p orbitals
When l ≠ 0 the wavefunctions are not spherically symmetric; they depend upon both angular coordinates q and f, and it is therefore much more difficult to convey size and shape on a printed page. For l = 1 (p orbitals) ml = 0, ±1, hence there are three p-orbitals for each value of n. The angular part of the p-orbitals is given by the three spherical harmonics Y1, 0, Y<sub>1, +1 and Y1, -1. The simplest of these is
Y1, 0 (θ, φ) = √3/4π cos θ
Angular functions such as this can be expressed in the form of a polar plot on polar graph paper, where the value of cos θ, for example, is plotted along the radial line labelled by angle θ, as in the figure on the right.
A polar plot does not give the shape of the orbital (this must include Rn, l (r)), but it shows the direction or orientation of the orbital. In particular we can see that Y1, 0(θ, φ) is directed along the z axis, hence the orbital with angular dependence Y1, 0 is called a pz orbital.
Another common representation of orbitals is as 3-D figures. Since Y1, 0 is independent of θ, a 3-D representation can be obtained from the polar plot by rotating it about the vertical axis in the figure.
The angular functions with ml ≠ 0 are more difficult to represent pictorially, as they not only depend upon q and f, but have imaginary parts as well:
Y1, +1 (θ, φ) = √3/8π sin θe+i, φand
Y1, -1 (θ, φ) = √3/8π sin θe-1, φ
Because Y1, +1 and Y1, -1 are degenerate, any linear combination of the two must also be a valid wavefunction with the same energy. It is customary to use the combinations given on p. 440 of the text which yield real wavefunctions.
3-D figures of all three real 2p orbitals are no doubt familiar to you- they feature prominently in the first chapters of most organic chemistry textbooks, for instance. Such plots show the surfaces of the orbitals within which the probability that the electron will be located is 90%, or some other high value. They are actually approximations to these 90% probability surfaces, and you will find quite different plots in various texts which claim to be representations of the same surface! Note that Engel and Reid do not actually give any 3-D figures: instead, in Figure 20.7, they show a two-dimensional contour plot of the 2py and some other orbitals. This is a good reminder that orbitals really don’t have ‘surfaces’- any atomic orbital you care to name is non-zero everywhere in the universe.
