Interaction energy, many-body

Table 10 Some of the correlated parts of the many-body interaction energy terms obtained for the studied systems using different basis sets. The values are given in Hartree. (Distances He-He = 5.67bohr, Ne-Ne = 5.0 bohr)...

II. Many-body interaction energies and correlation contributions in the PT formalism... 

The analytical expressions for many-body interaction energies are defined in a recurrence manner ... 

Relative contributions of electron correlation into many-body interaction energies for Be v and Li r clusters and 7 are defined by Eqs. (38) and... 

Figure 5.2 Structures and many-body interaction energies (kcal/mol) in ...

The monomer energies from Tables 1 and 2 have been used to determine the Edist values (Eq. 12]) shown in Table 4. (Again, a conversion factor of lEh 2625.5kJmor has been adopted.) For these symmetric cyclic (HF) n = 3 — 5) clusters, Edist is simply n x (E[HF ] — [HF]) = x (—Erlx). The many-body interaction energy can then be calculated from i t and Edist via Fq. [17]. Recall that within the RMA, Edist = 0 so that in the bottom half of Table 4 is the same as Ejnt. 

The 2-body through 5-body contributions to the many-body interaction energy shown in Table 4 are relatively simple to compute because there are only a few symmetry-unique terms. As mentioned earlier, there exist at most two unique 2-body energies [in (HF)4 and (HF)5j and two unique 3-body energies [in (HFlsj. Furthermore, all monomers in a given cluster are identical, and the cor-... 

Xantheas, S. S. (1996a). Singificance of higher-order many-body interaction energy terms in water clusters and bulk water. Philosophical Magazine Part B, 73,107-115. 

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

Molecular dynamics calculations have been performed (35-38). One ab initio calculation (39) is particularly interesting because it avoids the use of pairwise potential energy functions and effectively includes many-body interactions. It was concluded that the structure of the first hydration shell is nearly tetrahedral but is very much influenced by its own solvation. 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

It is evident that for Bes and 804 clusters the 3-body energy is not only the dominant term of the m-body decomposition but it is the single stabilization factor. The extremely large magnitude of Q 3(2,3) for Bes does not follow from physics of many-body interactions. It is due to the almost zero value of because the equilibrium distance in the Bea triangle is located in... 

Bc3 cluster the 3-body forces cannot be approximated solely by the Axilrod-Teller term. The reasons for the satisfactory approximation of many-body energy by the Axilrod-Teller term in the bulk phases of the rare gases were discussed by Meath and Aziz . As follows from precise calculations of the 3-body interaction energy in the Hcg , Neg and Ara trimers, both the Axilrod-Teller and the exchange energies are important. Nevertheless, in some studies of many-body interactions, the exchange effects are still neglected and the many-body contribution is approximated by only dispersion terms, for example see... 

The so-called Hartree-Fock (HF) limit is important both conceptually and quantitatively in the quantum mechanical theory of many-body interactions. It is based upon the approximation in which one considers each particle as moving in an effective potential obtained by averaging over the positions of all other particles. The best energy calculated from a wavefunction having this physical significance is called the Hartree-Fock energy and the difference between this and the exact solution of the non-relativistic wave equation is called the correlation energy. 

Inspection of Fig. 1(c) reveals that there are a few pairs of atoms with a preferred distance. Analysis of many such images in terms of site occupation probabilities as a function of adatom distances revealed significant deviations from a random distance distribution, and the existence of adsorbate interactions which indeed oscillate with a wave vector of 2kp [16]. The decay followed the l/r2-prediction only for large distances, while significant deviations were observed at distances below 20 A and interpreted as a shortcoming of theory [16]. However, an independent study, carried out in parallel, focused on two body interactions only, i.e., the authors counted only those distances r from a selected atom to a nearby atom where no third scatterer (adatom or impurity) was closer than r [17]. This way, many body interactions were eliminated and the interaction energy E(r) yielded perfect... 

The application of perturbation theory to many-body interactions leads to pairwise-additive and non-pairwise-additive contributions. For example, in the case of neutral, spherically symmetric systems which are separated by distances such that the orbital overlap can be neglected, the first non-pairwise-additive term appears at third order of the Rayleigh-Schrodinger perturbation treatment and corresponds to the dispersion energy which results from the induced-dipole-induced-dipole-induced-dipole78 interaction... 

In Sections 17.2 and 17.3, we have reviewed the QM/MM approach based on the real-space grids [40,41,58,59,60,61,62] and the novel theory of solutions [14,15,16], respectively. As has been suggested, the theory of energy representation is readily applicable to a solute that is quantum chemically described. The present section is devoted to the details of the methodology, referred to as QM/MM-ER, developed by combining the QM/MM approach with the theory of energy representation [19]. The point of the method is to divide the total solvation free energy into the contributions due to the pairwise additive interaction between the solute and the solvent and the residual contribution due to the electron density fluctuation. A focus will be placed on the treatment of the many-body interaction inherent in the quantum chemical object. 

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