  The exchange splitting energy $J$ of the lowest \emph{gerade} and
\emph{ungerade} states of the H$_2^+$ molecular ion was calculated using a
volume integral expression of symmetry-adapted perturbation theory and standard
basis set techniques of quantum chemistry. The performance of the proposed
expression was compared to the well known surface integral formula. Both
formulas involve the primitive function which we calculated employing either
the Hirschfelder-Silbey perturbation theory or the conventional
Rayleigh-Schr\"odinger perturbation theory (the polarization expansion). Our
calculations show that very accurate values of $J$ can be obtained using the
proposed volume integral formula. When the Hirschfelder-Silbey primitive
function is used in both formulas the volume formula gives much more accurate
results than the surface integral expression. We also show that using the
volume integral formula with the primitive function approximated by
Rayleigh-Schr\"odinger perturbation theory, one correctly obtains only the
first four terms in the asymptotic expansion of the exchange splitting energy.
