Expectation radial density

As the IRC proceeds from reactants to product, molecular radial density is seen to move from the tt bond systems of the ethene and butadiene molecules into the regions of space between the ethene and butadiene carbon atoms to form sigma bonds. In the transition state, there is molecular radial density between these carbon atoms that is about half of the molecular radial density found in the product of the reaction. However, the ethene bond in the transition state has maintained much of its double bond character in the transition state as compared to the reactant complex, which is consistent with an early transition state for the reaction as expected through the Hammond postulate. 

The former corresponds effectively to a one-component compressible polymer solution, while the character of a compressible binary mixture becomes more apparent at higher pressures in the vicinity of the triple line. The composition is held constant, and the temperature is varied. From Fig. 8 (b) we conclude that the composition of the coexisting phases remains almost constant in the temperature interval 0.75 < kiTje < 0.82 for both pressures. At low pressure, the nucleation barrier decreases monotonously with temperature as expected. At higher pressure, however, the nucleation barrier exhibits a non-monotonous dependence on temperature AG exhibits both a maximum and a minimum upon increasing temperature at fixed molar fraction. The inset compares the radial density distributions of the critical bubbles and planar interfaces at ksT/e = 0.7573. In both cases the solvent density at the center of the bubble is higher than at coexistence and there is an enrichment of solvent at the interface of the bubble. However, there are no qualitative differences in the structure, in agreement with the observation of Talanquer and co-workers [196] for binary Lennard-Jones mixtures. 

This subroutine calculates the three radial distribution functions for the solvent. The radial distribution functions provide information on the solvent structure. Specially, the function g-AB(r) is die average number of type B atoms within a spherical shell at a radius r centered on an aibitaiy type A atom, divided by the number of type B atoms that one would expect to find in the shell based cm the hulk solvent density. 

By means of a modification of the TFD method in the near nnclear region for the electron and energy densities, which introduces exact asymptotic properties, radial expectation values and the atomic density at the nucleus are evaluated, comparing fairly closely to the HF results, with a large improvement of the TF estimates. In addition to this, momentum expectation values can be estimated from semiclassical relations. 

Among the average properties which play a special role in the study of quantum fermionic systems are the radial expectation value (r ), the momentum expectation value (j9 ) and the atomic density at the nucleus p(0) = <5(r)>. These density-dependent quantities are defined by... 

As we said previously, the radial expectation values constitute a test of the description of the density in different regions depending on a. For a smaller than -1 the most important region is the near nuclear one. Now we show in the Table 3 the values from a = -2 to a = 2 obtained with the present method and with HF. In Figures 2 and 3 are illustrated a = -2 and a = - 1 compared with the //F values. 

The pressures developed in the deton reaction zone in condensed expls are of the order of 103 to 10 atm. Material at such pressures cannot in general be contained, so that the flow behind the front has a component radially outward. Gases, which develop much lower deton pressures (of the order of 10 atm), can be confined in a tube, and for them the one- dimensional approximation is good. The diverging flow is expected and is found experimentally to result in lower pressures and densities within the steady wave, and consequently in lower detonation velocities. 

Thus, the r and components of the velocity gradient are completely disregarded and Vq1 depends on those coordinates only through the r and dependence of the (radial) force density and the ground state shear velocity. The quality of this model increases with increasing kR. It is expected that the velocity perturbation is overestimated in this model and thus the hypothetical instability threshold is lowered, which makes the model appealing at least as a first attempt. 

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