# Extended Comment on "One-Range Addition Theorems for Coulomb Interaction Potential and Its Derivatives" by I. I. Guseinov (Chem. Phys. Vol. 309 (2005), pp. 209 - 213)

## TL;DR

This paper critically examines Guseinov's derivation of one-range addition theorems for the Coulomb potential, highlighting mathematical flaws and emphasizing the need for proper interpretation as generalized functions.

## Contribution

The paper provides a rigorous critique of Guseinov's methods, demonstrating the invalidity of his addition theorems for the Coulomb potential and clarifying their proper mathematical context.

## Key findings

- Guseinov's addition theorems are mathematically flawed.
- The Coulomb potential does not belong to the Hilbert spaces used.
- Proper interpretation requires viewing these as generalized function expansions.

## Abstract

Addition theorems are principal tools that express a function $f (\bm{r} \pm \bm{r}')$ in terms of products of other functions that only depend on either $\bm{r}$ or $\bm{r}'$. The best known example of such an addition theorem is the Laplace expansion of the Coulomb potential which possesses a characteristic two-range form. Guseinov [Chem. Phys. {\bf 309}, 209 - 213 (2005)] derived one-range addition theorems for the Coulomb potential via the limit $\beta \to 0$ in previously derived one-range addition theorems for the Yukawa potential $\exp \bigl(-\beta | \bm{r}-\bm{r}'| \bigr) /| \bm{r}-\bm{r}'|$. At first sight, this looks like a remarkable achievement, but from a mathematical point of view, Guseinov's work is at best questionable and in some cases fundamentally flawed. One-range addition theorems are expansions in terms of functions that are complete and orthonormal in a given Hilbert space, but Guseinov replaced the complete and orthonormal functions by nonorthogonal Slater-type functions and rearranged the resulting expansions. This is a dangerous operation whose validity must be checked. It is shown that the one-center limit $\bm{r}' = \bm{0}$ of Guseinov's rearranged Yukawa addition theorems as well as of several other addition theorems does not exist. Moreover, the Coulomb potential does not belong to any of the Hilbert spaces implicitly used by Guseinov. Accordingly, one-range addition theorems for the Coulomb potential diverge in the mean. Instead, these one-range addition theorems have to interpreted as expansions of generalized functions in the sense of Schwartz that converge weakly in suitable functionals.