Hello and welcome back to The Deep Dive.
I'm your host and today we are tackling a subject that I think
really rewires the way you look at the physical world.
It's a fundamental shift in perspective, isn't it?
It really is. You know, we spend so much time
in school, in chemistry class, looking at these diagrams of
molecules. You open a textbook and what do
you see? These ball and stick models.
Right, they're static. They're rigid, frozen in place.
Exactly. They look like little
architectural statues sitting in a void.
Very clean, very peaceful. It's what I like to call the
museum view of chemistry. Everything is perfectly still,
perfectly posed under glass. A museum view.
I love that. But the source material we're
covering today, this is Lecture 4 from Professor Pedro Camargos
course on structural methods in inorganic chemistry.
It basically takes a sledgehammer to that entire
museum. It really does.
It shatters the illusion of stillness.
It tells us that if you could actually shrink yourself down to
the nanoscale and, you know, look at the molecule, you would
not see a statue, you'd see mosh pit.
A mosh pit is a very evocative way to put it, but yes, that is
honestly closer to the truth than the statue.
It feels that way. The atomic world is just defined
by constant, rhythmic, almost chaotic motion.
Bonds are stretching, angles are bending, entire molecules are
tumbling and twisting through space.
It is a frantic dance that never, ever stops.
And that is the hook for today's deep dive.
We're exploring what we're calling the mathematics of the
wiggle. The Wiggle.
I like it. Specifically, how do scientists
actually see this dance? Because let's be real, we don't
have a microscope powerful enough to just record a video of
a silicon atom doing the cha cha.
No, we certainly do not. The wavelength of visible light
is far, far too large to resolve individual atomic bonds moving
in that way. So.
You can't watch the dance directly.
You can't. So we have to act like
detectives. We have to listen to the music,
so to speak. We measure the energy of those
vibrations and this entire field, this science, is called
vibrational spectroscopy. Which sounds incredibly
technical, and I won't lie to you, our listener, We're going
to get into some pretty heavy weeds today.
We've got quantum mechanics, we have matrix algebra, and we have
something called group theory. The trifecta of scary sounding
topics. Right.
But the mission here is to break this down so that by the end of
the hour, you can look at a spectrum, you know, those
squiggly lines on a graph, and understand that it's not just
noise, it's actually a map of symmetry and quantum physics.
That's the goal, and it's a beautiful journey because it
connects two very different worlds.
We start with very simple, intuitive physics, like weights
on a spring, something you can picture in your garage.
OK, very tangible. And we end up with some pretty
abstract mathematics things called character tables and
irreducible representations to predict exactly how light is
going to interact with matter. So let's lay out the road map
for everyone. First, we're going to talk about
the bond itself. Why on earth do we treat atoms
like balls on a spring? What's the physics there?
The foundational analogy. Then we're going to look at what
the source calls the great divide in this field.
This is the 2 main ways we can detect these vibrations.
Infrared spectroscopy or IR and ramen spectroscopy.
The two detectives, they're working the same case, but
they're looking for completely different clues.
I like that. And then we get to the main
event. We're going to walk through,
step by step, the actual algorithm that chemists use to,
well, predict the future. It really is a form of
prediction. We're going to take a specific
molecule, silicon dichloride dihydride, or CCL 2H2, and use
symmetry to calculate exactly what its spectrum should look
like before we even do an experiment.
It's a classic example. It's complex enough to be
interesting, but simple enough that we can actually do the math
together if we just take it slow.
And finally, we'll talk about why this all matters.
I mean, why do we care about the wiggle in the 1st place?
So let's dive in. Section 1.
The bond is a spring. The lecture notes start right
here with this fundamental analogy, the harmonic
oscillator. This is the bedrock of the
entire field of vibrational spectroscopy.
To even begin to understand a chemical bonds vibration, you
have to strip away the complexity for a moment.
Forget about electrons and orbitals and just think about
mechanics. OK, pure physics.
Imagine 2 heavy balls, like 2 steel ball bearings, and they're
connected by a metal spring. OK, I'm visualizing it. 2 steel
balls, 1 spring connecting them. Now if you pull them apart and
just let go, they don't just snap back to the middle and
stop. Right now, they keep going.
They have momentum, they overshoot the center, compress
the spring and they get pushed back out.
And they overshoot again. They oscillate back and forth,
back and forth. And they oscillate at a specific
speed, don't they? A specific frequency?
It wouldn't be random. Precisely.
And that frequency is determined by just two things.
First, the mass of the balls. Heavier atoms are more sluggish,
they move more slowly. It's just harder to reverse
their direction. Mass makes sense.
What's the second thing? The stiffness of the spring.
In chemistry, we call this the force constant of the bond.
It's just a measure of how strong that bond is.
So let me see if I've got this. A single bond between two atoms
would be like a loose kind of slinky spring, and a triple bond
would be like a really stiff suspension coil from a car.
That is the perfect analogy. A triple bond between 2 carbon
atoms is incredibly stiff. It snaps back and forth very,
very quickly. That's a high frequency
vibration. Well, a single bond is looser,
so it vibrates at a lower frequency.
Exactly. This all makes perfect sense in
a classical physics way. If I have a spring in my garage,
I can understand this intuitively, but the lecture
notes immediately pivot to the quantum reality and this is
where things start to get weird. The garage analogy breaks down.
It has to because in your garage you can pull that spring any
distance you want. You can give it any amount of
energy in a continuous spectrum. You can pull it 1mm or 1.1mm or
1.11. Right, it's analog.
But a molecule is digital. It is quantized.
Meaning you can't vibrate at just any energy level.
It can't. The molecule has specific
discrete energy states that it is allowed to inhabit.
We call these vibrational levels, and we label them with a
quantum number V. So you have the ground state
which is V-0, the first excited state V1, the 2nd V2 and so on
up. So it's a ladder.
You can stand on rung one, or you can stand on rung 2, but you
cannot hover in the space between the rungs.
That's it. You cannot.
And when a molecule absorbs energy to vibrate more
vigorously, it doesn't ramp up at speed gradually.
It essentially teleports from, say, rung 0 to to run one.
It's an instantaneous quantum jump.
Now there is an equation in the source material for the energy
of these rungs. It says EV equals H Omega times
and then in ( v + 12. I want to stop and look at that
part in the parentheses, the V + 1/2.
It's a very specific and a very, very important term.
Because if I'm at the absolute bottom of the ladder, the lowest
possible energy, the ground state, where V is 0, the energy
isn't 0. No, it's not.
The math says the energy is 1/2 times any Omega.
What is that? You've just found the zero point
energy. This is one of the most profound
and frankly bizarre concepts in all of quantum mechanics.
So a molecule can never stop moving.
It can never ever have 0 energy. It can never sit perfectly
still. Not even at absolute 0 if we
could freeze the entire universe down to 200 two 173.15°C.
Even then, if you could somehow cool a single molecule to
absolute zero, it would still be vibrating with that 1/2 unit of
energy. It would still be wiggling.
Why is nature just fundamentally jittery?
It's because of the Heisenberg Uncertainty principle.
Quantum mechanics simply forbids us from knowing both the exact
position and the exact momentum of a particle at the same time.
If a molecule stopped vibrating completely, what would we know?
We know exactly where the atoms were, their position, and we
know their speed. Their momentum was 0.
We would know both things perfectly.
Know too much? We would violate the fundamental
laws of physics. So to keep the universe legal,
nature forces everything to keep moving just a little bit.
It's the zero point wiggle. That is slightly unsettling, the
idea that this chair I'm sitting on, the air I'm breathing, the
very neurons in my brain, nothing is ever truly at rest.
It's all shimmering. It is a shimmering universe.
Now we have to complicate the spring model just a little bit
more. The harmonic oscillator.
That perfect spring is what we call a useful lie A.
Useful lie. I like that.
It works really well for small vibrations right at the bottom
of that energy ladder, but real bonds aren't perfect springs.
The lecture calls this the reality check or the an harmonic
oscillator. Right.
Think about your garage spring again.
If you hook one end to a solid wall and the other into a truck,
and you start to drive the truck away, what happens?
Oh, the spring stretches, then it deforms permanently and
eventually it just snaps. Exactly.
A perfect harmonic oscillator would never snap.
Its energy just goes up and up forever in a perfect parabolic
curve. But real chemical bonds have a
breaking point. And we describe this with a
different curve, the Morse potential.
The Morse potential. It sounds like something from a
spy movie, doesn't it? We've activated the Morse
potential, Mr. Bunk? Yeah.
Exactly, but it's really just a graph of energy versus the
distance between the atoms. On the left side of the graph,
where you try to push the atoms too close together, the energy
just shoots up wildly. Because the nuclei are both
positive, they repel each other. They don't want to touch.
Great. There's a steep wall of
repulsion. But on the right side of the
graph, as you stretch the bond out, the curve doesn't go up
forever. It starts to flatten out.
It approaches a plateau. And that plateau is the bond
breaking. That is the dissociation energy.
We call it day. Once you put more energy into
the bond than that value, the spring snaps.
The atoms just fly apart. They're no longer a molecule.
So the Morse potential is the map of a breakable spring.
What does that do to our ladder of energy levels?
The rungs? Well, in the perfect spring
model, the harmonic oscillator where the rungs on the ladder
are all evenly spaced, the energy jump from step one to two
is the exact same as the jump from step 10:50.
OK, perfectly regular. But in the Morse world, the real
world, the steps get smaller as you go up.
They get squished together. Exactly as you go higher up the
ladder, as you get closer and closer to that breaking point,
the rungs get squeezed closer and closer together.
This effect is called in harmonicity.
There's not a perfect ladder. It's not.
It complicates things for higher energy vibrations, but for our
purposes today, we usually focus on that very first biggest jump
from the ground state V-0 to the first excited state V1.
That's the fundamental vibration and it's the one we see most
often, OK. So we've established our
physics. We have atoms on breakable
springs, they're vibrating on quantized rungs of a ladder, and
they can never, ever stop. Now let's talk about detection.
How do we actually measure this dance?
The lecture introduces the two methods, infrared or IR and
Roman. The two detectives.
The lecture calls this the Great Divide.
Why do we need two different ways to measure what is
essentially the same thing, a vibration?
Because they don't see the same things, they interact with the
molecule through entirely different physical mechanisms.
They're looking for different clues, so they are
complementary. You need both to solve the case.
Let's start with infrared spectroscopy IR.
I feel like this is the one most people encounter first in a
chemistry course. It's the standard.
The workhorse IR spectroscopy is based on a simple mechanism
absorption. You can think of it as a theft.
A theft of energy. Recisely you shine a beam of
infrared light which is basically just radiant heat
through your sample of molecules.
Now if the frequency of a photon of that light exactly matches
the frequency of a bonds vibration.
The molecule eats the photon. It eats the photon.
It absorbs that specific packet of energy to jump up 1 rung on
its vibrational ladder. OK, that seems straightforward
enough, but there's a catch, isn't there?
The lecture talks about a selection rule.
Yes, the molecule is picky. It can't just absorb any photon
just because the frequency matches.
It needs a handle to grab onto the photon's energy.
And for IR the rule is there must be a change in the
molecules dipole moment. Correct.
A dipole moment is just a measure of charge separation in
a molecule. Is there a positive end and a
negative end? Let's take a simple molecule
like carbon monoxide, CEO. Oh yeah, CEO Oxygen is what we
call very electronegative. It's an electron hog.
It pulls the bonding electrons towards itself, making the
oxygen end slightly negative. Which leaves the carbon end
slightly positive. Exactly.
So the molecule is like a tiny little magnet, but for electric
charge instead of magnetism, it has a permanent dipole.
Now imagine that bond stretching and compressing.
The distance between the positive and negative ends is
changing. Right, the strength of the
dipole is oscillating. It's getting bigger and smaller,
bigger and smaller. And light itself is an
oscillating electromagnetic wave.
Bingo. Because the molecule now has an
oscillating electric field of its own.
That changing dipole it can couple with the oscillating
electric field of the light wave.
The light has a handle to grab onto.
It can transfer its energy. So if a vibration causes the
charge distribution to shift, IR can see it.
But what if there's no handle? Let's consider oxygen gas O22
identical oxygen atoms. They pull on the electrons with
equal strength. Perfectly equal.
There's no positive end, no negative end.
The dipole moment is 0. And if they stretch apart and
come back together? It's still perfectly symmetric.
0 is still zero. There is no change in the dipole
moment, no handle. So you can shine as much
infrared light as you want at a sample of pure oxygen gas and it
will never absorb it. It's just transparent.
Correct, it is completely transparent to IR radiation.
This is actually a very very good thing for life on Earth.
Also, nitrogen N2 and oxygen O2 make up 99% of our atmosphere.
Neither of them has a changing dipole when it vibrates.
Neither absorbs IR. If they did, they would trap
heat like crazy and the greenhouse effect would turn
Earth into Venus. It's the trace gases that are
the problem. It's the lopsided ones, the ones
with dipoles, carbon dioxide, water vapor, methane.
They have the handles to grab that outgoing heat radiation.
That's a fascinating connection to climate science.
So IR is the detective that looks for electrical imbalance.
Now what about the other detective, Ramon?
Spectroscopy. Rahman Rahman is the
sophisticated, slightly more mysterious cousin.
It does not rely on absorption at all.
It relies on scattering. Scattering just means the light
bounces off. Right.
Essentially, yes. In a ramen experiment you don't
use infrared light. You hit the molecule with a
sledgehammer of light, usually a very bright monochromatic laser,
often in the visible range, like a green or red laser.
OK, much more energetic approach.
Much more now 99.999% of those laser photons will simply hit
the molecule and bounce off with the exact same energy they came
in with. That's called Rayleigh
scattering, I think. It is.
It's elastic scattering, it's boring, it's the reason the sky
is blue. But it tells us nothing about
the vibration. Just filter that light out.
We are looking for the one in a million photon that does
something different. The special one, the rum and
photon. Exactly.
Sometimes a photon will hit the molecule and interact with its
electron cloud, and in that brief interaction it loses a
tiny bit of energy to the molecule.
The molecule steals a quantum of energy enough to jump up 1
vibrational rum. So the photon bounces away, but
it's tired. It has less energy than it
started with. Which means it's frequency has
shifted. It's color has shifted slightly.
It becomes redder. This specific process is called
stoke scattering. By measuring exactly how much
energy the photon lost, we know exactly how much energy the
vibration required. It's a very indirect way of
measuring. So what's the selection rule
here? IR needed a changing dipole
moment. What does Rahman need?
The notes say it needs a change in polarizability.
Polarizability. It's a mouthful.
I know. Essentially, it's a measure of
how squishy the electron cloud of the molecule is.
Squishy. Think of the cloud of electrons
around the molecules atoms like a balloon.
Can you distort it? Can you stretch it?
Can you compress it? In an electric field?
That's polarizability. Big floppy molecules with lots
of electrons far from the nuclei are very polarizable.
OK, so how does a vibration change the squishiness?
Imagine a perfectly symmetric molecule like that O2 molecule.
Again, imagine it breathing the bond, stretching and contracting
symmetrically. When it expands, the electrons
are held a bit further from the nuclei.
They're looser, more squishy. And when it contracts, they're
pulled in tighter, less squishy. So the squishiness itself, the
polarizability is oscillating with the vibration.
And that's the handle for Rahman.
That's a handle. So as a general rule, Rahman
spectroscopy loves symmetry. It loves big symmetric breathing
motions that change the overall size or shape of the electron
cloud. IR, on the other hand, loves
asymmetry, lopsided pulling and bending that changes the charge
balance. So that really is the great
divide. IR sees the lopsided stuff,
Rahman sees the symmetric stuff. And that is why we need both.
If you only used IR, you would be completely blind to the
symmetric vibrations of many, many molecules.
If you only used Rahman, you might miss some very important
asymmetric stretches. Together they give you the
complete picture of the dance. OK, so we have our tools, we
know the physics, we have our two detectives.
But now we run into the problem of complexity.
If I have a simple molecule like AO or HCLI, can guess the
vibration, it just stretches 1 vibration.
Easy. But molecules are rarely that
simple. Right.
As soon as you add more atoms, the number of possible wiggles
just explodes. The notes give us a formula for
this three and the six. It's a simple but powerful
counting rule. N is just the number of atoms in
your molecule. Each of those N atoms lives in a
3D World, so it can move in three directions, XY and Z.
That gives us a total of three times north, or three N total
degrees of freedom. Every possible motion is in that
number. But not all of those are
vibrations. That's a key point.
It's a critical .3 of those motions correspond to the whole
molecule just flying through the room.
That's translation along the XY and Z axis.
We don't care about that. It's not a wiggle.
It's not an internal wiggle, and for a non linear molecule 3 more
of those motions correspond to the whole thing tumbling end
over end like a thrown football. That's rotation around the XY
and Z axis. We don't care about that either.
So we start with three in total motions, and we subtract those
six whole body motions. 3 translations, 3 rotations.
And what's left? The three N 6 must be the
internal vibrations, the wiggles.
So let's introduce our case study for today.
The molecule is silicon dichloride dihydride, that's
Secl 2H2. Let's visualize it.
You have a silicon atom sitting in the center.
Attached to it, pointing sort of forward, are two chlorine atoms,
and pointing sort of backward are are two hydrogen atoms.
It's shaped like a distorted tetrahedron.
So we have 5 atoms total, the silicon, 2 chlorines, 2
hydrogens. Let's do the math. 3 * 5 is 15
total motions. 15° of freedom. Subtract the six for translation
and rotation. That leaves 9. 9 distinct
vibrational modes. 9 fundamental wiggles.
And trying to visualize 9 different ways a pyramid can
wiggle is really difficult. I can imagine a stretch, maybe a
bend, but nine of them all happening and coupled together.
It's impossible to rely on intuition alone.
You might guess 3 or 4 the obvious ones, but you will
absolutely miss the others and more importantly, you won't have
any idea which ones are IR active and which are ramen
active. This is where we need the
algorithm. We need a systematic, foolproof
way to predict all nine vibrations and their activity.
This brings us to Section 3, Symmetry and Group theory.
This is the part that usually scares chemistry students.
It involves matrices and Greek letters and character tables,
but at its core it is just a very rigorous way of quantifying
shape. We start by assigning the
molecule to something called a point group.
For our molecule CCL 2H2, the group is called C2V.
What does that label actually mean?
The label C2V is basically shorthand.
It's a code that describes the complete set of symmetry
operations that the molecule possesses.
And a symmetry operation is a move you can make to the
molecule that leaves it looking unchanged.
It's like a magic trick. If you, the observer, close your
eyes, I perform the operation and you open your eyes again.
You can't tell that I did anything.
The molecule looks identical to how it started.
OK, that makes sense. For this C2V group, there are
four specific moves allowed and we need to understand them
because we're going to use them as our tools in the calculation,
OK? Let's walk through the four
moves. The first one listed is EE.
Stands for the identity operation.
It comes from the German word Einheit, which means unity.
It basically means do nothing. Wait, why is do nothing a move?
It seems kind of pointless. It seems that way, but in
mathematics, and especially in Group theory, you always need a
unit. You need a one.
It's the baseline against which everything else is measured.
If you do nothing, every atom by definition stays exactly where
it is. OK, I can accept that move
number 2 is C2. This is a rotation.
Right. Imagine an axis, a skewer
running vertically right through the central silicon atom.
This is our main Z axis. The C2 operation means we rotate
the entire molecule 180° around the skewer.
So if I visualize our molecule, the silicon is on the skewer, so
it just spins in place, it doesn't move.
Correct. But the two chlorine atoms,
let's say 1 is front left and one is front right, they swap
places, and the two hydrogen atoms at the back also swap
places with each other. But because a chlorine atom
looks exactly like another chlorine atom, the molecule as a
whole appears unchanged. Correct.
So C2 is a valid symmetry operation for this molecule.
OK, moves number three and four are both reflections or mirror
planes. They have the symbol Sigma.
We have two of them. The first one is a vertical
mirror plane that slices right through the silicone atom and
the two chlorine atoms. We can call this the subsplane.
So the silicon and the two chlorines are sitting inside the
glass of the mirror. Exactly.
They don't move, but the two hydrogens are sticking out on
either side of the mirror. So when you reflect, the left
hydrogen swaps places with the right hydrogen.
And the second mirror plane. That is the city Slane.
It's also vertical, but it's perpendicular to the first one.
This mirror slices through the silicon atom and the two
hydrogen. OK, so in this case the silicon
and the hydrogen sits still and the two chlorine swap places
with each other. You've got it.
Those are our four tools. Do nothing E spin 180° C two,
mirror one and mirror 2. Now comes the matrix part.
We're going to use these four tools to generate a special
code. This is Section 4 Calculating E3
N gamma 3 N. This sounds intimidating, but
it's really just a counting exercise.
We want to know how the total motion of the molecule, all
fifteen of those degrees of freedom we talked about,
responds to each of these 4 symmetry operations.
We need to generate a set of four numbers, one for each
operation, and to get each number.
The notes say we ask 2 questions for each move.
First, how many atoms stayed in the exact same spot?
We'll call that new for unshifted atoms.
And 2nd, for those atoms that stayed put, what happened to
their XY and Z coordinates? This gives a character checks a
gaze. OK, let's run the numbers first.
Operation E do nothing. How many atoms stay?
Put all five of them, nothing moves, so new is 5.
Easy enough. Now for the coordinates.
If you do nothing, X stays X, that's a + 1 contribution, Y
stays Y that's another plus one, and Z stays Z another plus one.
So the sum is 3. So the calculation is simple, 5
unshifted atoms times the character of 3 = 15.
That's our first number, and it makes sense, right?
It represents the 15 total degrees of freedom that we
calculated with three NS. It's a good sanity check.
OK, next operation C2, the 180° spin.
Who stays put? Well, the chlorine swap and the
hydrogen swap, so only the central silicon atom, it just
spins in place, so new 1. Correct.
Now the coordinates. This is a little harder to
visualize. We're spinning 180° around the Z
axis. So the Z axis itself doesn't
change, it stays put that contributes a + 1.
Right, but think about the X&Y axis if you have a point at
position X and you rotate it 180°.
You end up on the other side at Nikex.
And similarly Y becomes may Y, so the total character
contribution is plus one for Z -, 1 for X and -1 for Y.
So 1 - 1 - 1 = - 1. Exactly.
So the math is 1 unshifted atom times the character of -1, which
equals -1. That's our second number.
OK, our code is 15 -, 1 not not next mirror one.
The one called sucks. This is the one that contains
the silicon and the two chlorines.
So how many atoms are sitting inside the mirror unshifted?
The silicon and the two chlorines, that's three atoms,
so NU, not 3. And the coordinates.
If an atom is sitting in the XE plane, it's X coordinate stays X
+ 1 and it's Z coordinate stays Z + 1, but it's Y coordinate
which sticks out from the mirror gets reflected to the other
side. To Max.
So that's a Max 1. So the total character is plus 1
+ + 1 -, 1 which is +1. The map is 3 unshifted atoms
times 1 = 3. And finally, mirror 2, this is
the one that holds the silicon and the two hydrogens.
Same logic, right? The silicon and the two
hydrogens sit in the mirror, so that's three unshifted atoms.
The character for reflection is plus one.
The result is 3 * 1 which is 3. So we've done it.
We have generated our code vector.
It is 15 Nanish 133. This feels like we've found a
secret key, but what does it unlock?
What is this code? Right now, that vector is what
we call a reducible representation.
It's a jumble. Imagine you made a smoothie.
You've blended up translations, rotations, and vibrations all
together into this one messy list of numbers.
So our job now is to unblend this movie to figure out what
the original ingredients were. Exactly this is Section 5, the
reduction. This is where we find the
fundamental symmetry species. And to do this we need a recipe
book, the character table for the C to V point group.
The character table is our key. It lists all the possible
fundamental ingredients for this specific symmetry.
We call them irreducible representations, or ear reps for
short. For the C2V group, the
ingredients have simple labels A1A2B1 and B2.
In these labels, they aren't just random, right?
They describe different types of symmetry, like a one is usually
the totally symmetric one where nothing changes sign.
Correct. A2 usually involves something
antisymmetric with respect to rotation, and B1 and B2 are
antisymmetric with respect to the different mirror planes.
They're just labels for fundamental patterns of
symmetry. So our goal is to figure out how
many units of A1A2B1 and B2 are hidden inside our messy
smoothie. Vector 15 make it 133.
And for that we use the reduction formula.
It looks scary in a textbook, but it's essentially just a
systematic way of comparing our vector to the vectors for each
ingredient in the character table.
I want to walk through just one of these calculations so we can
all feel the mechanism. Let's try to find how many A1
units are in there. OK, first we look at the
character table. The row for a one has its own
code vector. It's 1111.
OK, so the code for A1 is 1111. Our code for the molecule is
15/9. It was 133.
The formula says we multiply them together pair by pair.
Let's do it first. Pair 15 from our vector times
one from the A1 vector is 15. Second pair -1 from our vector
times one from A1 is -1. Third pair 3 * 1 is 3.
The fourth pair 3 * 1 is also 3. Now we sum those results up 15 +
-, 1 + 3 + 3. That's 15 -. 1 is 14 + 3 is 17 +
3 is 20. Perfect.
The final step of the formula is to divide that sum by the order
of the group, which is just a fancy way of saying the number
of operations we used. We use 4 operations EC2 and the
two Sigma, so we divide by 4. And 20 / 4 is 5.
A clean integer that is deeply satisfying.
If we'd gotten 5.2 or something, we'd know we made a mistake
somewhere. Exactly.
Quantum mechanics and group theory count in whole numbers.
So there it is. Inside our messy smoothie of 15
motion, there are exactly 5 distinct units that have A1
symmetry. That's amazing.
So if we were to repeat this exact process for the other
ingredients. You would you'd take our vector
15 monotis 133 and compare it to the vectors for A2B1 and B2 from
the character table. I won't make us do all the
multiplication out loud, but the source material shows the
result. For a 2 the map gives us 2, for
B1 it gives us four, and for B2 it also gives us 4.
That's right. So the full recipe of our
smoothie, the total motion of the molecule is described AS5A1
plus 2A2 plus 4B1 plus 4B2. If you add up the coefficients 5
+ 2 + 4 + 4, it equals 15. It all checks out.
But we are not done. Remember, we only want the
vibrations. We've successfully categorized
everything, but now we have to subtract the junk.
This is section 6, isolating the vibrations, the junk being the
whole molecule flying through space, translation and tumbling
end over end rotation. To do this, we go back to our
recipe book, the character table.
On the right hand side of most tables, there are columns that
tell us explicitly which symmetry labels correspond to
the simple XYZ axis, which represent translation and the
rotation axis RXYRZ. OK, I'm looking at the C2V
character table right now. It shows that translation along
the Z axis T's is listed in the row for A1.
So one of our 5A1 motions is just the whole molecule moving
up and down the Z axis. We need to throw it out.
Translation along the X axis TX corresponds to B1 and
translation along Y tie corresponds to B2.
So we have to throw away one of our B ones and one of our B twos
as well. That takes care of the flying.
Now for the spinning. The table shows rotation around
the Z axis. RZ corresponds to A2.
So one of our 2A2 motions is just the whole molecule spinning
like a top. Useless for vibration.
Get rid of it. And rotation around Y rye is B1
and rotation around XRX is B2. B1 and B2 each lose another
motion to rotation. OK, let's do the final tally.
Let's subtract everything and see what's left.
We started with five units. We subtracted one for
translation along Z. We are left with 4A1 vibrational
modes. For a two, we started with two
units, we subtracted one for rotation around Z we are left
with one A2 vibrational mode. For B1, started with four, we
subtract 1 for translation TX and one for rotation rye.
That leaves 2B1 vibrational modes.
And for B2 we also started with four.
We subtract 1 for translation tie and one for rotation RX.
That leaves 2B2 vibrational modes.
So the final true vibrational fingerprint of this molecule is
4A1 plus one A 2 + 2 B 1 + 2 B2. Let's do our sanity check.
Let's add them up. 4 + 1 + 2 + 2 of equals 9.
And our three and six prediction was 9.
The math works perfectly. We have successfully found and
categorized every single one of the 9 wiggles of this molecule.
This is amazing. We've gone from a shape to a
list of abstract symmetry labels.
This brings us to Section 7. The final verdict, The selection
rules. We know we have 9 vibrations,
but can our detectives actually see them?
This is where it all pays off. We bring back our two
detectives, IR and Ramen. Let's start with IR
spectroscopy. OK.
The rule for IR, as we said, is that the vibration must cause a
change in the dipole moment. And in the language of group
theory, that means the symmetry of the vibration must be the
same as the symmetry of one of the linear axis XY or Z.
So we just have to look at the character table again.
The Z axis we said belongs to A1, the X axis belongs to B1 and
the Y axis belongs to B2. So any vibration that has one of
those 3 symmetry labels will be IR active.
It will show up as a peak on the IR spectrum.
Let's look at our list. We have 4A1 vibrations since a
one matches the Z axis. All four of those are IR active.
That's. Four peaks.
We have two B1 vibrations. B1 matches the X axis, so both
of those are also IR active. That's another two peaks, 6
total so far. And we have two B2 vibrations.
B2 matches the Y axis, so those two are also active.
So our final prediction, 4 + 2 + 2, we expect to see 8
fundamental bands or peaks in the IR spectrum of this
molecule. But wait, we have 9 vibrations
in total. What happened to the A2
vibration? Look at the character table
again. In the row for A2, does it list
XY or Z next to it? No.
It just has arza for rotation. That's the key.
That means the A2 vibration, which is a kind of twisting
motion of the hydrogens against the chlorines, does not create a
net change in the dipole moment along any of the XY or Z axis.
So it's invisible to IR. It is IR inactive.
The IR detective walks right past that clue, completely blind
to it. OK, so if I run an IR scan of
this molecule, I should see eight peaks.
Now what about our other detective ramen?
Ramen looks for a change in polarizability, the squishiness.
In the language of the character table, this corresponds to the
quadratic functions, things like XEQUZWECDDD double.
So we just checked the list again.
I'm looking at the table. The A1 row has X Phi 1Y row and
Z of fire. So check A1 is ramen active.
The B1 row has XC. Check B2 row has YZ.
Check B2. Is ramen active?
And what about our missing child A2?
The A2 row it has guy Z listed next to it, so check it's active
too. In the C2V point group, every
single one of the symmetry species is ramen active.
So ramen sees all nine vibrations.
Exactly. Exactly.
So here's our final concrete prediction based on nothing but
the molecule shape. The IR spectrum will show 8
bands. The Ramen Spectrum will show 9
bands. And that one missing band in the
IR, The difference between 8:00 and 9:00?
That's the smoking gun. It's the fingerprint of the
geometry. Imagine you're a chemist and you
have a vial of some unknown gas. You suspect it might be CCL 2H2.
You run both Spectra. If you see that 8 versus 9
pattern, you can be extremely confident that the molecule has
C2V symmetry. And if the molecule had a
different shape, a different arrangement of atoms.
The math would have spit out completely different numbers.
The symmetry would be different, the character table would be
different, the selection rules would be different.
The pattern of peaks is a direct consequence of the shape.
That is incredibly powerful. We essentially just derive the
blueprint for identifying matter using light without ever
touching the substance itself. That is the power of group
theory. It turns these qualitative ideas
we have about shape into hard quantitative predictions about
experimental data. Before we move on, I want to
touch on one last concept from the notes, the rule of mutual
exclusion. It sounds very definitive.
It is. It's a very useful shortcut for
chemists. It applies only to molecules
that have a very specific type of symmetry element called a
center of inversion. A center of inversion?
What is that? It means that if you started any
atom in the molecule, draw a straight line through the exact
center of the molecule, and keep going that same distance out the
other side, you hit an identical atom.
Like carbon dioxide, CO2, you have oxygen on the left, carbon
in the middle, oxygen on the right.
If I start at the left oxygen and go through the central
carbon, I hit the right oxygen. Exactly.
CO2 has a center of inversion. Our molecule sexy L2H2 does not.
If you start at a chlorine and go through the silicon, you hit
empty space between the hydrogens.
OK, So what is the rule for molecules that have this
feature? The rule of mutual exclusion
states if a molecule has a center of inversion, no
vibrational mode can be active in both IR and Roman.
It's strictly either no overlap. Strictly, if the IR detective
sees it, the ramen detected is blind to it.
If ramen sees it, IR is blind. They are mutually exclusive.
So let's connect that back to our case for sexy L2H2.
We saw that eight of its vibrations, all the A1B1 and B2
modes, appeared in both lists. They were active in both IR and
Roman. Which immediately confirms what
we already knew from looking at it.
It does not have a center of inversion.
The fact that the peaks overlap proves the lack of that specific
symmetry element. It's a binary check that
chemists use all the time. It's like a logic puzzle.
I see a peak at 500 wave numbers in both my IR and ramen Spectra.
Therefore I can definitively say my molecule cannot be Centro
symmetric. Precisely.
It's an immediate piece of structural information.
So this brings us to the final section, why this matters.
We've done a lot of math. We've calculated gammas and
subtracted rotations. But practically speaking on the
lab bench, what does this actually tell a working chemist?
At the most fundamental level, it tells us structure.
This whole process is one of our primary tools for identifying
what things are made of. Because different bonds wiggle
at different speeds. Exactly.
Every type of bond has a characteristic frequency.
For example, a carbon oxygen double bond, a CEO carbonyl
group is a very stiff, strong spring.
It almost always vibrates somewhere around 1700 inverse
centimeters. It's a dead giveaway.
And an OH bond, like in water or an alcohol.
That has a very light hydrogen on the end, so it vibrates much,
much faster. You see a big broad peak way up
around 3400 inverse centimeters. So the Spectra is like a parts
list for the molecule. I see a peak at 1700.
OK, I have a carbonyl group. I see a peak at 3400.
I must have an alcohol. Yes, but the group theory part
we just did takes it to the next level.
It tells us how those parts are arranged in 3D space.
Are the 20 bonds on a water molecule symmetric?
Are they bent? Are they linear?
The number of peaks and which ones are IR active versus ramen
active tells us the architecture.
So it connects the parts list to the actual blueprint of the
molecule. That is a perfect way to put it.
You know, when we started this section, the lecture notes
mentioned something called symmetry coordinates.
This is the idea that the atoms don't really vibrate on their
own. That's right, it's a crucial
concept. We found that the vibrations
were things we labeled A1 or B2. That label doesn't say the
silicon chlorine bond vibrates, it says the entire molecule
vibrates in a way that has B2 symmetry.
It's teamwork. It is.
The two chlorine atoms might stretch together in phase.
At the same time the two hydrogens might bend together in
phase. It's a collective motion.
The atoms are mechanically coupled through the bonds.
Like a series of connected pendulums, if you start one
swinging, it affects all the others.
It's a great analogy. If one chlorine tried to move
completely independently, it would drag the silicon atom,
which would in turn drag the other fluorine and both
hydrogens. The only stable, sustainable
ways for the molecule to move are these normal modes of
vibration, where everyone moves in a perfectly synchronized
pattern. So the molecule is a
synchronized swim team. And group theory is just the
choreography notes for the routine.
It tells you who moves where and when.
Well, we have covered a massive amount of ground today.
We started with a simple spring in a garage.
We found out that the entire universe is shivering with 0
point energy, that nothing is ever truly still.
We met our two detectives, IR and Rahman, and learned that one
looks for electrical lopsidedness and the other looks
for molecular squishiness. We entered the matrix, so to
speak, to calculate the degrees of freedom first Segal to H2 and
generated that reducible representation.
We then unblended the smoothie, breaking that complex data down
into its simple symmetry ingredients.
We subtracted the flying and the spinning to isolate the nine
true vibrations, the actual wiggles.
And finally, we use the rules of to predict exactly which
detective would see which dance move and which ones would be
invisible. It is a complete journey from
abstract concept like shape all the way to a concrete observable
prediction you could test in a lab.
It really does change how I look at my coffee cup now, the
caffeine molecules in there, they are doing this complex
three and six dance right now billions and billions of times a
second and. The selection rules we just
worked out are determining right now which photons of light from
the room are being absorbed by your coffee and which are being
scattered. It's happening constantly.
Symmetry is nature's filter. That is the ultimate take away.
Symmetry isn't just about things looking pretty or balanced.
In physics and chemistry, symmetry is the rule book.
It dictates what interactions are allowed to happen and which
ones are strictly forbidden. I think that is a perfect place
to leave it and a provocative thought for you, our listener,
to take away. If symmetry is the rule book for
how light and matter interact, what other fundamental rules of
the universe are secretly just consequences of a deeper, hidden
symmetry? It's a question physicists are
still wrestling with. Professor Camargo's lecture
notes were a beast, but I think we managed to tame them.
We certainly gave it our best shot.
The math holds up. Thank you so much for guiding us
through the matrices and the character tables.
It was a pleasure to explore the molecular dance with you.
And thanks to you, the listener, for sticking with us through the
group theory. Keep vibrating, keep learning,
and we'll see you on the next deep dive.