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 Jsurf[Φ], the volume-integral formula of the symmetry-adapted perturbation theory JSAPT[Φ], and a variational volume-integral formula Jvar[Φ]. The calculations are based on the multipole expansion of the wave function Φ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j0 in the large-R asymptotic series J(R) = 2e(-R-1)R(j0 + j1R(-1) + j2R(-2) + ⋯) converge with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the Jvar[Φ], Jsurf[Φ], and JSAPT[Φ] formulas are used, respectively. Additionally, we observe that also the higher jk coefficients are predicted correctly when the multipole expansion is used in the Jvar[Φ] and Jsurf[Φ] formulas. The symmetry adapted perturbation theory formula JSAPT[Φ] predicts correctly only the first two coefficients, j0 and j1, gives a wrong value of j2, and diverges for higher jn. 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.