Angular Wavefunctions

The wavefunctions for any system that is free to move in the spherical polar angles theta and phi (i.e., that has no dependence on these angles in the particle's potential energy function, as in the hydrogen atom or the 3-D rigid rotor) are called Spherical Harmonic functions Y_l,m. These graphs plot the real and imaginary parts of the spherical harmonics for values of the quantum numbers i and m that you choose. They are the building blocks for atomic wavefunctions in general, and their shapes and orientations in space are important to learn. Don't worry about their specific mathematical forms. Instead, concentrate on size, shape, and direction as the quanum numbers l and m change. You can change the values for l and m as you wish, but don't make l bigger than about 6 in order to keep the expression for Y on the page!

Since these are in general complex number functions with real and imaginary parts, it takes two graphs to show them fully, one for each part. The real part of the function is on the left, and the imaginary part (if there is one!) is on the right. The x, y, and z Cartesian axes are in black, and the theta and phi spherical polar angular axes are in red.
Here's a question for you to answer. Do the functions change when m is replaced by -m for any given l value? Can you give a physical reason for your answer?

Chemists generally use linear combinations - hybrids - of the spherical harmonic functions to describe atomic orbital angular shapes and orientations. These hybrids are purely real functions and thus easier to visualize than the spherical harmonics themselves. You have seen some or all of these shapes before in nearly every chemistry course you have taken, and they should be familiar to you. Concentrate here on their shapes, orientations, and nodal properties as you change l and m.
These graphs plot the chemist's hydrogen atom angular functions introduced on page 643 and in Table 18.1. The graphs are spherical polar plots, and you can change the values for l and m as you wish. For high l values, the preset resolution will not be great enough for a super-smooth display, but the symmetry will be clear. Remember that m must be less than or equal to l and only positive m matters for these hybrids.