Symmetry-adapted perturbation theory (SAPT) is a theory of intermolecular interactions. When computing intermolecular interaction energies one typically computes the energy of two molecules infinitely separated and in contact, then computes the interaction energy by subtraction. SAPT, in contrast, is a perturbative expression for the interaction energy itself. The various terms in the perturbation series are physically meaningful, and this decomposition of the interaction energy can aid in the interpretation of the results. A brief overview of the theory is given below; for additional technical details, the reader is referred to Jeziorski et al..432, 433 Additional context can be found in a pair of more recent review articles.943, 392
In SAPT, the Hamiltonian for the dimer is written as
where and are Møller-Plesset fluctuation operators for fragments and , whereas consists of the intermolecular Coulomb operators. This part of the perturbation is conveniently expressed as
The quantity is the nuclear interaction energy between the two fragments and
describes the interaction of electron with nucleus .
Starting from a zeroth-order Hamiltonian and zeroth-order wave functions that are direct products of monomer wave functions, , the SAPT approach is based on a symmetrized Rayleigh-Schrödinger perturbation expansion432, 433 with respect to the perturbation parameters , , and in Eq. (12.47). The resulting interaction energy can be expressed as432, 433
Because it makes no sense to treat and at different orders of perturbation theory, there are only two indices in this expansion: for the monomer fluctuations potentials and for the intermolecular perturbation. The terms are known collectively as the polarization expansion, and these are precisely the same terms that would appear in ordinary Rayleigh-Schrödinger perturbation theory, which is valid when the monomers are well-separated. The polarization expansion contains electrostatic, induction and dispersion interactions, but in the symmetrized Rayleigh-Schrödinger expansion, each term has a corresponding exchange term, , that arises from an anti-symmetrizer that is introduced in order to project away the Pauli-forbidden components of the interaction energy that would otherwise appear.433
The version of SAPT that is implemented in Q-Chem assumes that , an approach that is usually called SAPT0.392 Within the SAPT0 formalism, the interaction energy is formally expressed by the following symmetrized Rayleigh-Schrödinger expansion:432, 433
The anti-symmetrizer in this expression can be written as
where and are anti-symmetrizers for the two monomers and is a sum of all one-electron exchange operators between the two monomers. The operator in Eq. (12.53) denotes all of the three-electron and higher-order exchanges. This operator is neglected in what is known as the “single-exchange” approximation,432, 433 which is expected to be quite accurate at typical van der Waals and larger intermolecular separations, but sometimes breaks down at smaller intermolecular separations.535
Only terms up to in Eq. (12.52)—that is, second order in the intermolecular interaction—have been implemented in Q-Chem. It is common to relabel these low-order terms in the following way [cf. Eq. (12.51)]:
The electrostatic part of the first-order energy correction is denoted and represents the Coulomb interaction between the two monomer electron densities.433 The quantity is the corresponding first-order (i.e., Hartree-Fock) exchange correction. Explicit formulas for these corrections can be found in Ref. 432. The second-order term from the polarization expansion, denoted in Eq. (12.54), consists of a dispersion contribution (which arises for the first time at second order) as well as a second-order correction for induction. The latter can be written
where the notation , for example, indicates that the frozen charge density of polarizes the density of . In detail,
and . The second term in Eq. (12.55), in which polarizes , is obtained by interchanging labels.422 Finally, the second-order dispersion correction has a form reminiscent of the MP2 correlation energy:
The similarity between Eq. (12.58) and the MP2 correlation energy means that SAPT jobs, like MP2 calculations, can be greatly accelerated using resolution-of-identity (RI) techniques, and an RI version of SAPT is available in Q-Chem. To use it, one must specify an auxiliary basis set. The same ones used for RI-MP2 work equally well for RI-SAPT, but one should always select the auxiliary basis set that is tailored for use with the primary basis of interest, as in the RI-MP2 examples in Section 6.6.1.
It is common to replace and in Eq. (12.54) with their “response” (“resp”) analogues, which are the infinite-order correction for polarization arising from a frozen partner density.432, 433 Operationally, this substitution involves replacing the second-order induction amplitudes, in Eq. (12.56), with amplitudes obtained from solution of the coupled-perturbed Hartree-Fock equations.777 (The perturbation is simply the electrostatic potential of the other monomer.) In addition, it is common to correct the SAPT0 binding energy for higher-order polarization effects by adding a correction term of the form433, 392
to the interaction energy. Here, is the counterpoise-corrected Hartree-Fock binding energy for . Both the response corrections and the correction are optionally available in Q-Chem’s implementation of SAPT.
It is tempting to replace Hartree-Fock MOs and eigenvalues in the SAPT0 formulas with their Kohn-Sham counterparts, as a low-cost means of introducing monomer electron correlation. The resulting procedure is known as SAPT(KS),1048 and does offer an improvement on SAPT0 for some strongly hydrogen-bonded systems.375 Unfortunately, SAPT(KS) results are generally in poor agreement with benchmark dispersion energies,375 owing to incorrect asymptotic behavior of approximate exchange-correlation potentials.660 The dispersion energies can be greatly improved through the use of long-range corrected (LRC) functionals in which the range-separation parameter, , is “tuned” so as to satisfy the condition , where is the HOMO energy and “IE" represents the ionization energy.538 Monomer-specific values of , tuned using the individual monomer IEs, substantially improve SAPT(KS) dispersion energies, though the results are still not of benchmark quality.538 Other components of the interaction energy, however, can be described quite accurately SAPT(KS) in conjunction with a tuned version of LRC-PBE.538 Use of monomer-specific values is controlled by the variable DFT-LRC in the $xpol section, as described in Section 12.12.3, with monomer-specific values of that must be specified in a $lrc_omega input section. As an example, some values of for various monomers obtained using the IE-tuning condition and the LRC-PBE functional are listed in Table 12.2. Clearly there is a non-trivial variation amongst the optimally-tuned values of .
|S22 monomers||S66 monomers||ions|
Finally, some discussion of basis sets is warranted. Typically, SAPT calculations are performed in the so-called dimer-centered basis set (DCBS),1049 which means that the combined basis set is used to calculate the zeroth-order wave functions for both and . This leads to the unusual situation that there are more MOs than basis functions: one set of occupied and virtual MOs for each monomer, both expanded in the same (dimer) AO basis. As an alternative to the DCBS, one might calculate using only ’s basis functions (similarly for ), in which case the SAPT calculation is said to employ the monomer-centered basis set (MCBS).1049 However, MCBS results are generally of poorer quality. As an efficient alternative to the DCBS, Jacobson and Herbert422 introduced a projected (“proj”) or “pseudocanonicalized” basis set, borrowing an idea from dual-basis MP2 calculations.910 In this approach, the SCF iterations are performed in the MCBS but then Fock matrices for fragments and are constructed in the dimer () basis set and then pseudocanonicalized, meaning that the occupied-occupied and virtual-virtual blocks of these matrices are diagonalized. This procedure does not mix occupied and virtual orbitals, and thus leaves the fragment densities and zeroth-order fragment energies unchanged. However, it does provide a larger set of virtual orbitals that extend over the partner fragment. This larger virtual space is then used to evaluate the perturbative corrections. All three of these basis options (MCBS, DCBS, and projected basis) are available in Q-Chem.