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

Abstract

The exchange splitting $J$ of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula $J_{\text{surf}}[\phi ]$, the volume-integral formula of the symmetry-adapted perturbation theory $J_{\text{SAPT}}[\phi ]$, and a variational volume-integral formula $J_{\text{var}}[\phi ]$. The calculations are based on the multipole expansion of the wave function $\phi$, which is divergent for any internuclear distance $R$. Nevertheless, the resulting approximations to the leading coefficient $j_0$ in the large-$R$ asymptotic series $J(R)=2e^{-R-1}R(j_0+j_1{R}^{-1}+j_2{R}^{-2}+\cdots )$ converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the $J_{\text{var}}[\phi ]$, $J_{\text{surf}}[\phi ]$, and $J_{\text{SAPT}}[\phi ]$ 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 $J_{\text{var}}[\phi ]$ and $J_{\text{surf}}[\phi ]$ formulas. The SAPT formula $J_{\text{SAPT}}[\phi ]$ 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.

11 pages, 3 figures, submitted to J. Chem. Phys