Chapter 10: Molecular Symmetry

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement not replaced the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome back to the Deep Dive.

You know, if you've ever opened a physical chemistry textbook, maybe like Atkins, DePaula, and Keeler, there's that bit where shape suddenly gets, well, really mathematical.

We're talking about molecular symmetry, and today we're going way beyond just VSCBR theory.

We're diving into the core system that chemists use to predict pretty much everything, bonding, spectroscopy, properties.

Our mission to walk you through topic 10 from Atkins, molecular symmetry, and pull out that predictive framework all without needing the textbook diagrams right in front of you.

That's the plan.

We're taking that intuitive idea of shape and sharpening it into the precise language of, well, symmetry and group theory.

It's a three -step process, really.

First, we lay the groundwork.

That's topic 10A, classifying molecules using symmetry.

Then, topic 10B, we quantify it using group theory.

Right, the math part, exactly.

Yeah.

And finally, the payoff in 10C, applying it all to quantum chemistry to make real predictions.

Okay, let's start with the basics.

Symmetry.

There are two terms that sound similar but mean different things.

Symmetry operation and symmetry element.

What's the deal there?

Yeah, that distinction is crucial.

A symmetry operation is the action you perform.

Think rotation, reflection, that sort of thing.

Like actually doing something to the molecule.

Precisely.

Whereas the symmetry element is the geometrical thing, the action relates to the point, the line, or the plane.

So you have the act of rotating, the operation, and the axis of rotation, the element.

Got it.

Okay, so there are, what, five core types of these operations and elements we need to know.

That's right, five fundamental building blocks.

First up is the n -fold rotation, which we label C -on.

Okay.

This just means rotating the molecule by 360 degrees divided by n around a specific axis.

If it looks the same after the rotation, it has a stod axis.

Like water, 2H2 axis, you spin it 180 degrees.

Exactly, 180 is 360 divided by 2, so water has a C -tility axis.

Ammonia, NH3 -3 dollars, needs a 120 -degree turn, so that's a C -2 axis.

And an important point, if there are multiple rotation axes, the one with the highest in value, we call that the principal axis, it sets the orientation for other elements.

Okay, principal axis is the main one.

What's next?

Reflection.

Yep.

Reflection, symbol sigma -dor.

This happens across a mirror plane, and because that principal axis is so important, we define different types of mirror planes relative to it.

Ah, so there's subtypes here.

Three main ones.

First is sigma -dollar for vertical.

This plane contains the principal axis.

Think of the planes in water or ammonia again, slicing through the molecule along the main rotation axis.

Okay.

Then there's sigma -horizontal.

This plane is perpendicular to the principal axis.

Benzene is a classic example.

The plane of the molecule itself is a sigma.

Right, slicing it flat.

Exactly.

And finally, sigma -dollar for dihedral.

These are a bit trickier.

They are vertical planes, but they specifically bisect the angle between two axes that are perpendicular to the principal axis.

You see these in more complex shapes.

Okay, down rotation sigma -reflection.

Yeah.

What else is in the toolkit?

Inversion, symbol allotter.

This operates through a single point right in the center of the molecule called the center of symmetry.

How does that work?

Imagine taking every point, 6 y z in the molecule, drawing a line through that central point, and going an equal distance out the other side to x y z z z.

If the molecule looks identical after doing that to all its points, it has an inversion center.

So like a perfect cube has one.

Perfect example.

Benzene also has one.

But water, ammonia, methane, they don't.

If you try to invert water, the oxygen would stay put, but the hydrogens would end up somewhere else entirely.

Right.

Okay, that's three.

What's the fourth?

You mentioned one that's a bit weird.

Ah, yes.

The n -fold improper rotation.

This one's kind of a composite operation, a two -step process.

Okay.

First, you perform a rotation, 316 degrees.

Then immediately after, you reflect everything through a plane that's perpendicular to that rotation axis, basically, a sigma reflection relative to this.

Oh, so rotate, then reflect.

Exactly.

And here's the kicker.

Neither the signal rotation nor the sigma reflection, taken alone, has to be a symmetry operation of the molecule for the combined sigma and operation to be valid.

Yeah.

Wait, really?

Really.

Methane, CH4 is the classic example.

As sorosera 4 -axis, you rotate by 90 degrees,

then reflect perpendicularly.

Methane doesn't have a C4 -dollar axis on its own nor the required sigma, but the combined sor 4 operation leaves it looking the same.

It's a purely mathematical symmetry sometimes.

Huh.

That is improper.

Okay.

That was the last one.

Number five, the simplest one.

Identity symbol E -dollar.

It means do nothing.

Every single molecule possesses this operation because, well, doing nothing obviously leaves it unchanged.

Okay, fair enough.

So sigma, sigma, CDACACD, that's the full set.

And once you figure out all the symmetry elements a molecule has.

You classify it into a specific point group.

It's like a family name based on its complete collection of symmetry elements.

Examples you might hear are C -supa -dollar for water, D4 -boron trifluoride, or T -dollars for methane.

And this is where it gets powerful, right?

Just knowing the point group, like CDA or TED, tells you something immediately.

Before any complex calculations.

Absolutely.

Two huge things right off the bat.

First,

polarity.

Okay.

A molecule can only have a permanent electric dipole moment be polar if it belongs to one of just a few point groups.

C and Ds, these are groups with lower symmetry.

So if it's in a different group.

If it has higher symmetry, particularly if it has a center of symmetry, $80, it cannot be polar.

Think about it.

If you invert the molecule through its center and it looks the same, the dipole vector would have to point in the opposite direction, but it also has to be unchanged.

The only way that works is if the vector is zero.

So carbon dioxide, CO32 dollars, linear, symmetrical.

It belongs to D -lay -acent dollar, which has an inversion center.

So boom, non -polar.

But ozone, O3 -3 dollars, which is bent, is CT2 dollars.

No inversion center.

It can be polar.

And it is.

Okay.

That's a neat shortcut.

What's the second big prediction?

Chirality.

You know, whether a molecule is handed, like our left and right hands, non -superimposable mirror images.

This determines if it can rotate plane polarized light, its optical activity.

Right.

Crucial in biology and pharma.

Hugely important.

And the symmetry rule is surprisingly simple.

A molecule is chiral only if it does not possess any axis of improper rotation.

Wait, only $7?

What about mirror planes or inversion centers?

Well, think about the definitions.

A mirror plane, sigma, is actually mathematically equivalent to an ax double one operation, rotate 360, reflect.

And an inversion center is equivalent to an ax toe two operation, rotate 180, reflect.

Ah.

So the symmetry rule automatically includes those two.

Exactly.

If a molecule has any sesonoxis, including a mirror plane or inversion center, it cannot be chiral.

It will be superimposable on its mirror image.

Glycine has a mirror plane, sigma, so it's acral.

Alanine doesn't have any sesad element, so it is chiral.

Okay.

So classifying the shape gives us these cool qualitative predictions, polar or not, chiral or not.

But chemistry needs numbers, right?

How do we take a label like CC dollar and use it for, say, predicting which orbitals can overlap or which spectral lines show up?

That's where we move from just classifying shapes to quantifying symmetry using group theory.

It's the mathematical machinery behind it all.

Sounds intense.

It sounds more intimidating than it is.

A mathematical group just has to follow four basic rules.

Think of the set of symmetry operations for a given molecule, like water.

CC2 dollar has sigma, CZ2W, sigma VOT.

Okay.

What are the rules?

The set must include the identity operation, AUL.

Water has EU.

Check.

Two.

Every operation three dollars must have an inverse, also in the set, such that doing three dollar gets you back to EO.

For two, CC you're rotating 180 degrees twice.

CT2, C2 dollar.

Reflecting twice across the same plane also gives EO.

So check.

Makes sense.

The group must have closure.

This means if you combine any two operations in the set, the result must be equivalent to another single operation that's also in the set.

For water, doing the C22 rotation and then reflecting across one sigma plane is exactly the same as just reflecting across the other sigma plane.

So sigma C2 equals sigmas.

All combinations work out.

Check.

Okay.

And the fourth.

The operations must be associative, which means if you combine three operations, say A, B, and C, it doesn't matter if you combine A and B first, then C, or if you combine B and C first, then A, like ABC equals ABC.

Symmetry operations always obey this.

Check.

So the set of symmetry operations for a molecule forms a mathematical group.

How does that help us quantify things?

The real power comes when we represent these abstract operations using matrix representations, usually written as DR for an

matrices.

Uh oh.

Stay with me.

It's about how things transform.

Pick a set of basis functions, maybe the atomic orbitals on the molecule, like the oxygen 2p orbitals and hydrogen 1s orbitals in water.

Okay.

Now perform a symmetry operation, say the 22202 rotation.

What happens to those orbitals?

Some might stay in the same place, but flip their sign.

Others might swap places with each other.

We can represent these changes, the sign flips, the swaps using a matrix, a big square matrix, D222.

So for water, with say five relevant orbitals, you'd have a five by five matrix for each of the four symmetry operations.

That sounds complicated again.

It can be, but here's the clever bit.

We often don't need the whole matrix.

What we really care about is a single number derived from it.

The character of the representation, symbol GGA.

Okay.

What's the character?

It's simply the sum of the numbers on the main diagonal of the matrix DR.

This is called the trace of the matrix.

This one number magically captures the essential information about how the entire basis set transforms under that operation R without needing the full matrix.

Much simpler.

Ah, a shortcut within the matrix.

I like shortcuts.

We all do.

Now this leads us to the absolute core concept of applying group theory and chemistry.

Irreducible representations or irreps.

Okay.

Think of any representation matrix you build from your basis set, like that five by five matrix for water's as being reducible.

It can be mathematically broken down or reduced into a sum of simpler, more fundamental component representations.

These fundamental building blocks are the irreps.

They're unique for each point group and are listed in what chemists call character tables.

So the big matrix is made of smaller standard pieces, the irreps.

Essentially.

Yes.

Any representation gamma can be written as a sum of irreps like gamma one plus gamma two plus gamma three three, meaning he contains two blocks of irrep one, one of irrep two and one of irrep three.

How do we know how many irreps there are and what they look like?

There are strict rules based on the group structure.

First, we group symmetry operations into classes.

Operations are in the same class if they are related to each other by some other symmetry operation within the group.

For ammonia, the C three, the three 120 degrees and C three to 240 degrees rotations are in one class.

The three sigma reflections are in another class.

C does is always in a class by itself.

Okay, so classes group similar operations.

Right.

And the key mathematical theorem is

the number of irreps for a point group is exactly equal to the number of classes in that group.

Ah, a direct link.

And these irreps have standard labels, often called symmetry species labels.

They usually depend on the irreps dimensionality.

If T a one by one matrix, just a number, the label is usually a dollars.

If T two two, the label is a dollars.

If T two three, it's dollars.

And there's always one special irrep called a dollars or sometimes just a dollar, the totally symmetric one where the character is plus one for every operation dimensionality.

You mean like one D to D three D.

How does that relate to the molecule itself?

It relates directly to degeneracy.

The maximum dimensionality found among the irreps of a molecule's point group tells you the maximum possible degree of orbital degeneracy allowed for that molecule.

You mean like if orbitals have the Exactly.

If a point group like waters TC Bayer only has one D irreps, A's and B's, then the molecule cannot have any inherently degenerate orbitals.

All orbitals must be non -degenerate by symmetry.

But what about something like methane?

Methane is T D dollars.

The T to point group does have three dimensional irreps labeled T D dollar and T two dollar two.

Because a three D irrep exists, methane can have triply degenerate orbitals.

And indeed, it's three two P orbitals form a triply degenerate set belonging to the T two dollar irrep.

The geometry through group theory dictates the possible quantum energy levels.

Okay, this is really starting to connect.

We classify the shape, quantify it with group theory and irreps.

And that tells us about degeneracy.

Now, how does this become that ultimate shortcut you mentioned for calculations?

It comes down to predicting when integrals must be zero.

So much of quantum chemistry involves calculating integrals, often very complex ones.

Group theory gives us a simple, powerful rule to know if the answer is zero without doing the calculation.

Zero integrals.

Why is that useful?

Because it tells us when things cannot happen or cannot interact.

The fundamental rule is this.

An integral over all space, like F to F D tau,

can only be non -zero if the function F inside the integral, the integrand, belongs to or contains the component that belongs to the totally symmetric irreducible representation of the molecule's point group.

Only if the thing inside the integral have a dollar one symmetry.

Or contains a dollar one symmetry if F is complex.

If the overall symmetry of the integrand transforms purely as some other irrep like B dollar or A dollars, the integral must be exactly zero.

Symmetry forces it to cancel out.

Wow.

Okay, how does that apply to say bonding?

Think about the overlap integral S between two atomic orbitals, 201 and 202.

The integral is T psi 2 D tau.

This tells us if the orbitals can constructively interfere to form a molecular orbital.

The integrand here is the product tan psi 2 high on E2.

Okay, so we need to know the symmetry of that product.

Precisely.

We find the symmetry species of the product K psi 2 tau by taking the direct product of the symmetry species of T psi 2 into 2 sin.

You essentially multiply the characters of their respective irreps, gamma and gamma.

And the crucial result.

The direct product times gamma will contain the totally symmetric representation A 20 no, only if gamma one tile is the same as gamma, only if one is 200 dollars.

Wait, so the overlap integral is only non -zero if the two orbitals belong to the exact same irreducible representation.

Exactly.

That's the massive insight for bonding.

Only orbitals of the same symmetry species can have non -zero overlap and thus contribute to the same molecular orbital.

Orbitals of different symmetry are inherently orthogonal by symmetry they cannot mix.

Let's use methane T again.

You said the four H ones orbitals combine to give over O's plus T two two symmetry combinations.

Right.

And on the central carbon, the twos orbital has O one symmetry.

The three two P orbitals together have two total one symmetry.

So the carbon twos can only overlap with the hydrogen combination that also has a two one symmetry.

Yes.

And the carbon two P orbitals T two two O's can only overlap with the hydrogen combinations that have T total two symmetry.

The A one orbitals ignore the T total orbitals and vice versa.

Symmetry dictates the bonding partners.

No wasted effort calculating zero overlaps.

That's incredibly efficient.

Does this apply to spectroscopy too?

Like which transitions we see?

Perfect application.

It governs the selection rules.

The intensity of light absorption or emission for a transition between an initial state psi and a final state psi depends on something called the transition dipole moment.

And that's an integral.

It is.

It looks like when you feed interstep U tau.

Here doubts represents the dipole moment operator, which corresponds to the coordinates six Y or zero dollars, depending on the polarization of the light interacting with the molecule.

So now we have three things in the integral final state, initial state and the light interaction operator Q.

Right.

And for the transition to be allowed, meaning it can happen and have non zero intensity, the symmetry of that entire integrand must contain the totally symmetric representation eight L one.

So we need the triple direct product Gamso one time to contain eight L one.

Exactly.

You look up the symmetry species for the initial state, the final state, and for the operators, six dollars and zero one.

They're also listed in character tables.

You calculate the triple product.

If it contains eight L one, the transition to symmetry allowed for that specific polarization, conducts.

Well, as a rule, if not, it's symmetry forbidden.

So group theory tells you not just if a transition can happen, but also which direction the light needs to be polarized.

Precisely.

It gives you very specific experimental predictions.

OK, one last piece mentioned in Atkins.

Symmetry adapted linear combinations,

SLCs.

What role did they play?

SLCs are basically the what a framework, but it seems like we've gone from just looking at shapes to having this really powerful quantitative predictive engine.

We classified molecules using just five operations,

used group theory to put numbers characters on those operations and derive these fundamental rules about what zero and what's not specifically the eight dollar one rule for integrals, leading to overlap rules for bonding and selection rules for spectra.

That's the essence of it.

And maybe a final thought to leave you with connecting back to how things change.

Symmetry arguments are fantastic for telling us when something must be zero.

They provide strict yes, no answers based on the perfect point group.

But molecules aren't always perfect, right?

Exactly.

They vibrate, they react.

They might be slightly distorted by their environment.

Consider methane again, perfectly tetrahedral tallers.

But what if, say, you pull on one C -H bond, making it slightly longer, the molecule isn't C anymore.

Its symmetry is lower.

Yes, it might drop to C symmetry.

Or if you distort two bonds, maybe C two dollars.

When the point group changes, the set of irreducible representations changes.

Degeneracies might be lifted.

Orbitals that couldn't mix before because they had different symmetry species in Tevinet might suddenly find they have the same symmetry species in the lower symmetry group.

Ah, so changing the shape, even slightly, changes the rules for bonding and reactivity.

Precisely.

If the point group dictates the properties, what happens when the point group itself is subtly perturbed?

How do the electronic structure and the chemical possibilities shift?

It makes you think about chemical change not just as atoms moving, but as a change in the fundamental symmetry landscape of the molecule.

That's a fascinating perspective.

Symmetry breaking is a driver of chemistry.

A great place to wrap up.

Thank you for joining us for this deep dive into the elegant world of molecular symmetry.