Exchange splitting of the interaction energy and the multipole expansion of the wave function
The exchange splitting $J$ of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula $J_{\textrm{surf}}[\varphi]$, the volume-integral formula of the symmetry-adapted perturbation theory $J_{\textrm{SAPT}}[\varphi]$, and a variational volume-integral formula $J_{\textrm{var}}[\varphi]$. The calculations 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 when the 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 two coefficients, $j_0$ and $j_1$, gives a wrong value of $j_2$, and diverges for higher $j_n$. Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.