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The Anharmonic Oscillator

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • The Anharmonic Oscillator 0:09
    • Vibration-Rotation Interaction & Centrifugal Distortion
    • Making Corrections to the Harmonic Oscillator
    • Selection Rule for the Harmonic Oscillator
    • Overtones
    • True Oscillator
    • Harmonic Oscillator Energies
    • Anharmonic Oscillator Energies
    • Observed Frequencies of the Overtones
    • True Potential
    • HCl Vibrational Frequencies: Fundamental & First Few Overtones
  • Example I: Vibrational States & Overtones of the Vibrational Spectrum 22:42
    • Example I: Part A - First 4 Vibrational States
    • Example I: Part B - Fundamental & First 3 Overtones
  • Important Equations 27:45
    • Energy of the Q State
    • The Difference in Energy between 2 Successive States
    • Difference in Energy between 2 Spectral Lines

Transcription: The Anharmonic Oscillator

Hello, welcome back to, welcome back to Physical Chemistry.0000

Today, our discussion of molecular spectroscopy continues.0004

I think I will go back to black here.0015

We corrected for the rigid rotator energies by adjusting for both the vibration rotation interaction and for centrifugal distortion.0019

The rigid rotator there, the energy was J × J + 1.0070

The adjusted centrifugal distortion gave us F of J = B × J × J + 1 - this J² J + 1².0085

If we also decide to include the vibration rotation interaction + the centrifugal distortion,0112

we end up with this longer and not altogether necessary equation.0144

F of J = E × J × J + 1 - Α E × R + ½ × J + 1 - D J² × J + 1².0154

We must now make a correction to the harmonic oscillator to allow for the fact0185

that the potential energy curve is not harmonic, it is not a parabola.0190

We must now adjust, make corrections to the harmonic oscillator,0199

to adjust for the fact that the oscillator deviates from harmonic behavior for higher vibrational states.0231

As R increases, it starts to deviate more from the parabola, from higher vibrational states.0261

In other words, as the vibration of quantum number R increases.0271

Let us go ahead and take a look at what that looks like.0282

The green line is the harmonic oscillator.0286

The blue line is the real potential energy curve.0294

We see that as we go from R0 to R1 to R2 to R3, there are deviations.0297

Notice that the energies, the blue line, here the energies of the harmonic oscillator are evenly spaced.0304

Notice that the energies here is almost the same but is slightly below the harmonic energy.0313

Here it is below the harmonic energy, more below the energy, more below the energy, more below the energy.0320

Because that is the case, the spacing is actually decreasing as it is going up.0329

The spacing between energy levels is actually going up.0335

In standard temperatures, most diatomic molecules are in the ground vibrational state or R = 0 vibrational state.0341

In other words, most molecules are actually there.0377

They are not distributed so much as you might expect.0380

The harmonic oscillator, the harmonic potential V is equal to ½ KX² is a good approximation to the actual potential.0383

The green line is very very close to the blue line.0405

It is a good approximation to the true potential which is the blue line.0409

The harmonic oscillator, the energy is G of R is equal to fundamental frequency × R + ½.0432

And we said that this ν is equal to 1/2 π C × K / M ^½.0452

There is my fundamental vibration frequency at the ground level.0461

Now, we are talking about just the harmonic oscillator.0471

The selection rule, as we said is that δ R = + or -1.0480

Therefore, we already did this.0489

This δ, this ν observed which is equal to G of R + 1 - G of R.0491

When I do that, I end up with the frequency that I'm going to observe is actually just ν, the fundamental vibration frequency.0505

The 0 to 1 transition, I’m going to get absorption at ν, the fundamental vibration frequency.0533

From 1 to 2, the absorption at ν, the fundamental vibration frequency.0544

It is a constant, it is not based on the quantum number.0548

The quantum number goes away.0550

From 2 to 3, the absorption is at ν.0553

Again, this is what we mean by the spacing of the energy levels is the same.0557

There is no difference, that is all the spacing in the harmonic oscillator.0564

They are all the same spacing.0567

From this 0 to 1, 1 to 2, 2 to 3, since they are absorbing at ν, we just expect one line in the vibrational spectrum.0571

We expect one line in the spectrum and that line is going to be at the fundamental vibration frequency.0586

And we do see one line, we do see one very strong line, one strong dominant line.0605

We should see anything else but we do.0629

The real spectra show other winds of much weaker intensity.0637

These lines are called overtones .0667

These weaker lines are called overtones and they represent the following transitions.0676

And they represent the transitions 0 to 2, 0 to 3, 0 to 4, and so on.0691

The true oscillator, the blue line.0708

The true oscillator, what we call the anharmonic oscillator.0716

In other words, the blue line of the image that you saw, you are going to see in a minute.0722

It does not adhere to the δ R = + or -1 selection rule.0733

It is not adhere to δ R = + or -1.0739

We will also see δ R = + or -2, + or -3, + or -4.0744

Thus, giving rise to these overtones.0756

We do not just see one transition frequency.0759

Let us see what we have here.0767

Let me move forward.0775

You see the blue line, the anharmonic oscillator, it shows not just transition from 0 to 1, 1 to 2, 2 to 3,0780

it also shows transitions from 0 to 2, 0 to 3, 0 to 4.0789

Those are all overtones .0794

The harmonic oscillator energies G of R is equal to the fundamental vibration frequency ν~ × R + ½.0797

The energy adjusted for anharmonic behavior, I will just say anharmonic energies.0815

The anharmonic oscillator, the true oscillator energy is G of R is equal to sub E R + ½ fundamental vibration0833

frequency – X sub E ν sub E × ν + R + ½² +, there is cubic terms, there is quartet terms but they are very insignificant.0853

These two are the only ones that we need.0878

This X sub here, this is called the anharmonicity constant.0882

It is another one of those parameters that is tabulated.0898

As we said earlier, the overtones are transitions from 0 to R.0904

The observed frequencies of these absorption, the observe frequencies of these transitions,0935

they are the G of R - G of 0 is equal to N sub E R + ½ - X sub E N sub E R + ½² - ν ½ - X E ν sub E × ½².0950

What you will end up getting is observed frequency for these overtones is0994

fundamental vibrations frequency × R - this C sub E ν sub E × R × R + 1.1000

R takes on the value of 1, 2, 3, and so on.1011

These are the frequencies of the overtones that we observe.1016

These are the observed frequencies of the overtones.1023

Let us see what we have here.1041

Notice for the true potential, the blue line, the energies are less than the energies for the harmonic oscillator.1044

That is what the equation says.1057

The G of R energy equation = equilibrium × R + ½.1060

These are the energy for the harmonic oscillator adjusted for anharmonic behavior.1078

The adjustment ½².1087

As R increases, this term increases.1091

It deviates more from harmonic behavior.1098

The blue line is less than the green line.1101

This blue line is more less than the green line.1104

This blue line is more less than the green line.1107

This blue line is more less than the green line.1110

For the true potential, the energies are less than whatever the harmonic oscillator predictions.1117

In each case, the blue is less than the green, for R = 0, R =1, R =2, R=3.1153

As R increases, the spacing between these lines of energy also decrease.1159

In other words, G of R + 1 – G of R, you end up with the fundamentals frequency -2 × XE ν of E × R + 1, R = 0, 1, 2, and so on.1188

The energies are less than what the harmonic oscillator predicts.1215

Not only that, the harmonic oscillator predicts that the energy spacing between the levels is constant, it is not.1222

As you go up, the energy, the spacing between the blue lines actually get smaller and smaller until it goes up,1227

until it goes finally to 0.1234

At that point, the molecules dissociate, they come apart.1239

Which is what this DE means, that is what D sub 0 mean.1244

They are the dissociation energy.1248

The energy between R = 0 state and infinite distance between them.1250

They are completely apart.1254

Let us go ahead and take a look at a data table for the overtone spectral lines.1258

Instead of taking a look at the spectrum which tends to be very broad when I take a look at a data table.1264

These are the hydrogen chloride vibrational frequencies.1271

They are the fundamental in the first few overtones.1274

The fundamental is the transition from 0 to 1.1277

The first overtone is 0 to 2, second overtone is 0 to 3, third overtone is 0 to 4, and so on.1280

The frequencies that we observe, the actual spectra that we look at, we get the 288656688347 to 10923.1287

If we stick with just the harmonic oscillator approximation, we get a 288657728658 11544,1298

if we use the equation with the adjustment for the anharmonic oscillation, we get these numbers.1309

Notice that these numbers in the anharmonic oscillator are a lot better,1319

are a lot closer to the real values than just the harmonic oscillation.1326

Here, the fundamental vibration frequency is 2886.1330

The ν sub E is 2991.1336

The X sub E is 53.1340

And again, these values are tabulated for the different molecules they are available.1343

If values for the spectroscopic parameters have been tabulated from various molecules,1348

these parameters are calculated from real spectral data.1353

Let us go ahead and do a quick example here so we can all work a little bit with this information.1363

The following data for HF is taken from a table of collective spectroscopic parameters.1369

This is one line.1373

On the table for the molecule hydrogen fluoride, notice we have the rotational constant B,1375

we have the centrifugal distortion constant.1383

We have this Α sub E which was the vibration rotation interaction of adjustment constant.1389

We have the ν sub E, we have the X sub E, and we have the equilibrium bond length.1395

These are in picometers, the rest of these are all in inverse cm.1401

Use the appropriate parameters to calculate under the anharmonic oscillator1405

the energies of the first of four vibrational states and the fundamental in the first three overtones of the vibrational spectrum.1410

What is that we are going to need here?1420

Let us do A, A is the energies.1424

The equation for the energy is G of R and we are under the anharmonic oscillator.1429

It is the N sub E × R + ½ - X sub E N sub E × R + ½.1436

They want the first 4 vibrational states.1451

We are going to do the G of 0.1454

We are going to do G of 1, G of 2, and G of 3.1458

0, 1, 2, 3, that is the value R that we are going to put in.1463

The parameters that we need are that one and that one.1466

When we use these parameters, just 4138.3 and 0.0217 inverse cm1475

and put it into this equation for the different values of R, what we get is the following.1482

We get 2046 inverse cm, we get 6005.2 inverse cm.1488

We get 9783.9 inverse cm, and we get 13383.1497

These are the energies of the first 4 vibrational states of hydrogen fluoride.1508

The ground state and then the first R =1, R =2, R =3.1518

We need to find the fundamental and the first 3 overtones.1524

Remember the fundamental was the 0 to 1 transition.1528

Part B, we are looking for the equation, the calculated, the observed, what is that we like to see is ν sub E × R – X sub E ν sub E.1532

We are just applying the equation R × R + 1.1546

Did I copied that correctly?1552

R = 1, 2, 3, 4.1554

R starts at 1, for the observed frequency R starts at 1.1557

The 0 to 1 transition, when I put 1 in to here, I end up with ν calculated is equal to 3958 inverse cm.1565

I will see a peak at that frequency.1583

I put 2 in for R and I will see 7737.2 inverse cm.1588

I will see a peak at that, I will see a line at that.1600

It will be a much weaker line, very weak but it will be there.1603

The 0 to 3 transition, I will see a frequency at 11,336 inverse cm.1612

And my 4th overtone I should see a line at around 14755.1630

This is called the fundamental.1641

That is the one that I'm going to see, dominant.1648

These others are the overtones.1652

They are of weaker intensity.1654

Let me go ahead and actually put it on the next page.1660

Let me put it in black.1664

Please be very certain that you understand the following differences.1667

The energy of the quantum state.1698

The difference in energy between two successive states.1708

And 3, the difference in energy between two spectral lines.1729

They are very different things.1745

I know it is very confusing with molecular spectroscopy.1748

The energies are the energies of each individual state.1758

The difference in energy between two successive states is the energy of the upper state - the energy of the state right below it.1768

It is E sub I + 1 – E sub I.1776

In other words, the δ E.1778

The difference in energy between the two spectral lines is ν of I + 1.1783

It is the observed frequency of absorption for one transition - the observed frequency1791

of absorption for the transition just before it.1798

That is the difference.1803

E sub I, the energy that is the important equation.1808

Everything else comes from that.1816

The energy is the important equation.1817

If you understand what is happening, if you understand what is going on spectroscopically,1828

what has to happen from one transition to another transition,1833

the energy will allow you to derive any other equation that you might need, depending on what you might need.1837

The energy is the important equation.1842

It is the important equation, the others can be derived.1844

Thank you so much for joining us here at

We will see you next time, bye.1851