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.