SAPT theory

The first-order energy in the SAPT theories is given by... 

SAPT theories are continuously refined and extended. Readers of this chapter surely are not interested to find here a synopsis of a very intricate subject that could be condensed into compact and elegant formulations, hard to decode, or expanded into long and complex sequences of formulas. This subject can be left to specialists, or to curious people, for which the above given references represent a good starting point. 

The various EDA techniques discussed so far were based on the variational principle. Another route to partition the INT is by employing the perturbation method. " Among the various perturbational tools used to analyze the individual energy components contributing to the stabilization of intermolecular complexes, many-body SAPT has received widespread attention.Jeziorski et al. have reviewed various aspects of wavefunction-based SAPT theory. 

The methods of electronic structure calculations can be divided into size extensive and size nonextensive ones. The latter methods should not be used to compute intermolecu-lar interactions since the errors resulting from violation of size extensivity are usually large compared to the interaction energies. The SAPT theory is size extensive since all interaction energy formulae can be given a connected diagrammatic representation. ... 

Obviously, the RS method is obtained from the iterative process expression defined by Eq. 11 with no symmetrization performed. The simplest SAPT theory with symmetrization, taking into account the exchange part of the interaction energy in finite order, is the SRS expansion formulated in [40]. hi the SRS method, the wave function corrections are taken directly from the RS theory, = V Rs > perturbation energies are calculated from the... 

It is quite instructive to look in more detail into the lowest-order energy corrections and their asymptotics for the SAPT theories discussed so far. For this purpose, we introduce a simplified notation (X) = (po X[Pg.56]

As a first step towards construction of a regularized SAPT theory that includes both Vp and Vt, let us note that if V p and Sp are known, the remaining part of the interaction energy can be recovered by means of a perturbation expansion in powers of Vt. Since this expansion does not influence the, already correct, asymptotics of 8, we can now employ a method that is convergent despite the presence of the Pauli-forbidden continuum, i.e. the EL-HAV, AM, or JK theory. Unlike the case of non-regularized expansions, there is no asymptotics-related reason to expect that the JK method will perform better than the other two. If we choose the ELHAV theory, the successive corrections to the energy and the wave function, referred to as the regularized ELHAV (R-ELHAV) corrections, are obtained from the formulae [40]... 

A fiirther diflfieulty arises beeause the exaet wavefiinetions of the isolated moleeules are not known, exeept for one-eleetron systems. A eoimnon starting point is the Hartree-Foek wavefiinetions of the individual moleeules. It is then neeessary to inelude the eflfeets of intramoleeular eleetron eorrelation by eonsidering them as additional perturbations. Jeziorski and eoworkers [M] have developed and eomputationally implemented a triple perturbation theory of the syimnetry-adapted type. They have applied their method, dubbed SAPT, to many interaetions with more sueeess than might have been expeeted given the fiindamental doubts raised about the method. SAPT is eurrently both usefiil and praetieal. A reeent applieation [ ] to the CO2 dimer is illustrative of what ean be aehieved widi SAPT, and a rieh soiiree of referenees to previous SAPT work. 

However, due to the availability of numerous techniques, it is important to point out here the differences and equivalence between schemes. To summarize, two EDA families can be applied to force field parametrization. The first EDA type of approach is labelled SAPT (Symmetry Adapted Perturbation Theory). It uses non orthogonal orbitals and recomputes the total interaction upon perturbation theory. As computations can be performed up to the Coupled-Cluster Singles Doubles (CCSD) level, SAPT can be seen as a reference method. However, due to the cost of the use of non-orthogonal molecular orbitals, pure SAPT approaches remain limited... 

As the exchange energy, the polarization-exchange energy (.poi-txch is also nonadditive. The standard PT cannot be applied to the calculation of the poi-exch- The reason is that the antisymmetrized functions of zeroth order (Ai/>o. ..) are not eigenfunctions of the unperturbed Hamiltonian Ho as long as the operator Ho does not commute with the antisymmetrizer operator A. Many successful approaches for the symmetry adapted perturbation theory (SAPT) have been developed for a detailed discussion see chapter 3 in book, the modern achievements in the SAPT are described in reviews . 

Binary Interaction-Induced Dipoles. Ab initio quantum chemical calculations of interaction-induced dipole surfaces are known for some time (Section 4.4, pp. 159 ff.) Such calculations were recently extended for the H2-He [17] and H2-H2 [18] systems, to account more closely for the dependencies of such data on the rotovibrational states of the H2 molecules [19]. New calculations of the kind and quality are now available also for the H-He system [20], the H2-H system [21], and the HD-He system [22]. New computational methods, called symmetry adapted perturbation theory (SAPT), were shown to be also successful for calculating interaction-induced dipole surfaces of such simple, binary van der Waals systems [23-26]. 

Contrary to the previously described supermolecular approach, perturbation theory treatment allows for the partition of the interaction energy into physically interpretable components. The most frequently used method for this purpose is symmetry-adapted perturbation theory (SAPT) [13]. More recently, great effort has also been invested in the development of DFT-SAPT [14-16], In the present contribution, we use the variational-perturbational scheme [17-20], In this approach, the intermolecular interaction energy components are determined based on the wave functions of the subsystems evaluated in the dimer-centered basis set. Thus, both interaction energy and its components are BSSE-free. More details about this scheme can be found elsewhere [21-23]. The total intermolecular interaction energy at the MP2 level of theory can be expressed as follows ... 

In principle, the theory reviewed in Sections 4-6 can be applied to interactions of arbitrary systems if the full configuration interaction (FCI) wave functions of the monomers are available, and if the matrix elements of H0 and V can be constructed in the space spanned by the products of the configuration state functions of the monomers. For the interactions of many-electron monomers the resulting perturbation equations are difficult to solve, however. A many-electron version of SAPT, which systematically treat the intramonomer correlation effects, offers a solution to this problem. 

A general approach to the intramonomer correlation problem is known as the many-electron (or many-body) SAPT method88,141 213-215. In this method the zeroth-order Hamiltonian H0 is decomposed as H0 = F + W, where F = FA + FB is the sum of the Fock operators, FA and FB, of monomer A and B, respectively, and W is the intramonomer correlation operator. The correlation operator can be written as W = WA + WB, where Wx = Hx — Fx, X = A or B. The total Hamiltonian can be now be represented as H = F + V + W. This partitioning of H defines a double perturbation expansion of the wave function and interaction energy. In the SRS theory the wave function is obtained by expanding the parametrized Schrodinger equation as a power series in and A,... 

Application of the conventional wave function approach in the symmetry-adapted perturbation theory (SAPT) has been shown to give very accurate description of the dispersion interaction and has provided intermolecular potentials which performed... 

A symmetry-adapted perturbation theory approach for the calculation of the Hartree-Fock interaction energies has been proposed by Jeziorska et al.105 for the helium dimer, and generalized to the many-electron case in Ref. (106). The authors of Refs. (105-106) developed a basis-set independent perturbation scheme to solve the Hartree-Fock equations for the dimer, and analyzed the Hartree-Fock interaction energy in terms of contributions related to many-electron SAPT reviewed in Section 7. Specifically, they proposed to replace the Hartree-Fock equations for the... 

It should be stressed that the relation between the SCRF and SAPT approaches is not obvious, as the former describes the solvation energetics in terms of the free energy of solvation at a finite temperature T, while in the latter one considers the interaction energy between the molecule of the solute and all molecules of the solvent at T = 0 K. One should also note that in the SCRF theory the solvent is modeled by a polarizable continuum, so the SCRF Hamiltonian is semiempirical. Still, by assuming a discrete equivalent of the SCRF Hamiltonian one can get approximate relations between SAPT and SCRF at T = 0 K. A SAPT analysis of the free energy of solvation AG within the SCRF method was reported in Ref. (234). It was shown that the free energy of solvation AG is given by,... 

All these comparisons between theory and various high precision experimental data show that the ab initio SAPT potential for He-C02 reproduces various physical properties of the complex. This suggests that the potential is accurate not only around the van der Waals well, but also its anisotropy in the attractive and repulsive regions, as well as the volume of the van der Waals well are correct. 

The SAPT potential for the He-C02 complex was also used in the calculations of the rovibrational spectra of the He -CC clusters 366. High resolution experimental data were also reported in this paper. Comparison of the theoretical and experimental effective rotational constants B and other spectroscopic characteristics as functions of the cluster size N is shown on Figure 1-9. Again, the agreement between the theory and experiment is impressive showing that theory can describe with trust spectroscopic characteristics of small clusters He -CO This especially true for the effective rotational constant and the frequency shift of the C02 vibration due to the solvation by the helium atoms. One may note in passing that the clusters HeA,-C02 with the number of helium atoms N around 20 do not exhibit all the properties of the C02 molecule in the first solvation shell of the (quantum) liquid helium at very low temperatures. 

Simulations of the liquid water properties have been the subject of many papers, see Ref. (374) for a review. Recently a two-body potential for the water dimer was computed by SAPT(DFT)375. Its accuracy was checked375 by comparison with the experimental second virial coefficients at various temperatures. As shown on Figure 1-16, the agreement between the theory and experiment is excellent. Given an accurate pair potential, and three-body terms computed by SAPT376, simulations of the radial 0-0, 0-H, and H-H distribution functions could be... 

Jeziorski B, Moszynski R, Ratkiewicz A, Rybak S, Szalewicz K, Williams HL (1993) SAPT A program for many-body symmetry-adapted perturbation theory calculations of intermolecular interaction energies. In Clementi E (ed) Methods and techniques in computational chemistry METECC-94, vol B. STEF, Cagliari, pp 79-129... 

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