Additivity, pair-wise

The continuum electrostatic approach has been considered also appropriate for studying biomolecular systems when electronic polarisation effects, typically neglected by the additive pair-wise potentials commonly use in MD and MC, are dominant in relation to the conformational flexibility, and when changes in protonation states of tritable sites [193-198] or electron-proton coupling phenomena [199-204] occur. 

The initial cube measures 17.9 A along one edge, and contains 455 atoms. Accepting the approximation of pair-wise additivity of interactions, we are faced with the requirement for the evaluation of 455x454/2 - 103,285 pair interactions in order to once estimate the total energy of the system. Fortunately, only roughly 20% of these pairs interact significantly. 

Fig. 4.10. The left side of the scheme represents the starting situation of pure Zeeman splitting, as described by (4.48) and shown before in Fig. 4.9. In this example, the field B = (0,0,B), which defines the quantization axis, is chosen as the z-direction. The additional quadrupole interaction, as shown on the right side of Fig. 4.10, leads to a pair-wise shift of the Zeeman states with mj = 3/2 and mi = 1/2 up- and down-wards in opposite sense. In first order, all lines are shifted by the same energy as expected from the m/-dependence of the electric...

Despite these modifications there remain a number of well-documented problems with the AM1/PM3 core-repulsion function [37] which has resulted in further refinements. For example, Jorgensen and co-workers have developed the PDDG (pair-wise distance directed Gaussian) PM3 and MNDO methods which display improved accuracy over standard NDDO parameterisations [38], However, for methods which include d-orbitals (e.g. MNDO/d [23,24], AMl/d [25] and AMI [39,40]) it has been found that to obtain the correct balance between attractive and repulsive Coulomb interactions requires an additional adjustable parameter p (previously evaluated using the one-centre two-electron integral Gss, Eq. 5-7), which is used in the evaluation of the two-centre two-electron integrals (Eq. 5-8). 

In view of this and in line with the DFT-D approach described by Grimme [118], we have added an atom-atom pair-wise additive potential of the form Csemi-empirical energy [19-21] in order to account for dispersion effects [43], Thus the dispersion corrected semi-empirical energy ( Pm3-d) is now given by ... 

In a hard-sphere system, the trajectories of particles are determined by momentum conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. For not too dense systems, the hard-sphere models are considerably faster than the soft-sphere models. Note that the occurrence of multiple collisions at the same instant cannot be taken into account. 

Reductive elimination is simply the reverse reaction of oxidative addition the formal valence state of the metal is reduced by two (or one in a bimetallic reaction), and the total electron count of the complex is reduced by two. While oxidative addition can also be observed for main group elements, this reaction is more typical of the transition elements in particular the electronegative, noble metals. In a catalytic cycle the two reactions always occur pair-wise. In one step the oxidative addition occurs, followed for example by insertion reactions, and then the cycle is completed by a reductive elimination of the product. 

With a pairwise distribution of reagents it is impossible to find separately the parameters ve and ae from the kinetic curves in the majority of practically important situations. Only combinations of these parameters can be found. This is, first of all, connected with the fact that, for real systems, the form of the pair-wise distribution function is, as a rule, unknown. In addition, with the pair-wise distribution it is usually also impossible to determine the initial concentration n(0). 

An introduction to the modeling methods can be found in refs. [22,231. The classical MD simulations reported here were performed with the modified AMBER software/241 in which the potential energy consists of harmonic deformations of bond and angles, dihedral energies, plus non-bonded interactions represented by a sum of pair wise additive coulombic and van der Waals contributions ... 

Bader, Novaro, and Beltran-Lopez135 have calculated the potential surface for various geometries of three and four helium atoms near to the Hartree-Fock limit to determine the deviation of the Hartree-Fock energies of interaction from pair-wise additivity. 

Void [11] has established Hqn(I) for the attractive van dcr Waals interactions between two spherical particles of radius R covered with a shell of a different material of thickness 5, by assuming pair-wise additivity of molecular interactions. 

Assuming pair-wise additivity of the potential, the Helmholtz free energy F of a system of N particles of diameter d in a volume V can be expressed, to second order, as [21] ... 

To compute the dependence of the free energy on the thickness h of the film, pair-wise additivity is assumed and the 6-12 Lenard-Jones potential... 

The thermodynamic variables of liquid systems, such as energy, pressure, etc., can be expressed in terms of the radial distribution function if one makes the assumption that the total potential energy is pair-wise additive, i.e.,... 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

The expressions for and E contain both pair-wise additive and three-body... 

Table 10. Intermolecular energies in unpolar systems using pair-wise additive potentials...

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. 

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