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SAPT(KS) calculations and their many-body extension, XSAPT(KS), uses a
Kohn-Sham DFT description of the monomers in order to introduce
*intra*molecular electron correlation in a low-cost way, then described
the *inter*molecular 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,^{538} 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.^{375} 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 $\mathcal{O}({N}^{4})$ and $\mathcal{O}({N}^{5})$,
respectively, with respect to system size. Other terms in SAPT0 scale no
worse than $\mathcal{O}({N}^{3})$ and can be computed efficiently for large monomers using
an atomic orbital (AO)-based implementation of the non-dispersion terms in
SAPT.^{541}

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 (12.54) and
(12.58), 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 5.7.2.
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
“+*ai*D”,^{541} to distinguish it from the “+D” dispersion
corrections of DFT-D, although we simply called it “+D” in earlier
work.^{536, 537, 539} The composite method is called
XSAPT(KS)+*ai*D; see Ref. 537 for an overview and
Ref. 541 for an efficient implementation in the AO basis.
The latter version exhibits $\mathcal{O}({N}^{3})$ scaling without significant memory
bottlenecks, and is applicable to supramolecular complexes whose monomers
contain $\gtrsim 100$ atoms.^{541}

To request an XSAPT(KS)+*ai*D calculation, set JOBTYPE =
XSAPT in the *$rem* section to perform XSAPT, with an appropriate
choice of SCF method (Hartree-Fock or DFT). The +*ai*D part of the
algorithm is invoked by two keywords in the *$sapt* input section: first, set
Algorithm to AO to select the $\mathcal{O}({N}^{3})$ AO-based version of
XSAPT; and second, set EmpiricalDisp equal to 1, 2, 3, or 4. The latter
choices correspond, respectively, to the “first generation" (+*ai*D1)
potential,^{536} the second-generation (+*ai*D2)
potential,^{537}, the third-generation (+*ai*D3) dispersion
potential,^{539} or the many-body dispersion (+MBD) potential.^{127}
All four versions exhibit similar performance for
total interaction energies in small molecules,^{539, 127} but
unlike its successors, the
+*ai*D1 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 +*ai*D1 benefits from some error
cancellation amongst energy components,^{537, 539} and as such
its use is not recommended.) The difference between +*ai*D2 and
+*ai*D3 is a larger training set for the latter, which was designed to
afford better coverage of $\pi $-stacked systems. As such, the +*ai*D3 correction
is the superior choice out of the pairwise potentials in the +*ai*D suite of methods.

The first three generations of +*ai*D 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,

$${E}_{disp}=-\sum _{i\in A}\sum _{\begin{array}{c}j\in B\\ A\ne B\end{array}}\left[{f}_{6}({R}_{ij})\frac{{C}_{6}^{ij}}{{R}_{ij}^{6}}+{f}_{8}({R}_{ij})\frac{{C}_{8}^{ij}}{{R}_{ij}^{8}}\right],$$ | (12.61) |

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 +*ai*D3 potential (the best of the pairwise +*ai*D potentials)
suffers from this approximation in large systems,^{541} and a correction based on the difference
between XSAPT and SAPT dispersion energies was proposed.^{542}
While this correction performs well, all of the pairwise dispersion potentials (+*ai*D1, +*ai*D2, and +*ai*D3)
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.*(see Section 5.7.5 for details)
in order to naturally account for nonadditive dispersion effects,^{127}
and because the +MBD method is based on the electron density it is much more
connected to the *ab initio* method being used.
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 (EmpiricalDisp 4) is recommended,
but the +*ai*D3 potential (EmpiricalDisp 3) remains quite good for smaller systems.

As with XPol, the XSAPT and XSAPT(KS)+*ai*D 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)+*ai*D
are asked to cite Ref. 536 for +*ai*D1,
Ref. 537 for +*ai*D2, Ref. 539 for
+*ai*D3, or Ref. 127 for +MBD, along with Ref. 541 for the AO-based version of XSAPT.

EmpiricalDisp

Requests a +*ai*D dispersion potential.

INPUT SECTION: *$sapt*

TYPE:

INTEGER

DEFAULT:

3

OPTIONS:

$n$
Use version $n=1$, 2, 3, or 4.

RECOMMENDATION:

Use +MBD. The second-, third-, and fourth-generation versions were
parameterized using *ab initio* dispersion data and afford accurate energy
components, in addition to accurate total interaction energies. The
third-generation version was parameterized using an expanded data set designed
to reduce some large errors observed for $\pi $-stacked complexes using
+*ai*D2. The fourth-generation version accounts for many-body dispersion
effects that are very important even in moderately large systems.

$molecule 0 1 -- 0 1 O -1.551007 -0.114520 0.000000 H -1.934259 0.762503 0.000000 H -0.599677 0.040712 0.000000 -- 0 1 O 1.350625 0.111469 0.000000 H 1.680398 -0.373741 -0.758561 H 1.680398 -0.373741 0.758561 $end $rem JOBTYPE xsapt EXCHANGE gen BASIS aug-cc-pVTZ MEM_TOTAL 46000 MEM_STATIC 4000 AO2MO_DISK 35000 CHELPG_DX 5 CHELPG_HEAD 30 CHELPG_H 110 CHELPG_HA 590 $end $xpol embed charges charges CHELPG DFT-LRC print 3 $end $sapt algorithm ao ! for use with +aiD dispersion order 2 ! 2nd-order SAPT basis projected EmpiricalDisp 3 print 3 $end $xc_functional x wPBE 1.0 c PBE 1.0 $end $lrc_omega 502 502 $end