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

Abstract

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.20181451055117] 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) [FreyJ. A. Frey, J. A. Chem. Rev.201611656145641] 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.

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Recent developments in symmetry‐adapted perturbation theory

“…The development of SAPT‐F12 is an ongoing research direction also in my group, and very recently our collaborators and we have performed the second step of this development by extending the work of Przybytek in two directions. First, we have eliminated the bottlenecks of the full amplitude optimization, which leads to an N 8 scaling with the system size and often ill‐conditioned numerical equations, by introducing two efficient approximations.…”

“…The first one, the optimized diagonal Ansatz (ODA), sets all off‐diagonal amplitudes to zero but computes the diagonal ones in an unrestricted optimization, and the second, the fixed‐amplitude Ansatz (FIX), sets all these diagonal amplitudes to the same value obtained by a simple minimization of a quadratic function. Both approximations scale like N 5 with the system size similar to conventional (albeit with a larger prefactor), are free from numerical instabilities, and lose very little accuracy compared to Przybytek's Ansatz with fully optimized amplitudes. Even the simplest possible F12 correction to dispersion, the ‐F12(MP2) Ansatz utilizing a suitable partitioning of the local MP2‐F12 pair functions, provides substantial improvement over conventional although the individual results are a little more erratic .…”

“…Both approximations scale like N 5 with the system size similar to conventional (albeit with a larger prefactor), are free from numerical instabilities, and lose very little accuracy compared to Przybytek's Ansatz with fully optimized amplitudes. Even the simplest possible F12 correction to dispersion, the ‐F12(MP2) Ansatz utilizing a suitable partitioning of the local MP2‐F12 pair functions, provides substantial improvement over conventional although the individual results are a little more erratic . Second, we have employed the F12 amplitudes obtained from either Ansatz to derive and implement an F12 correction to , which was shown to significantly speed up the basis set convergence of the exchange‐dispersion contribution, as illustrated in Figure on the example of the CH 4 ─CH 4 complex.…”

“…The rough grouping of different approaches according to their relative complexity has been illustrated by symbols used to rate the difficulty of North American ski trails. (Reprinted with permission from Reference . Copyright 2019 American Chemical Society)…”