Components of the Interaction Energy

Because of its importance in the water dimer, as well as in a number of other H-bonded systems, the electrostatic interaction was partitioned into a multipole series in powers of 1/R, consisting of terms corresponding to interactions between dipole, quadrupole, and so 

Szczesniak et al. considered the factors leading to the degree of linearity of the H-bond in the water dimer and the pyramidalization of the proton acceptor oxygen. The dependence of the Hartree-Fock interaction energy was calculated as a function of both a and (3 (see earlier), as were the dispersion energy, and second-order Mpller-Plesset correlation 

An alternate approach, and one which is more commonly taken, is to partition the dispersion into interactions between atoms on the two. subunits. Probably the simplest example as.sumes an inverse sixth power dependence upon interatomic separation. 

The CjjY are parameters fit to each pair of atoms X and Y on molecules A and B, and is the distance separating these atoms. Szczesniak et al. ° fit these parameters by a least-squares method so as to reproduce their calculated dispersion as closely as possible. 

The reason for the discrepancy between the angular dependence of e can be traced by examining the distance-dependence of these two terms. At long distance the former remains negative while the latter becomes repulsive. This repulsive character results from another term that appears in the second-order correlation. Since the multipole moments of the water monomers are lower in the correlated wave function than SCF, the attractive electrostatic interaction becomes weaker when correlation is included. Hence, the correlation correction to the electrostatic interaction, shows up as a repulsive term. 

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Aguilar M A and Olivares del Valle F J 1989 A computation procedure for the dispersion component of the interaction energy in continuum solute solvent models Ohem. Rhys. 138 327-36... 

Intermolecular Energy decomposition analyses (EDA) are very useful approaches to calibrate force fields. Indeed, an evaluation of the different physical components of the interaction energy, especially of the many-body induction, is a key issue for the development of polarisable models. 

Table 6-2. Effect of dimer basis set on the components of the interaction energy...

In order to have an insight into the three-body effect,we continue the study of the He-clusters. Fortunately, there are published examples for several He-clusters, as cited above. All of these studies, however, were performed in the canonical representation. The use of the localized representation allows us to separate the dispersion and the charge transfer components of the interaction energy for the three-body effects as it was similarly done for the two-body effects. The calculation of the interaction energy in the SMO-LMBPT fiumework has been discussed in detail in several papers [8-10] The formulae given at the correlated level, however, were restricted to the two-body interaction. 

With the superscript R we indicate that the corresponding operator is related to the solvent reaction potential, and with the subscripts r and rr the one- or two-electron nature of the operator. The convention of summation over repeated indices followed by integration has been adopted, p is the electron density operator and is the operator which describes the two components of the interaction energy we have previously called t/en and f/ne. In more advanced formulations of continuum models going beyond the electrostatic description, other components are collected in this term. yR is sometimes called the solvent permanent potential, to emphasize the fact that in performing an iterative calculation of P > in the BO approximation this potential remains unchanged. 

In 1971 Morokuma258 proposed a simple partitioning of the Hartree-Fock interaction energy into some physically interpretable contributions, hopefully related to the components of the interaction energy as defined by SAPT. In this method one removes from the Fock matrix and from the energy expression the integrals (in the atomic basis) which are assumed to be unrelated to the considered type of... 

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... 

More sensitive to the level of theory is the vibrational component of the interaction energy. In the first place, the harmonic frequencies typically require rather high levels of theory for accurate evaluation. It has become part of conventional wisdom, for example, that these frequencies are routinely overestimated by 10% or so at the Hartree-Fock level, even with excellent basis sets. A second consideration arises from the weak nature of the H-bond-ing interaction itself. Whereas the harmonic approximation may be quite reasonable for the individual monomers, the high-amplitude intermolecular modes are subject to significant anharmonic effects. On the other hand, some of the errors made in the computation of vibrational frequencies in the separate monomers are likely to be canceled by errors of like magnitude in the complex. Errors of up to 1 kcal/mol might be expected in the combination of zero-point vibrational and thermal population energies under normal circumstances. The most effective means to reduce this error would be a more detailed analysis of the vibration-rotational motion of the complex that includes anharmonicity. 

The previous components of the interaction energy can be derived in the independent particle approximation and so appear within the context of Hartree-Fock level calculations. Nevertheless, inclusion of instantaneous correlation will affect these properties. Taking the electrostatic interaction as an example, the magnimde of this term, when computed at the SCF level, will of course be dependent on the SCF electron distributions. The correlated density will be different in certain respects, accounting for a different correlated electrostatic energy. The difference between the latter two quantities can be denoted by the correlation correction to the electrostatic energy. 

One of the problems of this technique lies in the mixing term which provides no physical insights. Furthermore, this term grows to uncomfortably large proportions when the interaction strengthens. It was also found that the various components of the interaction energy are even more sensitive to basis set choice than is the total interaction energy. The... 

Some of these points can be illustrated with simple examples. Table 1.4 lists the components of the interaction energy of the water dimer computed by the Morokuma-Kitaura scheme for the water dimer, taken in its experimental geometry, with an interoxygen separation of 2.98 The electrostatic term is clearly highly sensitive to the basis set chosen. 

Table 1.7 Comparison between Kitaura-Morokuma and perturbational components of the interaction energy of ------, nh a...

The authors go on to conclude that the red shift of the v, band in this H-bonded complex can be directly attributed to the lengthening of the Oj—H bond. By partitioning the interaction energy into various components, they show how the stretch of this bond makes it both more polar and polarizable, which in mrn, increases the induction and charge transfer components of the interaction energy. Although the authors did not include correlation in their treatment, the same could be said for dispersion energy which is directly related to polarizabilities of the individual monomers. It is for this reason that a nearly linear relation-.ship is observed between Av and Ar. Zilles and Person have reached a similar conclusion that the polarity and polarizability of the O—H bond increases upon formation of the H-... 

POL Polarization energy. The component of the interaction energy that results when the electric field of one subunit perturbs the electron density on its partner. Also abbreviated as PL. 

Having described the components of the interaction energies, let us consider a number of specific examples in detail (Table 4.1). Unlike the total interaction energy, which can be measured experimentally, the individual energy components cannot. The theoretical estimate of these quantities is often dependent on the method of calculation, but their qualitative features are usually independent of methodology. 

The second additivity problem concerns the question whether the interaction potential in Van der Waals trimers or multimers (or molecular solids or liquids) is a sum of pairwise intermolecular (A—B) potentials. This question can be considered for each component of the interaction energy. The (first order) electro-... 

Figure 3 The components of the interaction energy between pyridazine and methanol for a coplanar, linear N H—O hydrogen bond of length d(N H) = 1.85 A.

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