Radial probability density

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

Draw a plot of the radial probability density (e.g., r2[Rjjj(r)]2 with R referring to the radial portion of the STO) versus r for eaeh of the orthonormal Ei s orbitals found in Exereise 1. 

The magnitude and "shape" of sueh a mean-field potential is shown below for the Beryllium atom. In this figure, the nueleus is at the origin, and one eleetron is plaeed at a distanee from the nueleus equal to the maximum of the Is orbital s radial probability density (near 0.13 A). The radial eoordinate of the seeond is plotted as the abseissa this seeond eleetron is arbitrarily eonstrained to lie on the line eonneeting the nueleus and the first eleetron (along this direetion, the inter-eleetronie interaetions are largest). On the ordinate, there are two quantities plotted (i) the Self-Consistent Field (SCF) mean-field... 

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...

Fig. 8 Radial probability density for the hydrogen atom in ns states.

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 2.1 Radial probability density plots for Is and 2s orbitals of hydrogen atom...

They are given by the dashed line curves in Fig. 2.12. A conceptually useful quantity is the probability of finding the electron at some distance, r, from the nucleus (in any direction) which is determined by the radial probability density, P ,(r) = r2R (r). We see from the full line curves in Fig. 2.12 that there is a maximum probability of locating the electron at the first Bohr radius, aly for the Is state and at the second Bohr radius, a2, for the 2p state. 

Plot these radial functions and the corresponding radial probability densities. By differentiation of the radial probability density show that the maximum probability of locating the electron a distance r from the nucleus occurs at the first and second Bohr radii for the Is and 2p states respectively. Where does the maximum probability occur for the 2s state Why does the 2s radial function have one node but the 2p radial function is nodeless (outside the origin) ... 

Fig. 9. Radial probability density distribution, derived from quantum-mechanical predictions. Ibr rubidium, along with Bohr planetary model for the same atom...

The probability of finding the electron in the ground state of the hydrogen atom between radii r and r + dr is given by D(r)Ar, where D(r) is the radial probability density function shown in Figure 4.5. The most probable distance of the electron from the nucleus is found by locating the maximum in D(r) (see Problem 4.12 below). It should come as no surprise to discover that this maximum occurs at the value r = ao, the Bohr radius. 

Figure 4.5 The radial probability density function for the 1s atomic orbital of the hydrogen atom...

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

Consider the radial probability density function, D(r), for the ground state of the hydrogen atom. This function describes the probability per unit length of finding an electron at a radial distance between r and r + dr (see Figure 6.5). 

Explain the difference between the probability density distribution for an orbital and its radial probability. 

In all the radial probability plots, the electron density, or probability of finding the electron, falls off rapidly as the distance from the nucleus increases. It falls olf most quickly for the 1 orbital by r = Sa, the probability is approaching zero. By contrast, the 3d orbital has a maximum at r = 9ao and does not approach zero until approximately r = 20ao- All the orbitals, including the s orbitals, have zero probability at the center of the nucleus, because Anir R = 0 at r = 0. The radial probability ftinctions are a combination of which increases rapidly with r, and R, which may have maxima and minima, but generally decreases exponentially with r. The product of these two factors gives the characteristic probabilities seen in the plots. Because chemical reactions depend on the shape and extent of orbitals at large distances from the nucleus, the radial probability functions help show which orbitals are most likely to be involved in reactions. 

Fig. 10. Radial probability density, 4 ry V, in ground state hydrogen 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. 

FIGURE 5.4 Four representations of hydrogen s orbitals, (a) A contour plot of the wave function amplitude for a hydrogen atom in its Is, 2s, and 3s states. The contours identify points at which i//takes on 0.05, 0.1, 0.3, 0.5, 0.7, and 0.9 of its maximum value. Contours with positive phase are shown in red those with negative phase are shown in blue. Nodal contours, where the amplitude of the wave function is zero, are shown in black. They are connected to the nodes in the lower plots by the vertical green lines, (b) The radial wave functions plotted against distance from the nucleus, r. (c) The radial probability density, equal to the square of the radial wave function multiplied by 1. (d) The "size" of the orbitals, as represented by spheres whose radius is the distance at which the probability falls to 0.05 of its maximum value. 

FIGURE 5.6 Radial wave functions for np orbitals and the corresponding radial probability densities r Rnt-... 

FIGURE 5.12 Dependence of radial probability densities on distance from the nucleus for Hartree orbitals in argon with n = 1, 2, 3. The results were obtained from self-consistent calculations using Hartree s method. Distance is plotted in the same dimensionless variable used in Figure 5.10 to facilitate comparison with the results for hydrogen. The fact that the radial probability density for all orbitals with the same value of n have maxima very near one another suggests that the electrons are arranged in "shells" described by these orbitals. 

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)... 

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