# 12.13.1 Theory

(May 16, 2021)

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.. Additional context can be found in a pair of more recent review articles. ,

In SAPT, the Hamiltonian for the $\rm A\cdots B$ dimer is written as

 $\hat{H}=\hat{F}^{A}+\hat{F}^{B}+\xi\hat{W}^{A}+\eta\hat{W}^{B}+\zeta\hat{V}\;,$ (12.49)

where $\hat{W}^{A}$ and $\hat{W}^{B}$ are Møller-Plesset fluctuation operators for fragments $A$ and $B$, whereas $\hat{V}$ consists of the intermolecular Coulomb operators. This part of the perturbation is conveniently expressed as

 $\hat{V}=\sum_{i\in A}\sum_{j\in B}\hat{v}(ij)$ (12.50)

with

 $\hat{v}(ij)=\frac{1}{\bigl{|}\vec{r}_{i}-\vec{r}_{j}\bigr{|}}+\frac{\hat{v}_{A% }(j)}{N_{A}}+\frac{\hat{v}_{B}(i)}{N_{B}}+\frac{V_{0}}{N_{A}N_{B}}\;.$ (12.51)

The quantity $V_{0}$ is the nuclear interaction energy between the two fragments and

 $\hat{v}_{A}(j)=-\sum_{I\in A}\frac{Z_{I}}{\bigl{|}\vec{r}_{j}-\vec{R}_{I}\bigr% {|}}$ (12.52)

describes the interaction of electron $j\in B$ with nucleus $I\in A$.

Starting from a zeroth-order Hamiltonian $\hat{H}_{0}=\hat{F}^{A}+\hat{F}^{B}$ and zeroth-order wave functions that are direct products of monomer wave functions, $|\Psi_{0}\rangle=|\Psi_{A}\rangle|\Psi_{B}\rangle$, the SAPT approach is based on a symmetrized Rayleigh-Schrödinger perturbation expansion with respect to the perturbation parameters $\xi$, $\eta$, and $\zeta$ in Eq. (12.49). The resulting interaction energy can be expressed as

 $E_{\mathrm{int}}=\sum_{i=1}^{\infty}\sum_{j=0}^{\infty}\left(E^{(ij)}_{\mathrm% {pol}}+E^{(ij)}_{\mathrm{exch}}\right)\;.$ (12.53)

Because it makes no sense to treat $\hat{W}^{A}$ and $\hat{W}^{B}$ at different orders of perturbation theory, there are only two indices in this expansion: $j$ for the monomer fluctuations potentials and $i$ for the intermolecular perturbation. The terms $E^{(ij)}_{\mathrm{pol}}$ 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 $E^{(ij)}_{\mathrm{pol}}$ has a corresponding exchange term, $E^{(ij)}_{\mathrm{exch}}$, that arises from an anti-symmetrizer $\hat{\cal A}_{AB}$ that is introduced in order to project away the Pauli-forbidden components of the interaction energy that would otherwise appear.

The version of SAPT that is implemented in Q-Chem assumes that $\xi=\eta=0$, an approach that is usually called SAPT0. Within the SAPT0 formalism, the interaction energy is formally expressed by the following symmetrized Rayleigh-Schrödinger expansion:

 $E_{\mathrm{int}}(\zeta)=\frac{\langle\Psi_{0}|\zeta\hat{V}\hat{\cal A}_{AB}|% \Psi(\zeta)\rangle}{\langle\Psi_{0}|\hat{\cal A}_{AB}|\Psi(\zeta)\rangle}\;,$ (12.54)

The anti-symmetrizer $\hat{\cal A}_{AB}$ in this expression can be written as

 $\hat{\cal A}_{AB}=\frac{N_{A}!N_{B}!}{(N_{A}+N_{B})!}\;\hat{\cal A}_{A}\hat{% \cal A}_{B}\bigl{(}\hat{1}+\hat{\cal P}^{AB}+\hat{\cal P}^{\prime}\bigr{)}\;,$ (12.55)

where $\hat{\cal A}_{A}$ and $\hat{\cal A}_{B}$ are anti-symmetrizers for the two monomers and $\hat{\cal P}^{AB}$ is a sum of all one-electron exchange operators between the two monomers. The operator $\hat{\cal P}^{\prime}$ in Eq. (12.55) denotes all of the three-electron and higher-order exchanges. This operator is neglected in what is known as the “single-exchange” approximation, which is expected to be quite accurate at typical van der Waals and larger intermolecular separations, but sometimes breaks down at smaller intermolecular separations.

Only terms up to $\zeta=2$ in Eq. (12.54)—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. (12.53)]:

 $E_{\mathrm{int}}^{\rm SAPT0}=E^{(1)}_{\mathrm{elst}}+E^{(1)}_{\mathrm{exch}}+E% _{\mathrm{pol}}^{(2)}+E_{\mathrm{disp}}^{(2)}\;.$ (12.56)

The electrostatic part of the first-order energy correction is denoted $E^{(1)}_{\mathrm{elst}}$ and represents the Coulomb interaction between the two monomer electron densities. The quantity $E^{(1)}_{\mathrm{exch}}$ is the corresponding first-order (i.e., Hartree-Fock) exchange correction. Explicit formulas for these corrections can be found in Ref. 505. The second-order term from the polarization expansion, denoted $E_{\mathrm{pol}}^{(2)}$ in Eq. (12.56), 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

 $E_{\mathrm{ind}}^{(2)}=E_{\mathrm{ind}}^{(2)}(A\leftarrow B)+E_{\mathrm{ind}}^% {(2)}(B\leftarrow A)\;,$ (12.57)

where the notation $A\leftarrow B$, for example, indicates that the frozen charge density of $B$ polarizes the density of $A$. In detail,

 $E_{\mathrm{ind}}^{(2)}(A\leftarrow B)=2\sum_{ar}t_{ar}(w_{B})_{ra}$ (12.58)

where

 $(w_{B})_{ar}=(\hat{v}_{B})_{ar}+\sum_{b}(ar|bb)$ (12.59)

and $t_{ar}=(w_{B})_{ar}/(\epsilon_{a}-\epsilon_{r})$. The second term in Eq. (12.57), in which $A$ polarizes $B$, is obtained by interchanging labels. Finally, the second-order dispersion correction has a form reminiscent of the MP2 correlation energy:

 $E_{\mathrm{disp}}^{(2)}=4\sum_{abrs}\frac{(ar|bs)(ra|sb)}{\epsilon_{a}+% \epsilon_{b}-\epsilon_{r}-\epsilon_{s}}\;.$ (12.60)

The induction and dispersion corrections both have accompanying exchange corrections (exchange-induction and exchange-dispersion).

The similarity between Eq. (12.60) 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 $E_{\mathrm{ind}}^{(2)}$ and $E_{\mathrm{exch\mbox{-}ind}}^{(2)}$ in Eq. (12.56) with their “response” (“resp”) analogues, which are the infinite-order correction for polarization arising from a frozen partner density. Operationally, this substitution involves replacing the second-order induction amplitudes, $t_{ar}$ in Eq. (12.58), with amplitudes obtained from solution of the coupled-perturbed Hartree-Fock equations. (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 form ,

 $\delta E_{\mathrm{int}}^{\rm HF}=E_{\mathrm{int}}^{\rm HF}-\Bigl{(}E_{\mathrm{% elst}}^{(1)}+E_{\mathrm{exch}}^{(1)}+E_{\mathrm{ind,resp}}^{(2)}+E_{\mathrm{% exch\mbox{-}ind,resp}}^{(2)}\Bigr{)}$ (12.61)

to the interaction energy. Here, $E_{\mathrm{int}}^{\rm HF}$ is the counterpoise-corrected Hartree-Fock binding energy for $\rm A\cdots B$. Both the response corrections and the $\delta E^{\rm HF}_{\rm int}$ 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), and does offer an improvement on SAPT0 for some strongly hydrogen-bonded systems. Unfortunately, SAPT(KS) results are generally in poor agreement with benchmark dispersion energies, owing to incorrect asymptotic behavior of approximate exchange-correlation potentials. The dispersion energies can be greatly improved through the use of long-range corrected (LRC) functionals in which the range-separation parameter, $\omega$, is “tuned” so as to satisfy the condition $\epsilon_{\rm HOMO}=-\mbox{IE}$, where $\epsilon_{\rm HOMO}$ is the HOMO energy and “IE” represents the ionization energy. Monomer-specific values of $\omega$, tuned using the individual monomer IEs, substantially improve SAPT(KS) dispersion energies, though the results are still not of benchmark quality. Other components of the interaction energy, however, can be described quite accurately SAPT(KS) in conjunction with a tuned version of LRC-$\omega$PBE. Use of monomer-specific $\omega$ values is controlled by the variable DFT-LRC in the $xpol section, as described in Section 12.12.3, with monomer-specific values of $\omega$ that must be specified in a$lrc_omega input section. As an example, some values of $\omega$ for various monomers obtained using the IE-tuning condition and the LRC-$\omega$PBE functional are listed in Table 12.2. Clearly there is a non-trivial variation amongst the optimally-tuned values of $\omega$.

Finally, some discussion of basis sets is warranted. Typically, SAPT calculations are performed in the so-called dimer-centered basis set (DCBS), which means that the combined $A+B$ basis set is used to calculate the zeroth-order wave functions for both $A$ and $B$. 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 $|\Psi_{\!A}\rangle$ using only $A$’s basis functions (similarly for $B$), in which case the SAPT calculation is said to employ the monomer-centered basis set (MCBS). However, MCBS results are generally of poorer quality. As an efficient alternative to the DCBS, Jacobson and Herbert introduced a projected (“proj”) or “pseudocanonicalized” basis set, borrowing an idea from dual-basis MP2 calculations. In this approach, the SCF iterations are performed in the MCBS but then Fock matrices for fragments $A$ and $B$ are constructed in the dimer ($A+B$) 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.