H atom orbitals in 3D
The various pictures of H atom orbitals in the text or elsewhere on our web site are important places to turn to learn how atomic orbital quantum numbers control the size, shape, orientation, and nodal patterns of orbitals in general. It is important to remember, however, that orbitals really are threedimensional objects, and it is often helpful to see threedimensional pictures of them to reinforce their shapes and nodal patterns. Since the World Wide Web's technology is not yet capable of projecting a 3D holographic image into space à la Star Wars movies, we'll have to settle for animations of 3D images. A few of these are collected here in the form of QuickTime movies or static pictures of 3D drawings.
We start with the three 2p orbitals, p_{x}, p_{y}, and p_{z}. The movie below spins these around the z axis. The flat nodal planes, one per orbital because l = 1 and the number of nodal planes equals l, are shown in gray. The orbitals (the wavefunctions) are shown in red and blue to distinguish their algebraic signs. The red parts have the opposite algebraic sign of the blue parts. As you play the movie, you will notice that the p_{z} orbital in the middle doesn't seem to move. That's because this orbital has m = 0, and all m = 0 orbitals are cylindrically symmetric about the z axis; they look the same as they spin around the z axis of symmetry.
If we look at similar pictures for a d and an f orbital, we can see how these pick up one extra nodal plane for each step up in l.
The orbitals shown here have only flat planar nodes, but among the five d and seven f orbitals, one of each type has m = 0, and these are cylindrically symmetric with two (for d) or three (for f) planar nodes, as shown below in comparison to the p_{z} orbital. For the p_{z} orbital, the one nodal plane is the flat xy plane. For the d orbital, the two nodal planes are cones (shown in two different shades of gray). For the f orbital, two of the three nodal planes are cones, but the third is a flat plane, the xy plane again.
The p_{z} orbital (l = 1, m = 0), cylindrically symmetric about the z axis.
The d_{z}2 orbital (l = 2, m = 0), cylindrically symmetric about the z axis.
The f_{z}3 orbital (l = 3, m = 0), cylindrically symmetric about the z axis.
Spherical nodes (also called radial nodes) are difficult to see in a 3D animation of the types above: these nodes look the same from all angles. But we can take advantage of the radial probability function (discussed in the page on H atom sizes) to illustrate and animate these type of nodes. Consider the 3s orbital. With n = 3 and l = 0, it has two spherical nodes. If we start at the origin (at the nucleus) and walk radially outward in any direction, we will see the electron probability rise at first, then fall to zero as we reach the first node, then it rises again and falls when we reach the second node, then it rises rapidly to a maximum value and finally falls to zero as we approach infinite radius. The animation below illustrates this behavior. It plots the radial probability distribution for the 3s orbital superimposed on a sphere with a radius that varies as the animation progresses. The sphere represents the orbital (which is, of course, spherically symmetric), and its color is a shade of green with an intensity proportional to the value of the radial probability funtion at each animation radius. Rather than playing this movie in the usual way, try stepping slowly from frame to frame (use the arrow keys in the controller) to follow the action in better detail.
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