N-body decomposition

Chapman RD (2007) Organic Difluoramine Derivatives. 125 123-151 Christie RA, Jordan KD (2005) n-Body Decomposition Approach to the Calculation of Interaction Energies of Water Clusters 116 27-41... 

In the n-body decomposition procedure, the total binding energy is written as... 

Interestingly, the n-body decomposition procedure also appears to be advantageous for deahng with the basis set superposition error problem. Specifically, calculations on small water clusters reveal that much of the error due to... 

Christie RA, Jordan KD (2005) -Body Decomposition Approach to the Calculation of Interaction Energies of Water Clusters 116 27-41 Clot E, Eisenstein O (2004) Agostic Interactions from a Computational Perspective One Name, Many Interpretations 113 1-36 Conley B, Atwood DA (2003) Fluoroaluminate Chemistry 104 181-193 Contreras RR, Su4rez T, Reyes M, Bellandi F, Cancines P, Moreno J, Shahgholi M, Di BUio AJ, Gray HB, Fontal B (2003) Electronic Structures and Reduction Potentials of Cu(II) Complexes of [N,N -Alkyl-bis(ethyl-2-amino-l-cyclopentenecarbothioate)] (Alkyl = Ethyl, Propyl, and Butyl) 106 71-79... 

As one can see, the non-iterative correction 8qA Eq. (30), is defined in terms of the n-body components of 3 with n > in, t. These components originate from the following decomposition of the operator. ... 

Mg , and Ca (n =2, 3) clusters at the Mpller-Plesset electron correlation level (MP4) and the SCF level, using a reasonably large basis set [6-311 + G(3df)]. The 2-and 3-body decompositions of the interaction energy at the MP4 and SCF levels, the NBO population analysis and the electron density difference maps allow to elucidate the nature of bonding in alkaline-earth clusters 2001 by Academic Press. 

Our analysis is based on accurate calculations performed in Ref. [22] at the Moller-Plesset electron correlation level of the interaction energy and its many-body decomposition for Be , Mgn, and Ca (n= 2 and 3) clusters using a reasonably large basis set [6-311 + G (3df)]. All calculations were also carried out at the SCF level which allowed to study separately the SCF and electron correlation contributions and give a physical analysis of each term in the dimer and trimer energy decompositions. 

The well-known " decomposition of the N-body potential into component (or n-point) potentials... 

The nonadditivity tends to increase for more strongly coupled systems (sometimes dramatically), and, consequently, the quality of the 2-body approximation deteriorates.In clusters of HF and/or H2O, the nonadditivity can account for more than half of i t, which necessarily implies that the error associated with the 2-body approximation can exceed 50%. This section of the tutorial will use (HF)3, (1TF)4, and (HFjs to demonstrate the procedure for calculating these 2-body interactions as well as higher order (3-body,. .. N-body) contributions via a many-body decomposition of Eim. 

The most common rigorous many-body decomposition scheme for weakly bound clusters is based upon the approach introduced by Hankins, Moskowitz, and Stillinger in 1970. Two lucid descriptions of the procedme can be found in Ref. 104 and 105. Technically, a many-body decomposition of Eint decomposes the energy of the cluster E /i/2 -/n] into 1-body (Ej),... 

Because the cyclic (HF) clusters n = 3 — 5) used in this tutorial are symmetric, the number of computations required to perform a many-body decomposition of the interaction energy is reduced dramatically. In general, application of the decomposition procedure to a pentamer could require as many as 25 additional calculations (4) = 5 for the tetramer subsets, (3) = 10 for the trimer subsets, and (2) = 10 for the dimer subsets. For (HF)5, however, symmetry reduces this to 5 calculations (1 unique tetramer computation, 2 unique trimer computations, and 2 unique dimer computations). 

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. 

However, if there is a lower enetgy decomposition pathway available, then an alternative degradation reaction dominates. There is a growing body of evidence to suggest that cydization reactions to form small stable ring compounds are one such decomposition pathway, especially for polyamides containing monomers with four to six carbon atoms (77,78) (eq. 7). The first example of this is the formation of cyclic amines, which is the principal decomposition pathway in nylon-4,6 (in eq. 7, n = 1 and R = H) (79) this has also been observed in MPMD-containing polyamides (in eq. 7, n = 2,... 

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