Higher body interactions

It is convenient to discuss the three- and higher body interaction energies with the clusters of water molecules because for these systems a series of ab initio calculations was reported. The first of... 

Semi-empirical Hamiltonians and operators are taken to be state independent [56] and have the same Hermiticity as their true counterparts. Consequently, the valence shell effective Hamiltonians and operators they mimic must also have these two properties. Table I shows that the effective Hani iltonian and operator definitions H and A, as well as H and either A or a fulfill these criteria. Thus, these definition pairs may be used to derive the valence shell effective Hamiltonians and operators mimicked by the semi-empirical methods. Table III indicates that the commutation relation (4.12) is preserved by all three definition pairs. Hence, the validity of the relations derived from the semi-empirical version of (4.12) depends on the extent to which the semi-empirical Hamiltonians and operators actually mimic, respectively, exact valence shell effective Hamiltonians and operators. In particular, the latter Hamiltonians and operators contain higher-body terms which are neglected, or ignored, in semi-empirical theories. These nonclassical higher body interactions have been shown to be nonnegligible for the valence shell Hamiltonians of many atoms and molecules [27, 145-149] and for the dipole moment operators of some small molecules [56-58]. There is no a... 

These data can be used in conjunction with high-level ah initio quantum mechanical calculations to identify the source and magnitude of contributions to two- and higher-body interaction.An accurate reflection of these types of interactions in model potentials is essential for computationally efficient methods of determining complex interactions in larger cluster systems. 

In contrast to SW approach, which is originally derived by expanding the total potential energy into contributions from one, two, three, and higher-body interactions, the Tersoff [88Terl] potential abWons the n-body form and instead expresses potential in nominally two-body form ... 

The RIS model (1.57) gives Coo = 3.1, but the experimental value is C o = 6.7, almost twice as large. This discrepancy is attributed to the effect of two-body and higher body interactions. 

Such cancellation effects have been seen for higher-body interactions in CS2 (9b,32). 

Here Ut is the potential of atom i due to some external field (such as an applied electric field or due to confinement in a finite domain such as within the walls of a nanotube). The i-j pair interactions describe the force between atoms i and j in the absence of all other atoms. This however is only an approximation to the true force, because the presence of a third atom in the vicinity would generally cause some distortion in the electron cloud around atoms 1 and 2 and therefore change the interaction force. Such three and higher body interactions are however relatively small and are usually neglected. That is, the series Eq. (26) is usually truncated at the second term. We now discuss the most common types of pair interactions. 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

In applying this equation to multi-solute systems, the ionic concentrations are of sufficient magnitude that molecule-ion and ion-ion interactions must be considered. Edwards et al. (6) used a method proposed by Bromley (J7) for the estimation of the B parameters. The model was found to be useful for the calculation of multi-solute equilibria in the NH3+H5S+H2O and NH3+CO2+H2O systems. However, because of the assumptions regarding the activity of the water and the use of only two-body interaction parameters, the model is suitable only up to molecular concentrations of about 2 molal. As well the temperature was restricted to the range 0° to 100 oc because of the equations used for the Henry1s constants and the dissociation constants. In a later study, Edwards et al. (8) extended the correlation to higher concentrations (up to 10 - 20 molal) and higher temperatures (0° to 170 °C). In this work the activity coefficients of the electrolytes were calculated from an expression due to Pitzer (9) ... 

We do not have to consider the projections of the CCSD equations on higher-than-hextuply excited configurations, since for Hamiltonians containing up to two-body interactions the 2 (2) moments with k> 6 vanish. Once the generalized moments of the CCSD equations are known, we can define the quantities M (2) d>), using eq (18), to calculate the non-iterative MMCC correction... 

With increasing density, N-body interactions with N = 3, 4,. .. may have a discernible effect on the total intensities as well as on the shape of the absorption profile. One may expect a ternary component, and at higher densities perhaps four-body, etc., spectral components that are superimposed with the binary spectrum. At the highest densities (e.g., liquids and pressurized fluids) every monomer may be assumed in permanent interaction with a substantial number of near neighbors. At intermediate densities, that is well below liquid densities, one may be... 

Above we have stated that over a substantial range of gas densities, essential parts of the profiles of collision-induced absorption spectra are invariant if normalized by density squared, a/q2, in pure gases, or by the product of densities, cl/q Q2, in mixed gases. Induced spectra that show this density-squared dependence may be considered to be of a binary origin. Above, we have seen examples that at very low frequencies many-body effects may cause deviations from the density-squared behavior at any pressure, over a limited frequency band near zero frequency (intercol-lisional effect). Furthermore, with increasing densities, a diffuse N-body effect with N > 2 more or less affects most parts of the observable spectra. It is interesting to study in some detail how the three-body (and perhaps higher-order) interactions modify the binary profiles. 

Spectral moments may be computed from expressions such as Eqs. 5.15 or 5.16. Furthermore, the theory of virial expansions of the spectral moments has shown that we may consider two- and three-body systems, without regard to the actual number of atoms contained in a sample if gas densities are not too high. Near the low-density limit, if mixtures of non-polar gases well above the liquefaction point are considered, a nearly pure binary spectrum may be expected (except near zero frequencies, where the intercollisional process generates a relatively sharp absorption dip due to many-body interactions.) In this subsection, we will sketch the computations necessary for the actual evaluation of the binary moments of low order, especially Eqs. 5.19 and 5.25, along with some higher moments. 

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