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The exchange splitting $J$ of the interaction energy of the hydrogen atomwith a proton is calculated using the conventional surface-integral formula$J_{\textrm{surf}}[\varphi]$, the volume-integral formula of thesymmetry-adapted perturbation theory $J_{\textrm{SAPT}}[\varphi]$, and avariational volume-integral formula $J_{\textrm{var}}[\varphi]$. Thecalculations are based on the multipole expansion of the wave function$\varphi$, which is divergent for any internuclear distance $R$. Nevertheless,the resulting approximations to the leading coefficient $j_0$ in the large-$R$asymptotic series $J(R) = 2 e^{-R-1} R ( j_0 + j_1 R^{-1} + j_2 R^{-2} +\cdots) $ converge, with the rate corresponding to the convergence radii equal to 4,2, and 1 when the $J_{\textrm{var}}[\varphi]$, $J_{\textrm{surf}}[\varphi]$,and $J_{\textrm{SAPT}}[\varphi]$ formulas are used, respectively. Additionally,we observe that also the higher $j_k$ coefficients are predicted correctly whenthe multipole expansion is used in the $J_{\textrm{var}}[\varphi]$ and$J_{\textrm{surf}}[\varphi]$ formulas. The SAPT formula$J_{\textrm{SAPT}}[\varphi]$ predicts correctly only the first twocoefficients, $j_0$ and $j_1$, gives a wrong value of $j_2$, and diverges forhigher $j_n$. Since the variational volume-integral formula can be easilygeneralized to many-electron systems and evaluated with standard basis-settechniques of quantum chemistry, it provides an alternative for thedetermination of the exchange splitting and the exchange contribution of theinteraction potential in general.

Exchange splitting of the interaction energy and the multipole expansion of the wave function