Radial distribution function for

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

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. 

The radial distribution function for the population of the Earth, for instance, is zero up to about 6400 km from the center of the Earth, rises sharply, and then falls back to almost zero (the almost takes into account the small number of people who are on mountains or flying in airplanes). 

FIGURE 1.32 The radial distribution function tells us the probability density for finding an electron at a given radius summed over all directions. The graph shows the radial distribution function for the 1s-, 2s-, and 3s-orbitals in hydrogen. Note how the most probable radius icorresponding to the greatest maximum) increases as n increases. 

FIGURE 1.42 The radial distribution functions for s-, p-, and cf-orbitals in the first three shells of a hydrogen atom. Note that the probability maxima for orbitals of the same shell are close to each other however, note that an electron in an ns-orbital has a higher probability of being found close to the nucleus than does an electron in an np-orbital or an nd-orbital. 

Fig. 1.—Radial distribution function for carbon tetrachloride. The part of the curve beyond 4 A. is without significance.

Carbon Tetrachloride.—By the usual visual method and by other methods involving microphotometer records, we have assigned8 to the carbon tetrachloride molecule the value 1.760 0.005 A. for the C-Cl distance, a value supported by other recent work.9 The radial distribution function for this molecule calculated by Equation 6, using the ten terms for which data are given in Table I, is shown in Fig. 1. 

The six-term radial distribution function for ethane (Fig. 1, curve A) shows maxima at 1.16,... 

W(Xy yy z)y W r) Density and radial distribution functions for the end-to-end coordinates of a polymer chain (usually Gaussian functions). 

In Fig. 5.12, the radial distribution functions for the neutral iron-atom are plotted. It is evident that the orbitals with the same main quantum number occupy similar regions in space and are relatively well separated from the next higher and next lower shell. In particular, the 4s orbital is rather diffuse and shows its maximum close to typical bonding distances while the 3d orbitals are much more compact. 

Fig. 5.12 Normalized radial distribution function for the neutral Mn atom (3d 4s as calculated by the UHF method...

Higher-order functions are readily determined from Table 6.1. The radial distribution functions for the Is, 2s, 2p, 3s, 3p, and 3d states are shown in Figure 6.5. 

The results depicted in the figure are averages for 10 snapshots at 0.75 g/cm3. Similar positions and amplitudes for the first maximum and first minimum were obtained in MD simulation for a C44H90 melt at 400 K and 0.76 g/cm3 [165], The kink near 4 A does not appear in this MD simulation, but a similar kink does appear in the site-site intermolecular radial distribution function for PE reported by Honnell et al. [166],... 

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... 

Fig. 52. The excess (above a smooth parabolic background) radial distribution function for HaO(as) weighted appropriately for neutron scattering from D20(as) (from Ref.82 )...

Fig. 55. Radial distribution function for liquid water synthesized as a mixture of ice I, ice II and ice III (from Ref. 75>)...

In this equation g(r) is the equilibrium radial distribution function for a pair of reactants (14), g(r)4irr2dr is the probability that the centers of the pair of reactants are separated by a distance between r and r + dr, and (r) is the (first-order) rate constant for electron transfer at the separation distance r. Intramolecular electron transfer reactions involving "floppy" bridging groups can, of course, also occur over a range of separation distances in this case a different normalizing factor is used. 

In the conventional Debye-Huckel treatment the equilibrium radial distribution function for a pair of reactants g(r) is simply equal to exp(-w/RT) with w given by (15)... 

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.)... 

Fig. 17. Gd-aqueous proton radial distribution function for the aqueous solution of the Gd(III)(DOTP) complex (after Borel, A. Helm, L. Merbach, A.E. Chemistry - A European Journal 2001, 7, 600-610).

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

Figure 9.4 Radial distribution functions for liquid (left) and amorphous (right) InP. In each case, distribution functions for In In, P P, In P atom pairs and for all atoms are shown. The zeros for the P P, In P, and total distribution functions are displaced vertically by 1, 2, and 3 units, respectively. (Reprinted by permission from the source cited in Fig. 9.2.)...

Figure 4.3-2 Radial distribution function for dimethylimidazolium and chloride ions relative to chloride. Full line cation-anion dashed line anion-anion.

Figure 2.1 Radial distribution functions for an NaCl melt at 1148 K, as found by neutron diffraction. Concentric, mutually exclusive shells of anions and cations can clearly be seen extending to large distances relative to the sizes of the ions. (From Edwards, F.G., Enderby, J.E., Howe, R.A., and Page, D.I., /. Phps. C Solid State Phys., 8, 3483-3490, 1975. With permission.)...

FIGURE 2.33 Co edge EXAFS radial distribution function for Co (CO) 4. 

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