Chapter 11: Molecular Spectroscopy

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Welcome to the Deep Dive.

Today we're really getting into how light interacts with molecules.

It's fascinating stuff.

Our topic is molecular spectroscopy.

We'll be unpacking the physical chemistry behind those spectra we see.

So, our mission today is to understand where molecular spectra actually come from.

We know atoms absorb photons and change electronic states,

but molecules are a whole different ball game, right?

Exactly.

It gets more complex, but also much more informative.

With molecules, it's not just electrons jumping around.

The energy changes involve the molecule's rotation and its vibration too.

All of this involves a photon being absorbed or emitted or sometimes even scattered.

Right, and that complexity is actually the key point because more complexity means more information.

Atomic spectra might be simpler, cleaner maybe, but these molecular spectra, they let us measure things like bond strengths, bond lengths, even angles between bonds, dipole moments.

Incredible detail.

It really is like using light as a super precise ruler.

We're measuring energy changes to figure out the molecule structure and how it behaves.

Okay, so for you listening, we're going to walk through this step by step.

We'll start with the general features, then look at rotation, vibration, and finish up with electronic changes and what happens next.

Sound good?

Sounds like a plan.

Let's start with the basics, maybe the different energy scales involved.

Yeah, good idea.

The type of radiation we need depends entirely on how much energy the molecule needs to absorb for a particular change.

How different are these energy jumps for rotation, vibration, and electronic transitions?

Oh, they're hugely different.

Rotational changes are tiny, barely any energy needed, so we use very gentle low -energy microwave radiation for that.

Okay, microwaves for rotation.

Then for vibrations, think bonds, stretching, and bending.

That takes quite a bit more energy.

Now we're talking infrared radiation, IR.

Right, IR for vibrations.

And then the big one.

Yeah, the electronic transitions.

Those are the real energy hogs.

They need the biggest jump, which means we need high -energy UV or visible light.

So this energy ladder rotation, way down low, vibration in the middle, electronic way up high, explains why we need different kinds of spectroscopy, different instruments for each one.

Absolutely, and it also ties into how we measure them.

You mentioned Einstein's processes, stimulated absorption, stimulated emission, and spontaneous emission.

Right.

Why is it that we usually measure IR by absorption, but for UVVs, we often look at emission?

Ah, that's a really neat consequence of the physics.

The rate of spontaneous emission, where the molecule just gives off a photon on its own compared to stimulated emission, that ratio goes up incredibly fast with the frequency of the transition, like frequency cubed.

Cubed, yeah.

Every three hours.

So electronic transitions happen at really high frequencies, right?

UVVs light.

This means spontaneous emission is super likely.

The excited molecule just, well, glows pretty quickly.

So we monitor the light it emits.

Makes sense.

High frequency makes it likely to just drop back down and release light.

Exactly.

But vibrational transitions, much lower frequency,

infrared.

So spontaneous emission is really slow, quite unlikely comparatively.

The molecule doesn't just glow easily on its own time scale.

So we have to force the issue.

Pretty much.

We shine IR light through the sample and measure how much gets absorbed.

That's why IR is mostly an absorption technique.

Okay, that clarifies a lot.

Now, before any transition happens, absorption or emission, it has to be, well, allowed, right?

Yeah.

These selection rules come into play.

What's the fundamental physical thing a molecule needs to interact with light?

It needs what's called a transition dipole moment.

It sounds fancy, but the idea is simple.

During the transition, the distribution of charge in the molecule has to change in a way that creates an oscillating dipole moment.

Crucially, that oscillation has to be at the same frequency as the light wave.

So the light's electromagnetic field needs something to sort of push and pull on.

Precisely.

If there's no oscillating charge it can interact with, no handle to grab, the transition is forbidden.

It just won't happen effectively.

We split these rules up, don't we, into gross and specific?

Yeah.

Gross selection rules are the big picture requirements.

Does the molecule need a permanent dipole moment for this type of spectroscopy, or does the dipole need to change during a vibration, that kind of thing.

And the specific selection rules.

Those get down to the nitty gritty of quantum numbers.

They tell you exactly which changes in rotational quantum number J or vibrational quantum number V are allowed, like delta J or delta V or delta V equals PM 100 and L01.

Okay.

So if a transition is allowed, we then want to know how much light is absorbed.

That brings us to the Beer -Lambert law.

Yes, the Beer -Lambert law.

It's fundamental for quantitative absorption spectroscopy.

It basically says the intensity of light, I, goes down exponentially as it passes through the sample.

And that decrease depends on the path length, L, and the concentration of the absorbing molecule, J.

Right.

The equation is $1, I set, I, 10 epsilon, J, L, L.

Where $10 is the initial intensity.

And that crucial term, epsilon, epsilon, that's the molar absorption coefficient.

Epsilon tells us how strongly the molecule absorbs light at that specific wavelength.

Exactly.

You can sort of think of it as the molecule's effective cross -sectional area for capturing a photon of that energy.

Big epsilon means it's really good at absorbing that light.

Now, just quickly before we move into rotation modes,

spectral lines aren't perfectly sharp peaks.

They have some width.

What causes that broadening, especially in gases?

Good point.

There are a couple of main culprits.

First up is Doppler broadening.

Like the siren effect.

Exactly like the siren effect.

If a molecule is moving towards your detector as it absorbs or emits, the frequency looks slightly higher blueshifted.

If it's moving away, it looks lower redshifted.

And since molecules in a gas are whizzing around in all directions.

You get a spread of frequencies, a broadened line.

This effect gets bigger with higher temperatures because the molecules move faster and also with higher transition frequencies.

Okay, Doppler effect.

What's the other main one?

The other big one is lifetime broadening.

This comes from the Heisenberg uncertainty principle.

Uncertainty.

Linking energy and time.

That's the one.

If the lifetime of the excited state is very short,

then the uncertainty in its energy is large.

A shorter lifetime means a broader line.

And what shortens the lifetime in a gas?

Collisions.

If the excited molecule bumps into another one, it can lose its energy non -radiatively.

This is often called collisional broadening or pressure broadening.

Ah, because more pressure means more collisions.

Precisely.

So if you want sharper lines, one trick is to lower the pressure of the gas sample.

Fewer collisions, longer lifetimes, narrower lines.

Clever.

Okay, let's transition now up the energy ladder slightly.

Just focusing on pure rotational motion.

This is the realm of microwave spectroscopy.

How do we start classifying molecules based on how they rotate?

We look at their moments of inertia around their principal axes.

You get different categories.

There are linear rotors.

Obvious examples are things like CO2 or HTL.

Then you have spherical rotors where all three moments of inertia are identical.

Methane CH4 is a classic example.

Yeah, linear spherical.

What else?

And then symmetric rotors.

These have two identical moments of inertia and one different.

They can be prolate, sort of cigar -shaped, like methyl chloride.

Or they can be oblate, more like a pancake.

Think benzene.

And the energy of rotation depends on these moments of inertia?

Inversely, yes.

We define a rotational constant, usually written as tilde b darn, b tilde, which is inversely proportional to the moment of inertia.

So a large, heavy molecule with a big, I will have a small tilde b and rotate more slowly, energetically speaking.

And the energy levels aren't evenly spaced, are they?

No, the gap between adjacent rotational levels actually increases.

It's proportional to J plus one, where J is the rotational quantum number.

So the levels spread out as the molecule spins faster.

We often start by modeling these as rigid rotors, but that's an approximation, isn't it?

Yeah.

What happens when they spin really fast?

Yeah, rigid is just a starting point.

Real molecules aren't perfectly rigid.

When they spin fast, centrifugal force stretches the bond slightly.

Ah, centrifugal distortion, like spinning weights on elastic bands.

Exactly.

The bond stretches, which increases the moment of inertia.

I, just a tiny bit.

And since tilde b is inversely related to I, the effective rotational constant actually decreases slightly at higher J values.

We need to add a small correction term for that.

Right.

Otherwise our calculations of bond lengths from the spectra would be slightly off, especially for fast rotations.

Correct.

So focusing now on microwave spectroscopy, which measures these pure rotational transitions, what's the absolute must -have for a molecule to show up in a microwave spectrum?

The gross selection rule here is simple.

It must have a permanent electric dipole moment.

Other dollars.

That's the key.

So perfectly symmetric non -polar molecules like hydrogen H2, nitrogen N2, or carbon dioxide CO2, they're invisible to microwave spectroscopy.

No permanent dipole.

And if it does have a permanent dipole, the specific selection rule is?

It's delta J, 81, NaO 500.

The molecule absorbs a microwave photon and jumps up one rotational level.

And the spectrum looks like?

A series of lines.

And because of that delta J of PM 101 rule and the energy level spacing, these lines are very nearly equally spaced with a separation of two altality.

Okay.

But what about all those important molecules like H2 or N2 or even methane that don't have a permanent dipole?

How do we study their rotation?

For those, we need a different technique.

Rotational Raman spectroscopy.

Raman involves scattering light, not absorbing it.

What's the requirement here?

Instead of a permanent dipole, the molecule must be anisotropically polarizable.

Meaning?

Meaning an electron cloud is easier to distort or polarize in some directions than others.

Take H2, no permanent dipole, so microwave inactive, but you can distort its electron cloud more easily along the bond axis than perpendicular to it.

That anisotropy makes it Raman active.

So even spherical rotors like methane, which are perfectly symmetric.

They're not anisotropically polarizable.

Their electron cloud distorts the same way, no matter the direction.

So spherical rotors are inactive in a rotational Raman too.

You need that directional difference.

And the specific selection rule for rotational Raman is different too, isn't it?

It's not delta J PM 11?

Correct.

For linear molecules, it's delta J equals zero PM 22.

That jump of two levels is characteristic.

Why plus or minus two?

That seems odd.

It's actually got a neat classical picture behind it.

Imagine the oscillating electric field of the light inducing a dipole in the molecule through polarization.

As the molecule rotates 360 degrees, the alignment that causes maximum interaction, and thus the induced dipole comes back to its original state twice, once every 180 degrees.

Ah, so that twice per rotation, max onto the delta J equals two P two rule.

Exactly.

And because that rule, the lines you see in a rotational Raman spectrum, the Stokes and anti -Stokes lines scattered off the main frequency are separated by $4, not $2 tiller.

Clever.

Okay, one last thing on rotation, nuclear statistics.

This sounds complicated.

It can be, but the core idea relates to the Pauli principle and the symmetry of the total wave function, including the nuclear spins.

For molecules with identical nuclei, like O2 or CO2 with identical oxygen isotopes, certain rotational states might be forbidden depending on the nuclear spin.

So for CO2 with oxygen 16 nuclei, which have zero spin, you find that all the odd J rotational levels are simply missing.

It dramatically affects the appearance of the spectrum, leading to alternating intensities or missing lines.

It's a direct quantum mechanical effect tied to the nuclei.

Fascinating.

Okay, let's climb the energy ladder again.

Moving on to vibrational spectroscopy, typically done with infrared light.

What's the simplest model we use for bond vibration?

We start with the harmonic oscillator, basically treating the bond like a perfect spring obeying Hooke's law.

The potential energy curve is a parabola.

And the frequency of that vibration depends on?

Two things.

The force constant, Kfd, which tells you how stiff the bond is, stiffer spring, higher frequency, and the effective mass of the two atoms involved.

Lighter atoms vibrate faster.

So stiff bonds between light atoms give the highest vibrational frequencies.

Now, for a vibration to absorb IR light, what's the gross selection rule?

The vibration must cause a change in the molecule's electric dipole moment.

Not necessarily a permanent dipole, but a change during the motion.

Exactly.

So take CO2 again, the symmetric stretch where both CO bonds stretch out and in together.

That preserves the symmetry, no dipole change, so it's IR inactive.

But the anti -symmetric stretch, where one bond stretches while the other compresses, that creates a temporary dipole, so it is IR active.

Same for the bending motions.

And for the simple harmonic oscillator model, the specific selection rule is very strict.

Yes, it's delta V PMHL1.

You can only go up or down one vibrational energy level at a time.

That's why the strongest absorption is usually the fundamental transition from V dollars up to

But we know bonds aren't perfect springs.

They can break.

The harmonic model doesn't allow for dissociation.

Right.

That's its biggest flaw.

Real bonds show anharmonicity.

The potential energy well isn't a perfect parabola.

It's steeper at short distances due to repulsion.

And it flattens out at long distances as the bond breaks.

We use the Morse potential often as a better approximation, right?

Yes, the Morse potential captures that anharmonicity much better.

It includes a dissociation energy term.

And what are the consequences of this anharmonicity for the spectrum?

Two main things.

First, the vibrational energy levels are no longer equally spaced.

They get closer and closer together as the vibrational quantum number increases.

They converge towards the dissociation limit.

Okay.

Levels bunch up at high energy.

What's the second consequence?

Because the selection rules get relaxed slightly by anharmonicity, transitions with delta VPM2, PM33, and so on become weakly allowed.

These are called overtone transitions.

They're usually much weaker than the fundamental, but you can often see them.

And that convergence of energy levels.

Can we use that?

Absolutely.

We can use it to estimate the bond dissociation energy.

If you measure the frequencies of the fundamental and maybe a couple of overtones, you can see how the energy gap between levels is shrinking.

That's the Bird Spooner plot.

Exactly.

You plot the energy difference delta GG between adjacent levels versus the quantum number V.

It should be roughly a straight line going downwards.

Extrapolating that line back to where the energy gap would be zero gives you an estimate of the vibrational level where dissociation occurs, and thus the dissociation energy tilde dollars.

Very cool.

Now, in the gas phase, we don't just see a single line for the VGON -Leftero -dollar -dollar transition.

We see complex structure.

Why?

Because the molecule's rotating at the same time it's vibrating.

When it absorbs an IR photon to jump from V -dollar to V -dollar -dollar, its rotational state, J, can also change.

This gives us vibration -rotation spectra.

And this leads to different branches in the spectrum.

Yep.

If the rotational quantum number decreases by one, delta J, nanostand dollar one one, during the vibrational transition, that's the P branch.

If it increases by one, delta J plus one, that's the R branch.

What about delta J equals dollars?

If delta J is allowed, which it sometimes is, especially for more complex molecules or certain types of vibrations, that forms the Q branch, often appearing as a more intense pileup of lines near the center.

For simple linear diatomics, the Q branch is usually forbidden for infrared absorption.

And the spacing within the P and R branches tells us about the rotational constant.

But wait, does tilde change when the molecule vibrates?

Ah, good point.

Yes, it does.

Slightly.

Because the average bond length is usually a tiny bit longer in the excited vibrational state compared to the ground state.

A longer bond means a larger moment of inertia, which means a smaller rotational constant.

So we actually have tilde B1 and tilde B4 VLR, and they're slightly different.

How do we figure out both values accurately from the spectrum?

There's a neat trick called the method of combination differences.

You look at pairs of lines in the P and R branches that start from, or end in, the same rotational level in one of the vibrational states.

By taking specific differences in their frequencies, you can isolate the rotational constants for the upper and lower vibrational states independently and with high precision.

Okay, it's a mathematical way to tease apart the rotational constants in the two different vibrational states.

Exactly.

It lets us see how vibration affects the molecule's rotational properties, which relates back to how the bond length changes.

Now, let's zoom out quickly to polyatomic molecules.

Things get complicated fast.

How many ways can a bigger molecule vibrate?

The number of vibrational modes is given by a simple formula.

For a linear molecule with n atoms, it's 3 on 0, 5, 5 dollars.

For a non -linear molecule, it's 3 on 60 segs.

So water, H2O non -linear with 3 atoms.

3 dollars times 3, 6 equals 3, 3 modes.

Right.

A symmetric stretch, an asymmetric stretch, and a bending motion.

These independent, synchronous motions of all the atoms are called normal modes.

Each normal mode behaves kind of like its own harmonic or anharmonic oscillator with a characteristic frequency.

Predicting which of these many modes are IR active or Raman active seems like it could be a nightmare.

It would be, without help from symmetry.

Symmetry analysis is incredibly powerful here.

One of the most useful shortcuts is the exclusion rule.

What does that say?

It applies specifically to molecules that have a center of symmetry, also called the center of inversion.

For such molecules, the rule is no vibrational mode can be both IR active and Raman active.

They are mutually exclusive.

So if you see a vibration in the IR spectrum, you won't see it in the Raman and vice versa.

For a centrosymmetric molecule, yes, it's a fantastic diagnostic tool.

If you see bands appearing in both the IR and Raman spectra at the same frequency, you can immediately say the molecule does not have a center of symmetry.

How does symmetry determine activity more generally?

We use group theory.

Basically, each normal mode transforms according to a specific symmetry species within the molecule's point group.

A mode is IR active if its symmetry species transforms in the same way as the Cartesian coordinates X, Y, or Z, because that corresponds to a dipole moment change.

They're Raman active.

A mode is Raman active if its symmetry species transforms like one of the quadratic forms, like XW or XY or ZW, because that corresponds to a change in polarizability.

So symmetry gives us definite yes -no answers for IR and Raman activity for every single mode.

Powerful stuff.

Incredibly powerful.

Saves a ton of guesswork.

OK, final section.

Let's jump up to the highest energies.

Electronic spectra, usually in the UV or visible range.

Here, we're exciting electrons from one orbital to another.

Right.

And just like with atoms, we use term symbols to label these electronic states, especially for linear molecules.

You see symbols like sigma, pi, delta related to the orbital angular momentum along the bond axis.

There are also labels for parity, Giefer -Girard symmetric for undurayed anti -symmetric with respect to inversion,

and reflection symmetry, sigma plus or sigma depth.

And symmetry gives us selection rules here, too.

Absolutely.

One very important one for centrosymmetric molecules is the Laporte rule.

It states that allowed electronic transitions must involve a change in parity.

So $1 left -right arrow G transitions are allowed, but the left -right arrow G and $1 left -right arrow G transitions are forbidden.

Is that why the famous D or T transitions in many transition metal complexes, which are often the R left -right arrow G, are technically forbidden and usually quite weak?

Precisely.

They often borrow intensity by coupling with molecular vibrations that break the center of symmetry temporarily.

These are called vibronic transitions.

The most crucial concept for understanding the intensity patterns within an electronic transition is the Franck -Condon principle.

What's the core idea?

The core idea is that electronic transitions are fast, virtually instantaneous compared to the much slower motion of the heavy nuclei.

Think of it like taking a snapshot.

So when the electron jumps, the nuclei are basically frozen in place.

Exactly.

The transition occurs vertically on a potential energy diagram.

The molecule finds itself in the new electronic state, but with the same nuclear geometry and momentum it had just before the transition.

And how does that determine the intensity of different vibrational peaks within the electronic band?

The intensity of a transition from a vibrational level five dollars in the lower electronic state to five dollars in the upper electronic state is proportional to the square of the overlap integral between their vibrational wave functions, often written as two up to twos.

So maximum intensity happens when the vibrational wave functions of the initial and final states overlap the most.

Yes.

If the upper electronic state has a very similar bond length to the lower state, the vertical transition from five dollars will likely land near the minimum of the upper potential well, overlapping best with V dollars.

So the zero dollar transition will be strongest.

But if the bond length changes significantly upon excitation, then the vertical transition from V dollars might land higher up on the wall of the upper potential well, where the wave function overlaps better with a higher vibrational level,

say V2 or V43 dollars.

In that case, that transition will be the most intense one in the vibrational progression you see.

It tells you visually about the geometry change.

And we also see rotational fine structure on these electronic transitions, just like in IR.

We do, but often with a twist.

Because the bond length can change quite significantly between electronic states, the rotational constants till V can be very different.

What does that lead to?

This difference can cause the lines in either the P branch or the R branch to get closer and closer together, eventually pile up, and then seem to reverse direction.

This pile up creates a sharp edge in the spectrum called a bandhead.

Seeing whether the bandhead is in the P or R branch tells you whether the bond length increased or decreased upon excitation.

Okay, so the molecule gets excited electronically.

It can't stay there forever.

How does it get back down?

What are the main decay pathways?

The most common radiative pathway back down from the first excited singlet state to the ground singlet state is fluorescence.

Singlet to singlet.

Spin is conserved.

Is it fast?

Yes, fluorescence is typically quite fast happening on nanosecond timescales.

Usually before fluorescence occurs, the molecule quickly loses any excess vibrational energy within the sector barter state through collisions.

This is called vibrational relaxation.

So it emits from the bottom of the sector, or what?

Pretty much.

And since it lost that vibrational energy first, the emitted fluorescent light always has lower energy, longer wavelength, than the light that was initially absorbed.

That energy difference is called the Stokes shift.

Okay, fluorescence is fast, singlet to singlet, the Stokes shift.

What about that slow afterglow effect we sometimes see?

Oh, that's phosphorescence.

This is emission from an excited triplet state down to the ground singlet state, triplet to singlet.

That involves a change in spin.

Is that allowed?

It's spin -forbidden, which means it's much, much slower than fluorescence.

Lifetimes can be milliseconds, seconds, or even longer.

For phosphorescence to happen, the molecule first has to get from the excited singlet state, S $1, to the excited triplet state, T $1.

How does it do that?

Through a non -radiative process called intersystem crossing.

It's a radiationless flip between states of different spin multiplicity, often helped by heavy atoms.

Once the molecule is in the T $1 state, it's kind of trapped because the direct drop back to sea -all over is so slow.

T $1 acts like a reservoir.

So fluorescence is fast.

$7 right arrow, S cell 1.

Phosphorescence is slow, T $1 right arrow are requiring intersystem crossing first.

Are there non -radiative ways to decay too?

Yes.

The molecule might simply lose its energy as heat through collisions,

internal conversion or intersystem crossing followed by vibrational relaxation.

Or if the excited state is unstable, the energy might cause a bond to break, leading to dissociation.

If this happens from a state that overlaps with the dissociative state, it's called pre -dissociation.

And how would dissociation show up in the spectrum?

It often leads to a blurring or complete loss of the fine vibrational and rotational structure, sometimes transitioning into a continuous absorption band because the fragments can fly apart with any amount of kinetic energy.

And all this understanding of stimulating emission and managing excited states leads us to lasers.

Absolutely.

Lasers are all about harnessing stimulated emission.

The key is to create a population inversion, where you manage to get more molecules into the upper energy state than are left in the lower state.

Which is the opposite of the normal thermal distribution.

Exactly.

Once you have that inversion, a single incoming photon of the right frequency is much more likely to trigger stimulated emission producing an identical photon than it is to be absorbed.

This leads to light amplification.

Many lasers use clever four -level systems to achieve and maintain this population inversion efficiently.

Wow.

Okay, so looking back, we've covered a huge amount of ground.

We started with the general principles, energy scales, Einstein's coefficients, selection rules, the Beer -Lambert law for quantifying absorption, and why spectral lines have width.

Then we dove into the specifics of motion.

Rotation, characterized by moments of inertia, and the rotational constant tilby.

Studied by Microwave, two -taller Laib spacing needs permanent dipole.

And Raman, $4 tilby spacing, needs anisotropic polarizability.

We touched on nuclear statistics too.

Then vibration.

Model first as a harmonic oscillator, depending on force, constant, catable, and effective mass,

but needing anharmonicity, Morse potential, to explain overtones and dissociation via Burj -Spooner plots.

We saw how vibration and rotation couple, giving P, Q, R branches, and how symmetry dictates IRI and activity via the exclusion rule for centrosymmetric molecules.

And finally, electronic transitions in the UV beam.

Governed by selection rules like LaPorte, with intensities dictated by the Froum -Condon principle, vertical transitions, wave function overlap.

We saw how rotational structure leads to bandheads, and how excited states decay via fluorescence, stellar right aerofacetal, or phosphorescence.

$T right R West slow needs intersystem crossing, or dissociation, all underpinning laser action via stimulated emission and population inversion.

It's a really comprehensive picture.

And it's not just theory, is it?

The source material points out direct applications.

Astrophysicists use these unique spectral fingerprints to identify molecules light years away in space.

Definitely.

And closer to home, environmental scientists rely heavily on understanding the vibrational spectra of gases like CO2 and methane.

Those specific vibrations are exactly how greenhouse gases absorb infrared radiation and contribute to climate change.

The spectroscopy is fundamental to the models.

It really connects the microscopic quantum world to large -scale phenomena.

Okay, time for one final thought for you, our listener, to ponder.

We've seen that different spectroscopic techniques probe different molecular properties.

IR needs a changing dipole.

Microwave needs a permanent dipole.

Raman needs changing polarizability.

So think about that, especially the exclusion rule for symmetric molecules.

What's an example of a non -polar molecule that also is a center of symmetry, where you'd really need to rely almost exclusively on Raman spectroscopy to get information about its bond lengths via rotation, and bond stiffness via vibration?

Something to mull over.

That's a good one to think about.

It ties a lot of the concepts together.

Thank you so much for joining us on this deep dive into the world of molecular spectroscopy.

We hope you feel you've got a much clearer picture of how we use light to understand the intricate dance of molecules.

We'll see you next time.