Pair-correlation length

The value of the jump distance in the )0-relaxation of PIB found from the study of the self-motion of protons (2.7 A) is much larger than that obtained from the NSE study on the pair correlation function (0.5-0.9 A). This apparent paradox can also be reconciled by interpreting the motion in the j8-regime as a combined methyl rotation and some translation. Rotational motions aroimd an axis of internal symmetry, do not contribute to the decay of the pair correlation fimction. Therefore, the interpretation of quasi-elastic coherent scattering appears to lead to shorter length scales than those revealed from a measurement of the self-correlation function [195]. A combined motion as proposed above would be consistent with all the experimental observations so far and also with the MD simulation results [198]. 

Analysis of the correlation functions demonstrates also impressive general agreement between the superposition approximation and computer simulations. Note, however certain overestimate of the similar particle correlations, X r,t), at small r, especially for d = 1. In its turn the correlation function of dissimilar particles, Y(r,t), demonstrates complete agreement with the statistical simulations. Since the time development of concentrations is defined entirely by Y(r, t), Figs 5.2 and 5.3 serve as an additional evidence for the reliability of the superposition approximation. An estimate of the small distances here at which the function Y (r, t) is no longer zero corresponds quite well to the earlier introduced correlation length o, equation (5.1.47) as one can see in fact that at moment t there are no AB pairs separated by r < o-... 

AB pairs separated by the distance r < have recombined Y(r < ) w 0. In its turn, on the same distance interval a number of similar AA and BB pairs is in excess (compared to the Poisson distribution), X (r, t) 1. Therefore, a system structure could be symbolically presented as a pattern in Fig. 2.8 where, however, the diffusion length Id should be replaced now by the correlation length . 

The pair correlation function is a short range quantity in liquids, decaying to unity after a few molecular diameters, the correlation length However, in supercritical fluids g(r) has a much longer range and t, becomes considerably larger than the mean inter-molecular separation at the critical density. For instance, for carbon dioxide , = 5.5 nm at Tc compared to the mean intermolecular separation of 0.55 nm (Eckert, Knutson and Debenedetti 1996). 

Fig. 8 Monomer-monomer (gmmir)) pair correlation function for the packing fractions 77=1(T3, 10-4, and 10-5 (left to right) of rod-like (...) and flexible chains (—). The chain length is N=63 and the Bjerrum length ZB=0.5 b...

We start to follow the relative motion between two particles of mass mp separated by a specified initial distance. This initial distance is chosen to be large compared to the correlation length Xr (the average distance over which the relative velocity of the particle pair is correlated). Therefore, the initial relative motion of the particle pair, which is considered as the initial condition in the calculations, is uncorrelated and its initial relative velocity distribution is Maxwellian. The correlation length Xr is given by (1)... 

The pair-correlation function curves from these experiments show peaks corresponding to the covalent O-H bond length of 1.00 A, the hydrogen bond... 

The total G(rz) can be expressed as a neutron-weighted sum (dependent on the scattering lengths of the constituent atoms) of the partial pair correlation functions 8ifrz) according to... 

As the wavefunction approaches the Hartree-Fock limit one would expect the Ti-Cl bond distance to be shorter than the experiment because of the lack of bond-pair correlation. The bond-pair correlation added by the GVB wavefunction lengthened the Ti-Cl bond 0.021 A, because the GVB wavefunction adds only limited left-right correlation and none of the dynamical correlation. For most A-B bonds, the calculated bond lengths at the SCF level are too short, and the correlation added by a GVB calculation accounts for a major portion of the non-dynamical correlation error in the SCF wavefunction. But for Ti-Cl bonds, both the SCF and GVB calculations predict too long a bond distance because they do not include necessary dynamical atomic correlation of the Cl atoms. 

The LDA has been widely applied to problems in atomic, molecular, surface, and solid-state physics with surprisingly good results overall (7). Apparently, the uniform gas model describes pair correlations even in inhomogeneous systems fairly well. In this communication, we shall focus our attention on molecular binding energies and related properties such as equilibrium bond lengths and vibrational frequencies. In Tables I, II, and III, we... 

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