Supermolecular approach

For quantitative description of H-bond interactions in solution or in gas phase, it is a common practice to define the H-bond energy by intermolecular interaction energy. In the supermolecular approach, H-bonding energies are described by the free energy of the complexation reaction (3.1), which is estimated according to the following equation  

In summary, the supermolecular approach provides a reasonable good accuracy of H-bond strength prediction and could be used together with the experimental scales as a benchmark for descriptor-based predictions discussed in the following section. 

The energy necessary to overcome the barrier from the trap side is equal to 

In the supermolecular method the interaction energy is calculated from its definition (13.1) using any reliable method of electronic energy calculations. For the sake of brevity we will consider the interaction of two subsystems A and B. 

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Solvation effects on the molecular vibrations of 128 were studied by SCRF methods and by supermolecular approaches of 128 with one water molecule [97JPC(B) 10923, 98JPC(A)6010]. Correlations between the N—H (uracil) and O—H (water) bond elongations and the corresponding frequency shifts of the stretching vibrations are reported as... 

As is apparent from the above definitions, each of these effective matrices depend on basis sets and molecular orbitals of both fragments. It is also important to observe that these matrices possess a correct asymptotic behavior as at large interfragment distances they become the usual overlap and Fock matrices of the separate fragments, while the paired secular systems uncouple and converge to the separate Roothaan equations for the single monomers. Finally, as it is usual in a supermolecular approach, the interaction energy is expressed as... 

The major interest of this paper is the calculation of the polarizability, a, of the anion F in an aqueous solution. Using the supermolecular approach, we calculated for several configurations obtained from the simulation, the polarizability of F surrounded by the first solvation shell (n water molecules) and subtracted the polarizability of these n water molecules without the F, i.e.. 

The potential energy surface for HeBr2 complex is examined using the supermolecular approach. In a supermolecular calculation, the interaction energy between a pair of atoms or molecules, is given by... 

The supermolecular approach is used for the determination of the intermolecular energies, AE ... 

Nucleophilic vinylic substitutions of 4//-pyran-4-onc and 2,6-dimethyl-4//-pyran-4-one with a hydroxide ion in aqueous solution were calculated by the density functional theory (B3LYP) and ab initio (MP2) methods using the 6-31+G(d) and 6-31G (d) basis sets. The aqueous solution was modelled by a supermolecular approach, where 11 water molecules were involved in the reaction system. The calculations confirmed a different addition-elimination mechanism of the reaction compared with that in the gas phase or non-polar solution. Addition of OH- at the C(2) vinylic carbon of the pyranone ring with an activation barrier of 10-11 kcalmol-1 (B3LYP) has been identified as the rate-determining step, in good quantitative and qualitative agreement with experimental kinetics. Solvent effects increase the activation barrier of the addition step and, conversely, decrease the barrier of the elimination step.138... 

From the point of view of the Buckingham formula (Equation (2.23)) only the effect of long-range electrostatic and induction interactions crE of the solvent molecule with the reaction field is included in the traditional methods of the (n) group (continuum models). Contrary to that, the supermolecular approach (I) or combined MD/QM methods (III) includes the short-range term cra and the long-range crw and some of the [Pg.132]

Within the supermolecular approach, the total intermolecular interaction energy (AE) of a complex A---B is evaluated as the difference between the energy of the complex (EA" B) and the sum of the energies of its subsystems (EA, EB) ... 

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 ... 

The details of SAPT are beyond the scope of the present work. For our purposes it is enough to say that the fundamental components of the interaction energy are ordinarily expanded in terms of two perturbations the intermonomer interaction operator and the intramonomer electron correlation operator. Such a treatment provides us with fundamental components in the form of a double perturbation series, which should be judiciously limited to some low order, which produces a compromise between efficiency and accuracy. The most important corrections for two- and three-body terms in the interaction energy are described in Table 1. The SAPT corrections are directly related to the interaction energy evaluated by the supermolecular approach, Eq.(2), provided that many body perturbation theory (MBPT) is used [19,28]. Assignment of different perturbation and supermolecular energies is shown in Table 1. The power of this approach is its open-ended character. One can thoroughly analyse the role of individual corrections and evaluate them with carefully controlled effort and desired... 

In order to obtain more information on the solvation process Yang and Cui performed a so called natural energy decomposition analysis (NEDA) on monomethyl phosphate ester (MMP) solvated in water. They used a supermolecular approach where the solute plus a number of water molecules (up to 34) were treated quantum-mechanically. A further set of water molecules was treated with a force-field model. Their results indicate that there is a substantial charge transfer between the solute and the nearest solvent molecules. The interaction energy due to this transfer was found to amount to some 70-80% of that of the electric interactions. Since MMP forms hydrogen bonds with the water molecules, all results together suggests that for such a system it is important to include the nearest solvent molecules in the quantum-mechanical treatment, whereas a continuum approximation or a force field may not be sufficiently accurate. 

Another drawback of interaction energies obtained in the supermolecular approach is the so called basis-set-superposition-error (BSSE). This is a consequence of using a finite set of basis functions to describe the atomic orbitals the molecular orbitals are made of. A finite set of basis functions lead to an error. A, in the three energies that enter the rhs of equation (10). In fact, the BSSE can be defined as... 

There are several studies [15] in the literature that attempt to evaluate the BSSE for a given basis as the difference between the interaction energy obtained with that basis and with the largest possible for the system of interest (the Hartree-Fock limit). In some cases, as with the minimal basis sets, the BSSE can be much larger than the interaction energy [15,16] and the supermolecular approach might become risky. 

When used with methods that include electronic correlation, the supermolecular approach accounts for all interaction terms, charge transfer included. The value of the various contributions, however, is not separately known and decomposition procedures [25-28] have been proposed to this end, the most widely applied being due to Morokuma [25]. Generally speaking, these procedures rely on the calculation of expectation values of the energy in states described by eigenfunctions that are products of eigenfunctions of the separated species. 

The two main advantages of these methods compared to the supermolecular approach are that, first, the various contributions are separately know and their physical meaning is apparent, and, second, the results are not affected by BSSE. However, interaction energies computed this way are still rather expensive, either because they require very large basis sets with polarization functions or because they imply multiconfiguration calculations. 

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