Atomic orbitals radial probability function

How many maxima would you expect to find in the radial probability function for the 4s orbital of the hydrogen atom How many nodes would you expect in this function ... 

Consider the discussion of radial probability functions in A Closer Look in Section 6.6. (a) What is the difference between the probability density as a function of rand the radial probability function as a function of r (b) What is the significance of the term 47rr in the radial probability functions for the s orbitals (c) Based on Figures 6.18 and 6.21, make sketches of what you think the probability density as a function of r and the radial probability function would look like for the 4s orbital of the hydrogen atom. 

Fig. 8.4. Constant probability density contours of spz hybrid atomic orbitals as a function of their p-characters. Top line p-characters ranging from 0 to 0.500. Bottom liner p-characters ranging from 0.500 to 1.000. Since both s andp orbitals have cylinder symmetry about the z-axis, so do the sp hybrids. The hybrids have been formed from s and pi orbitals with identical radial wavefunctions R r) according to equation (8.20) with positive values for the constant a. The shading covers areas where the electron density is greater than 1/5 of the maximum density for the s orbital.

Let s examine the difference between probability density and radial probability function more closely. Figure 6.22 shows plots of [ilf r)f as a function of r for the Is, 2s, and 3s orbitals of the hydrogen atom. You will notice that these plots look distinctly different from the radial probability functions shown in Figure 6.19. 

Figure 1.31 (a) The radial part of the wavefunction for the 2p and 3p hydrogen atomic orbitals, (b) The radial probability function for the orbitals shown in (a). 

Using the radial probability function, calculate the average values of r for the Is and 2s orbitals of the hydrogen atom. How do these compare with the most probable values (Hint You will need to consult a standard table of integrals for this problem.)... 

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... 

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. 

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

FIGURE 1.25 The radial wave-functions of the first three s-orbitals of a hydrogenlike atom. Note that the number of radial nodes increases (as n — l), as does the average distance of the electron from the nucleus. Because the probability density is given by ijr2, all s-orbitals correspond to a nonzero probability density at the 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. 

Let us now consider how we might represent atomic orbitals in three-dimensional space. We said earlier that a useful description of an electron in an atom is the probability of finding the electron in a given volume of space. The function ij (see Box 1.4) is proportional to the probability density of the electron at a point in space. By considering values of at points around the nucleus, we can define a surface boundary which encloses the volume of space in which the electron will spend, say, 95% of its time. This effectively gives us a physical representation of the atomic orbital, since ij may be described in terms of the radial and angular components R r) and A 0,... 

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. 

Often, it is more meaningful physically to make plots of the radial distribution function, P(r), of an atomic orbital, since this display emphasizes the spatial reality of the probability distribution of the electron density, as shell structure about the nucleus. To establish the radial distribution function we need to calculate the probability of an electron, in a particular orbital, exhibiting coordinates on a thin shell of width, Ar, between r and r - - Ar about the nucleus, i.e. within the volume element defined in Figure 1.6. 

Radial nodes (spherical nodes) result when R = 0. They give the atom a layered appearance, shown in Figure 2.8 for the 3 and orbitals. These nodes occur when the radial function changes sign they are depicted in the radial function graphs by R(r) = 0 and in the radial probability graphs by = 0. The lowest energy orbitals of each clas-... 

FIGURE 3.7 The representations of the electronic probability density of existence (wave-functions) for Hydrogenic atoms, for the superior (excited) levels (or shells, quantified by the number n) with the respective sub-sheUs (or orbitals, quantified by the mixed numbers nl), employing the derived radial wave functions in terms of respective Laguerre polynomials of Table 3.1. 

Each HE molecular orbital is written as a linear combination of functions describing atomic orbitals. The entire set of such equations for the atomic orbitals in a molecule is called a basis set, and each equation is called a basis set function. Ideally, these basis set fimctions would have the properties of hydrogenic wave functions, particularly with regard to the radial dependence of electron density probability as a function of distance of the electron from the nucleus, r. A type of basis set function proposed by Slater uses a radial component incorporating and such functions are called Slater t)rpe orbitals (STOs). Gaussian type orbitals (GTOs) have radial dependence and are easier to solve analytically, but they do not describe the radial dependence of electron density as well as do the STOs. 

According to Table 1.3, an s orbital, defined as any atomic orbital with 1 = 0, has no dependence on 6 or . It depends only on r. Such a function is said to be spherically symmetric and is defined by the radial function R (r). To get a sense of where the electrons are likely to be in these orbitals, we plot the probability of finding the electron in a spherical shell of thickness dr at a distance of r from the nucleus (see ThPDS abiBn ] 0jgg jrj is given by the expression... 

Figure 3.2 shows the radial distribution functions for the hydrogen 2s and 2p orbitals, from which it can be seen that the 2s orbital has a considerably larger probability near the nucleus than the 2p orbital. When an electron in a polyelectronic atom occupies the n = 2 level, it would be more stable in the 2s orbital than in the 2p orbital. In the 2s orbital it would be nearer the nucleus and be more strongly attracted than if it were to occupy the 2p sub-set. 

In practice, one would not specify the values (since these do not determine the symmetry of the atomic orbital), but would simply choose as many basis functions as practical to adequately model the shape of the radial probability density. The maximum 1 value, on the other hand, is normally specified for each atom, and does not need to be limited to the atom s ground state 1 value. For example, the atomic orbital functions on a hydrogen nucleus typically include up to 1 = 1 functions, while a carbon atom in the same calculation might be given basis functions up through 1=2. 

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