Radial probability distribution function

Figure 1.7 Plots of (a) the radial wave function (b) the radial probability distribution function and (c) the radial charge density function 4nr Rl( against p...

For visualization purposes we have made plots of pair distribution functions, defining the electron-nuclear radial probability distribution function D(ri) by the formula... 

Fig. 1-2.—The wave function u, its square, and the radial probability distribution function 47rrVi fo the normal hydrogen atom.

Figure 10.2. The radial probability distribution functions for hydrogen orbitals Is (gray) 2s (black) and 3s (heavy black).

Figure 10.4. Use of the radial probability distribution function to calculate probability.

We saw in Chapter 6 that the probability of finding an election in three-dimensional space depends on what orbital it is in. Look back at Figures 6.19 and 6.22, which show the radial probability distribution functions for the s orbitals and contour plots of the 2p orbitals, respectively, (a) Which orbitals, 2s or 2p, have more electron density at the nucleus (b) How would you modify Slater s rules to adjust for the difference in electronic penetration of the nucleus for the 2s and 2p orbitals ... 

The radial probability distribution functions for 2s and 2p electrons are shown in Figure 2.5a and b, respectively. Note that there is a small probability of finding the 2s electron at the first Bohr radius, but the most probable location is around 5 Bohr radii (recall the Bohr model predicts the electron shells to be spaced according to n ). On the other hand the 2p electron has the highest probability density at 4 Bohr radii. 

The structure of the adsorbed ion coordination shell is determined by the competition between the water-ion and the metal-ion interactions, and by the constraints imposed on the water by the metal surface. This structure can be characterized by water-ion radial distribution functions and water-ion orientational probability distribution functions. Much is known about this structure from X-ray and neutron scattering measurements performed in bulk solutions, and these are generally in agreement with computer simulations. The goal of molecular dynamics simulations of ions at the metal/water interface has been to examine to what degree the structure of the ion solvation shell is modified at the interface. 

Figure 8. The structure of hydrated Na and CP ions at the water/Pt(IOO) interface (dotted lines) compared with the structure in bulk water (solid lines). In the two top panels are the oxygen ion radial distribution functions, and in the two bottom panels are the probability distribution functions for the angle between the water dipole and the oxygen-ion vector for water molecules in the first hydration shell. (Data adapted from Ref. 100.)...

Fig. 1. Electron-electron distribution functions for single-configuration He wavefunction (a) radial probability distribution P(ri2) (b) intracule function h(ri2)- In both graphs, the curve with largest maximum is for the closed-shell wavefunction that of intermediate maximum is for the split-shell wavefunction that of smallest maximum is for the wavefunction containing exp( —yri2).

Fig. 3. Z-scaled electron-nuclear distribution functions for H, He, Li, and Ne (a) radial probability distribution D(r ) Z (b) radial density /o(ri)/Z. The curves can be identified from the fact that higher maxima correspond to higher Z.

Fig. 4. Backfolding in dendrimers as predicted by analytical theory [12]. Free end probability distribution function of the radial distance for generations 2-7. All data has been calculated assuming a realistic excluded volume parameter of the segments of the dendrimer (see [12] for further details). Reproduced with permission from [12]...

The radial probability density function is sometimes called the radial distribution function. 

Radial Probability Distribution Curves A curve that shows the variation of wave function with distance from the nucleus is known as radial probability distribution curve. 

The plots for hydrogen-like wave functions of radial function R(r) versus r, the distance from the nucleus and the probability distribution function 4jrr2[R(r) 2 versus r are shown... 

Solution of the wave equation for these conditions, would give three expressions for the wave function /, and we could again plot radial probability distributions These are not shown, but in all cases, the probability is zero at the origin, rises to a maximum value and decreases as r becomes large. We may again construct surfaces which will enclose nearly all the probability of finding an electron with the above values of the quantum numbers. 

So far we have discussed the electron density for the ground state of the H atom. When the atom absorbs energy, it exists in an excited state and the region of space occupied by the electron is described by a different atomic orbital (wave function). As you ll see, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

The electron s wave function (iK atomic orbital) is a mathematical description of the electron s wavelike behavior in an atom. Each wave function is associated with one of the atom s allowed energy states. The probability density of finding the electron at a particular location is represented by An electron density diagram and a radial probability distribution plot show how the electron occupies the space near the nucleus for a particular energy level. Three features of an atomic orbital are described by quantum numbers size (n), shape (/), and orientation (m/). Orbitals with the same n and / values constitute a sublevel sublevels with the same n value constitute an energy level. A sublevel with / = 0 has a spherical (s) orbital a sublevel with / = 1 has three, two-lobed (p) orbitals and a sublevel with / = 2 has five, multi-lobed (d) orbitals. In the special case of the H atom, the energy levels depend on the n value only. 

Distinguish between i / (wave function) and i (probability density) understand the meaning of electron density diagrams and radial probability distribution plots describe the hierarchy of quantum numbers, the hierarchy of levels, sublevels, and orbitals, and the shapes and nodes of s, p, and d orbitals and determine quantum numbers and sublevel designations ( 7.4) (SPs 7.4-7.6) (EPs 7.35-7.47)... 

We are also interested in knowing the total probability of finding the electron in the hydrogen atom at a particular distance from the nucleus. Imagine that the space around the hydrogen nucleus is made up of a series of thin spherical shells (rather like layers in an onion), as shown in Fig. 12.17(a). When the total probability of finding the electron in each spherical shell is plotted versus the distance from the nucleus, the plot in Fig. 12.17(b) is obtained. This graph is called the radial probability distribution, which is a plot of Atrr R versus r, where R represents the radial part of the wave function. 

A FIGURE 6.18 Radial probability distributions for the Is, 2s, and 3s orbitals of hydrogen. These graphs of the radial probability function plot probability of finding the electron as a function of distance from the nucleus. As n increases, the most likely distance at which to find the electron (the highest peak) moves farther from the nucleus. 

Fig. 4 Cation radial probability density functions. Each line is the average of 5 plots from different GCMC runs where the random A1 locations were all the same but the seed numbers were different. Note that the distributions do not differ substantially between when carbon dioxide is present and when it is not. In both cases, there is a peak near ztao corresponding to the S8R site and another much further away corresponding to cations on the S6R site. However, while the distribution of cations in the S6R is essentially unchanged, there is a small shift in the distribution of cations in the S8R when carbon dioxide is present The jnobability of a cation being between 1-2 A from the center of the ring is smaller while around 3 A the probability is largta than in the absence of the adsorbate...

This effect must not be confused with the cybotactic effects we have mentioned, nor with the hole in the solute-solvent correlation function gMs(t) (see Figure 8.5). The hole in the radial correlation function is a consequence of its definition, corresponding to a conditional property, namely that it gives the radial probability distribution of the solvent S, when the solute M is kept at the origin of the coordinate system. Cybotactic effects are related to changes in the correlation function gMs(t) (or better gMs(r> )) with respect to a reference situation. Surface proximity effects can be derived by the analysis of the gMs(r,fi) functions, or directly computed with continuum solvation methods. It must be remarked that the obtention of gMs(r) functions near the surface is more difficult than for bulk homogeneous liquids. Reliable descriptions of gMs(ri re even harder to reach. 

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