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The exchange splitting energy $J$ of the lowest \emph{gerade} and\emph{ungerade} states of the H$_2^+$ molecular ion was calculated using avolume integral expression of symmetry-adapted perturbation theory and standardbasis set techniques of quantum chemistry. The performance of the proposedexpression was compared to the well known surface integral formula. Bothformulas involve the primitive function which we calculated employing eitherthe Hirschfelder-Silbey perturbation theory or the conventionalRayleigh-Schr\"odinger perturbation theory (the polarization expansion). Ourcalculations show that very accurate values of $J$ can be obtained using theproposed volume integral formula. When the Hirschfelder-Silbey primitivefunction is used in both formulas the volume formula gives much more accurateresults than the surface integral expression. We also show that using thevolume integral formula with the primitive function approximated byRayleigh-Schr\"odinger perturbation theory, one correctly obtains only thefirst four terms in the asymptotic expansion of the exchange splitting energy.

Asymptotics of the exchange splitting energy for a diatomic molecular ion from a volume integral formula of symmetry-adapted perturbation theory