Interaction energy, many-body expansion

Interaction Energy Represents the Non-additivity of the Total Energy Many-body Expansion of the Rigid Interaction Energy What Is Additive, and What Is Not ... 

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

Electronic structure calculations have recently become capable to provide interaction potentials for medium-size dimers that can be used to predict properties of such dimers with accuracy approaching, and in a few favorable cases even surpassing, experimental accuracies [1-4]. The dimer (pair) potentials are the basic building blocks of potential surfaces for larger clusters represented in the form of many-body expansions. Such expansions decompose each surface into intramonomer contributions, i.e. the potentials within single molecules (monomers), the pair potentials, and the so-called nonadditive potentials. Since derivatives of potential energy surfaces define forces, one may alternatively use the term force helds equivalently with potentials . 

Obviously, the many-body expansion of the interaction energy can be defined only when the quantum states of all subsystems can be unambiguously specified. For strongly interacting systems, such as metals or chemically bound molecules, this condition is not fulfilled and the suitability of the many-body expansion can be questioned. In most applications of Eq. (2), the number of molecules N will be fixed and therefore the index N in [A, Af] will be omitted from now on. 

Many-body Expansion of the Rigid Interaction Energy... 

The non-additivity is highlighted in what is called the many-body expansion of the interaction energy, where the interaction energy is expressed as the sum of two-body, three-body, etc. energy contributions. The body... 

We may conclude that in our (fictitious) example, at the given configuration, the many-body expansion of the interaction energy jnt = -10 kcal/mol represents a series decaying rather quickly A7 (2,4) = —17 kcal/mol for the two-body, A (3,4) = -1-5 for the three-body and A7 (4,4) = +2 for the four-body interactions. 

The pair potential functions for the description of the intermolecular interactions used in molecular simulations of aqueous systems can be grouped into two broad classes as far as their origin is concerned empirical and quantum mechanical potentials. In the first case, all parameters of a model are adjusted to fit experimental data for water from different sources, and thus necessarily incorporate effects of many-body interactions in some implicit average way. The second class of potentials, obtained from ab initio quantum mechanical calculations, represent purely the pair energy of the water dimer and they do not take into account any many-body effects. However, such potentials can be regarded as the first term in a systematic many-body expansion of the total quantum mechanical potential (dementi 1985 Famulari et al. 1998 Stem et al. 1999). 

In order to obtain a consistent higher-order operator, the first idea is to refrain from expansion in 1/c. This approach leads in most cases to energy-dependent or non-hermitian operators, which prevent the formulation of a set of mutually orthogonal zero-order solutions. These could in turn be used as the basis for methods making use of configuration interaction or many-body perturbation theory. ... 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

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