Convergence properties of the multipole expansion of the exchange contribution to the interaction energy

ABSTRACT

The conventional surface integral formula J_surf[Φ] and an alternative volume integral formula J_var[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 J_surf[Φ_N] and J_var[Φ_N], where Φ_N is the multipole expansion of Φ truncated after the 1/R^N 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 J_var[Φ]. At low order, the observed convergence of J_var[Φ_N] is, however, geometric and switches to harmonic only at certain value of N = N_c dependent on K. An equation for N_c 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.

GRAPHICAL ABSTRACT

KEYWORDS:

• Molecular interactions
• symmetry-adapted perturbation theory
• multipole expansion
• exchange energy
• intermolecular forces

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.

Additional information

Funding

This work was supported by the National Science Centre, Poland [project number 2014/13/N/ST4/03833].