Orbital radial densities/distribution function

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 ... 

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. 

The square ot the radial part of th(e wavefunction of an orbital provides information about how tho electron density within the orbital varies as a function of distance from the nucleus. These radial distribution functions show that, in a given principle shell, the maximum electron density is reached nearer to the nucleus as the quantum number I increases. However, the proportion of the total electron density which is near to the nucleus is larger for an electron in an s orbital than in a p orbital. 

Figure 7.6 Density p r) and radial distribution function D r) for a hydrogen Ij orbital.

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)... 

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. 

As previously mentioned, in the approach that correlates atomic radii with the electronegativity it is of fundamental importance to know the atomic electron density of a given system. In this respect, a suitable treatment is based on the Slater orbital electronic picture that produces the normalized distribution functions under the radial form (Slater, 1964) ... 

Figure 7.18 (a) Plot of electron density in the hydrogen 1s orbital as a function of the distance from the nucleus. The electron density falls off rapidly as the distance from the nucleus increases, p) Boundary surface diagram of the hydrogen 1s orbital, p) A more realistic way of viewing electron density distribution is to divide the Is orbital into successive spherical thin shells. A plot of the probability of finding the electron in each shell, called radial probability, as a function of distance shows a maximum at 52.9 pm from the nucleus. Interestingly, this is equal to the radius of the innermost orbit in the Bohr model. 

It is of interest to compare the radial wave functions between 4f and 5f elements. O Figure 18.19 shows relativistic calculations of and Am valence wave functions of the radial part. This figure indicates that the 4f subshell of Eu is core like because of the lower density around the ionic radius of the ion. In the corresponding 5f ion, Am, 5f subshell penetrates the 6s and 6p orbitals. The wider distribution of the 5f orbital allows much participation in bonding for the actinides than for the lanthanides. The 4f orbitals of lanthanide atoms are mainly localized in the core and do not contribute to the chemical bonding. 

The radial distribution function R j r) describes the probability density of the electron as a function of r, added over all angles. Similarly, we may write a polar distribution function, which gives the probability density of the electron as a function of the polar angle 0, summing over all values of r and . Sketch a polar graph of the polar distribution functions for the 3s, 3po, and 3do orbitals. Include the proper normalization. 

Our individual one-electron HF or KS wavefunctions represent the individual molecular orbitals, and the square of the wavefunction gives us the probability distribution of each electron within the molecule. We do not know the form of the real multi-electron wavefunction a priori, nor the individual one-electron HF or KS functions, but we can use the mathematical principle that any unknown function can be modeled by a linear combination of known functions. A natural choice for chemists would be to use a set of functions that are similar in shape to individual atomic orbitals. To do this, we need to consider atomic radial distribution functions, such as the ones shown in Figure 3.2 for hydrogen. These are plots of how the electron density varies at any given distance away from the nucleus. 

Figure 1.2 Radial distribution functions of the valence orbitals in the (a) s-(Mg), (b) p-(Sn), (c) d-(Cr), and (d) f-(Eu) blocks of the periodic table. Black lines correspond to the core density.

Figure 1.1 shows typical plots of the orbital radial functions. However, in order to interpret this information as electron density distribution it is often more useful to consider the surface density functions S(r) plotted in Fig. 1.2... 

Let s take a moment to add just a little more detail to this argument. Figure 14.4 shows a schematic representation of electron density (or electron probability ) versus atomic radius for the Af,Sd, and 6s orbitals. (Electron density here is defined as the function ATTr tf/, sometimes known as the radial distribution function. It gives the probability of finding the electron on the surface of a series of concentric spheres of radius r. You may have studied this in previous chemistry courses but you need not know the mathematical details to understand the following argument.) Note that most of the time the 6s electron, as expected, is found farther away from the nucleus (located at r = 0) than the 4f and 5d electrons. We say the 6s electron is shielded from the nucleus by the intervening 4f and 5d electrons. 

A schematic representation of the electron densities (radial distribution functions, RDFs) of the 4/, Sd, and 6s orbitals. The 6s electron has five small maxima of electron density (or probability) mostly within the 4/ and 5d RDFs. These enable the 6s orbital to penetrate through the filled 4/and 5d electron clouds and therefore experience a greater-than-expected effective nuclear charge. The 6p electron (not shown for reasons of clarity) is similar to the 6s but has one fewer small maximum of electron density. 

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