 We are discussing absorption spectrum and as is not very common in our course we start with the same kind of opening slide that is a last lecture. But now perhaps you understand it little better. Here is an absorption spectrum of benzene dissolved in cyclohexane. Now you know what is molar extinction coefficient and why the unit is centimeter inverse per molar. Now you know why the x axis is in wavelength rather than in electron volt or something like that and now you can sort of guess that this structure comes because these are vibronic transitions and this different heights or different values of molar extinction coefficient for different wavelength tells us that the probability of these vibronic transitions are not all the same. So, this entire spectrum is of the very well known benzenoid band that is available in all organic well aromatic organic molecules. It shows up nicely if the solvent is non aromatic but non polar also. Now what we are going to learn is why is it that this third band is most intense why not the first band? When will the first band become most intense? Because if you remember we had shown a spectrum of some other compound earlier that the first lowest energy band was the most intense band. That depends on the relative values of the bond lengths in the ground and the excited states. Now see I have taken this figure from Wikipedia and I have done a little bit of surgery. I have cut the top part and I have moved it around. So please neglect this blue vertical arrow wherever it comes I have drawn my own arrow wherever required. Now structured absorption spectra like the one that you have seen arise out of vibronic transitions v equal to 0 to v dash equal to 0 1 2 3 whatever. This is governed by something called Frank Condon principle. There are two things to understand about Frank Condon principle. Actually Frank Condon principle started with the classical formulation and it essentially said that the transitions are all vertical. What is the meaning of that? In this diagram y axis is energy x axis is nuclear coordinate very easy to understand if you are talking about diatomic molecules it is simply inter nuclear separation. So when it is vertical what it means is that when there is a transition between two electronic states there is no change in inter nuclear separation. So no inter nuclear rearrangement takes place during electronic transition. This is the classic formulation of Frank Condon principle and the quantum formulation is that the intensity of transition is governed by Frank Condon factor which is integral of chi v dashed chi v. What are this chi v dashed? Well chi v chi v dash this dot is a lot of ink please neglected. Well these are the vibrational wave functions. Energy is a quantized vibrational quantum numbers range from 0 to whatever 0 1 2 3 so on and so forth. And what we are showing here is an anharmonic oscillator. This is essentially the kind of potential energy surface you get for an actual diatomic molecule. So the energy caps keep decreasing decreasing decreasing until here they become a continuum and this is where if you promote the molecule from here to here then the bond is going to break. Another point to note is that there is a minimum nonzero value of energy corresponding to v equal to 0. This is called 0 point energy for our course it is sufficient if you know this we will not go into further discussion. Coming back to wave functions. The v equal to 0 wave function is essentially a Gaussian. Higher wave functions are this Gaussian multiplied by what are called hermite polynomials forms of which we do not need to know at this point. But this is what they look like. So for v equal to 0 there is no node. For v equal to 1 there is a node at the center. For v equal to 2 there are 2 nodes equispaced from the center. For v equal to 3 there are 1 2 3 nodes the middle one is at the center and so on and so forth. As you go higher up the number of nodes increases pretty much like your particle in a box problem. But these wave functions are not signed functions. And also these wave functions go a little beyond this potential energy surface because you might remember our discussion of how quantization arises out of boundary conditions. Boundary conditions for these wave functions is that they have to vanish at inter nuclear separation of infinity and minus infinity. That is why it is okay if they go beyond the potential energy surface. Now, Frank-Condon factor means the integral of the product of the vibrational wave function from which the transition originates and the vibrational wave function of the level to which the transition goes. So essentially it is a numerical integration we will see how it is done. So now let us say this is the excited state of the molecule the atomic molecule let us say. So essentially this kind of diagram means what is what does this minimum indicate the minimum indicates equilibrium bond length. So in this situation bond length is the in the excited state is greater than that in the ground state that is what we are discussing right now we will discuss other situations when they are equal or when the excited state bond length is smaller than the ground state bond length also. But let us see what this means. So this is the wave function of the target state. Let me just draw it here this is what it is something like this and goes up goes on something like this. So what is this Frank-Condon factor it is basically this function multiplied by this function this function multiplied by this function for a given value of nuclear coordinate and then you add them up. So essentially you multiply the two plots and find the area under the curve this is how you can find out Frank-Condon factor. Another thing to remember is that all upward transitions start for nuclear coordinate equal to equilibrium bond length because for v equal to 0 that is where the maximum is. So this is where the maximum of psi is that is why psi star is maximum and dx is equal everywhere. So the maximum probability of finding this wave function is at the equilibrium bond length for v equal to 0. So all transitions start for at the center of this wave function and we have 5. So this is where it is going to go. So what do we see if we work out the Frank-Condon factors of all other wave functions actually the Frank-Condon factor will be maximum for v dash equal to 2. So now what will the spectrum look like I am plotting against energy. So not wave length energy please remember. So let us say this is the 00 dash transition it will have some intensity some absorbance. The next one will actually have a little more because if you work out Frank-Condon factor will be greater for 02 dash 01 dash transition. For 02 dash it will be maximum. For 03 dash it will fall again and it will keep decreasing. So it is going to go through a maximum. If you remember the absorption spectrum of benzene it does go through a maximum because this is what the situation is like for benzene. Of course benzene is not a diatomic molecule it is a polyatomic molecule. So the nuclear coordinate there is not simply an internuclear separation. It is something else we do not need to get into that now. Next 00 dash transition there is a name for it it is called the band origin. What is the meaning of band origin? Suppose there is no vibronic structure which is a pure electronic transition then the only transition you would see is 00 dashed. So that is called the band origin that is the smallest energy vibronic band that you can hope to observe. In a situation like this many times what happens is that the intensity of the band origin is so small that you do not even see it. So finding the band origin may be a non-trivial problem depending on what kind of a system we are handling. Great. Now let us go to the situation where the bond length in excited state is equal to that in the ground state. Now obviously the transition that will be most intense the transition that will have the largest rank condom factor is 00 dash transition. Please neglect this small blue arrow I have told you why that has a reason. So rank condom factor will be largest for 00 dash transition. So what will the spectrum be like? As you go higher up in the energy the vibronic lines become smaller and smaller and smaller in intensity. This is the situation in I think naphthysine that is the spectrum we are seeing in the previous module. What happens if the bond length in the excited state is smaller than in the ground state? Well then again actually the same thing happens as what happens in the situation where bond length is in the excited state is greater than that in the ground state. Because once again rank condom factor would be maximum for 02 dash transition it would fall off both ways. So the spectrum would go through a maximum. And sometimes the bond length in the excited state may be very much larger in that of the ground state with the limit that the bond length in the excited state may be infinity which means you would not even see this minimum anywhere it will just keep decreasing decreasing decreasing as you increase the nuclear coordinates we will come to that also. But before that if the bond length in excited state is much greater than in ground state then the frank condom factor is going to be maximum for some very high energy vibrational level which can be in this continuum range. So now what will happen? 02 dash transition will have some very very small intensity 01 dash will have little more 02 dash will have a little more it will keep going like this until you reach this level after which what will happen is that the molecule is fragmented. So it can have any energy. So suddenly after a certain level you are going to get not discrete structure anymore but rather a continuum. So that continuum so this wavelength or energy at which the continuum sets in is called the continuum limit it can be related to many other things which we will not going to discuss today whoever is interested please read this from one of these textbooks Banwell and Banwell is a good textbook for this but you are going to have this lower energy will have structure at higher energy after a certain limit you are going to have continuum this is the signature that the molecule is dissociating by a transition and a transition to a bound state photoelectron spectroscopy is something that was known when you give say X-ray or something the electron is expelled so basically provide ionization energy here we are not providing ionization energy is a much lower energy photon than the ionization energy yet one can make the molecule dissociate and actually this was the question from which Frank and Condon started working out their principle. So what we learn is that even by a relatively low energy photon one can bring about dissociation of a diatomic molecule also polyatomic molecule but that is more complicated let us stick to diatomic molecules at the moment one can bring about dissociation by transition to a state that energy state that goes to minimum provided the energy takes it to continuum that is the first important thing. Next we can think of that other extreme we are talking about if the bond length is so large that it is close to infinity then we will never reach a minimum right then the excited state is going to look like this it will be a it will just fall off exponentially asymptotically and there will be no vibration energy level associated with it because it is an unbound state in this case so I hope this reminds you of the energy of anti-bonding orbitals for example. So in this case what will happen is that your spectrum is going to be a continuum all the way there will be no structure you are going to have a transition to a dissociative state this is called a dissociative state it does not go through a minimum so you can cause a dissociation even without providing energy and there is another very interesting phenomenon called pre-dissociation let us say our transition states are like our sorry energy levels are like this the bond lengths are different but not so different that you can cause dissociation unless you provide this much of energy however many times what happens is that energy levels cross these potential energy surfaces cross and crossing can be of two types diabetic and diabetic that is another important topic but without getting into that let us say that this excited state that goes through a minimum crosses another excited state which is dissociative then what will happen you excite to higher energy levels you are going to get lines in the spectrum excite to lower energy levels you will get lines excite to energy levels close to say 2 here then there can be a crossover from the bound energy state to a dissociative energy state and that is where you will get a continuum so the spectrum will be something like this you have lines at low energy you have lines at high energy and in the middle you have this kind of a continuum this is a signature of pre-dissociation so this is what we wanted to discuss about spectroscopy but before closing I would like to point out a very elegant and very important experiment that has been performed by considering this phenomena that you can bring about dissociation by excitation by UV visible light and potential energy surfaces can cross and that example is something that we encounter every day in labs sodium we know flame test right why is it that we get yellow flame for sodium you take an ion Na plus I minus but you get the characteristics spectrum of sodium atom that is because this here this figure is from Atkins physical chemistry book but I will show you the original work so Na plus plus I minus the ionic state actually goes to a minimum Na plus I covalent state is dissociative however these two energy states cross and if you excite you can actually excise you can have a situation like this you excite regular sodium plus I minus giving the right energy you can do a transition to this dissociative sodium and iodine covalent so that is your neutral sodium atom by photo excitation you can make this sodium iodide dissociate this experiment which was already known was performed by using a short pulse high intensity lasers when I say high intensity short pulse I mean lasers that are on for a few femtosecond and this is a seminal work done by Professor Amel Zuel and his group for which he got Nobel Prize in 1999 here you see Zuel with his Nobel Prize and what they did was that they excited this and will not get into the technique but let us just just believe me when I say that they could work out the time evolution of the population of Na plus and time evolution of population of Na. So, Na plus is reactant Na is product what would happen the moment you trigger the reaction by a pulse of light then over time sodium gets depleted and Na plus gets formed so population of Na plus would increase or I minus for that matter and population of Na or I for that matter would decrease without going into experimental detail that is what was observed the top curve where you see a rise that is a measure of the time evolution of population of the neutral state the neutral Na I the bottom one is a decay that is for the Na plus I minus situation note the x axis x axis is time and time is in picosecond and femtosecond if you look at this curve it gets saturated in about 2 or 3 picosecond so if you fit this curve you get a time constant of some 200 femtosecond or so so the reason why this was a Nobel Prize winning work was that this was this is the first experiment that determine how much time does it take to break a bond and the answer was something like 200 femtosecond or so not only that this experiment was what Zuhel called a snapshot of the bond breaking does not just tell us how much time it takes it tells us how it happens actually I would prefer a video recording of bond breaking you see we said that this tells us how the population grows of the covalent form it does not grow smoothly does it there are oscillations in the signal and these oscillations are much more prominent for Na plus I minus what are these oscillations see it goes down that means there is a decay then it forms again to some extent then it goes down forms again to some extent and then gradually it goes away it is like a damped oscillation what is happening here what is happening is you have Na plus and I minus they are together give the laser pulse they start breaking do not break completely come back start breaking this time they go a little further come back start breaking and after the 3 or 4 oscillations they dissociate completely so this is how a bond breaks and so what we are saying is that not only can you say whether it is an n pi star transition not only can you determine the concentration of the solute not only can you say fingerprint a molecule using electronic spectroscopy but you can also by using advanced laser spectroscopic technique try to get an intricate idea about dynamics and this becomes even more so when we look at not only just absorption but also we start worrying about what happens after absorption by light absorption we have created a molecule in its excited state then what happens does the molecule just come down while coming down how does it emit the excess energy in the form of heat or light can it do some reaction in the excited state that it cannot do in the ground state this these are the questions that we will touch upon very briefly in the next module