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

TL;DR

This paper analyzes the convergence behavior of multipole expansions in calculating exchange energy contributions, revealing different convergence rates and patterns for surface and volume integral formulas, with implications for many-electron diatomic systems.

Contribution

It provides closed-form asymptotic formulas and new insights into the convergence properties of multipole expansions in exchange energy calculations.

Findings

Surface integral formula converges with radius 2.

Volume integral formula converges with radius 4.

Convergence switches from geometric to harmonic at a critical N_c.

Abstract

The conventional surface integral formula $J_{\rm surf}[\Phi]$ and an alternative volume integral formula $J_{\rm var}[\Phi]$ are used to compute the asymptotic exchange splitting of the interaction energy of the hydrogen atom and a proton employing the primitive function $\Phi$ in the form of its truncated multipole expansion. Closed-form formulas are obtained for the asymptotics of $J_{\rm surf}[\Phi_N]$ and $J_{\rm var}[\Phi_N]$, where $\Phi_N$ is the multipole expansion of $\Phi$ 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 results 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, $K$th order polarization function is used to approximate the primitive function the convergence radius becomes equal to unity in the case of $J_{\textrm{var}}[\Phi]$. At low order the observed convergence of $J_{\rm var}[\Phi_N]$ 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 SAPT expansion used in applications to many-electron diatomics.