12.10.3 Dispersion Models‣ 12.10 The XPol+SAPT (XSAPT) Method ‣ Chapter 12 Fragment-Based Methods ‣ Q-Chem 7.0 User’s Manual

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, ⁷⁶¹ Lao K. U., Herbert J. M.
J. Chem. Phys.
(2014), 140, pp. 044108. Link 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. ⁵⁶⁰ Herbert J. M. et al.
Phys. Chem. Chem. Phys.
(2012), 14, pp. 7679. Link 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 ${\cal{O}}({N^{4}})$ and ${\cal{O}}({N^{5}})$ , respectively, with respect to system size. Other terms in SAPT0 scale no worse than ${\cal{O}}({N^{3}})$ and can be computed efficiently for large monomers using an atomic orbital (AO)-based implementation of the non-dispersion terms in SAPT. ⁷⁶⁴ Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link

In view of this, both the efficiency and the accuracy of XSAPT(KS) calculations is improved if second-order dispersion, i.e., $E_{\mathrm{disp}}^{(2)}+E_{\mathrm{exch\mbox{-}disp}}^{(2)}$ in Eqs (12.61) and (12.65), is replaced by an ad hoc atom–atom dispersion potential of the $-C_{6}/R^{6}-C_{8}/R^{8}-\cdots$ 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”, ⁷⁶⁴ Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link to distinguish it from the “+D” dispersion corrections of DFT-D, although we simply called it “+D” in earlier work. ⁷⁵⁹ Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2012), 3, pp. 3241. Link ^, ⁷⁶⁰ Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link ^, ⁷⁶² Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link The composite method is called XSAPT(KS)+aiD; see Ref. 760 Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link for an overview and Ref. 764 Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link for an efficient implementation in the AO basis. The latter version exhibits ${\cal{O}}({N^{3}})$ scaling without significant memory bottlenecks, and is applicable to supramolecular complexes whose monomers contain $\gtrsim 100$ atoms. ⁷⁶⁴ Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link

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 ${\cal{O}}({N^{3}})$ AO-based version of XSAPT; and second, set Dispersion in the $sapt section according to one of the choices that is described below. Available models include pairwise empirical dispersion (+aiD) models, ⁷⁵⁹ Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2012), 3, pp. 3241. Link ^, ⁷⁶⁰ Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link ^, ⁷⁶² Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link the many-body dispersion (+MBD) potential ²⁰⁰ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 2706. Link and a revised version of it (+MBDrev), ¹⁴⁰⁶ Villot C., Lao K. U.
J. Chem. Phys.
(2024), 160, pp. 184103. Link and “optimized” $C_{6}$ - $C_{8}$ MBD models, as described below. Results among these models are very similar for total interaction energies in small molecules, ⁷⁶² Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link ^, ²⁰⁰ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 2706. Link but different in their performance for large molecules and ions.^{200, 557, Paice:2026b}

Dispersion
       Requests a dispersion potential for use with (X)SAPT.
INPUT SECTION: $sapt
TYPE:
       STRING
DEFAULT:
       aiD3
OPTIONS:
       aiD First-generation empirical pairwise dispersion potential aiD2 Second-generation empirical pairwise dispersion potential aiD3 Third-generation empirical pairwise dispersion potential MBD Many-body $C_{6}$ and $C_{8}$ dispersion potential MBDrev Revised MBD dispersion, refit to SAPT10K data OC8AIMMBD Optimized $C_{6}$ - $C_{8}$ MBD with nonrelativistic free-atom reference data OC8RAIMMBD Optimized $C_{6}$ - $C_{8}$ MBD, relativistic reference data RAIMVDW Pairwise atoms-in-molecules dispersion only (no MBD term), relativistic references
RECOMMENDATION:
       The MBD-based models are generally better than the aiD models, especially for large systems where MBD effects (beyond pairwise additivity) become important. The optimized $C_{6}$ - $C_{8}$ MBD models perform even better for ions and extend MBD to heavy elements via relativistic reference data.

The +aiD potentials make a pairwise approximation for dispersion, in which 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,

E_{\text{disp}}=-\sum_{i\in A}\sum_{\begin{subarray}{c}j\in B\\ A\neq B\end{subarray}}\Bigg{[}f_{6}(R_{ij})\frac{C_{6}^{ij}}{R^{6}_{ij}}+f_{8}% (R_{ij})\frac{C_{8}^{ij}}{R^{8}_{ij}}\Bigg{]}\;,

(12.68)

where $i$ and $j$ are nuclei in molecules $A$ and $B$ , respectively. The original +aiD1 model ⁷⁵⁹ Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2012), 3, pp. 3241. Link 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 and is not recommended. ⁷⁶⁰ Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link ^, ⁷⁶² Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link The difference between +aiD2 ⁷⁶⁰ Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link and +aiD3 ⁷⁶² Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link 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. That said, the pairwise approximation for dispersion 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. ⁷⁶⁴ Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link

Although a correction to this pairwise approximation has been proposed, ⁷⁶⁵ Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 5128. Link based on the difference between XSAPT and SAPT dispersion energies, all of the +aiD models are ad hoc and their corrections do not depend on the applied level of theory. The +MBD potential uses a modified version of the many-body dispersion potential of Ambrosetti et al. ⁴³ Ambrosetti A. et al.
J. Chem. Phys.
(2014), 140, pp. 18A508. Link ^, ⁴⁸³ Gray M., Herbert J. M.
Annu. Rep. Comput. Chem.
(2024), 20, pp. 1. Link (see Section 5.7.6) in order to account for nonadditive dispersion in a natural way. ²⁰⁰ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 2706. Link Because the +MBD method is based on the electron density, it is much more connected to the ab initio method being used. ¹⁹⁹ Carter-Fenk K., Lao K. U., Herbert J. M.
Acc. Chem. Res.
(2021), 54, pp. 3679. Link 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 is recommended in most cases. To improve the transferability and accuracy of dispersion interactions across diverse chemical environments, the four parameters contained in the MBD model have been re-optimized against the comprehensive SAPT10K benchmark data set of SAPT2+(3)(CCD)/aug-cc-pVTZ dispersion energies. ¹⁴⁰⁶ Villot C., Lao K. U.
J. Chem. Phys.
(2024), 160, pp. 184103. Link The resulting revised model, denoted MBDrev, is used in conjunction with (X)SAPT for non-dispersion contributions and delivers systematically improved performance relative to the original (X)SAPT+MBD approach, including for chemically realistic systems comprising hundreds of atoms. ⁷⁶⁷ Lao K. U.
J. Chem. Phys.
(2024), 161, pp. 234103. Link However, the Hirshfeld partition that is inherent to MBD-based dispersion models is ill-defined for ions, for which MBD sometimes gives poor results,⁵⁵⁷ even relative to +aiD3. ⁴⁸¹ Gray M., Herbert J. M.
J. Chem. Theory Comput.
(2022), 18, pp. 2308. Link

For this reason, an optimized family of atoms-in-molecules dispersion models is also available, in which the per-atom response properties $(\alpha_{0}$ , $C_{6}$ , and $R_{\mathrm{vdW}})$ are obtained by Tkatchenko–Scheffler scaling of free-atom reference data, ¹³⁶¹ Tkatchenko A., Scheffler M.
Phys. Rev. Lett.
(2009), 102, pp. 073005. Link ^, ⁴⁸³ Gray M., Herbert J. M.
Annu. Rep. Comput. Chem.
(2024), 20, pp. 1. Link using an iterative version of Hirshfeld volume partition that is better behaved for ions. As in the original (X)SAPT+MBD approach, ²⁰⁰ Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 2706. Link ^, ⁸⁴⁵ Liu K.-Y., Carter-Fenk K., Herbert J. M.
J. Chem. Phys.
(2019), 151, pp. 031102. Link ^, ¹⁹⁹ Carter-Fenk K., Lao K. U., Herbert J. M.
Acc. Chem. Res.
(2021), 54, pp. 3679. Link these models combine both $C_{6}/R^{6}$ and $C_{8}/R^{8}$ dispersion terms, with the former computed using the Drude oscillator Hamiltonian of Tkatchenko’s MBD model. ⁴³ Ambrosetti A. et al.
J. Chem. Phys.
(2014), 140, pp. 18A508. Link However, this new “optimized” approach uses $C_{8}$ parameters that are obtained in an optimal way from the Drude oscillator invariants from the $C_{6}$ MBD calculation. ⁴⁶⁴ Góger S. et al.
J. Phys. Chem. Lett.
(2023), 14, pp. 6217. Link This new family of models for (X)SAPT is collectively called “oC8aimMBD” (optimized atoms-in-molecules MBD), and there are three variants that differ only in their free-atom reference data, as indicated in the description of the Dispersion keyword above. The oC8aimMBD approach uses non-relativistic Hartree–Fock reference data for the free atoms, whereas oC8(r)aimMBD uses four-component Dirac–Hartree–Fock references in uncontracted Dyall basis sets, and is recommended whenever elements with $Z>36$ are present. Finally, the (r)aimVDW model retains only the pairwise $C_{6}/R^{6}$ correction, dropping the many-body term. The per-fragment Hirshfeld evaluation is enabled automatically; set print to 3 in the $xpol section reports the per-atom Hirshfeld volume ratios used to scale the dispersion parameters.

As with XPol, the XSAPT and XSAPT(KS)+aiD methods do not function 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. 759 Lao K. U., Herbert J. M.
J. Phys. Chem. Lett.
(2012), 3, pp. 3241. Link for +aiD1, Ref. 760 Lao K. U., Herbert J. M.
J. Chem. Phys.
(2013), 139, pp. 034107. Link for +aiD2, Ref. 762 Lao K. U., Herbert J. M.
J. Phys. Chem. A
(2015), 119, pp. 235. Link for +aiD3, Ref. 200 Carter-Fenk K. et al.
J. Phys. Chem. Lett.
(2019), 10, pp. 2706. Link for +MBD, Ref. 1406 Villot C., Lao K. U.
J. Chem. Phys.
(2024), 160, pp. 184103. Link for +MBDrev, and Ref. for the optimized aimMBD models. All of these are based on the AO-based version of XSAPT that was introduced in Ref. 764 Lao K. U., Herbert J. M.
J. Chem. Theory Comput.
(2018), 14, pp. 2955. Link .

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12.10.3 Dispersion Models