Furthermore, we can make all the analogies for the thermodynamic state

functions that we had made previously. Thus for instance, the expectation value

of the vibrational energy will be the probability averaged accessible energies,

and that'll be minus the partial derivative of the log of the partition

function now indexed with a subscript that it's the vibrational partition

function we're talking about with respect to beta.

And if you'd rather work with temperature instead of beta, then it's kT squared

partial log q, partial T. And the only difference then between all

these various partition functions is just the subscript.

Of course, the energies are different as well that are available.

But the formalism of working with the partition functions is the same.

There's an electronic energy. There's a vibrational energy.

There's a rotational energy. And there's a translational energy.

So, let's take a moment to think about the relationship between energies and

partition functions. I'll let you take a look at that, and

then we'll come back. Okay.

So, just a quick review. That the energy of a molecule is a sum of

its translational energy, its rotational energy, its vibrational energy, and its

electronic energy. And that leads, then, to a molecular

partition function. So when I take e to the minus beta times

the total energy, which is this sum here, then, because it's an exponential of a

sum, I can break it up into a product of exponentials.

So here's the partition function for translation, e to the minus beta, all the

possible translational levels. All the possible rotational levels.

And I'm emphasizing with different indices here, I, j, k, l.

It certainly can be any combination of these various energy levels.

Because in this partition function it can be any combination of energy levels.

So the molecular partition function is the product of the individual

translational, rotational, vibrational, and electronic partition functions.

Let's also talk about degeneracy. So, when we look at partition functions.

We've, up 'til now, discussed them as a sum over states.

I've talked about state being a possible energy.