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 three-dimensional objects,
and it is often helpful to see three-dimensional
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,
px, py, and
pz. 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
pz 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
pz orbital. For the
pz 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 pz orbital (l = 1, m = 0),
cylindrically symmetric about the z axis.
The dz2
orbital (l = 2, m = 0), cylindrically symmetric
about the z axis.
The fz3
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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