13.13 The XPol+SAPT (XSAPT) Method

13.13.1 Theory

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. (13.39)] less the intermolecular interactions contained in the XPol fragment Fock operators. A standard SAPT0 correction (see Section 13.12) is then computed for each pair of monomers, using Eq. (13.44) in conjunction with the modified perturbation. This affords the dimer interaction energy, EintAB. The total XSAPT energy is369

EXSAPT=A(a[2ϵaA-𝐜a(𝐉A-12𝐊A)𝐜a]+EnucA+B>AEintAB), (13.50)

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 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). 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 Herbert525, 527 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 δintHF 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 13.11 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 13.12. Researchers who use Q-Chem’s XSAPT code are asked to cite Refs. 414, 369. The latter contains a thorough discussion of the theory; a briefer summary can be found in Ref. 415.

Example 13.27  Example of an XPol + SAPT0 calculation using ChElPG charges for the XPol calculation and computing Eind,resp(2) and Eexch-ind,resp(2) by solving CPHF equations as discussed in Section 13.12.

$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.

Example 13.28  XSAPT(KS) calculation on water tetramer using the LRC-ωPBEh 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. 369).

$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