 So, now we see that for an anharmonic oscillator, so not just a harmonic oscillator, which has energy levels of n plus a half h nu, but for an anharmonic oscillator that exhibits some non-harmonicity, some anharmonicity, we have this correction to the energy levels. Since if we're talking about spectroscopy, it's more convenient rather than talking about the energy levels themselves to talk about them in units of wave numbers. So if I take the energies, divide by hc, so this is now in units of wave numbers, that equation looks very similar, we've just changed the frequencies or h times nu for energies, we've changed those into units of wave numbers. Likewise for the anharmonicity correction, divide by h and c, dividing by c turns the frequency into a frequency in units of wave numbers. So that's a more convenient version of the same equation when we're talking about spectroscopy, and this equation is going to help us understand why it is the diatomic molecules have the spectra that they do in the infrared portion of the spectrum. So to understand why that's true, let's calculate, for example, the difference in energy when I make a transition from one level to another in a diatomic molecule that's behaving like a harmonic or an anharmonic oscillator. So that energy difference is if I'm making a transition from one level to the next level up, difference between the n plus first energy and the nth energy, or if I do that instead in units of wave numbers, then that is the difference in energy between this equation with an n plus one and this equation with an n. So that's going to work out to be the n plus first energy level. Instead of n, I'm going to use n plus one. So n plus one plus another factor of a half gives me n plus three halves, and I'm in units of wave numbers, so I don't need the Planck's constant. And the anharmonicity correction looks like n plus three halves squared times the anharmonicity correction. So that's the energy of the n plus first energy level. If I subtract from that the energy of the nth level n plus a half for the harmonic oscillator and n plus a half squared minus this negative, so that gets a positive sign for the anharmonicity correction. So there's some cancellation that happens here, n times new tilde minus an n times new tilde those cancel and I'm left with just three halves minus one half or one factor of new tilde. Little bit of algebra is needed to take n plus three halves quantity squared minus n plus one half quantity squared times this x e new tilde sub e. So I've got n plus three halves quantity squared that looks like n squared twice three halves is three and three halves squared is nine over four. That's what I get by squaring n plus three halves multiplying x e new e and then I've got a positive opposite sign and plus one half quantity squared that's going to be n squared plus n plus one over four x sub e new sub e with a tilde on top. The cancellation I've got a minus n squared and a positive n squared those can cancel completely. I've got a minus three n and a positive one n those almost canceled but not quite or partially canceled but don't. So minus three n plus two n is minus two n that's what's okay minus two n and I've also got nine fourths with a negative sign one fourth with a positive sign so those leave me a minus two. That minus two n and minus two those are all multiplying x sub e new sub e tilde. So if I rewrite that as fundamental vibrational frequency in units of wave number plus let's say minus twice n plus one so minus two n minus two that's the same as minus twice n plus one that's the size of the anharmonicity correction. So if we make some sense out of that equation the energy in units of wave numbers needed to go from one level to the next one up isn't just the harmonic oscillator frequency but it's the harmonic oscillator frequency minus a little bit and that anharmonicity correction depends on which level I'm in that gets larger the correction gets larger as n gets larger. So what that means is if we draw an energy ladder the harmonic oscillator has energy levels at one half h new three halves h new five halves h new and so on evenly spaced for the perfectly harmonic oscillator the anharmonic oscillator with this anharmonicity correction those energy levels have all decreased the lowest one is decreased a little the upper one is decreased more the higher ones decrease even more so that anharmonicity correction is those downwards arrows I've drawn that have just lowered the energy by a little bit. The fact that the upper levels get corrected by more for the anharmonicity correction than the lower levels mean that every time I make a transition if I make a transition from the zero state up to the first state it's not h new it's a little bit less than that. If I go from the first state up to the second state it's not h new it's not even h new minus that small correction it's h new minus an even bigger correction. So if we look at what that means for an actual molecule if I bring up a graph here of the spectrum for an actual diatomic molecule that's going to be the absorption in the infrared for the carbon monoxide molecule and remember what we talked about before is that the fundamental vibrational frequency we expected this spectrum when we were only thinking about the harmonic oscillator we expected it to be centered around the fundamental vibrational frequency of 2170 wave numbers what we actually see is it centered a little bit lower it's red shifted the center of the spectrum is redshifted from the fundamental vibrational frequency we can explain why that's true now so for carbon monoxide with this fundamental vibrational frequency 2170 the anharmonicity correction x sub e times the fundamental vibrational frequency that is 13.3 inverse centimeters so if I want to know what is delta e in units of wave numbers to go from let's say the zero state up to the first state according to the equation we just figured out we could just plug into this expression calculate the energy of the first minus the energy of the second I'm sorry the energy of the first minus the energy of the zero state or we could use this expression we've just derived it's going to be 2170 is the fundamental vibrational frequency as a wave number if I subtract so if I'm going from n equals zero n plus one is equal to one so I'm going to subtract twice the anharmonicity correction so 2170 minus 27 leads me 2143 wave numbers and that is in fact you can see from this diagram 2150 is right here 2140 would be about there this is in fact right at 2143 inverse centimeters so the cause of this red shift in the absorption spectrum is because of the anharmonicity that the decrease in the energy levels because of the anharmonicity is what causes the spectrum to be redshifted and in fact it helps explain another mystery about this spectrum let's say I were to do the same calculation not for the zero to one but for the one to two level so I'm going not from n equals zero to n equals one but from n equals one to n equals two we expect that to be redshifted a little further because n has gotten larger or because the upper states are decreased in energy even more by anharmonicity so in fact that's true if we repeat that calculation with the numbers we've already got that's going to work out to be subtracting another 26 wave numbers from this value that's going to end up with a value of 2117 wave numbers so we'd expect to see the same sort of spectrum but centered not at 2143 centered at 2117 centered about here and in fact you may have noticed this especially if you're looking at this on a large enough screen there is in fact very small if you look closely down here there's a whole extra series of peaks that has the same shape as the spectrum we've already seen but much smaller and it is in fact centered around 2117 this would be the n equals one up to n equals two transitions you can't see that so the n equals one up to n equals two transitions occur here the reason they're so small they're much less intense than the main sequence of these absorption peaks is because there aren't many molecules remember the populations of the vibrational states for diatomic molecules are primarily overwhelmingly in the n equals zero state in the ground state there's not very many molecules in the n equals one state sitting around waiting able to accept a photon to jump up to n equals two state there are a few there's a small number of them so they can absorb a little bit of the light that comes to them but that light isn't centered at 2143 it's it's again red shifted by a little bit more so it's less intense and redshifted further so and harmicity helps us explain many of the features now of this absorption spectrum in the infrared for diatomic molecules like carbon monoxide it actually is going to help us explain another mystery about these infrared spectra as well and that's what we'll talk about next