Many-body decomposition

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 many-body decomposition of the interaction energy at different approximation is performed according to the general definitions, see Refs [36, 6]. For trimers we have only 2- and 3-body interaction energies. In the ho-moatomic case they are represented by the following formulae ... 

A more detailed analysis of the nature of binding is based on the many-body decomposition of the interaction energy... 

Table V Interaction energy and the many-body decomposition at the equilibrium geometry for the C3 symmetry, in kcal/mol. 

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),... 

By expanding A2E, A E,..., A E, the expressions for the components of the cluster energy (and therefore interaction energy via Eq. [10] and [13]) can be simplified. In the following form, it is easier to see a connection between this many-body decomposition and the inclusion-exclusion principle (also known as the sieve principle) from combinatorial mathematics ... 

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). 

CP-corrected many-body decomposition. However, that would increase the time of the computations for this tutorial substantially. 

Mala RA, Cabral BJC (2010) QM/MM approaches to the electromc spectra of hydrogenbonding systems with connection to many-body decomposition schemes. In Sabin JR, Brandas, E, Canute S (eds) Advances in quantum chemistry. Elsevier Acadtanic, San Diego... 

Intermolecular Energy decomposition analyses (EDA) are very useful approaches to calibrate force fields. Indeed, an evaluation of the different physical components of the interaction energy, especially of the many-body induction, is a key issue for the development of polarisable models. 

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... 

Using cumulant reconstruction functionals A3[Ai, A2] and A4[Ai, A2], one can certainly derive closed, nonlinear equations for the elements of Ai and A2, which could be solved using an iterative procedure that does not exploit the reconstruction functionals at each iteration. Of the RDM reconstruction functionals derived to date, several [7, 8, 11] utilize the cumulant decompositions in Eqs. (25c) and (25d) to obtain the unconnected portions of D3 and D4 exactly (in terms of the lower-order RDMs), then use many-body perturbation theory to estimate the connected parts A3 and A4 in terms of Aj and A2, the latter essentially serving as a renormalized pair interaction. Reconstruction functionals of this type are equally useful in solving ICSE(l) and ICSE(2), but the reconstruction functionals introduced by Valdemoro and co-workers [25, 26] cannot be used to solve the ICSEs because they contain no connected terms in D3 or D4 (and thus no contributions to A3 or A4). 

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. 

Many-body energy decomposition of the interaction energy of Be and Li at clusters, calculated at the MP4 level , in a.u.. 

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... 

J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, ]. Chem. Phys., 94, 4334 (1991). A Direct Product Decomposition Approach for Symmetry Exploitation in Many-Body Methods. I. Energy Calculations. 

The question now arises how all the above-mentioned phenomenologically classified interactions can be quantified. Of course, theory can yield unambiguous results if the additive decomposition of the overall interaction pattern into individual contributions is a suitable approximation in a certain case. It is, however, clear from the outset that many-body effects make a decomposition difficult, although this may be circumvented by a direct reference to the electronic wave function, which automatically adjusts to a given nuclear configuration, i.e. to a given arrangement of atoms. In Ref. [214], for example, an attempt is made to monitor the cooperative action of electrostatics in crown ether hydration via maps of the electrostatic potential. 

Direct Product Decomposition Approach for Symmetry Exploitation in Many-Body Methods. 1. Energy Calculations. 

Many bodies have the property of exerting on other bodies an action which is very different from chemical affinity. By means of this action they produce decomposition in bodies, and form new compounds into the composition of which they do notenter. This new power, hitherto unknown, is common both in organic and inorganic nature—/ shall call it catalytic power. I shall also call catalysis the decomposition of bodies by this force. 

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