Localized pair correlation

Typical results for a semiconducting liquid are illustrated in figure Al.3.29 where the experunental pair correlation and structure factors for silicon are presented. The radial distribution function shows a sharp first peak followed by oscillations. The structure in the radial distribution fiinction reflects some local ordering. The nature and degree of this order depends on the chemical nature of the liquid state. For example, semiconductor liquids are especially interesting in this sense as they are believed to retain covalent bonding characteristics even in the melt. 

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

Finally, we relate the gradient of the local density ViPq(fi) to the pair correlation functions. For this purpose we take the gradient of the expansion (5)... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

One important use of X-ray probes is in the study of local order and displacements, but this is not within the scope of the present book. The recent availability in intense synchrotron sources with selectable X-ray energies permits high-precision measurements of chemically specific atomic-pair correlations in solid solution alloys. A recent review of the technique is given by G.E. Ice and C.J. Sparks (Modern Resonant X-ray studies of alloys local order and displacement) in Annual Reviews of Materials Science 1999, 29, 25-52. 

Fig. 53. Fit to the pair correlation function goo for the model consisting of a local pentamer plus continuum (from Ref. 5>).-------neutron diifraction data,-------model calculation...

Chapter 4 deals with the local dynamics of polymer melts and the glass transition. NSE results on the self- and the pair correlation function relating to the primary and secondary relaxation will be discussed. We will show that the macroscopic flow manifests itself on the nearest neighbour scale and relate the secondary relaxations to intrachain dynamics. The question of the spatial heterogeneity of the a-process will be another important issue. NSE observations demonstrate a subhnear diffusion regime underlying the atomic motions during the structural a-relaxation. 

In the previous section it has been shown that the temporal evolution of the pair correlations at the interchain peak is governed by the structural relaxation. If we move now towards more local scales - i.e. higher Q-values - we see that the static correlations observed in Spair(Q) correspond to pair correlations along a given chain. It is then natural to think that their time dependence might relate to dynamic processes other than the structural relaxation. 

A close estimate of the dynamic correlation energy was obtained by a simple formula in terms of pair populations and correlation contributions within and between localized molecular orbitals. The orbital and orbital-pair correlation strengths rapidly decrease with the distance between the orbitals in a pair. For instance, the total valence correlation energy of diamond per carbon atom, estimated as 164 mh, is the result of 84 mh from intra-orbital contributions, 74.5 mh from inter-orbital closest neighbors contributions, and 6.1 mh from interorbital vicinal contributions. The rapid decay of the orbital correlation contributions with the distance between the localized orbitals explains the near-... 

The zero differential overlap approximation can be applied in the localized representation. This was demonstrated by calculating for C H, CioTfio and C14//14, respectively the total energy corrections and the pair correlation energies through second and third order in different approximations. When the strongly local contributions were only... 

Taking into account that the electrons of the systems investigated are only weakly localizable the results are not discouraging the intra-pair correlation energies are not much affected. It is shown that even for these weakly localizable systems more than 50% of the correlation energy consist of (strictly) local contributions (Kapuy et ah, 1984). 

The partitioning of the pair-correlations, could thus be done according to a so-called law or order of neighborhood . This low is further applied in (Kapuy et al, 1990). respectively. The application of this law of neighborhood is useful for studying the transferable properties ofthe correlation energy contributions. The results clearly show, that the local and non-local effects can be separated. The contributions of distant (virtual) orbitals can be omitted to a rather good approximation. 

Table 5 Intra- and inter-pair correlation energy contributions for the localized molecular orbitals in the super-molecule + H2O...

An alternative, already implicit in van der Waals theory of the surface tension [237], exploits an interconnection between the coefficient Co and the pair correlation function g(ra, rp) of the inhomogeneous system. As the local density distribution determining g(ra, rp) is unknown, one has to resort to a... 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

The insulating phase which appears as the result of such SIT is rather specific it contains pair correlations between the localized electrons as the remnant of the superconducting pairing. Such insulator is called the Bose-insulator [11] and the correlated electrons are called localized electron pairs. Of course, existence of such phase should be confirmed experimentally. 

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