Efficient density-fitted explicitly correlated dispersion and exchange dispersion energies

The leading-order dispersion and exchange-dispersion terms in symmetry-adapted perturbation theory (SAPT), $E^{(20)}_{\rm disp}$ and $E^{(20)}_{\rm exch-disp}$, suffer from slow convergence to the complete basis set limit. To alleviate this problem, explicitly correlated variants of these corrections, $E^{(20)}_{\rm disp}$-F12 and $E^{(20)}_{\rm exch-disp}$-F12, have been proposed recently. However, the original formalism (M. Kodrycka et al., J. Chem. Theory Comput. 2019, 15, 5965-5986), while highly successful in terms of improving convergence, was not competitive to conventional orbital-based SAPT in terms of computational efficiency due to the need to manipulate several kinds of two-electron integrals. In this work, we eliminate this need by decomposing all types of two-electron integrals using robust density fitting. We demonstrate that the error of the density fitting approximation is negligible when standard auxiliary bases such as aug-cc-pVXZ/MP2FIT are employed. The new implementation allowed us to study all complexes in the A24 database in basis sets up to aug-cc-pV5Z, and the $E^{(20)}_{\rm disp}$-F12 and $E^{(20)}_{\rm exch-disp}$-F12 values exhibit vastly improvement basis set convergence over their conventional counterparts. The well-converged $E^{(20)}_{\rm disp}$-F12 and $E^{(20)}_{\rm exch-disp}$-F12 numbers can be substituted for conventional $E^{(20)}_{\rm disp}$ and $E^{(20)}_{\rm exch-disp}$ ones in a calculation of the total SAPT interaction energy at any level (SAPT0, SAPT2+3, ...). We show that the addition of F12 terms does not improve the accuracy of low-level SAPT treatments. However, when the theory errors are minimized in high-level SAPT approaches such as SAPT2+3(CCD)$\delta$MP2, the reduction of basis set incompleteness errors thanks to the F12 treatment substantially improves the accuracy of small-basis calculations.