One of the important goals of this paper is to scrutinise how S 4 corrections improve over S 2 one. To this end we have tested the efficacy of S 2 and S 4 corrections with a particular focus on systems anticipated to exhibit difficulty converging in terms of S integrals, such as metals interacting with molecules and π-stacked ethylenes. Our findings reveal that, in the first-order, the full exchange energy at the S 4 approximation substantially ameliorates the S 2 approximation reducing the S 2 error by 4 orders of magnitude for Ar-H 2 O system (to 10 -4 % in the minimum), and by nearly 3 orders of magnitude at the repulsive wall. Specifically, in the challenging case of the Be-NH 3 system, the improvement exceeds one order of magnitude, decreasing to about 1% relative error. Regarding the second-order induction energy, the S 2 approximation performs a few percent worse than in the first-order as in Refs. 2 and 3.