Difference pair-distribution function

Figure 1. Experimental difference pair-distribution functions g(r) (circles) for solvation shell of lithium ion in a LiCl solution in DMF. The curve corresponding to the octahedral coordination, resulted in the best fit, is drawn by solid lines. Contribution calculated with an assumption of tetrahedral coordination is shown by dots.

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

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. 

The experimental evidence, then, suggests that quantum interference is absent in liquid metals. At first sight this might seem to contradict Ziman s (1961) theory of liquid metals, in which waves scattered by different atoms do interfere this, however, is not so. Following arguments of Baym (1964), Greene and Kohn (1965) and Faber (1972), one should not use in that theory the Fourier transform S(k) of the instantaneous pair-distribution function, but rather... 

We mention the obvious fact that for a given system and temperature the number of partial waves needed is a function of the separation R for which g(R) is to be evaluated. Large R require many partial waves at large R the pair distribution function g(R) is, therefore, largely classical (correspondence principle). The classical and quantal pair distribution functions g(R) differ most significantly at separation less than Rm, the position of the potential minimum. Quantum computations of the pair distribution function may, therefore, be restricted to relatively small separations. For larger separations classical or semi-classical expressions may be employed which will be sketched below. In this way, the number of partial waves required for pair distribution functions need not be much greater than for line shape computations. 

Figure 5.1 shows as an example the low-density limit of the pair distribution function for He-Ar pairs at 295 K. The solid line is based on Eq. 5.36 and the dashed line is the classical approximation, g(R) — exp (—V(R)/kT). The two agree closely in the example shown, but at lower temperatures or for less massive systems the classical and quantal pair distribution functions differ strikingly. 

Results. In Table 5.1 we compare a few results of classical, semi-classical and quantum moment calculations. An accurate ab initio dipole surface of He-Ar is employed (from Table 4.3 [278]), along with a refined model of the interaction potential [12]. A temperature of 295 K is assumed. The second line, Table 5.1, gives the lowest three quantum moments, computed from Eqs. 5.37, 5.38, 5.39 the numerical precision is believed to be at the 1% level. For comparison, the third line shows the same three moments, obtained from semi-classical formulae, Eqs. 5.47 along with 5.37 with the semi-classical pair distribution function inserted. We find satisfactory agreement. We note that at much lower temperatures, and also for less massive systems, the semi-classical and quantal results have often been found to differ significantly. The agreement seen in Table 5.1 is good because He-Ar at 295 K is a near-classical system. 

For the exciton mechanism of defect production in alkali halides the Frenkel pairs of well correlated defects are known to be created [35], the mean distance between defects inside these pairs is much smaller than that between different pairs. The geminate pair distribution function could often be approximated as... 

Although this relationship looks similar to Eq. (3.257) for irreversible transfer, the Stern-Volmer constant of the latter (ko = k,) is different from Kf, which accounts for the reversibility of ionization during the geminate stage. The difference between Kg = R (0) and its irreversible analog K from (3.372) is worthy of special investigation based on the analysis of pair distribution functions obeying Eqs. (3.359). 

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

Fig. 1.2. Schematic diagram of the atom pair distribution functions for a crystalline and amorphous solid and a gas, scaled to the average separation of nearest neighbor atoms, showing the different degree of structural order.

In spite of the great success of the computer simulation methods in the determination of the microscopic properties of the solutions, the capacity of the traditional MD and MC simulations is always limited by the choice of the suitable potential functions to describe the interatomic interactions. The potentials are most often checked by comparison of the structural properties calculated from the simulation with those determined experimentally. The reverse Monte Carlo (RMC) method, developed by McGreevy and Pusztai [41] does not rely upon knowledge of any interaction potential, instead it generates a large set of atomic configurations on the condition that the difference between the experimental and calculated structure functions (or pair-distribution functions) should be minimum. The same structural... 

The use of difference methods offers a means whereby a detailed picture of ionic hydration can be obtained 22). For neutron diffraction, the first-order isotopic difference method (see Section III,A) provides information on ionic hydration in terms of a linear combination of weighted ion-water and ion-ion pair distribution functions. Since the ion-water terms dominate this combination, the first-order difference method offers a direct way of establishing the structure of the aquaion. In cases for which counterion effects are known to occur, as, for example, in aqueous solutions of Cu + or Zn +, it is necessary to proceed to a second difference to obtain, for example, gMX and thereby possess a detailed knowledge of both the aquaion-water and the aquaion-coun-terion structure. 

By use of a single isotopic substitution of ion (I = M or X), a first-order difference function Gi(r) can be obtained by direct differencing of all G(r) that contain all ten pair-distribution functions of the solution. Giir) is then a linear combination of only the four radial distribution functions specific to I and can be written as in Eq. (2) (74). 

Ewing used a different approach from Skapski to calculate the entropy contribution to a, in which the configurational entropy of the atoms in the surface layer was found by assuming a Boltzmann distribution of liquid particles in the field of the solid and approximating that distribution by the bulk liquid pair distribution function. This is equivalent to taking the distribution of atoms near a solid as the same as that around a single fixed atom. The results showed a significant entropic contribution to o, . 

In addition to the repulsive part of the potential given by Eq. (4), a short-range attraction between the macroions may also be present. This attraction is due to the van der Waals forces [17,18], and can be modelled in different ways. The OCF model can be solved for the macroion-macroion pair-distribution function and thermodynamic properties using various statistical-mechanical theories. One of the most popular is the mean spherical approximation (MSA) [40], The OCF model can be applied to the analysis of small-angle scattering data, where the results are obtained in terms of the macroion-macroion structure factor [35], The same approach can also be applied to thermodynamic properties Kalyuzhnyi and coworkers [41] analyzed Donnan pressure measurements for various globular proteins using a modification of this model which permits the protein molecules to form dimers (see Sec. 7). 

It can be seen from Eq. (18) that the improvement for the potentials Ujj(r) depends only on the difference between the predicted and experimentally observed pair distribution functions instead of any properties related to the reference state. Therefore, the iterative method does not face the reference state problem encountered by traditional mean-force/knowledge-based scoring functions. 

In the case of neutron diffraction, the radiation is scattered by the atomic nuclei, not by the electrons. It turns out that nucleons such as H and have very different scattering amplitudes. This means that isotope effects are very important in developing experimental strategies. Soper and Phillips [8] used data for the structure function obtained in mixtures of normal and heavy water to extract values of the partial structure factors for water. In this way they were able to determine all of the pair distribution functions for water from their diffraction data. These are gHnW. g oHW) and gooW- More details of their experimental results are given in section 2.10. 

To define a phase in a cluster is not a straightforward task [4]. We speak about phase-hke forms, with specific pair-distribution functions [5], that help to distinguish between different thermodynamic states in small systems. In what follows we use terms phase change and phase transformations for small systems, preserving the term phase transitions for bulk matter. 

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