ABSTRACT
The conventional surface integral formula Jsurf[Φ] and an alternative volume integral formula Jvar[Φ] are used to compute the asymptotic exchange splitting of the interaction energy of the hydrogen atom and a proton employing the primitive function Φ in the form of its truncated multipole expansion. Closed-form formulas are obtained for the asymptotics of Jsurf[ΦN] and Jvar[ΦN], where ΦN is the multipole expansion of Φ truncated after the 1/RN term, R being the internuclear separation. It is shown that the obtained sequences of approximations converge to the exact result with the rate corresponding to the convergence radius equal to 2 and 4 when the surface and the volume integral formulas are used, respectively. When the multipole expansion of a truncated, Kth order polarisation function is used to approximate the primitive function, the convergence radius becomes equal to unity in the case of Jvar[Φ]. At low order, the observed convergence of Jvar[ΦN] is, however, geometric and switches to harmonic only at certain value of N = Nc dependent on K. An equation for Nc is derived which very well reproduces the observed K-dependent convergence pattern. The results shed new light on the convergence properties of the conventional symmetry-adapted perturbation theory expansion used in applications to many-electron diatomics.
Acknowledgments
The authors thank Robert Moszyński for commenting on the manuscript. This article is dedicated to Professor Andreas Savin on the occasion of his 65th birthday.
Disclosure statement
No potential conflict of interest was reported by the authors.