The zeroth-order Hamiltonian for XSAPT is taken by the sum of
fragment Fock operators defined by the XPol procedure, and the perturbation is
the usual SAPT intermolecular perturbation
[Eq. (12.48)] less the intermolecular interactions
contained in the XPol fragment Fock operators. A standard SAPT0 correction
(see Section 12.13)
is then computed for each pair of monomers, using Eq. (12.53)
in conjunction with the modified perturbation. This affords the dimer interaction
energy, ${E}_{\mathrm{int}}^{AB}$. The total XSAPT energy is^{Herbert:2012}

$${E}_{\mathrm{XSAPT}}=\sum _{A}\left(\sum _{a}\left[2{\u03f5}_{a}^{A}-{\mathbf{c}}_{a}^{\u2020}({\mathbf{J}}^{A}-\frac{1}{2}{\mathbf{K}}^{A}){\mathbf{c}}_{a}\right]+{E}_{\mathrm{nuc}}^{A}+\sum _{B>A}{E}_{\mathrm{int}}^{AB}\right),$$ | (12.59) |

which is equal to the sum of the XPol monomer energies plus the pairwise SAPT corrections.
In this expression, we have removed the over-counting of two-electron
interactions present in Hartree-Fock theory, effectively taking the
*intra*fragment perturbation to first order. The generalization to a Kohn-Sham
description of the monomers is straightforward, which extends the SAPT(KS) approach to clusters
larger than dimers. This “XSAPT(KS)” approach is also available in Q-Chem.

The inclusion of many-body polarization within the zeroth-order Hamiltonian
makes the subsequent SAPT corrections less meaningful in terms of energy
decomposition analysis. For instance, the first-order electrostatic correction
in XSAPT is not the total electrostatic energy, since the former
corrects for errors in the approximate electrostatic treatment at zeroth order
(*i.e.*, the electrostatic embedding). In order to replenish some of the significance
of the XSAPT electrostatics, a “corrected” electrostatic energy is obtained
by subtracting the XPol embedding potential from the first-order electrostatic energy
obtained in SAPT, effectively removing the zeroth-order corrections from the first-order
electrostatics. The dispersion correction may be
less contaminated, since all of the XSAPT modifications to the traditional SAPT
perturbation are one-electron operators and therefore the pairwise dispersion
correction differs from its traditional SAPT analogue only insofar as the MOs
are perturbed by the electrostatic embedding. This should be kept in mind when
interpreting the output of an XSAPT calculation, although Lao and
Herbert^{Lao:2013, Lao:2015} later proposed a many-body energy
decomposition scheme for XSAPT that extends traditional SAPT energy
decomposition to systems containing more than two monomers. (The
aforementioned contamination problems are avoid through pairwise
${\delta}_{\mathrm{int}}^{\mathrm{HF}}$ corrections, comparing XSAPT results to
traditional SAPT based on gas-phase monomers.)

An XSAPT calculation is requested by setting JOBTYPE = XSAPT
in the *$rem* section. The choice of XPol charge embedding is controlled by
the embed and charges keywords in the *$xpol* input section;
see Section 12.12 and the example provided below. Additional job
control options for the SAPT part of the calculation are specified in the
*$sapt* input section as described in Section 12.13. Researchers who
use Q-Chem’s XSAPT code are asked to cite
Refs. Jacobson:2011, Herbert:2012. The latter contains a thorough
discussion of the theory; a briefer summary can be found in
Ref. Jacobson:2013.

$molecule 0 1 -- formic acid 0 1 C -1.888896 -0.179692 0.000000 O -1.493280 1.073689 0.000000 O -1.170435 -1.166590 0.000000 H -2.979488 -0.258829 0.000000 H -0.498833 1.107195 0.000000 -- formic acid 0 1 C 1.888896 0.179692 0.000000 O 1.493280 -1.073689 0.000000 O 1.170435 1.166590 0.000000 H 2.979488 0.258829 0.000000 H 0.498833 -1.107195 0.000000 $end $rem JOBTYPE XSAPT BASIS CC-PVDZ METHOD HF $end $xpol embed charges charges CHELPG ! charges derived from electrostatic potential $end $sapt basis projected ! use the pseudocanonicalized dimer basis CPHF ! solve CPHF equations for induction response $end

The latter example is simply a traditional SAPT0 (dimer) calculation but based
on zeroth-order monomer wave functions computed from a charge-embedded XPol
calculation. The following example corresponds to a truly “extended” SAPT
calculation, *i.e.*, one with more than two monomers.

$rem jobtype xsapt exchange gen basis 6-31G* $end $xpol embed charges charges chelpg $end $sapt algorithm mo ! could be ri-mo for RI approximation basis projected ! default choice; recommended 3b-ind ! include the 3-body induction couplings (optional) $end $xc_functional x wPBE 0.8 x HF 0.2 c PBE 1.0 $end $molecule 0 1 -- water 0 1 O -0.459965 1.488925 0.391165 H 0.442885 1.099622 0.558106 H -0.551255 2.236567 0.999244 -- water 0 1 O -1.111823 -1.126854 0.565807 H -1.153929 -0.145562 0.663733 H -2.016599 -1.451826 0.678719 -- water 0 1 O 1.661160 -0.139676 0.530681 H 1.455561 -0.313184 -0.421143 H 1.146044 -0.835459 0.974417 -- water 0 1 O 0.201725 -0.384036 -1.774045 H -0.394336 -0.876966 -1.168916 H -0.094680 0.533258 -1.645074 $end