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The radial Slater-type orbitals (STO) can be simply obtained by repeated parametric differentiation of the Yukawa Potential with respect to α. A new compact two-range addition theorem (AdT) for the STO is herein derived by explicit parametric differentiation of the well-known Yukawa AdT. The resulting addition formula is combined with the single-range AdT for solid spherical harmonics to present a new AdT for three-dimensional spherical coordinate STOs. We advance the proposition that this formula is “canonical” in the same sense that the Laplace expansion of the Coulomb potential is considered canonical. We demonstrate how this procedure can be employed for all exponential-type orbitals. © 2012 Wiley Periodicals, Inc.

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, $1/\sqrt{{x}^{2}+{\ensuremath{\beta}}^{2}}$, which becomes numerically intractable (because of extreme stiffness) as the softening parameter ($\ensuremath{\beta}$) approaches zero. We are able to maintain near machine accuracy for $\ensuremath{\beta}$ as low as $\ensuremath{\beta}={10}^{\ensuremath{-}8}$ using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as $\ensuremath{\beta}\ensuremath{\rightarrow}0$.

A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions: the Yukawa potential (e − αr ∕ r) and “evenly-loaded solid harmonics,” \(({r}^{2\nu +l}{Y }_{l}^{m}(\hat{r})\) for which translation formulas are available. The resulting two-range translation formula for the exponential orbital is presented and used to construct a new addition theorem for the Coulomb Sturmians.