For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

### 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
- The Anharmonic Oscillator
- 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
- Important Equations

- 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

### Physical Chemistry Online Course

### Transcription: The Anharmonic Oscillator

*Hello, welcome back to www.educator.com, 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 fact *0185

*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 vibration *0833

*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 is *0994

*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 oscillator*1405

*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 cm *1475

*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 frequency *1791

*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 www.educator.com.*1849

*We will see you next time, bye.*1851

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