Many-body expansion

Hermann, A., Rrawczyk, R.P., Lein, M., Schwerdtfeger, P Hamilton, I.P. and Stewart, J.J.P. (2007) Convergence of the many-body expansion of interaction potentials From van der Waals to covalent and metallic systems. Physical Review A, 76, 013202-1-013202-10. 

Sorbie and Murrell have developed a function for triatomic systems based on a many body expansion of PE. It has been applied to repulsive potentials as well and has become an attractive option for fitting ah initio PES. It has sufficient flexibility and extendable to higher polyatomic systems and to two-valued surfaces. The potential for triatomic system is decomposed into one-, two- and three-body terms. The potential is given as... 

A covenient starting point for developing covalent interactions is to write the energy as a many-body expansion of the form... 

While these potentials have been successful in modeling dynamic processes on silicon surfaces, the many-body expansion as applied in this case suffers from several drawbacks. Because all of the three potentials above have been fit to properties of the crystalline silicon soUd, they implicitly assume tetrahedral bonding. Atoms on the surface of silicon are known to exhibit nontetrahedral hybridizations, and so the results for surfaces are at best uncertain. Also, none of these potentials reproduce accurately the properties of the Sij diatomic molecule. This again inhibits a complete description of surface reactions. 

Eqs (46) and (49) are the basic equations of the ECC theory described in ref 124. The approximate ECC methods, such as ECCSD, are obtained by truncating the many-body expansions of cluster operators T and Z at some excitation level < N. so that T is replaced by eq (4), and Z is replaced by... 

Finally, we remark that, although the in the present study attention has been focussed on the excited states of some simple atomic systems, the method described here can be applied to molecular systems where they can be expected to support high accuracy as well as providing a suitable reference with respect to which well-founded many-body expansions for correlation effects can be developed. 

An expression of the type (7.101), which gives the bond order explicitly in terms of the positions of the neighbouring atoms, is called a bond order potential (BOP). Angularly dependent bond order potentials were first derived heuristically for the elemental semiconductors by TersofF (1988). We will see in the next chapter that a many-body expansion for the bond order may be derived exactly within the model. 

S.2.a. Many-body expansion to all orders, at all separations, no retardation... 

Topographical Aspects, Calculation, and Functional Representation via a Double Many-Body Expansion Method... 

In principle, the most general representation of the matrix elements of equation (68) includes a many-body expansion with one-, two-, and three-body terms. However, only two conditions [(a) and (b) in equations (66), (67)] may be used in the determination of the one- and two-body energy terms. If more conditions were to exist, the dimension of the matrix necessary to represent the adiabatic potential would be larger.16 In what follows we shall argue that the definition of V12 must depend on the type of conical intersection, namely on whether its locus is finite or infinite in extent (Section II.C). For the cases of HzO and 03 [equations (66), (67)] the conical intersection occurs along the C projection line, but only for finite values of the molecular perimeter (for H20, there is also a simple intersection at the O + H2 asymptotic channel, which is avoided for finite values of the OH distances117). For example, one gets for the linear dissociation of 03... 

The most general many-body expansion of the potential for the ground doublet state of an s3 system may, in principle, be obtained by adding a three-body term to the diagonal and nondiagonal elements of the hamiltonian matrix (75), say K(31), and V ... 

The partitioning of the potential energy into EHF and corr contributions proves to be most useful when used in conjunction with the many-body expansion in what has been called the double many-body expansion (DM BE).17-21 The three first terms are... 

Pastrana, M.R., Quintales, LA.M., Brandao, J., and Vatandas, A.J.C. (1990) Reeahbration of a single-valued double many-body expansion potential energy surface for ground-state HO2, J. Phys. Chan. 94, 8073-8080. 

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