The Supermolecule Approach

Interaction energies (Eim) on the basis of ab initio calculations are in general evaluated according to the so-called supermolecule approach  

E denotes on the right hand of the expression the total energy of the system specified. The interaction energy is by several orders of magnitude smaller than the individual total energies. Some consequences of this fact are discussed in the following sections. 

A large number of complexes of alkali metal, alkaline earth metal and ammonium cations has been studied using this approach (see Table 1). Much less effort has been made in the field of anions as guests. Some of these results are collected in Table 2. 

In case of complexes consisting of more than two constituents it has been shown that three-body terms are of significant magnitudes 120,121) i.e.  

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The simplest discrete approach is the solvaton method 65) which calculates above all the electrostatic interaction between the molecule and the solvent. The solvent is represented by a Active molecule built up from so-called solvatones. The most sophisticated discrete model is the supermolecule approach 661 in which the solvent molecules are included in the quantum chemical calculation as individual molecules. Here, information about the structure of the solvent cage and about the specific interactions between solvent and solute can be obtained. But this approach is connected with a great effort, because a lot of optimizations of geometry with ab initio calculations should be completed 67). A very simple supermolecule (CH3+ + 2 solvent molecules) was calculated with a semiempirical method in Ref.15). 

The continuum models represent a real alternative to the supermolecule approach. In this cases the solvation energy Esolv is assumed to be a sum of individual terms which can be calculated separately (see Eq. (6)). 

The basis set used for calculations were the STO-3G, the 3-21G, and the 6-31G basis sets, as implemented by the Gaussian-88 computer program. The energies of interaction were computed by using the supermolecule approach where the sum of the energies of the subsystems are substracted from the energy of the complex. All the species were geometry optimized. 

The alternative theoretical scheme for studying chemical reactivity in solution, the supermolecule approach, allows for the investigation of the solvation phenomena at a microscopic level. However, it does not enable the characterization of long-range bulk solvent forces moreover, the number of solvent molecules required to properly represent bulk solvation for a given solute can be so large that to perform a quantum chemical calculation in such a system becomes prohibitively expensive. ... 

The supermolecule approach was able to rationalize changes in the ligand-binding affinities for the 5-HT1A-R. The good linear correlations obtained through the Vin and Vdif indexes... 

A very similar QSAR approach for the modeling and prediction of selectivity of oq-AR antagonists has recently been carried out by Eric el al. It confirms the usefulness of the supermolecules approach and of the ad hoc shape descriptors in the rationalization of cq-ARs affinity and cq-AR subtype selectivity [98]. 

Cocchi, M. and De Benedetti, P.G. (1998) Use of the supermolecule approach to derive molecular similarity descriptors for QSAR analysis. Journal of Molecular Modelling, 4, 113-131. 

One point to address concerns the use of the words s pramolecular and supermolecule. The concept of supramolecular chemistry has become a unifying attractor, in which areas that have developed independently have spontaneously found their place. The word supramolecular has been used in particular for large multiprotein architectures and organized molecular assemblies [1.16]. On the other hand, in theoretical chemistry, the computational procedure that treats molecular associations such as the water dimer as a single entity is termed the supermolecule approach [1.34,1.35]. Taking into account the existence and the independent uses of these two words, one may then propose that supramolecular chemistry be the broader term, concerning the chemistry of all types of supramolecular entities from the well-defined supermolecules to extended, more or less organized, polymolecular associations. The term super molecular chemistry would be restricted to the specific chemistry of the supermolecules themselves. 

Space does not permit the inclusion of any of the papers dealing with hydrogen bonding188 or solvation phenomena,189 and the reader is referred to the above references for more details. It is clear, however, from recent work that it is feasible to include the interaction with several solvent molecules using the supermolecule approach, and very interesting and informative results have been obtained in this way. 

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. 

In the second family of approaches, explicit solvent molecules are placed around the gas phase stationary point structures. In some cases, the gas phase geometries are held constant and only the geometries and/or positions of the surrounding solvent molecules are optimized, and in other cases, the structure of the whole system (often called a supermolecule 32) is optimized. The supermolecule approach generally only involves explicit solvent molecules from the first (and occasionally second) solvation shell of the solute. 

Supermolecule model. By a "supermolecule" we imply a model consisting of the solute molecule surrounded by a certain number of solvent molecules. Pair complexes solute-solvent and solvent-solvent may be considered the simplest supermolecules. Since the cost of the supermolecule approach becomes prohibitive as the number of solvent molecules is increased, in most treatments only the first solvation shell is assumed. Such small clusters cannot of course provide a realistic model of a liquid but rather they give us a theoretical picture of what is referred as to "the solvation in the gas phase". As with the approach dealt with in the last paragraph, the ab initio calculation on the supermolecule should be followed by a statistical thermodynamic treatment. The use of the standard statistical thermodynamic is straightforward, in which case the supermolecule approach becomes e-quivalent to treatment of common chemical equilibria dealt with In Section 5.F. The calculations presented in Table 5.17 are just of this... 

In most cases, the solute consists of well-identified, neutral molecules that interact only weakly with each other. This means that reducing the solvent to consisting of just some few molecules around the solute may provide a reasonable approximation. In this case, the weak bonds i.e., hydrogen bonds or van der Waals bonds) to more distant solvent molecules are broken, but it is expected that this will lead to only insignificant electronic redistributions. Ultimately, the finite system consisting of the solute and the smaller number of solvent molecules can be treated with the accurate methods of the preceding subsection. This is the supermolecule approach. 

Since we are mainly interested in the properties of the solute, we may choose to use less accurate approaches for describing those part of the total electronic energy that originate from the solvent. Doing so we end up with a method that scales much more favourable with system size than the supermolecule approaches of section IIB. 

It came out immediately clear that the supermolecule approach cannot represent the method to be used in extensive studies of solvent effects. The computational costs increase in the ab initio versions with more than the fourth power of the number of basis set functions, at a given nuclear geometry of the supermolecule. Even more important it has been the recognition that, when the size of the solvation cluster exceeds some very low limits, the number of different nuclear conformations at an equivalent energy increases exponentially computational costs increase in parallel, and the introduction of thermal averages on these conformations becomes necessary. These facts, and some attempts to overcome them, are well summarized in a dementi s monograph (Clementi, 1976). The problem of multiple equivalent minima still plagues some discrete solvation models. 

This said for the role played by discrete models in the supermolecule approach in suggesting and supporting new versions of the continuum model, we may pass to consider other evolutions of models based on this approach. 

The interaction energy hypersurface for dimers (essentially described in a space with dimensionality < 6, being the internal monomer s geometry kept fixed) was used to get simplified analytical expression of AE(Rab) These simpler expressions were used either to reduce computational times in discrete models similar to the supermolecule approach, or in computer simulations of the liquid. 

An example of the first approach can be found in a clever paper by Noell and Morokuma (1976) (others will be discussed later), and one application of the second approach (probably the first application) in a paper by Popkie et al. (1973). The first approach belongs to the family of effective Hamiltonians (see Section 1) and introduces more chemistry in the description of the solute. However, it presents the same inconveniences of the supermolecule approach, in spite of the considerable reductions of computational times. In this Section we shall examine other more effective versions of this approach. The second approach has been largerly exploited in the following years. The Monte Carlo simulations of pure water given by Clementi and coworkers in the above quoted paper (Popkie et al., 1973) was followed by many other applications to solutions. The activities of dementi s group in this field during these initial years are well summarized in a monograph (Clementi, 1980). In the present Section we shall consider further extensions, more addressed to the study of chemical reactions. 

In this study, we have chosen the supermolecule approach and have used the semi-empirical quantum mechanical method called PCILO (Perturbative Configuration Interaction using Localized Orbitals) (16) to calculate intermolecular interactions. This method has recently been used successfully to calculate the intermolecular energies and geometries of hydrogen-bonded dimers of hydrocarbons and water (17,18). H-bonded complexes are particularly well characterized by this method (19). 

Among the few determinations of of molecular crystals, the CPHF/ INDO smdy of Yamada et al. [25] is unique because, on the one hand, it concerns an open-shell molecule, the p-nitrophenyl-nitronyl-nitroxide radical (p-NPNN) and, on the other hand, it combines in a hybrid way the oriented gas model and the supermolecule approach. Another smdy is due to Luo et al. [26], who calculated the third-order nonlinear susceptibility of amorphous thinmultilayered films of fullerenes by combining the self-consistent reaction field (SCRF) theory with cavity field factors. The amorphous namre of the system justifies the choice of the SCRF method, the removal of the sums in Eq. (3), and the use of the average second hyperpolarizability. They emphasized the differences between the Lorentz Lorenz local field factors and the more general Onsager Bbttcher ones. For Ceo the results differ by 25% but are in similar... 

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