Atomic orbital
Electrons don't orbit like planets; instead, their wave-like nature creates complex standing wave patterns. Explore how these patterns, visualized as orbitals, arise from fundamental wave properties and mathematical solutions. Discover the quantum numbers defining electron behavior and the surprising origins of familiar orbital shapes.
0:00
Although often depicted as little balls
0:02
orbiting the nucleus like planets in the
0:04
solar system, electrons actually exist
0:06
in these strange globules we call
0:09
orbitals, but they seem very strange.
0:12
Why would they just be like this
0:13
orbital? How can they possibly look like
0:16
this? That's what we're going to try to
0:18
tackle today.
0:20
To begin, the first thing I need you to
0:22
know is that everything is waves.
0:25
Electrons, protons, photons, you and me,
0:28
we're all waves.
0:31
But at large scales, these waves behave
0:33
more like particles. So it makes sense
0:35
why we don't really discuss the wavelike
0:38
nature of reality.
0:40
But at very small scales, like the
0:43
volume around an atom, we can no longer
0:45
ignore the wavelike nature of all
0:47
things. and waves behaves or there are
0:51
limitations to how a wave can manifest
0:53
in these very small environments.
0:58
Let's start by looking at a basic
1:00
one-dimensional wave. A wave is
1:02
characterized by its wavelength and
1:04
amplitude. But honestly, the amplitude
1:07
doesn't really change the wave itself.
1:09
If I play a 600Hz sine wave and another
1:12
with greater amplitude, I still call
1:14
them 600 htz. We really only care about
1:17
the wavelength.
1:18
A traveling wave can exist on a spectrum
1:21
of infinite possible wavelengths.
1:24
However, what if I clamp this edge of a
1:26
wave? As I start the oscillation,
1:29
everything looks normal. But when the
1:31
wavefront reaches this end, well, the
1:34
wave medium is clamped and can't move.
1:37
The wavefront wants to excite this point
1:39
but instead suddenly this excitation is
1:42
resisted which causes a snap or force
1:44
applied with the opposite amplitude.
1:47
This force propagates backwards down the
1:49
wave. This is often visualized as a
1:52
reflecting wave.
1:55
If we repeat and I keep the initial wave
1:57
visible, we can see that as the initial
2:00
wave reaches this point at its maximum
2:02
amplitude, the wave should be here. but
2:05
instead it's here. The only way for that
2:08
to happen is if an equally strong
2:10
negative force is exerted on the wave
2:12
medium. This strong negative force
2:15
propagates backwards. When it aligns
2:17
with a negative amplitude of the initial
2:19
wave, it constructs with the total
2:21
amplitude. As you would expect, when it
2:24
aligns with a strong positive amplitude,
2:26
they destruct.
2:28
If we keep this going, eventually we
2:30
stop seeing moving wave fronts and
2:31
instead a standing wave materializes.
2:35
Standing waves occur anytime you
2:37
constrict the boundaries for waves. We
2:39
still have infinite potential standing
2:41
waves as we can freely adjust the
2:43
wavelengths. But what if I also clamp
2:46
the other edge?
2:48
Now our wave is going to behave.
2:52
Because we have clamped both edges,
2:54
there are a finite number of waves that
2:56
can exist here. Mathematically, there
2:58
are still an infinite amount, but in
3:00
reality, it's very much a finite number.
3:04
Because confined waves become standing
3:06
waves, the only standing waves that will
3:08
exist in this space must be motionless
3:11
at the edges. That means all standing
3:14
waves will be some multiple of this wave
3:16
here or the fundamental frequency.
3:20
You can create half fourth and so on
3:22
wavelengths. Each one is a quantized
3:25
variation of this fundamental frequency.
3:27
But if you introduce a frequency that is
3:30
not a quanta of the fundamental
3:32
frequency, well, no you didn't.
3:34
Everything you can imagine here is just
3:36
some combination of quantized waves that
3:38
fit in this space. This is why a guitar
3:42
string always plays the same tones.
3:45
Along with wavelength, standing waves
3:47
possess two more characteristics to
3:49
describe them. These are called nodes
3:51
and anti-nodes.
3:54
Nodes are loces on the wave that don't
3:56
move at all. Whereas anti-inodes are the
3:58
locations where the most movement
3:59
occurs. The number of nodes is how we
4:02
characterize this wave's mode of
4:04
vibration. Adding more nodes changes the
4:07
mode of vibration and increases the
4:09
wave's total energy. For a
4:12
one-dimensional standing wave, mode of
4:14
vibration is characterized by a single
4:16
parameter. We'll call it n for nodes.
4:21
Let's move to the second dimension. In
4:24
the second dimension, we work with a
4:26
drum head. Just like in the
4:28
one-dimensional example, when we
4:29
constrain the vibration of our drum
4:31
head, any wave traveling towards the
4:33
clamped boundary will reflect and
4:35
interfere with itself. A drum head is
4:37
really just an infinite number of
4:39
one-dimensional waves. This means there
4:42
also exists a fundamental or base
4:44
frequency.
4:46
This is the easiest to conceptualize as
4:48
it is just the fundamental frequency of
4:50
a one-dimensional wave with radial
4:52
symmetry.
4:54
The biggest change between one and two
4:56
dimensions occurs with our nodes.
5:01
Nodes are one dimension behind the wave.
5:04
So in our one-dimensional wave, the
5:05
nodes were zero dimensional points. And
5:08
now in two dimensions, our nodes become
5:10
one-dimensional lines.
5:13
The vibrational modes of our 2D waves
5:16
are also defined by the number of nodes
5:18
or the parameter n. But when we increase
5:21
this parameter, we see we always get
5:23
radial symmetry. If a drum head truly
5:26
was an infinite amount of
5:27
one-dimensional waves, then we should be
5:29
seeing a wave like this somewhere. This
5:32
would mean the center of our drum head
5:34
would be motionless and a node. But we
5:37
can't have a point node at the center
5:38
since nodes in two dimensions are lines.
5:41
We can however draw a line straight
5:44
across. Now we find the waves we are
5:46
expecting. This new line is called an
5:49
angular or linear node. And just as we
5:52
can have multiple radial nodes, we can
5:54
have multiple angular nodes.
5:57
The vibrational mode of our 2D surface
5:59
is characterized by two parameters. N
6:02
dictates the number of radial nodes and
6:04
M dictates the number of angular or
6:07
linear nodes. I want to highlight that a
6:10
vibrating drum head doesn't just vibrate
6:12
in one of these patterns. In fact, it's
6:14
almost certainly a combination of dozens
6:16
of different patterns. But whatever is
6:19
happening in a drum head, you will
6:21
always be able to decompose those
6:22
vibrations into these modes of
6:24
vibration.
6:26
Here's a nice website that lets you
6:27
monkey around and see how that would
6:29
look like if you add different modes of
6:31
vibration together.
6:33
Let's move to the third dimension. Here
6:36
things get a little difficult to
6:38
conceptualize.
6:40
Earlier when we talked about
6:42
one-dimensional waves, we visualized
6:44
their excitations in the second
6:46
dimension. And for two-dimensional
6:47
waves, we visualize those excitations in
6:50
the third dimension. So with
6:52
three-dimensional waves, how do we
6:54
visualize those excitations?
6:58
As we have moved up a dimension, so too
7:01
have the nodes. Now nodes exist as
7:04
two-dimensional surfaces. And just like
7:06
with our drum head, there are two types
7:08
of nodes. radial and angular except
7:12
these are no longer the parameters for
7:14
the vibrational mode like we saw in the
7:16
previous dimensions. I'll come back to
7:19
that.
7:21
Radial nodes are simply shells. These
7:24
are easy to imagine as they're the
7:25
radial nodes of the drum head rotated
7:28
into a sphere. Each shell means a
7:31
spherical surface where the amplitude of
7:33
the wave is always zero. If we were to
7:36
cut our sphere in half and visualize
7:38
excitation with color, then this
7:40
behavior would look like a drum head.
7:42
However, this kind of radial behavior is
7:46
only found in acoustic spherical
7:48
harmonics or within a physical medium.
7:51
This wave oscillates between higher and
7:53
lower pressure as the particulates of
7:55
the medium compress against and expand
7:57
away from each other.
8:00
In quantum mechanics, we're not dealing
8:02
with a physical medium or collections of
8:04
particles. Instead, we're working with a
8:07
single quantum object or electron, which
8:10
does behave like a wave, but also
8:13
behaves like a particle. In order for us
8:16
to say the amplitude at a given point is
8:18
positive or negative, we need to know
8:20
about the phase of the electron's wave
8:23
function. However, when we measure an
8:26
electron, that interaction collapses
8:28
this wave function into a tiny region we
8:30
call a particle. This isn't really
8:33
useful for determining the phase of the
8:35
wave function. So, we can't.
8:38
For this reason, spherical harmonics in
8:41
quantum mechanics represents the
8:43
probability that at any given moment in
8:46
time, if we measure the location of the
8:48
electron, it will be in that location.
8:52
These values are called probability
8:54
densities.
8:57
If we look at an oscillating wave, we
8:59
see that it spends an equal amount of
9:01
time positive as it does negative. So,
9:04
if we can't know the phase of the wave,
9:07
we can just square our wave's maximum
9:09
amplitude.
9:11
Now, we know where the wave is
9:12
oscillating most even if we don't know
9:14
the phase. This represents the
9:17
probability density and is what we do
9:19
for electrons.
9:21
These waves or probability densities are
9:24
static and do not change with time. If I
9:27
now solve the radial component of the
9:29
Schroinger equation for hydrogen
9:31
electrons and spawn points based on
9:33
these probability densities, we can
9:36
visualize electron orbitals.
9:39
These spheres correspond to the s
9:41
orbitals around a nucleus. I've got s1
9:44
through 3 plotted here. We now have the
9:48
same dilemma with our drum head. We have
9:50
these clear nodes with radial symmetry,
9:53
but none that go through the middle.
9:56
Well, just like how we could trace a
9:58
line through the middle of our drum
9:59
head, we can place a plane or angular
10:02
node in the middle of our sphere.
10:05
Adding angular nodes is when spherical
10:08
harmonics start to get complicated.
10:12
Spherical harmonics or the solution to
10:14
these 3D waves is the product of the
10:17
radial and angular components for those
10:19
waves. Visualizations almost always
10:23
exclusively focus on this angular
10:25
component because as we saw the radial
10:28
components are kind of trivial. However,
10:31
if we look at this equation here, we see
10:34
that the angular component does not
10:36
depend on the radius which is a very
10:38
important caveat to highlight.
10:41
What do I mean by this?
10:43
Previously on a drum head, we can see
10:46
that as we move away from the center,
10:48
the amplitude of the wave changes with
10:51
the radius. We also see this in the
10:53
radial component of the spherical form.
10:56
The amplitude changes with radius.
10:59
However, what if I change the equation
11:01
to be solely based on the radial angle
11:03
or a variable called theta? Well, now
11:06
the value is constant as I move away
11:09
from the center. This function is no
11:11
longer dependent on the radius.
11:14
This is what is going on with the
11:15
angular component of spherical
11:17
harmonics. Solving the function here or
11:20
here gives you the same value. Which
11:23
means these visualizations are
11:25
confusing.
11:26
This style is nice for showing the
11:28
internal oscillation of the amplitudes.
11:30
But where is the sphere? And this style
11:33
is great because it shows the sphere.
11:36
except I thought the radius of a sphere
11:38
was a node. So why is the amplitude not
11:41
zero at the sphere's surface?
11:44
Let's answer this by visualizing the
11:46
angular component of this equation. We
11:49
have our sphere. We place an angular
11:52
node in the middle. This now means the
11:54
pressure or amplitude of our acoustic
11:56
wave will oscillate back and forth. Not
11:59
super exciting. animating this
12:01
oscillation in volumetrics isn't very
12:04
convenient and so no one does it.
12:06
Instead, what you do is take a sphere,
12:09
solve the angular component for this
12:10
location, then you either color that
12:13
part of the sphere based on that value,
12:16
displace that part of the sphere along
12:17
the normal by that value, or you can set
12:20
the radius of the sphere equal to the
12:22
magnitude of the angular component, but
12:25
color it to indicate if it's positive or
12:27
negative.
12:28
Now you can see why each style is valid
12:31
and also why they are misleading. They
12:34
represent how vibrations within a sphere
12:37
are oscillating along the azimuthal and
12:40
polar axes but don't actually illustrate
12:42
the internal 3D structure.
12:47
What about when we add additional
12:49
angular nodes?
12:51
The vibrational mode of electron
12:53
spherical harmonics is dictated by three
12:55
values or quantum numbers N, L and M.
13:00
These three numbers describe the angular
13:03
and radial nodes. N represents the
13:06
principal quantum number. We can think
13:08
of this as the energy level or number of
13:10
radial shells. You can also think about
13:13
this as a row on the table of elements.
13:16
L is the azimuthal quantum number and
13:18
represents angular momentum of the
13:20
electron. It defines the total number of
13:23
angular nodes and also corresponds to
13:25
the S, P, D, and F orbitals. You may be
13:28
familiar with m is the magnetic quantum
13:32
number and represents the degree to
13:34
which the angular momentum of the
13:35
electron is situated along the z-axis as
13:39
well as the number of angular nodes
13:40
along the z-axis.
13:43
I realize that probably doesn't make
13:45
sense. So, let's walk through each
13:47
number and I can explain along the way.
13:50
We're going to ignore the principal
13:52
quantum number since it doesn't change
13:54
the shape. We will just look at how
13:56
changing L and M affect the orbitals.
14:00
We begin with L equals 0. This is the S
14:03
orbital. As L represents the total
14:05
number of angular nodes, there are none.
14:08
As there are no angular nodes, M is also
14:11
zero. Since M represents the number of
14:13
angular nodes along the Zaxis
14:17
at L= 1, we now have one angular node.
14:20
This divides our sphere into two loes,
14:22
and our wave oscillates back and forth.
14:26
This is the P orbital.
14:28
Since we now have angular nodes, it's
14:31
possible they can be oriented
14:32
differently.
14:34
M represents the number of angular nodes
14:36
along the z-axis. So we can have either
14:39
one or zero z-axis aligned angular
14:42
nodes.
14:43
Now m equals 1.
14:46
But this value can equal positive or
14:48
negative 1. Why is that?
14:52
This is because the magnetic quantum
14:54
number represents the alignment of the
14:57
angular momentum along the zaxis.
15:00
If we trace an axis along an orbital, we
15:03
can see how similar that is to the
15:05
zaxis.
15:07
Imagine a yo-yo. If we provide no
15:10
angular momentum, the yo-yo dangles
15:12
straight down along the zaxis. This is m
15:15
equals zero.
15:17
If we provide some angular momentum, the
15:19
yo-yo starts to move and point away from
15:22
the zaxis.
15:24
Conceptually, this is what is happening
15:25
with the magnetic quantum number.
15:27
However, the amount of angular momentum
15:30
we can supply is quantized. So with L
15:33
equals 1 or one angular node, the yo-yo
15:36
either dangles or spins completely
15:39
horizontally. There's no in between
15:41
state. The reason this value is either
15:44
positive or negative is because although
15:46
this shape is symmetrical, the actual
15:49
equation says one lobe is positive and
15:51
one is negative. But this is arbitrary.
15:54
There's no reason it should be this way
15:56
or this way.
15:58
Despite being the same shape, these are
16:01
two different states. Each suggests the
16:04
electron should be located at opposite
16:06
poles which have completely different
16:08
angular momentum. Therefore, despite
16:10
occupying the same volume, this orbital
16:13
represents two different angular
16:15
momentum.
16:17
Why then call it the magnetic quantum
16:19
number when it describes angular
16:21
momentum?
16:23
This is because physicists measure m by
16:26
applying a magnetic field to the atom.
16:29
This would either increase or decrease
16:31
the energy of the electron in the
16:32
orbital. Hence, positive or negative
16:35
values. Orbitals aligned with the z-axis
16:38
would experience no change in energy. So
16:41
that is why m=0 refers to orbitals
16:44
aligned with the zaxis. Horizontal
16:47
orbitals experience the greatest change
16:49
in energy in a magnetic field and so the
16:52
maximum value of M refers to this
16:55
horizontal orientation.
16:58
Let's add another angular node with L
17:02
equ= 2. We have two angular nodes. This
17:05
state represents the d orbital. And
17:07
since we have two angular nodes, we can
17:09
now move one or both of these to align
17:12
with the z-axis.
17:14
As we move nodes to the Z-axis, the
17:16
orbitals start migrating out like our
17:19
spinning yo-yo. When all nodes are
17:21
aligned with the Z-axis, all of the
17:24
orbitals are found oriented
17:25
horizontally.
17:28
With L= 3, we have three angular nodes.
17:32
This is the F orbital. Again, moving
17:35
nodes to the Z-axis causes orbitals to
17:38
fan out.
17:40
If we continue this process, we see a
17:42
clear pattern emerge. The m=0 orbital is
17:46
simply this vertically oscillating
17:47
state. And as we increase m, the
17:50
orbitals spread out and align themselves
17:52
more with the horizontal plane.
17:55
But since in chemistry, natural elements
17:57
don't go past L= 3, this pattern isn't
18:01
clear and obvious, especially if
18:03
chemistry is the context in which you
18:05
are learning about these orbitals.
18:09
The last caveat to clear up is the
18:11
difference between complex and real
18:13
orbitals.
18:15
What you've seen so far, the familiar
18:17
dumbbell and cloverleaf shapes that are
18:19
used in every chemistry textbook are
18:21
what we call the real orbitals.
18:24
However, they aren't the direct
18:26
mathematical solutions of the Schroinger
18:29
equation.
18:31
Let's go back to our L= 1 or P orbitals.
18:35
We said m= 0 corresponds to an orbital
18:38
aligned along the zaxis. Simple enough.
18:42
But we also said m plus or minus1
18:45
corresponds to orbitals that are
18:47
oriented horizontally.
18:50
If you solve the math directly, the
18:52
orbitals for nonzero m values don't look
18:55
like dumbbells. Instead, they look like
18:58
donuts or tauruses.
19:00
These are the complex orbitals. The
19:03
reason they're called complex is because
19:05
they are described by complex numbers.
19:07
Numbers that have both a real and an
19:09
imaginary part.
19:12
You don't need to understand the math.
19:14
But when you rotate real waves or waves
19:17
described by s or cosine, their
19:19
amplitude changes at 90°. There is no
19:23
wave. There is no symmetry.
19:26
If we give a wave rotational symmetry,
19:29
then we can decompose that wave into two
19:32
complimentary components. We can use
19:34
these two waves to describe this wave no
19:37
matter the rotation.
19:40
Adding the y component of this wave to
19:42
this wave causes them to construct or
19:45
destruct into our original wave.
19:48
These complimentary components are the
19:50
real and imaginary parts of spherical
19:53
harmonics. Both are the exact same wave
19:56
but just rotated 90° from each other so
19:59
as to preserve symmetry. However, as you
20:02
may have noticed, rotating the wave
20:04
causes the phase to change.
20:07
Earlier I said we can't know the phase
20:10
of electrons. But with this solution,
20:12
since rotation represents a change in
20:15
phase, we can represent phase by
20:17
coloring a continuum rotating around the
20:19
orbital. positive values going one way
20:22
and negative values going the other. Now
20:25
that we can visualize the rotational
20:27
symmetry of electron orbitals, it's a
20:30
little easier to apply our yo-yo example
20:32
from earlier to describe how angular
20:34
momentum reshapes them.
20:37
These complex orbitals are a more
20:40
fundamentally accurate depiction of the
20:42
state of electrons around a nucleus.
20:45
So why don't we use these doughut shapes
20:48
in chemistry?
20:50
While they are mathematically pure, they
20:52
don't point in specific directions like
20:54
the x and y axis. This makes it hard to
20:57
visualize how they would overlap to form
20:59
chemical bonds.
21:01
Generally, there's more utility in
21:04
visualizing the orbitals like this.
21:08
So to come back to our original
21:11
question, why do electron orbitals look
21:13
so strange? It's because electrons are
21:17
waves and when you confine a wave to the
21:19
tiny 3D space around a nucleus, it can
21:22
only exist in specific standing wave
21:24
patterns. These patterns are dictated by
21:27
mathematical solutions called spherical
21:29
harmonics.
21:31
The weird lobes and spheres we see are
21:34
simply the regions in space where you
21:36
are most likely to find the electron for
21:38
a given standing wave pattern.
21:41
And the familiar shapes used in
21:43
chemistry are just convenient real
21:45
valued combinations of the more
21:47
fundamental complex rotating wave
21:50
solutions.
21:51
They are two different but equally valid
21:54
ways of describing the weird wacky world
21:57
of the electrons quantum nature.