12.14.3 Dispersion Models

SAPT(KS) calculations and their many-body extension, XSAPT(KS), uses a Kohn-Sham DFT description of the monomers in order to introduce intramolecular electron correlation in a low-cost way, then described the intermolecular interactions using second-order SAPT. As mentioned in The resulting interaction energies, however, are not of benchmark quality even when tuned LRC functionals are employed,⁶³² because although the use of DFT for the monomers often improves the description of hydrogen bonding (relative to Hartree-Fock-based SAPT0 calculations), the description of dispersion often deteriorates.⁴⁵⁶ In any case, SAPT0 dispersion is not of benchmark quality anyway, as it suffers from the usual MP2 overestimation of dispersion. At the same time the dispersion and exchange-dispersion terms are the most expensive parts of a SAPT0 or SAPT(KS) calculation, with a formal scaling of $𝒪(N^4)$ and $𝒪(N^5)$, respectively, with respect to system size. Other terms in SAPT0 scale no worse than $𝒪(N^3)$ and can be computed efficiently for large monomers using an atomic orbital (AO)-based implementation of the non-dispersion terms in SAPT.⁶³⁵

In view of this, both the efficiency and the accuracy of XSAPT(KS) calculations is improved if second-order dispersion, i.e., $E_{\mathrm{disp}}^{(2)}+E_{\mathrm{exch}\text{-}\mathrm{disp}}^{(2)}$ in Eqs (12.56) and (12.60), is replaced by an ad hoc atom–atom dispersion potential of the $-C_6/R^6-C_8/R^8-\mathrm{\cdots }$ variety. This is reminiscent of dispersion-corrected DFT or DFT-D, as described in Section 5.7.3. Unlike the situation in DFT, however, the dispersion energy is well-defined and separable within the SAPT formalism, so it can be replaced by atom–atom potentials without any fear of double counting of correlation effects, as there inevitably is in DFT-D. Moreover, in the present case the dispersion potentials can be fit directly to ab initio dispersion energies from high-level SAPT calculations [SAPT(DFT) and SAPT2+(3)], since the dispersion contribution is separable. As such, while the dispersion potentials that are described here are classical in form and do contain fitting parameters, they can nevertheless reasonably be described as ab initio dispersion potentials. We therefore describe this method as “+aiD”,⁶³⁵ to distinguish it from the “+D” dispersion corrections of DFT-D, although we simply called it “+D” in earlier work.^{630, 631, 633} The composite method is called XSAPT(KS)+aiD; see Ref. 631 for an overview and Ref. 635 for an efficient implementation in the AO basis. The latter version exhibits $𝒪(N^3)$ scaling without significant memory bottlenecks, and is applicable to supramolecular complexes whose monomers contain $\gtrsim 100$ atoms.⁶³⁵

To request an XSAPT(KS)+aiD calculation, set JOBTYPE = XSAPT in the $rem section to perform XSAPT, with an appropriate choice of SCF method (Hartree-Fock or DFT). The +aiD part of the algorithm is invoked by two keywords in the $sapt input section: first, set Algorithm to AO to select the $𝒪(N^3)$ AO-based version of XSAPT; and second, set Dispersion equal to aiD, aiD2, aiD3, or MBD. The latter choices correspond, respectively, to the “first generation" (+aiD1) potential,⁶³⁰ the second-generation (+aiD2) potential,⁶³¹, the third-generation (+aiD3) dispersion potential,⁶³³ or the many-body dispersion (+MBD) potential.¹⁶² All four versions exhibit similar performance for total interaction energies in small molecules,^{633, 162} but unlike its successors, the +aiD1 potential was fit to reproduce total interaction energies rather than being fit directly to ab initio dispersion data, and as a consequence does a much poorer job of reproducing individual energy components. (It was later discovered that the performance of +aiD1 benefits from some error cancellation amongst energy components,^{631, 633} and as such its use is not recommended.) The difference between +aiD2 and +aiD3 is a larger training set for the latter, which was designed to afford better coverage of $\pi$-stacked systems. As such, the +aiD3 correction is the superior choice out of the pairwise potentials in the +aiD suite of methods.

The first three generations of +aiD potentials make the pairwise approximation, where the interaction potential is assumed to be additive across all pairs of atoms. The pairwise dispersion approximation employs sums over atom pairs of the form,

where $i$ and $j$ are nuclei in molecules $A$ and $B$, respectively. The pairwise approximation breaks down in the limit of very large systems because the interactions between atom pairs are modulated by the local electrodynamic environment in the molecule. It was discovered that even the +aiD3 potential (the best of the pairwise +aiD potentials) suffers from this approximation in large systems,⁶³⁵ and a correction based on the difference between XSAPT and SAPT dispersion energies was proposed.⁶³⁶ While this correction performs well, all of the pairwise dispersion potentials (+aiD1, +aiD2, and +aiD3) are rather ad hoc and their corrections do not depend on the applied level of theory. The most recent +MBD potential uses a modified version of the many-body dispersion potential of Ambrosetti et al.³⁸ (see Section 5.7.6 in order to naturally account for nonadditive dispersion effects.¹⁶² Because the +MBD method is based on the electron density it is much more connected to the ab initio method being used, and this is presently the more accurate version of XSAPT.¹⁶¹ When combined with the XSAPT procedure, the XSAPT+MBD energy decomposition accounts for nonadditive polarization and dispersion effects. Due to its excellent performance regardless of system size, the +MBD potential (Dispersion MBD) is recommended, but the +aiD3 potential (Dispersion aiD3) remains quite good for smaller systems.

As with XPol, the XSAPT and XSAPT(KS)+aiD methods do not function with a solvation model or with external changes. Only single-point energies are available, and frozen orbitals orbitals are not allowed. Both restricted and unrestricted versions are available. Researchers who use XSAPT(KS)+aiD are asked to cite Ref. 630 for +aiD1, Ref. 631 for +aiD2, Ref. 633 for +aiD3, or Ref. 162 for +MBD, along with Ref. 635 for the AO-based version of XSAPT.