Radial distribution function, cluster

After this computer experiment, a great number of papers followed. Some of them attempted to simulate with the ab-initio data the properties of the ion in solution at room temperature [76,77], others [78] attempted to determine, via Monte Carlo simulations, the free energy, enthalpy and entropy for the reaction (24). The discrepancy between experimental and simulated data was rationalized in terms of the inadequacy of a two-body potential to represent correctly the n-body system. In addition, the radial distribution function for the Li+(H20)6 cluster showed [78] only one maximum, pointing out that the six water molecules are in the first hydration shell of the ion. The Monte Carlo simulation [77] for the system Li+(H20)2oo predicted five water molecules in the first hydration shell. A subsequent MD simulation [79] of a system composed of one Li+ ion and 343 water molecules at T=298 K, with periodic boundary conditions, yielded... 

Site-site radial distribution functions for the CNWS system (C carbon P polymer backbones W water H cluster containing hydronium). (Reprinted from K. Malek et al. Journal of Physical Chemistry C 111 (2007) 13627. Copyright 2007, with permission from ACS.)... 

It is found that the atomic arrangement, or a vacancy network, in a depleted zone in a refractory metal or a dilute alloy of a refractory metal, created by bombardment of an ion can be reconstructed on an atomic scale from which the shape and size of the zone, the radial distribution function of the vacancies, and the fraction of monovacancies and vacancy clusters can be calculated. For example, Wei Seidman108 studied structures of depleted zones in tungsten produced by the bombardment of 30 keV ions of different masses, W+, Mo+ and Cr+. They find the average diameters of the depleted zones created by these ions to be 18,25 and 42 A, respectively. The fractions of isolated monovacancies are, respectively, 0.13,0.19and0.28,andthe fractions of vacancies with more than six nearest neighbor vacancies (or vacancy clusters) are, respect-... 

The calculations of g(r) and C(t) are performed for a variety of temperatures ranging from the very low temperatures where the atoms oscillate around the ground state minimum to temperatures where the average energy is above the dissociation limit and the cluster fragments. In the course of these calculations the students explore both the distinctions between solid-like and liquid-like behavior. Typical radial distribution functions and velocity autocorrelation functions are plotted in Figure 6 for a van der Waals cluster at two different temperatures. Evaluation of the structure in the radial distribution functions allows for discussion of the transition from solid-like to liquid-like behavior. The velocity autocorrelation function leads to insight into diffusion processes and into atomic motion in different systems as a function of temperature. 

Figure 5. Radial distribution function(RDF) of whole Xe molecules and clusters for w = 0.90 and 1.00 nm pores at 75.5 kPa. Solid and dotted lines denote RDFs of the whole Xe molecules and clusters.

Despite the formation of clathrate-like clusters and complete 512 cages during these simulations, the increased ordering observed from the radial distribution functions and local phase assignments resulted in the authors concluding that their simulation results are consistent with a local order model of nucleation, and therefore do not support the labile cluster model. 

Korsunsky, V.I. (2000) The investigation of structure of heavy metal clusters and polynuclear complexes in powder samples with the radial distribution function method. Coord. Chem. Rev., 199, 55. 

Lennard-Jones potential. The new radial distribution function is estimated using the exponential approximation to the optimized cluster theory (31)... 

Considerable evidence exits of the survival of Zintl ions in the liquid alloy. Neutron diffraction measurements [5], as well as molecular dynamics simulations [6, 7], give structure factors and radial distribution functions in agreement with the existence of a superstructure which has many features in common with a disordered network of tetrahedra. Resistivity plots against Pb concentration [8] show sharp maxima at 50% Pb in K-Pb, Rb-Pb and Cs-Pb. However, for Li-Pb and Na-Pb the maximum occurs at 20% Pb, and an additional shoulder appears at 50% Pb for Na-Pb. This means that Zintl ion formation is a well-established process in the K, Rb and Cs cases, whereas in the Li-Pb liquid alloy only Li4Pb units (octet complex) seem to be formed. The Na-Pb alloy is then a transition case, showing coexistence of Na4Pb clusters and (Pb4)4- ions and the predominance of each one of them near the appropiate stoichiometric composition. Measurements of other physical properties like density, specific heat, and thermodynamic stability show similar features (peaks) as a function of composition, and support also the change of stoichiometry from the octet complex to the Zintl clusters between Li-Pb and K-Pb [8]. 

Using refined X-ray dilfraction techniques and the extraction of the radial distribution function from molecular X-ray scattering it has been possible to develop a model for a graphene layer [31]. This model is free of the difficulties mentioned above and predicts cluster sizes of between n = 3 and n — 5 for pericondensed rings (coronene, hexaperibenzo coronene) in full agreement with electron microscopic [25] and NMR [131] data which led to construction of Fig. 19. The model material [31] was a coal sample before carbonization containing, as well as the main fraction of sp2 centres, about 20% carbon atoms in aliphatic connectivity. This one-dimensional structure analysis represents a real example of the scheme displayed in Fig. 9(C). 

The dependence of thelC fOHj) interaction on distance is shown in Fig. 2.58. The ion-0 radial distribution functions for Na fHjO) and K fHjO) clusters are shown in Fig. 2.59. A histogram that illustrates the distribution of O-Na-0 angles in an NafHjO) cluster (simulated at 298 K) is shown in Fig. 2.60. Finally, Fig. 2.61 shows the number of water molecules in a sphere of radius r within the cluster. 

These diagrams indicate the limit of the hydration shell in the gas-phase ion as the first minimum in the radial distribution function. It is well pronounced for K, which has 8 molecules as the calculated coordination number on the cluster curiously, the sharpness of the definition for Na" is less atAl= 6 (and sometimes 7). The influence... 

Fig. 2.59. Ion-0 radial distribution functions in the ion-( 2 )199 cluster, (a) Na, (b) K. 1 Gj q (ordinate to the left). 2 Number of H2O molecules in the sphere of radius R (ordinate to the right). (Reprinted from G. G. Malenkov, Models for the structure of Hydrated Shells of Simple Ions Based on Crystal Structure Data and Computer Simulation, in The Chemical Physics of Solvation, Part A, R. R. Dogo-nadze, E. Kalman, A. A. Komyshev, and J. Ulstrup, eds., Elsevier, New York, 1985.)...

In cluster calculations, an element essential in solution calculations is missing. Thus, intrinsically, gas-phase cluster calculations cannot allow for ionic movement. Such calculations can give rise to average coordination numbers and radial distribution functions, but cannot account for the effect of ions jumping from place to place. Since one important aspect of solvation phenomena is the solvation number (which is intrinsically dependent on ions moving), this is a serious weakness. 

One of the typical minimized clusters 1 (methane) 10 (waters) is presented in Figure la,b. They show that the methane molecule is enclosed in a cavity formed by water molecules. The two spheres centered on a methane molecule, with radii of 3.6 and 5.35 A, correspond to the first maximum and the first minimum in the radial distribution function goo = goo(roc) in dilute mixtures of methane in water. It is worth noting that... 

Figure 1. Optimized methane (l) water (10) cluster, (a) The front view, (b) The view from the right. The two circles in Figure 1 correspond to the first maximum (3.6 A) and first minimum (5.35 A) of the radial distribution function goc = goc(roc).

Figure 17. (a) Projection of inner shell coordinates of 55-particle cluster onto plane passing through cluster s central atom and its two nearest neighbors. This projection is very nearly that obtained for an icosahedron, (b) Radial distribution function d taken about the atom nearest to cluster s center of mass, d = 1.7 is mean radial density T = 0.25. (From Nauchitel and Pertsin. )... 

Figure 21. Radial distribution functions calculated using a Fourier transform of scattering patterns produced with a Debye equation. Top Cuboctahedron (cluster with both octahedral 111 and cnbe 100 faces) and icosahedron (multiply twinned hep structure) clusters of the same size. Center Cuboctahedra of different sizes. Bottom Experimental and simulated cluster RDF of a Pt colloid. The fit is a 90 10 mixture of 55 and 147 cuboctahedral clusters, respectively. After Casanove et al. (1997).

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