Interaction effective pair-wise

In Equation (10), Vc represents a hard-core repulsion that is entropic in nature since it is linearly dependent on temperature in the expression for energy. Repulsion is generally associated with enthalpic interactions and we can consider the effect of an enthalpic interaction. Since Vc is associated with a single Kuhn unit we consider the average enthalpy of interaction per pair-wise interaction and the number of pair-wise interactions per Kuhn unit,... 

In the vast majority of MD applications a further simplification is made by using effective pair-wise additive potentials for atomic interactions. In simulations which contain flexible molecules, it is common practice to add terms which represent chemical bonds, bond angles, improper torsions and dihedrals. Interactions between atoms of molecules are represented by effective pair-wise additive potentials. This empirical approach splits the total potential energy of the system into a bonded (inter-molecule) and non-bonded (intra-molecular) part. 

The second example concerns the lithium ion, either considered in a cluster of water molecules or in aqueous solution. The idealized solution at infinite dilution of a lithium ion (without counter-ion) predicts six molecules of water in the first solvation shell if one uses pair-wise 2-body interactions, but the same type of computation predicts four molecules of water when 3-body effects are included. The computations were performed at room temperature. We have performed cluster computations for the Li fTO), system, with n = 1,2,3,4,5 and 6, using a density functional program developed in our laboratory. When we compute the most stable configuration for the pentamer complex Li+( starting from the most stable config-... 

When discussing the relative weights of the various types of interactions, simultaneously and not independently at work in a stable solid, it should be kept in mind that stabilizing interactions can be effective even in the presence of repulsive forces, as well as the opposite, namely destabilizing interactions may be observed in the presence of attractive forces. An appreciation of this conceptual distinction is crucial to the understanding of the effect of ionic charges on the nature of pair-wise non-covalent interactions. Particularly when the ions carry the same charge. 

Accordingly, a common further approximation for calculating the molecular orbitals tp, is to replace the Hel term containing the pair-wise electron-electron interactions by an effective potential that treats the interaction of a given electron with all other electrons in an average way. The resultant operator includes the one-electron terms of Hel ... 

The interpretation of the pharmacokinetic variables Cmax, AUCs and MRT of insulin glulisine was based on 95 % confidence intervals, after ln-transformation of the data. These 95 % confidence intervals were calculated for the respective mean ratios of pair-wise treatment comparisons. In addition, the test treatment was compared to the reference treatment with respect to the pharmacokinetic variables using an ANOVA with subject, treatment and period effects, after ln-transformation of the data. The subject sum of squares was partitioned to give a term for sequence (treatment by period interaction) and a term for subject within sequence (a residual term). Due to the explorative nature of the study, no adjustment of the a-level was made for the multiple testing procedure. 

The accuracy of the simulation results depends on a suitable choice of the parameters in the potential functions. On account of equation (23.1), an essential restraint of the calculation method is the pair-wise addition of atomic forces. Although effective pair potentials are used, three-body terms and interactions of higher order are neglected. Consequently, the major many-body contributions to the induced dipole interactions in aqueous ionic systems are not modelled accurately. A further simplification is a common application of... 

The first two terms in the right hand side (RHS) of Eq. (19.1) represent the combinatorial contribution to AGm which arises due to an increase in the number of possible chain configurations in the mixture relative to the pure components. The third term in the RHS of Eq. (19.1) represents noncombinatorial contributions to AGm- If we assume that this contribution arises from random, pair-wise contact between monomers, then it is proportional to a(1 4>a) and the x parameter is a measure of its strength. In this case X would depend only on T and would be independent of N, and However, additional constraints due to monomer architecture and connectivity, specific interactions, and finite compressibility can also give rise to noncombinatorial contributions. If the x parameter contains contributions from such effects then it may be a complicated function of Ni, A, and T. The expression for the combinatorial contribution was derived on the basis of several simplifications, and inadequacies of this expression are also Imnped into x-... 

Most classical simulations of molecular systems such as aqueous solutions employ pair-wise additive non-bonded potentials, which are often a combination of Lennard-Jones and coulombic interactions. When molecules possess internal degrees of freedom, additional intramolecular potentials such as bond, bond angle and dihedral angle potentials are typically employed. In aqueous ionic solutions, like sodium octanoate in water, there are strong many-body effects arising from the electrostatic polarization of the components in the system. A common... 

Figure 1 Pair-wise-interaction energy between spherical particles i and j in an electrostatically-stabilized dispersion. The potential Ujj(r) is plotted agairtst rja, where r is the centre-to-centre separation and a is the particle radius. Three regions of the potential are identified primary minimum (A), primary maximum (B), (height u J, and secondary minimum (C). The dotted line represents an effective hard-sphere potential...

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