 welcome to module 17 of chemical kinetics and transition state theory. In the last module, we were discussing partition functions. We looked at translational, rotational and vibrational partition function. Today, I just want to connect with another very related concept which is the quantum partition function. What we have done so far is it integral over x and p and x and p are classical concepts, Newtonian concepts and that is what we have been doing so far. Everything has been Newtonian, but as it turns out in some occasions, a little bit of quantum mechanical effects are important. We cannot ignore them. So, we will discuss quantum partition functions today and we will end up with a little bit of physical understanding of what partition function means really. What I am describing is from Atkins chapter number 15b of volume number 10 of 10th edition. But again, if your edition is different, do not worry the same information you can find in a different chapter number. So, just a quick recap what we have covered. We have defined the partition function as this integral over phase space divided by h to the power of 3n of e to the power of minus beta h. Then we looked at separating the variables and Hamiltonian into translation, rotation and vibration. And so, we get our partition functions also separated into translation, rotation and vibrations. So, we get 3 different qtr, q rotation and q vibration. And in the last module, we calculated these and showed that q translation is equal to this. For rotational, we did not provide the full proof, but just an intuition on how these come about. But you get importantly two different rotational partition functions for linear versus non-linear molecules. And for vibrational, we showed that the partition function is equal to this for one vibration. If you have multiple vibrations with different frequencies omega 1, omega 2, omega 3, so on, you just multiply the respective partition functions for vibrations. So, now we will move on to how to define quantum partition function. So, I will not get too much deep into what is quantum mechanics as much, that is not really a part of this course. The main thing to know is in quantum mechanics, there is no such thing as phase space. So, the notion of q comma p that we have been using so far, it disappears. What we instead have in place of q comma p is called wave functions. And I will not get into what wave functions really are. But we denote a wave function by some state i, which is given label. So, under point main point here is that this q comma p is completely continuous. I have q can take any number from minus infinity to plus infinity and so can p. But these states are discretized, they are quantized. That is the origin of the world quantum. So, q comma p gets replaced with state i, Hamiltonian of q comma p gets replaced with E i energies. And these E i's are the eigen energies of that system. So, we solve something called the Schrodinger equation and get these E i's. Again, we are not going to cover the Schrodinger equation really in this course. And the final thing is the integral over phase space that we have been doing so far divided by h to the power of 3n gets replaced by a discrete summation over all possible states. So, we had earlier shown that the equilibrium density matrix for classical mechanics is 1 over q e to the power of minus beta h where q was a partition function. I apologize, this should be a normalization constant. In quantum mechanics, that thing will become 1 over n into e to the power of minus beta E i. This will be rho equilibrium of state. So, wherever you see q comma p that gets replaced with state and wherever you see this h that gets replaced with energy. And finally, we have to define this partition function in quantum mechanics. So, in classical mechanics, we are doing this integral. In quantum mechanics, we replace this integral as a sum over all states e to the power of minus beta and h gets replaced with E i. So, in this slide, I am not really giving you proof on why you can do it. I am simply giving you a recipe because the proof of all of this is much beyond the scope of this course. We have to understand quantum mechanics much deeper to derive all of this. So, we will just use this recipe to calculate quantum partition functions now. So, we have now a quantum partition function defined as this q equal to sum over states of i e to the power of minus beta i. I will just provide one little comment. So, imagine you have these states. So, what happens in quantum mechanics is you have different states with different energies. So, let us say I have a bunch of states with different energy. This is let us say state 1 and its energy is E 1. This is state 2 and this is state 3. But interestingly, you can in quantum mechanics have states with equal energy. So, this is energy E 2 and this is energy E 3. Sorry, this is also energy E 2. So, state 2 and state 3, let us say equal energy and so on and so forth. You have different states and all states, some of the states have equal energy. So, if I am summing over states, what I will have is something like E to the power of minus beta E 1 plus E to the power of minus beta E 2 plus E to the power of minus beta E 2 will appear twice. One for one for state 2 and one for state 3 plus so on. So, you see that whichever states will appear twice will get a factor of 2. Similarly, if a state is occurring 3 times, it will get a factor of 3 then. So, in general, I can write this as sum over energy levels into G i into E to the power of minus beta E i where G i is the degeneracy. Degeneracy simply means number of states having energy E i. So, I can go from a summation over states to a summation over energy levels. So, now whichever version we write over states or energies, we want to do the same trick we did in classical mechanics. We do not want to deal with the full partition function which is much harder. We want to do divide and rule. We want to separate the states and the energies in terms of translational, rotational and vibrational levels. So, how do we do that? The point is it is a same trick as before. So, states i gets divided into a translational states j plus rotational states k plus vibrational states l. Now, in quantum mechanics, we actually get another one which is the electronic states as well, which we will discuss later. So, in quantum mechanics, we have an extra term other than this translation, rotation and vibration which is the electronic energy. So, that is the same thing as 1s energy versus 2s energy. That energies are different and so, we have a different electronic state energy as well. And the energy themselves get separated into translation, vibration, vibration and electronic. So, what we do is we divide this summation into the other different parts j, k, l, m and this also we separate a same thing as before. Instead of an integral, we have a summation. That is all. So, vibration and electronic. So, again we play the same trick and we separate the terms out. This is rotation, sum over l which will refer to vibration and sum over electronics, which will refer to electronic levels. So, this we then define to be q translation, this we define to be q rotation, q vibration and q electronic. So, the same decomposition we can also do in quantum mechanics. So, let us calculate the partition function for 1D translation. So, my system again is I have a box. This is particle is located in between 0 to some lx and the potential is 0 inside and infinite outside. So, we had shown for this potential the classical partition function is here and we will calculate it for the quantum part now. So, for this part actually the first thing we need to know are what are the energies and degeneracies Gis and Eis. So, we will see in this course simply provide that this comes from solving Schrodinger equation to be more precise, but we are not solving Schrodinger equation and simply providing you the final answer. So, the q will be sum over i G i e to the power of minus beta E i. So, this becomes sum over i G i is 1 e to the power of minus beta i square x square over 8 m l x square. Now, one important note about this quantum partition functions we do not include the 0 point energy in here by convention more or less. We can choose the energy whatever we want. So, we set the first ground state energy to be 0. So, my summation will start from 0 to infinity. Now, this summation is actually a very hard summation to do analytically, but if temperature is large that is I have a lot of these terms a lot of levels are populated. Then I can put this thing as an integral over dx e to the power of minus beta x square x square over 8 m l x square where I have replaced i with x and the summation with an integral dx. So, now what you get is a Gaussian integral and we know how to solve Gaussian integrals. So, again do not end up memorizing these integrals we will always provide you integrals. So, the answer for this is half root of pi over A and A here is the coefficient before x square which will be h square into 8 pi k B T m l x square. We will just simplify it a little bit. First thing we note 8 is 4 into 2 and 4 I take out of square root. So, I get a 2 that will cancel with this 2. I take x square out l x square out and get this and you can note that this is exactly the same as this. And the reason we get it the same is because we have made the approximation that temperature is large. We went from that summation to integral. The summation will not come out exactly equal to this. It will be slightly different but in the limit that T is large it will be equal. So, that is actually a general result. It is called classical quantum correspondence whenever temperature is large you will get the classical result back. So, now let us look at the rotational partition function. For classical we had given you two formulas one for linear and one for non-linear. Right now I will just derive it for linear and non-linear is slightly harder. It is not fundamentally harder it is just computationally harder. So, we are not going to do that derivation that we will assume to be true. So, we will look at the linear part. For linear part again the energies rotational energies are given by b into j into j plus 1 where b is your h bar square over 2 i and g j is given to be 2 j plus 1. This again comes from Schrodinger equation. So, there is some mathematics behind which is solving this rotational partition rotational Hamiltonian in 2D in linear for a linear molecule. What we have to do is sum over j from 0 to infinity g j which is 2 j plus 1 into e to the power of minus beta e j. Again this is a very hard summation to do. So, we are not going to attempt to solve this. We will approximate this is equal to integral from 0 to infinite dx 2 x plus 1 e to the power of minus beta b x square plus x. Now, this integral actually might look complex, but it is not really so. We are going to make a substitution y equal to x square plus x dy is 2 x plus 1. So, you know see that this is exactly what appears here. So, our life is simple. So, our integral becomes 0 to infinity dy e to the power of minus beta by which is an easy integral to do. This is equal to e to the power of minus beta b y divided by minus beta b from 0 to infinity. Once you put in the limits you can show this is equal to kbt over b. So, let us just put in the value of b from here h bar square h bar is h over 2 pi square divided by 2i. So, let us just simplify this. This 2 pi square will go in the numerator. So, I will get 8 pi square kbt kbti over h square and you will note that this is exactly the same as this. Once more the reason is because we went from a summation to an integral is the reason we get it the same. So, every time you make this high temperature approximation you will end up getting the classical answer and often classical answer is easier to calculate because we know simply how to do integrals better than we know how to do summations. So, the non-linear part I am not going to prove to you, but there as well at in the under high temperature limit this equation will also hold true quantum mechanically as well also true quantum mechanically. So, finally, let us look at the vibrational partition function and here we have a little bit of a surprise. So, this is not the same as translation and rotation we have to be more careful with vibrational. So, our q again is sum over i g i e to the power of minus beta e i where e i here are vibrational energies provided here and g i's are the degeneracies which is equal to 1 for vibration. So, let me substitute this energy is h bar omega 1 i plus half, but what we always do is as I have mentioned earlier we set the 0 point energy to be 0. So, we ignore the half part of it. So, the point is that our e 0 is h bar omega 1 over 2. So, that is our base we consider all energies now as e i minus e naught and what we are doing here is summation over this e i tilde. So, that 0 point energy gets removed away. It is a it is a matter of convention only it is nothing more I can I am free to choose my base energy where a energy is equal to 0. So, that is all I am doing I am not really saying that 0 point energy it is does not is not there I am simply calling that 0 point energy as 0 value that is my choice it is just mathematics. So, I will just provide you a summation sum over i or x to the power of i i equal to 0 to infinity is equal to 1 over 1 minus x if x is less than 1. So, I am going to call my x as e to the power of minus beta h bar omega 1 note that this is less than 1 you are taking exponent of some negative number beta h bar omega 1 is positive. So, e to the power of something negative which is less than 1. So, I can do this is equal to 1 over 1 minus x. So, this summation q has become sum over i x to the power of i vibration quantum mechanical. So, here we have to be more careful for vibration. First we can do the summation exactly that is good for us, but more importantly for vibration the high temperature limit does not really hold for rotation and translation high room temperature is usually high. So, the approximation that we made in the last few slides of calling the summation as integral and calling the high temperature limit is a good approximation. But here generally it is not and we will show it to you. So, we have looked at the quantum vibrational partition function and we have realized it is slightly different from the classical version. So, let us get back to our favorite example of HBR for which we have found all the rotational classical either rotational translation and vibration. And if you go back to your last module we showed the vibrational classical partition function was came out to be 0.078. So, let us see if the quantum version makes a difference. So, the quantum vibration partition function that we just derived is equal to 1 minus e to the power of minus H bar omega 1 divided by kT. So, this is equal to nothing but 1 minus e to the power of I will just use this number here 1 divided by 0.078 which is 1 minus roughly 12. What you realize is e to the power of minus 12 is actually a very very small number it is 0.000 something right it. So, we can almost neglect it. So, this is roughly equal to 1. You can of course, put it on a calculator and calculate this out exactly and what you are going to get is 1.000 something that is a lot of zeros after 1. And so, we do not care about that level of significant figures. And so, you see that the quantum partition function is very very different from the classical partition function more than a factor of 10. So, that is very important to know that the vibrational partition function quantum mechanical version can be very very different. So, unless otherwise specified for vibrational partition function we all always use the quantum version and not the classical version. So, finally, let us also look at the final component of the partition function which is the electronic part. So, remember we had divided our energies into this translation rotation vibration and electronic. And for each partition function we write this basically as sum over alpha e to the power of minus beta e electronic alpha. Now, the point is typically the electronic energies are much much more than temperature ok. Hundreds of time more than temperature thousands of time more than temperature. So, generally we approximate this thing as simply e to the power of minus beta e ground state only. I had forgotten the degeneracy I am sorry g of alpha into g of ground state. But by our convention this is chosen to be 0. Remember once more for a quantum partition function we always chose the e electronic ground state to be 0 or e ground state itself to be 0. So, we get this is equal to simply g of where g is the number of states with equal energy for the ground state ok. So, we will go let us look at a couple of examples. For example, if I want to think of hydrogen atom. So, for hydrogen atom you have 1 s orbital and you have 1 electron. But you note that this electron can also be in the down spin state. So, this is a little bit quantum mechanical each orbital can have up spin or down spin. So, with that basically your degeneracy is 2. So, every time you have an unpaired electron your degeneracy becomes a more than 1. So, your q electronic I am sorry electronic is equal to 2. The same thing I do for let us say h 2 molecule well h 2 molecule in the ground state has only one configuration ok. So, both electrons are paired when that happens your degeneracy is simply 1. There is only one possibility that the ground state can be in which is this 1 sigma 2. So, this is important how do you calculate the electronic partition function as well in quantum mechanics ok. So, now we have looked at the different partition functions. I want to just spend a few more minutes in explaining how do we understand partition functions what is the meaning of partition function. So, what we have been looking at is a partition function at a given temperature ok we had beta. Actually you can also define partition function at a given energy and if you do that that is simply the sum over states of that energy. So, quantum mechanically partition function at a given energy would simply mean the number of states at that energy which is actually exactly the same as the degeneracy. So, that is what partition function at constant energy means. And classically that would mean the area or the volume that that constant energy surface occupies. So, think of a contour plot of that particular energy in a high dimensional space in q, p space and find its volume that is your classical partition function at that energy. We can also define the partition function at a given temperature which is what we have calculated. So, the meaning changes slightly, but essence remains the same a quantum partition function is essentially the number of states that are accessible. So, you are at a given temperature which means you have a distribution of states. So, the quantum partition function is essentially telling you at this temperature these are the number of states that have some occupancy. This is a very broad and rough definition the exact definition is a summation. So, this is just an intuitive definition of partition function and classically that number of states becomes a volume in units of h. So, the accessible phase space volume at that temperature in units of h. So, that is the fundamental meaning of partition function. So, we have found the partition functions for HBr. So, the translational in the last module you can go and we calculated these numbers out. So, the translational partition function came out to be 7.2 into 10 to the power of 29. The rotational came out to be 24.6 and vibrational the classical number comes out to be 0.078 and today we calculated the quantum number and quantum number is very close to 1. So, what these numbers really mean is that the translational states you have roughly 10 to the power of 29 or 10 to the power of 30 states that are accessible at 300 Kelvin. The translational states are very, very closely the translational energy levels are very closely spaced and so you have a lot of translational states. Now, rotational states are somewhat far apart for HBr they are roughly like 5 to 10 wave numbers 20 wave numbers actually. So, at 300 Kelvin you have roughly 10 or 20 states that are accessible and in terms of vibrational quantum mechanically you have only one state that is accessible. So, what is the reason for that? The reason for that is that the vibrational frequency. So, the vibrational frequency is roughly 2500 wave numbers room temperature is roughly 200 wave numbers. So, you can do this calculation at 300 Kelvin what is it in wave numbers it is 200. So, you see the frequency is much more. So, the vibrational levels are very far apart vibrational levels are separated by 2500 wave numbers while temperature is rare this is 200 wave numbers. So, you do not have enough energy to access excited states at all and therefore quantum mechanically you get a quantum partition function of 1. So, that is how we understand partition functions. So, in summary, today we have looked at partition function and looked at what its physical meaning is it is essentially the number or the volume of accessible states at a given temperature. We have looked at the quantum partition functions today and we are going to use the same classical formula for translation and rotation, but for vibration we will use a different formula when it comes to quantum mechanics which is 1 minus e to the power of I apologize there is a little mistake this is beta h bar omega. Thank you very much.