Two-Body Interaction

The surface properties of metals are such that the surface tends to relax inwards bu systems described by two-body interactions tend to relax outwards. 

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). 

Comparatively little space will therefore be devoted to some rather recent approaches, such as the plasma model of Bohm and Pines, the two-body interaction method developed by Brueckner in connection with nuclear theory, Daudel s loge theory, and the method of variation of the second-order density matrix. This does not mean that these methods would be less powerful or less impor-... 

In the previous chapter we considered a rather simple solvent model, treating each solvent molecule as a Langevin-type dipole. Although this model represents the key solvent effects, it is important to examine more realistic models that include explicitly all the solvent atoms. In principle, we should adopt a model where both the solvent and the solute atoms are treated quantum mechanically. Such a model, however, is entirely impractical for studying large molecules in solution. Furthermore, we are interested here in the effect of the solvent on the solute potential surface and not in quantum mechanical effects of the pure solvent. Fortunately, the contributions to the Born-Oppenheimer potential surface that describe the solvent-solvent and solute-solvent interactions can be approximated by some type of analytical potential functions (rather than by the actual solution of the Schrodinger equation for the entire solute-solvent system). For example, the simplest way to describe the potential surface of a collection of water molecules is to represent it as a sum of two-body interactions (the interac-... 

The above preliminary comments are presented in order to appreciate problems which one faces in attempting to use quantum mechanics to obtain the interaction energy of two molecules of water or, differently stated, to obtain the two-body interaction potential. 

Beyond Two-Body Interaction Fragment-Localized Kohn-Sham Orbitals via a Singles-CI Procedure... 

From this point of view it is of interest to examine the consequences of full ther-malization of the classical Drude oscillators on the properties of the system. This is particularly important given the fact that any classical fluctuations of the Drude oscillators are a priori unphysical according to the Bom-Oppenheimer approximation upon which electronic induction models are based. It has been shown [12] that under the influence of thermalized (hot) fluctuating Drude oscillators the corrected effective energy of the system, truncated to two-body interactions is... 

As it will be explained in section 6, the usual way to evaluate the potential energy of a system simulated by Monte Carlo techniques, makes use of the pair potential approximation (although, as it will also be reviewed, several works have already appeared where nonadditivity corrections to the interaction potential have been included). In the pair potential approximation only two body interactions are taken into account. We will briefly explain here how to apply this approximation for the calculation of the potential energy, to the periodic system just described. The interaction potential energy under the pair potential approximation can be written as ... 

It should be noted that the results of VCS and BHF calculations using the same NN interaction disagree in several aspects. For instance for SNM the VCS and BHF calculations saturate at different values of density [5]. As for the SE using only two-body interactions the VCS approach yields a smaller value for 04 than BHF (see Table 1), and as a function of density in VCS the SE levels off at p 0.6 fm-3, whereas the BHF result continues to increase. Therefore it seems natural to ask whether the inclusion of more correlations by extending the BHF method (which is basically a mean field approximation) will lead to results closer to those of VCS. 

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) ... 

A related potential form, which was primarily developed to reproduce, structural energetics of silicon, was introduced by Tersoff and was based on ideas discussed by Abell . The binding energy in the AbeH-Tersoff expression is written as a sum of repulsive and attractive two-body interactions, with the attractive contribution being modified by a many-body term. 

One formalism which has been extensively used with classical trajectory methods to study gas-phase reactions has been the London-Eyring-Polanyi-Sato (LEPS) method . This is a semiempirical technique for generating potential energy surfaces which incorporates two-body interactions into a valence bond scheme. The combination of interactions for diatomic molecules in this formalism results in a many-body potential which displays correct asymptotic behavior, and which contains barriers for reaction. For the case of a diatomic molecule reacting with a surface, the surface is treated as one body of a three-body reaction, and so the two-body terms are composed of two atom-surface interactions and a gas-phase atom-atom potential. The LEPS formalism then introduces adjustable potential energy barriers into molecule-surface reactions. 

The substrate in these studies was restricted to be rigid, and Morse functions were used for the hydrogen-surface and two-body interactions. The parameters in the Morse functions were determined for single hydrogen atoms adsorbed on the tungsten surface by fitting to extended Huckel molecular orbital (EHMO) results, and the H2 Morse parameters were fit to gas-phase data. The Sato parameter, which enters the many-body LEPS prescription, was varied to produce a potential barrier for the desorption of H2 from the surface which matched experimental results. 

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... 

The procedures are similar to those used for the ground state energy. A general static external potential is treated in perturbation theory and the expansions are rearranged and resummed. The unperturbed system is of uniform density and fully extended and the thermodynamic limit is taken at the outset. Particular care is required to treat the chemical potential correctly. The result for D = 3 and arbitrary two-body interaction is [28,29]... 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

To gain an understanding of this mechanism, consider the Hamiltonian operator (H — Egl) with only two-body interactions, where Eg is the lowest energy for an A -particle system with Hamiltonian H and the identity operator I. Because Eg is the lowest (or ground-state) energy, the Hamiltonian operator is positive semi-definite on the A -electron space that is, the expectation values of H with respect to all A -particle functions are nonnegative. Assume that the Hamiltonian may be expanded as a sum of operators G,G,... 

Our discussion may readily be extended from 2-positivity to p-positivity. The class of Hamiltonians in Eq. (70) may be expanded by permitting the G, operators to be sums of products of p creation and/or annihilation operators for p > 2. If the p-RDM satisfies the p-positivity conditions, then expectation values of this expanded class of Hamiltonians with respect to the p-RDM will be nonnegative, and a variational RDM method for this class will yield exact energies. Geometrically, the convex set of 2-RDMs from p-positivity conditions for p > 2 is contained within the convex set of 2-RDMs from 2-positivity conditions. In general, the p-positivity conditions imply the (7-positivity conditions, where q < p. As a function of p, experience implies that, for Hamiltonians with two-body interactions, the positivity conditions converge rapidly to a computationally sufficient set of representability conditions [17]. 

The interaction energy between two molecules or molecular fragments is obtained as a sum over all pairwise atom-atom interactions. The atom-atom potential expressions implicitly assume that the interactions are two-body interactions, undisturbed by other bodies in the vicinity, and that they are isotropic about the atomic centers. 

So, Vij in Eq. (4) cannot be considered as a pure two-body interaction. But we can always represent the energy (4) as a sum of two-body interactions of isolated pairs and a remainder depending upon coordinates of three atoms,... 

This additional to the two-body interaction energies term originates from three-body interactions and is called the three-body interaction energy. 

There are many empirical and semi-empirical pair potentials which describe quite satisfactorily the properties of liquids and solids, see chapter 5 in book The parameters in these potentials are not real parameters of a true two-body interaction, their values depend upon properties of a medium. So these effective two-body potentials include nonadditive interactions through their parameters. The latter can not be directly related to the definite physical... 

Now we can decompose the m-body interaction energies, defined in previous section, into the PT series. For the two-body interaction energy, the expression directly follows from Eqs. (7), (8) and (13). 

The SMO-LMBPT method conveniently uses the transferability of the intracorrelated (one-body) parts of the monomers. This holds, according to our previous results [3-10], at the second (MP2), third (MP3) and fourth (MP4) level of correlation, respectively. The two-body terms (both dispersion and charge-transfer components) have also been already discussed for several systems [3-5]. A transferable property of the two-body interaction energy is valid in the studied He- and Ne-clusters, too [6]. In this work we focus also on the three-body effects which can be calculated in a rather straightforward way using the SMO-LMBPT formalism. 

The A scp term is calculated using the standard CP-method. At the correlated MP2 level, we have shown for several systems [7-10], that the AE terms are usually and systematically smaller than the dominant ( )+ Ecj) terms. The sum of these two terms provides a good approximation to the total interaction energy at the correlated level. It is important to emphasize that the AE values were obtained by making the difference with the values of the CP-corrected subsystems i.e. taking into consideration the "benefit effect" of the superposition of the basis set [3, 6]. As the charge-transfer components are of importance in the two-body interaction, (see a discussion in ref. 10), we will also investigate them separately for the three-body terms in the studied systems. 

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. 

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