 In the previous video we looked at how infrared light induces molecular vibrations. This is because as the electromagnetic field oscillates backwards and forwards it pushes and pulls the dipole moment of a molecule and if the frequency of the vibration and that change in the dipole matches the incoming radiation, it's absorbed. But why would this be quantized? Why can't it just produce any vibration whatsoever? For this we're going to look at the simplest case that we've already seen for rotational spectrums to be the case of the heteronuclear diatomic molecule. Just like with rotations we need to reach into our physics toolbox and dig out the reduced mass. This means that we can model two atoms vibrating against each other as just one mass vibrating against a fixed point. As the mass moves towards the fixed point the potential energy increases and the restoring force increases. As it moves away, energy also increases and the restoring force also increases. It's also worth pointing out that force and energy are related by integrating and differentiating over distance. This is known as the harmonic oscillator and the frequency associated with it is the harmonic frequency. And technically this is a very strong simplification because it implies the potential well is symmetrical. This isn't the case with the real system as we will see later. Again, the details of this are covered across both quantum mechanics courses particularly when we look at the creation and annihilation operators later on. Solving the Schrodinger equations for this quadratic potential well gives us quantized energy just as with particle in a box. The shapes of the waves are different but there's still quantization. Solving for the energy of each level shows us that there is an even progression in the energy levels where each is separated by the reduced Planck's constant multiplied by the fundamental frequency of the vibration. There's one more complicating factor to add to this model but it is important to remember. If you plug in zero for the vibrational quantum number that is v equals zero, you still get some energy. It's equivalent to half the reduced Planck's constant multiplied by the fundamental frequency. This comes down to some quantum mechanical properties namely the Heisenberg absurdity principle. If you remove all possible energy from atoms there is still some vibration of sorts there because there's still uncertainty in the position of these molecules so still a little bit of vibration. This is known as the zero point energy. Contrary to a lot of science fiction you can't actually extract this zero point energy there is no state lower in energy so you can't possibly extract it. Now we're going to look at how to extract force constants from vibrational data. Now the restoring force in the harmonic oscillator model is directly proportional to that distance of displacement so the force constant is that proportionality constant. As a result it's a sort of measure of the stiffness of a bond. The harmonic frequency is a function of both the force constant and the reduced mass. It's a fairly simple relationship. However it's absolutely essential to check the units on these equations as there are a number of very similar equations each expressing this energy in different units. Dividing the frequency by the speed of light gets us the wave number multiplying by the speed of light gets us the wavelength. So that's the main formula you'll need to work out a force constant. Obviously you'll need to do some rearrangement to yourself and also check units to make sure you're not accidentally calculating wavelength when you should be calculating wave number and so on. But next we'll look at the next level of complexity and with a little bit more maths on top. We've just seen the harmonic oscillator. What about the anharmonic oscillator?