6.6 Representations of Orbitals

In our discussion of orbitals we have so far emphasized their energies. But the wave function also provides information about the electron's location in space when it is in a particular allowed energy state. Let's examine the ways that we can picture the orbitals.

The s Orbitals

The lowest-energy orbital, the 1s orbital, is spherical, as shown in Figure 6.18. Figures of this type, showing electron density, are one of the several ways we use to help us visualize orbitals. This figure indicates that the probability of finding the electron around the nucleus decreases as we move away from the nucleus in any direction. When the probability function, , for the 1s orbital is graphed as a function of the distance from the nucleus, r, it rapidly approaches zero, as shown in Figure 6.20(a). This effect indicates that the electron, which is drawn toward the nucleus by electrostatic attraction, is unlikely to be found very far from the nucleus.

Figure 6.20 Electron-density distribution in 1s, 2s, and 3s orbitals. The lower part of the figure shows how the electron density, represented by , varies as a function of distance r from the nucleus. In the 2s and 3s orbitals the electron-density function drops to zero at certain distances from the nucleus. The spherical surfaces around the nucleus at which is zero are called nodes.

If we similarly consider the 2s and 3s orbitals of hydrogen, we find that they are also spherically symmetrical. Indeed, all s orbitals are spherically symmetric. The manner in which the probability function, , varies with r for the 2s and 3s orbitals is shown in Figure 6.20(b) and (c). Notice that for the 2s orbital, goes to zero and then increases again in value before finally approaching zero at a larger value of r. The intermediate regions where goes to zero are called nodes. The number of nodes increases with increasing value for the principal quantum number, n. The 3s orbital possesses two nodes, as illustrated in Figure 6.20(c). Notice also that as n increases, the electron is more and more likely to be located farther from the nucleus. That is, the size of the orbital increases as n increases.

One widely used method of representing orbitals is to display a boundary surface that encloses some substantial portion, say 90 percent, of the total electron density for the orbital. For the s orbitals, these contour representations are merely spheres. The contour or boundary surface representations of the 1s, 2s, and 3s orbitals are shown in Figure 6.21. They have the same shape, but they differ in size. Although the details of how the electron density varies within the surface are lost in these representations, this is not a serious disadvantage. For more qualitative discussions, the most important features of orbitals are their relative sizes and their shapes. These features are adequately displayed by contour representations.

Figure 6.21 Contour representations of the 1s, 2s, and 3s orbitals. The relative radii of the spheres correspond to a 90 percent probability of finding the electron within each sphere.

The p Orbitals

The distribution of electron density for a 2p orbital is shown in Figure 6.22(a). As we can see from this figure, the electron density is not distributed in a spherically symmetric fashion as in an s orbital. Instead, the electron density is concentrated on two sides of the nucleus, separated by a node at the nucleus; we often say that this orbital has two lobes. It is useful to recall that we are making no statement of how the electron is moving within the orbital; Figure 6.22(a) portrays the averaged distribution of the 2p electron in space.

Figure 6.22 (a) Electron-density distribution of a 2p orbital. (b) Representations of the three p orbitals. Note that the subscript on the orbital label indicates the axis along which the orbital lies.

Each shell beginning with n = 2 has three p orbitals: There are three 2p orbitals, three 3p orbitals, and so forth. The orbitals of a given subshell have the same size and shape but differ from one another in spatial orientation. We usually represent p orbitals by drawing the shape and orientation of their wave functions, as shown in Figure 6.22(b). It is convenient to label these as the px, py, and pz orbitals. The letter subscript indicates the axis along which the orbital is oriented. (We cannot make a simple correspondence between the subscripts [x, y, and z] and the allowed ml values [1, 0, and -1]. To explain why this is so would be beyond the scope of an introductory text.) Like s orbitals, p orbitals increase in size as we move from 2p to 3p to 4p, and so forth.

The d and f Orbitals

When n is 3 or greater, we encounter the d orbitals (for which l = 2). There are five 3d orbitals, five 4d orbitals, and so forth. The different d orbitals in a given shell have different shapes and orientations in space, as is apparent in Figure 6.23. Notice that four of the d orbitals have "four-leaf clover" shapes, and that each lies primarily in a plane. The dxy, dxz, and dyz lie in the xy, xz, and yz planes, respectively, with the lobes oriented between the axes. The lobes of the dx2 - y2 orbital also lie in the xy plane, but the lobes lie along the x and y axes. The dz2 orbital looks very different from the other four: It has two lobes along the z axis and a "doughnut" in the xy plane. Even though the dz2 orbital looks different, it has the same energy as the other four d orbitals. The representations in Figure 6.23 are commonly used for all d orbitals, regardless of principal quantum number.

Figure 6.23 Representations of the five d orbitals.

When n is 4 or greater, there are seven equivalent f orbitals (for which l = 3). The shapes of the f orbitals are even more complicated than those of the d orbitals. We will not present the shapes of the f orbitals. As you will see in the next section, however, you must be aware of f orbitals as we consider the electronic structure of atoms in the lower part of the periodic table.

In many instances later in the text you will find that a knowledge of the number and shapes of atomic orbitals is important to a proper understanding of chemistry at the molecular level. You will therefore find it useful to memorize the shapes of the orbitals shown in Figures 6.21 through 6.23.