Symmetry-adapted perturbation theory development

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 . 

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 ... 

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... 

Apart from these supramolecular approaches to the interaction energy, advanced perturbation theory techniques, which combine symmetry-adapted perturbation theory and DFT and are thereby able to capture dispersion interactions, are also under development [89-91]. 

Recently, a new theoretical method of calculating potential energy and dipole/polarizability surfaces for van der Waals molecules based on symmetry-adapted perturbation theory (sapt) of intermolecular forces (12)— (15) has been developed (16)-(24). In this method, referred to as many-body symmetry-adapted perturbation theory, all physically important contributions to the potential and the interaction-induced properties, such as electrostatics, exchange, induction, and dispersion are identified and computed separately. By making a perturbation expansion in the intermolecular interaction as well as in the intramolecular electronic correlation, it is possible to sum the correlation contributions to the different physical... 

The concept of calculating the interaction energy of two chemical systems A and B perturbatively is not at all a new idea. The first intermolecular perturbation expansion was proposed [22] just a few years after the foundations of quantum mechanics had been laid. Since then, numerous other expansions, now known under a common name of symmetry-adapted perturbation theory, have been introduced and the perturbation theory of intermolecular forces is now a fully mature approach. Thanks to the development of the many-body SAPT [23] and of a general-utility closed-shell SAPT computer code [24], the perturbative approach to intermolecular interactions has been successfully applied to construct PESs for numerous interacting dimers of theoretical and experimental interest [19-21,25-27]. One of the notable achievements of SAPT is an accurate description of the interactions between water molecules [21,28-32]. A recent paper by Keutsch et al. [33] compares the complete spectra of the water dimer with theoretical predictions obtained using an empirical potential fitted to extensive spectroscopic data, and with the predictions from a SAPT potential. These comparisons show that the latter potential probably provides the best current characterization of the water dimer force field. In another recent application, an SAPT PES for heUum in-... 

Clearly, standard Rayleigh-Schrodinger perturbation theory is not applicable and other perturbation methods have to be devised. Excellent surveys of the large and confusing variety of methods, usually called exchange perturbation theories , that have been developed are available [28, 65]. Here it is sufficient to note that the methods can be classified as either symmetric or symmetry-adapted . Symmetric methods start with antisymmetrized product functions in zeroth order and deal with the non-orthogonality problem in various ways. Symmetry-adapted methods start with non-antisymmetrized product functions and deal with the antisymmetry problem in some other way, such as antisymmetrization at each order of perturbation theory. 

A further difficulty arises because the exact waveflmctions of the isolated molecules are not known, except for one-electron systems. A common starting point is the Hartree-Fock wavefunctions of the individual molecules. It is then necessary to include the effects of intramolecular electron correlation by considering them as additional perturbations. Jeziorski and coworkers [66] have developed and computationally implemented a triple perturbation theory of the symmetry-adapted type. They have applied their method, dubbed SAPT, to many interactions with more success than might have been expected given the fundamental doubts [67] raised about the method. SAPT is currently both useful and practical. A recent application [68] to the CO2 dimer is illustrative of what can be achieved with SAPT, and a rich source of references to previous SAPT work. 

Charge transfer (CT) contributions are somewhat neglected by the formal elaborations of the perturbation theory. We know that they have an important role in the interpretation of many reactions (see. e.g. Fukui [17] and Klopman [18]) and represent an important factor in specific, albeit limited, classes of noncovalent interactions. The reason of this neglection of CT terms in the development of the symmetry-adapted version of the perturbation theory is due to the fact that attention has been focussed on small systems for which it was possible to consider the monomeric basis set as sufficient to describe all the aspects of the interaction. The definition of the charge-transfer contribution given by the K-M procedure is also open to criticism. An alternative decomposition scheme (Weinhold et al. [19]) in fact indicates this contribution in a modified version as one of the most important in determining the characteristics of non-covalent interactions. The Weinhold analysis in turn has been the object of several criticisms. In our opinion the occurrence of interpretative theories in competition is a positive fact science should benefit from this competition, unless one of the theories is decidedly inferior. 

In the present context, the term symmetry refers to the permutational symmetry of electrons, that is to the Pauh principle. This problem plays a central role in perturbation theories of intermolecular interactions since the antisymmetry of the perturbed wave functions has to be ensured. The symmetry-adapted PT is called also as exchange-PT , because the antisymmetry results exchange interactions between molecules A and B. Several formulations of the exchange-PT have been developed (Van der Avoird, 1967, Amos Musher 1967, Hirschfedler 1967, Murrel Shaw 1967, Salewicz Jeziorski 1979) which will not be discussed in detail. In the spirit of the present treatment, we shall focus on the application of second quantization to this problem. This formalism eo ipso guarantees the proper antisymmetry of any wave function expressed in terms of anticommuting fermion operators, thus the symmetry adaptation is done automatically and it does not require any further discussion. 

---