Third-order exchange-induction-dispersion energy in symmetry-adapted perturbation theory without single-exchange approximation

In symmetry-adapted perturbation theory (SAPT), the exchange corrections are often calculated utilizing the so-called single-exchange or S2 approximation, that is, by approximating the antisymmetrizer of the whole system by permutations of a single pair of electrons between weakly interacting subsystems. The error introduced in this approximation is typically tolerable; however, it is the reason for qualitatively wrong results of selected S2 SAPT corrections for some ionic systems. This work presents the third-order exchange-induction-dispersion correction (Eexch-ind-disp(30)) of SAPT, describing the quenching of the mixed induction-dispersion attractive energy by intermolecular exchange tunneling, derived without the single-exchange approximation in the second-quantization formalism. To obtain the explicit orbital expression for the Eexch-ind-disp(30) energy, the approach that was developed earlier by two of us B. Tyrcha, F. Brzęk, and P. S. Żuchowski, J. Chem. Phys. 160, 044118 (2024) was extended to the third order of SAPT. The quality of the S2 approximation to all available second- and third-order exchange corrections (including the newly developed Eexch-ind-disp(30) energy) has been studied on a selection of benchmark noncovalent interaction databases, including some ionic datasets.