Coordinate-scaling condition

Among the known properties of this functional are the coordinate scaling conditions first obtained by Levy and Perdew [55]... 

If one chooses a model system other than the uniform electron gas, the value of the constant Cp changes. However, the exponent 5/3 is required by the coordinate scaling condition Equation 1.42 and should not be changed. 

For the nonuniform coordinate-scaling, the following conditions are also established in the second-order scaling (p(x, y,z) pH = A p(Ax, Aj, z)) ... 

Of course, the coordinate scaling also applies to the axes of the simulation box, which are normally small multiples of the crystallographic cell constants a, b, c. Application of equation (11) may result in a rotation of the whole system, which will destroy the translational symmetry assumed for the periodic boundary conditions. So a compensating rotation R should be performed after the scaling ... 

In homopolymers all tire constituents (monomers) are identical, and hence tire interactions between tire monomers and between tire monomers and tire solvent have the same functional fonn. To describe tire shapes of a homopolymer (in the limit of large molecular weight) it is sufficient to model tire chain as a sequence of connected beads. Such a model can be used to describe tire shapes tliat a chain can adopt in various solvent conditions. A measure of shape is tire dimension of tire chain as a function of the degree of polymerization, N. If N is large tlien tire precise chemical details do not affect tire way tire size scales witli N [10]. In such a description a homopolymer is characterized in tenns of a single parameter tliat essentially characterizes tire effective interaction between tire beads, which is obtained by integrating over tire solvent coordinates. 

A family of cyclopentadiene(Cp)-containing iron-olefin complexes has been pioneered by Jonas [13-15]. The complexes 38-40 (Scheme 7) can be obtained in a large scale from ferrocene 37 under reducing conditions in the presence of suitable coordinating olefins. Complex 38 is a highly air-sensitive, crystalline material, whereas complexes 39 and 40 are more robust due to their cyclooctadiene (cod)... 

Rao and Singh32 calculated relative solvation free energies for alkyl- and tetra-alkylammonium ions using same conditions as described before for neutral molecules (except, atomic partial charges were not scaled for ions). The values obtained with coordinate coupling were in better agreement with... 

For many chemical reactions with high sharp barriers, the required time dependent friction on the reactive coordinate can be usefully approximated as the tcf of the force with the reacting solute fixed at the transition state. That is to say, no motion of the reactive solute is permitted in the evaluation of (2.3). This restriction has its rationale in the physical idea [1,2] that recrossing trajectories which influence the rate and the transmission coefficient occur on a quite short time scale. The results of many MD simulations for a very wide variety of different reaction types [3-12] show that this condition is satisfied it can be valid even where it is most suspect, i.e., for low barrier reactions of the ion pair interconversion class [6],... 

Even when forward reactions proceed rapidly at laboratory conditions, as is observed with Se(IV) and Cr(VI) reduction, evidence exists that chemical and isotopic equilibrium are not approached rapidly. Altman and King (1961) studied the kinetics of equilibration between Cr(III) and Cr(Vt) at pH = 2.0 to 2.5 and 94.8°C. Radioactive Cr was used to determine exchange rates, and Cr concentrations were greater than 1 mmol/L. Time scales for equilibration were found to be days to weeks. The mechanism of the reaction was inferred to involve unstable, ephemeral Cr(V) and Cr(IV) intermediates. Altman and King (1961) stated that the slowness of the equilibration was expected because the overall Cr(VI)-Cr(III) transformation involves transfer of three electrons and a change in cooordination (tetrahedral to octahedral). Se redox reactions also involve multiple electron transfers and changes in coordination. 

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