3 Short overview of theory

In SAPT, the total Hamiltonian for the dimer is partitioned as H = F + V + W, where F = FA + FB is the sum of the Fock operators for monomers A and B, V is the intermolecular interaction operator, and W = WA + WB is the sum of the Møller-Plesset fluctuation operators. The latter operators are defined as WX = HX - FX, where HX is the total Hamiltonian of monomer X. The interaction energy, Eint, is expanded as a perturbative series

 ∞ ∞ E = ∑ ∑ (E(nj) + E(nj)) int n=1j=0 RS exch

with the indices n and j denoting the orders in the operators V and W, respectively. The energies ERS(nj) are the corrections defined by the regular Rayleigh-Schrödinger perturbation theory. These terms were named “polarization” energies by Hirschfelder [44] and this terminology was used in earlier editions of SAPT, but was dropped later due to the confusion with the induction interactions often called polarization interactions. The exchange corrections, Eexch(nj), arise from the use of a global antisymmetrizer to force the correct permutational symmetry of the dimer wave function in each order, hence the name “symmetry adaptation”. Whereas the double perturbation theory expansion of Eq. (1) is very convenient for analyzing the results, SAPT is actually a triple perturbation theory as the WA and WB operators appear individually in SAPT expressions. The resulting triple index corrections, E(nij), with the consecutive indices referring to V , WA, and WB, respectively, will occasionally appear later on in this manual.

The RS corrections of the first order in V , ERS(1j), describe the classical electrostatic interaction and are denoted by Eelst(1j). An alternative to expanding the electrostatic energy in powers of the intramonomer correlation operator is to calculate monomer electron charge densities ρA and ρB at a certain level of correlation and then use these densities in the formula

 ∫ ∫ ∫ Ee(1l)st = ρA(r1) 1-ρB (r2)dr1dr2 + ρA(r)VB (r)dr + ρB(r)VA (r)dr+ V0, r12

where V A and V B denote the electrostatic potential of the nuclei of monomer A and B, respectively, and V 0 is the nuclear repulsion term. In sapt2012, the densities in Eq. (2) can be computed at the relaxed CCSD level. The quantity Eelst,resp(1)(CCSD) obtained in this way contains all the second- and third-order intramonomer correlation corrections as well as some other classes of diagrams (diagrams resulting from single and double coupled-cluster excitations) summed up to infinite order [45]. Similarly, the Eexch(1)(CCSD) correction sums up the respective exchange contributions [46].

The second-order corrections can be decomposed into the induction and dispersion parts:

E(R2Sj)= E (in2jd)+ Ed(2ijs)p and E (2exjc)h = E(e2jx)ch-ind + Ee(2xjc)h-disp.

The induction component is the energy of interaction of the permanent multipole moments of one monomer and the induced multipole moments on the other, whereas the dispersion part comes from the correlation of electron motions on one monomer with those on the other monomer. Similarly, the third-order polarization corrections are decomposed as

E (R3jS)= E (3injd)+ E(i3nj)d-disp + Ed(3ijsp)

and the same holds for the corresponding exchange corrections. A detailed discussion of the physical interpretation of various parts of the third-order polarization energy can be found in Refs. 1 and 38.

The SAPT interaction energy can be computed at different levels of intramonomer correlation and an approximate correspondence can be made between these levels and the correlation levels of the supermolecular methods. It can be shown [47], for example, that an appropriate sum of the polarization and exchange corrections of the zeroth order in W provides a good approximation to the supermolecular Hartree-Fock interaction energy, EintHF:

EHF = E (10)+ E(10) + E(20) + E (20) + δEHF , int elst exch ind,resp exch-ind,resp int,resp

where δEint,respHF, defined by the equation above, collects all third- and higher-order induction and exchange-induction terms. The subscript “resp” means that the coupled Hartree-Fock-type response of a perturbed system is incorporated in the calculation of this correction. Including the intramonomer correlation up to a level roughly equivalent to the supermolecular second-order MBPT calculation, we obtain the interaction energy referred to as SAPT2:

ESAinPt T2 = EHiFnt + E (1el2s)t,resp + ϵ(1e)xch(2)+ tE (2in2d)+ tEe(2x2c)h-ind + E(d20is)p + E (2ex0c)h-disp,

where the notation ϵ(n)(k) = j=1kE(nj) has been used, tEind(22) is the part of Eind(22) not included in Eind,resp(20), and tEexch-ind(22) is the estimated exchange counterpart of tEind(22):

t (22) (20) -tE(i2n2)d-- Eexch-ind ≈ E exch-ind,respE (20) . ind,resp

The highest routinely used level of SAPT, approximately equivalent to the supermolecular MBPT theory through fourth order, is defined by:

ESiAntPT = ESAinPt T2+ E (1el3s)t,resp + [ϵ(e1x)ch(CCSD )- ϵ(e1)xch(2)]+ ϵ(d2i)sp(2),

where ϵexch(1)(CCSD) = Eexch(1)(CCSD) - Eexch(10) is the part of ϵexch(1)() with intramonomer excitations at the CCSD level only.

The SAPT2 level of theory takes much less time than the full SAPT calculation and therefore it is recommended for large systems. If still faster calculations are required, the correction tEind(22) can be omitted, as it is usually fairly small.

The corrections Eind,resp(20), Eexch-ind,resp(20), Eelst,resp(12), and Eelst,resp(13) can also be computed in non-response versions, but these forms are not recommended and are not calculated unless explicitly requested.

In case the sum Edisp(20) + ϵdisp(2)(2) is not converged well enough, the CCD+ST(CCD) approach developed in Ref. 39 is also available in sapt2012. In this method, first, the dispersion energy is approximated at a level corresponding to the dimer CCD calculation. The energy Edisp(2)(CCD) obtained in this way can be shown [39] to contain the full corrections Edisp(20) and Edisp(21) and the so-called “DQ” part of Edisp(22). Next, to take into account the remaining, “S” and “T” contributions to Edisp(22), the expressions for these contributions (Eqs. (91) and (98), respectively, of Ref. 48) are evaluated with the converged CCD dispersion amplitudes replacing the first-order ones [hence the name CCD+ST(CCD)].

The corrections listed above constitute the set typically used in SAPT calculations. Recently, it has become possible to calculate also the corrections of the third order in V and zeroth order in W [38],

 (30) (30) (30) (30) (30) (30) (30) E SAPT = Eind + Eind-disp + Edisp + E exch-ind + Eexch- ind-disp + E exch-disp.

The first and fourth of these corrections constitute a part of the δEint,respHF quantity, however, for some systems it is advantageous to replace δEint,respHF by the sum Eind(30) + Eexch-ind(30) [38] or by their response versions [35]. Note that the third-order polarization and exchange corrections tend to cancel each other to a large extent, and one should not include any part of ERS(30) without including the corresponding exchange correction.

A few other corrections have been developed by the authors of SAPT but these are either not working in the current version of the program or for some other reasons are not recommended to be computed. These corrections include in particular various parts of Eelst,resp(14) [49].

The theory presented above was restricted to SAPT(MP/CC) for dimers. The SAPT(DFT) approach is actually simpler since the intramonomer correlation effects are accounted for by DFT and the only operator is V . The SAPT approach is quite similar for trimers, except that there is a total of six perturbation operators: V AB, V AC, V BC, WA, WB, WC in SAPT(MP/CC) and the three former operators in SAPT(DFT).

At intermonomer separations R large enough for the exchange effects to be negligible, the SAPT results become identical to those of the regular Rayleigh-Schrödinger perturbation theory. The calculation of the interaction energies in this region can be substantially simplified by neglecting the overlap effects and expanding V in the multipole series. The long-range part of the interaction energy becomes then expressed as a power series in R-1, with coefficients that can be obtained using only monomer properties (viz. multipole moments and polarizabilities). These monomer properties can be calculated ab initio at the correlation level consistent with finite-R SAPT calculations [5051] using the monomer parts of the basis set and the polcor suite of codes developed by Wormer and Hettema [3334] and distributed as a part of the package asymp_sapt.