  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.
