Probability distribution radial

Plot RI against p (or r), as shown in Figure 1.7(b). Since R dr is the probability of finding the electron between r and r + dr this plot represents the radial probability distribution of the electron. 

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...

A plot of radial probability distribution versus r/ao for a His orbital shows a maximum at 1.0 (that is, r = a0). The plot is shown below ... 

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. 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.

Figure 3. Analytical potential curves V(R,0,re) for HeBr2 complex with 0 = 0° (squares) and 0 = 90° (circles). The eigenvalues and radial probability distributions corresponding to the J = 0 lowest vibrational vdW level at each configuration, n = 0 (dotted line) and n = 2 (dashed line) respectively, are also included.

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

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 radial probability distribution curves for Is and 2s orbitals are shown in diagram below ... 

The radial probability distribution curve for 2 s orbital shows two maxima, a smaller one near the nucleus and bigger one at a larger distance. In between these two maxima, there is a maxima where there is no probability of finding the electron at that distance. The point at which the probability of finding the electron is zero is called a nodal point. 

Radial probability distributions for the 3d and 4s orbitals. Note that the most probable distance of the electron from the nucleus for the 3d orbital is less than that for the 4s orbital. However, the 4s orbital allows more electron penetration close to the nucleus and thus is preferred over the 3d orbital. 

Fig. 5.4 Radial probability distributions of hydrogen-like 3s, 3p and 3d orbitals for atom K.

The value of an ns orbital close to the nucleus is higher than that of an np orbital of the same atom s orbitals are more penetrating than p orbitals of equal quantum munber n (recall that p orbitals have a node at the position of the nucleus). As a consequence, there is more negative charge to be taken into account for efifect (b) in the case of a p orbital. The same is true of d orbitals with respect to p orbitals of equal n. Figure 5.4 shows the radial probability distributions of the degenerate hydrogen-like 3s, 3p and 3d orbitals for the potassium atom. 

The electron probability density along the line passing through the nuclei is graphically represented as in Fig. 4.5 for H2 and the isoprobability contours for a plane containing the intemuclear axis are similar to those of Fig. 4.7 for the same ion. If we seek a distribution similar to the radial probability distribution for atoms. Fig. 6.1 is obtained (ref. 65). It shows the circular distribution of electron density for different distances from the inter-nuclear axis. It is found that the electronic charge is concentrated in a circular doughnut around the H-H axis, with a maximum at about 37 pm from the axis and about 50-55 pm from each nucleus. 

The maximum of the radial probability distribution occurs for a higher r value in the SCF case, due to the inclusion of the (averaged) electronic repulsion. 

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.

Fig. 3.3. Radial probability distributions for various low-lying states of hydrogen (adapted from Condon and Shortley (1935)).

Radial wavefunctions depend on n and / but not on m thus each of the three 2p orbitals has the same radial form. The wavefunctions may have positive or negative regions, but it is more instructive to look at how the radial probability distributions for the electron depend on the distance from the nucleus. They are shown in Fig. 2 and have the following features. 

Fig. 2. Radial probability distributions for atomic orbitals with n=l-3.

The product ipy. 4mz is known as the radial probability distribution . We can interpret it thus The actual probability rises with the magnitude of r, and reaches a maximum when r has a value which we will call r. In other words, the most probable place where we could find the hydrogen electron would be on the surface of a sphere of radius r. This is illustrated in Figure 3. 

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. 

Figure 9.9 The radial probability distribution of an n 30 wavepacket in a hydrogen atom. The populations and binding energy of the eigenstates in the wavepacket are shown in the inset. Note that the figure shows the to probability distribution. A plot of the to wavepacket amplitude would show a single dominant feature in the region of the innermost lobe of the initial state (Is), with small wiggles at larger r (from Smith, et al., 2003b).

The peak of the radial probability distribution for the ground-state H atom appears at the same distance from the nucleus (0.529A, or 5.29x10 " m) as Bohr postulated for the closest orbit. Thus, at least for the ground state, the Schrddinger model predicts that the electron spends most of its time at the same distance that the Bohr model predicted it spent all of its time. The difference between most and all reflects the uncertainty of the electron s location in the Schrddinger model. 

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. 

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