Explicitly Correlated Dispersion and Exchange Dispersion Energies in Symmetry-Adapted Perturbation Theory

The individual interaction energy terms in symmetry-adapted perturbation theory (SAPT) not only have different physical interpretations but also converge to their complete basis set (CBS) limit values at quite different rates. Dispersion energy is notoriously the slowest converging interaction energy contribution, and exchange dispersion energy, while smaller in absolute value, converges just as poorly in relative terms. To speed up the basis set convergence of the lowest-order SAPT dispersion and exchange dispersion energies, we borrow the techniques from explicitly correlated (F12) electronic structure theory and develop practical expressions for the closed-shell E_disp⁽²⁰⁾-F12 and E_exch-disp⁽²⁰⁾-F12 contributions. While the latter term has been derived and implemented for the first time, the former correction was recently proposed by Przybytek [ J. Chem. Theory Comput. 2018 , 14 , 5105 - 5117 ] using an Ansatz with a full optimization of the explicitly correlated amplitudes. In addition to reimplementing the fully optimized variant of E_disp⁽²⁰⁾-F12, we propose three approximate Ansätze that substantially improve the scaling of the method and at the same time avoid the numerical instabilities of the unrestricted optimization. The performance of all four resulting flavors of E_disp⁽²⁰⁾-F12 and E_exch-disp⁽²⁰⁾-F12 is first tested on helium, neon, argon, water, and methane dimers, with orbital and auxiliary basis sets up to aug-cc-pV5Z and aug-cc-pV5Z-RI, respectively. The double- and triple-ζ basis set calculations are then extended to the entire A24 database of noncovalent interaction energies and compared with CBS estimates for E_disp⁽²⁰⁾ and E_exch-disp⁽²⁰⁾ computed using conventional SAPT with basis sets up to aug-cc-pV6Z with midbond functions. It is shown that the F12 treatment is highly successful in improving the basis set convergence of the SAPT terms, with the F12 calculations in an X-tuple ζ basis about as accurate as conventional calculations in bases with cardinal numbers (X + 2) for E_disp⁽²⁰⁾ and either (X + 1) or (X + 2) for E_exch-disp⁽²⁰⁾. While the full amplitude optimization affords the highest accuracy for both corrections, the much simpler and numerically stable optimized diagonal Ansatz is a very close second. We have also tested the performance of the simple F12 correction based on the second-order Møller-Plesset perturbation theory, SAPT-F12(MP2) [ Frey , J. A. ; Chem. Rev. 2016 , 116 , 5614 - 5641 ] and observed that it is also quite successful in speeding up the basis set convergence of conventional E_disp⁽²⁰⁾ + E_exch-disp⁽²⁰⁾, albeit with some outliers.