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# 12.14.2 Theory

(February 4, 2022)

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. (12.51)] 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. (12.56) in conjunction with the modified perturbation. This affords the dimer interaction energy, $E_{\mathrm{int}}^{AB}$. The total XSAPT energy is456

 $E_{\rm XSAPT}=\sum_{A}\left(\sum_{a}\Bigl{[}2\,\epsilon_{a}^{A}-\mathbf{c}_{a}% ^{\dagger}(\mathbf{J}^{A}-\tfrac{1}{2}\mathbf{K}^{A})\mathbf{c}_{a}\Bigr{]}+E_% {\mathrm{nuc}}^{A}+\sum_{B>A}E_{\mathrm{int}}^{AB}\right)\;,$ (12.62)

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 intrafragment perturbation to first order. The generalization to a Kohn-Sham description of the monomers is straightforward, which extends the SAPT0(KS) approach to clusters larger than dimers.

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 Herbert631, 633 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 $\delta_{\mathrm{int}}^{\rm HF}$ 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. 511, 456. The latter contains a thorough discussion of the theory; a briefer summary can be found in Ref. 512. Example 12.36 Example of an XPol + SAPT0 calculation using ChElPG charges for the XPol calculation and computing $E_{\mathrm{ind,resp}}^{(2)}$ and $E_{\mathrm{exch\mbox{-}ind,resp}}^{(2)}$ by solving CPHF equations as discussed in Section 12.13. $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
$end  View output The latter example is simply a traditional SAPT0 (dimer) calculation but based on zeroth-order monomer wave functions computed from a charge-embedded XPol calculation. The following example corresponds to a truly “extended” SAPT calculation, i.e., one with more than two monomers. Example 12.37 XSAPT calculation on water tetramer using the LRC-$\omega$PBEh functional. Includes the three-body induction couplings that arise at second order in perturbation theory when the number of monomers is greater than 2 (see Ref. 456). $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
$end$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


View output