Supermolecule method

The discussion of the preceding paragraphs clearly shows that indeed supermolecule method should be applied with great care. So we wish to end this section by saying that even though the supermolecule approach is conceptually very simple it cannot be used by simply running standard black box programs of quantum chemistry. 

It should be stressed that the analysis presented above is general, and applies to any system. However, for the majority of Van der Waals complexes the electrostatic term E will not be as important as it is for the CO dimer. On the other hand, this analysis shows that any supermolecule method should be applied with great care, and an understanding of the supermolecule results in terms of contributions as defined by the symmetry-adapted perturbation theory is necessary. 

At the outset of this type of calculation, the first question to be answered was whether the supermolecule method is liable to give reasonable H-bond energies, if any. The first CNDO/2 calculation yielded the values 5.8, 4.9 and 4.6 kcal/mole per hydrogen bond for the dimers IV, V and VI respectively, which was of a satisfactory order of magnitude. 

Above we referred to the development of the CC method by Cizek and Paldus [49-51], The CC method may be viewed as a consistent summation to infinite order of certain type of linked correlation (MBPT, MP) diagrams. Thus, there is a clear relationship between many-body permrbation theory [based on the MP operator of Eq. (Al) in Appendix 37A] and coupled cluster theory. Both are supermolecule methods that give size-extensive energies. 

The problem becomes more complex when studying solid phases because the microscopic NLO responses do not provide the full information about their macroscopic coimterparts, the second- and third-order nonlinear susceptibilities, and To make the transition between the microscopic and macroscopic, it is necessary to know the structure of the condensed phases as well as the nature and the effects of the intermolecular interactions in the bulk of the material. In both the Physics and Chemistry arena, several schemes have been proposed to characterize the NLO responses of solid phases. One of the authors has recently contributed to review these approaches [3] of which one of the extremes is occupied by the oriented gas approximation that consists in performing a tensor sum of the microscopic NLO properties to obtain the macroscopic responses of the crystal. The other extreme consists in performing a complete treatment of the solid by using the supermolecule method or by taking advantage of the spatial periodicity in crystal orbital calculations. In between these techniques, one finds the interaction schemes and the semi-empirical approaches. 

Another class of reactions whose understanding may require the inclusion of quantum effects consists of proton transfer reactions. The light mass of the proton indicates that such quantum effects might be quite important, but there have been attempts to simulate this process purely classically (primarily in the gas phase). An interesting method that lies in between gas phase calculations and full solution phase molecular dynamics is the supermolecule method used by Nagaoka et al. to calculate the dynamics of formamidine in water solvent. This system is quite interesting from the perspective of solution reaction dynamics because the transition state for this reaction incorporates a water molecule from the solvent. The overall process consists of two proton transfers, one from the formamidine molecule to the solvent water molecule and another one from the other end of the solvent water molecule back to the formamidine. 

Most popular in the ab initio calculation of intermolecular potentials is the so-called supermolecule method, because it allows the use of standard computer programs for electronic structure calculations. This method automatically includes all the electrostatic, penetration and exchange effects. If the calculations are performed at the SCF (self-consistent field) level the induction effects are included, too, but the dispersion energy is not. The latter, which is an intermolecular electron correlation effect, can be obtained by configuration interaction (Cl), coupled cluster (CC) calculations or many-body perturbation theory (MBPT). These calculations are all plagued... 

The need for size-consistency is an illustration of a fundamental difficulty of the supermolecule method. For a single system the variation principle ensures that the better the calculation, the better the energy. When we are interested in energy differences, however, the variational principle does not apply. In fact, there is a very serious source of error that can be attributed to the variational principle itself. 

As noted when introducing the Boys-Bernardi CP correction above, BSSE is a concern whenever the supermolecule method is used to compare the energies of fragments to the energy of the entire cluster [i.e., when computing the dissociation (Dg) or interaction (Tint) energy]. 

The supermolecule method is widely used for calculations of the intermolecular interaction energy. The total interaction energy ( totai) is calculated as the difference between the energy of the dimer [ (AB)j and the sum of the energies of monomers [E(A) and (B)j as shown in Eq. 1 ... 

The calculated interaction energy for the supermolecule method includes basis set superposition error (BSSE) [18]. The BSSE is corrected by the counterpoise method [19]. The energies of both dimer and monomers are calculated using the dimer s basis set in the counterpoise correction. The correction of BSSE is essential for accurate evaluation of weak intermolecular interactions, as the BSSE correction significantly changes the size of the calculated interaction energy. The effects of BSSE on the calculated interaction energy of the benzene dimer are illustrated in Fig. 1. 

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