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

ABSTRACTThe 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 harmo...