect to the one-particle basis set. We limit ourselves to an exposition of the idea, illustrated with some examples for the Hydrogen atom.