 Hello everyone, welcome to the module on Intermolecular Forces and Potential Energy Surfaces. We are currently discussing about potential energy surfaces and the potential energy surfaces of different kinds of molecules, either triatomic or even other kinds of systems. Before we get into looking at potential energy surfaces in a little more detail in this lecture, let us recap our memories on what we discussed in the previous lecture. In the previous lecture, we had started out by looking at what are called as potential energy surfaces and we had said that these are nothing but 3 dimensional or a n dimensional hypersurface in which by playing around with the coordinates or different modes or different degrees of freedom of a given molecule or atoms or ions, one can actually look at the potential energy of the system and such a diagram of potential energy versus the coordinates is what one would call a potential energy surface. And in case of where we are looking at chemical reactions, these coordinates are typically the vibrational degrees of freedom such as the bond lengths, bond angles and various other degrees of freedom. So we had seen that based on the kinds of the molecule, one could have either a hypersurface of 3n minus 5 or 3n minus 6 where n is the number of atoms or ions in the given system. Further, we had also said that these 3 dimensional surfaces can also be mapped onto 2 dimensional picture which is called as a reaction coordinate diagram and here what one does is to look at along a particular coordinate of interest that is either a particular bond length or a bond angle or even a combination of them and then draw them on a 2 dimensional representation. This is typically what one would use in looking at the transition states of a given reaction or even the transition state theory of a particular species. So having said this, we had also learnt or discussed what is called as a saddle point which is a very important aspect when we talk about potential energy surfaces and reaction coordinate diagrams. So if we take a diagram such as shown here, the one which we had looked at in the previous class as well, what we said is that this point where going from the one valley to the other, the hill one crosses is what is called as a saddle point and we had seen that it is called saddle because of the way it looks like that is on the 2 sides the energy is actually going down and on the orthogonal sides the energy actually increases. So an important point to note is that a saddle point is invariably highest energy point on a 2 dimensional reaction coordinate diagram however when it comes to potential energy surface of 3 or n dimensions, it is not the highest energy point. It is always somewhere in between because there are points of highest energy such as you could see here on the green parts of the surface are the ones which are actually having a higher energy compared to the saddle point. So a key distinction is on the 2 dimensional picture that is the reaction coordinate diagram saddle point represents the point of highest energy whereas on a n dimensional or n dimensional hyper surface which is called as a potential energy surface saddle point does not represent the point of highest energy. This is a very important distinction to keep in mind. So having sort of refreshed our memories on what are called as reaction coordinate diagrams and potential energy surfaces now let us start looking at a very simple case of a H3 system. So for that I am going to try and write down the reaction which we are interested in first. So let us say I have a hydrogen atom HA which is interacting with a hydrogen molecule. Just to distinguish the hydrogens I am writing them as HA, B and C and this would give me HA, HB plus HC. So if I am interested in this particular kind of a system where hydrogen atom is reacting with hydrogen molecule to give rise to similar product but with a slightly different connectivity. In this particular scenario what can happen is the 2 entities that is the HA and HB, HC can actually approach in 2 different ways. One is they can approach in a collinear fashion that is HA is approaching the HBC bond or it can also approach in a non collinear fashion that is at a different angle or at a particular angle. For the sake of the convenience and what why and it also turns out that that is the most reasonable way we shall look at only the collinear case that is the HA and HB, HC are actually interacting in a more of a head on fashion. So if we now restrict ourselves to the collinear way of interaction then in order to represent this we can actually choose 2 coordinates or 2 degrees of freedom. One is the bonding between the HA, HB that is we are going to form a new bond between HA and HB. So we can look at the how does the potential energy varies as a function of this that is RAB and we can also look at how does the potential energy varies as this bond actually gets ruptured that is RB and C. So these are the 2 parameters which we can vary and look at how does the total energy or the potential energy of the system varies. So if you now agree that these 2 parameters are good enough to describe the particular system we are interested in that is H3 in this particular case then one can come up with the potential energy surface which is shown on the left hand side and invariably such potential energy surfaces are constructed by what are called as electronic structure calculations which are typically based on quantum mechanics based computations where both these parameters that is the distance between the HA, HB and the distance between the HB and the C are varied and at each of the coordinates that is each of these points the total energy of the system is or the potential energy of the system is calculated and that is plotted as a function of these 2 coordinates. And if one does it we would for the H3 system we would end up in a diagram which would look like this. So we will try and dissect this a bit more and try to understand this. So let us start by looking at this diagram in a bit more detail. Okay, here I am trying to look at HB plus HC, so HB, HC which is bonded and I do not yet have HA which is still interacting with it. So if I just take the reactant that is HB and HC then I would see that the one which is one we can see on the left hand side is the one you can look at it as a one dimensional potential energy surface of HB, HC correct. That is you have an equilibrium bond distance where the HB and HC are bonded and now in this particular case the HA is somewhere here, this is at an infinite distance and only once the HA starts approaching the HB, HC the energy of the system varies and then they interact and ultimately lead to the product where HA and HB are bonded and HC is left out free and that is what one would find if you look at this particular part of the potential energy surface here you have HA, HB which are bonded and HC is now free. So I hope you see or you can imagine this where I am going from a one two dimensional kind of a potential energy surface of one hydrogen molecule to another that is the product and what is interesting is how or what path does it take that is something which is very important and we shall look at it in a bit more detail. But to look at it in a more of a bird's eye view what we have is we have two degrees of freedom in the HC system that is the two bond lengths which can be varied that is RAB and RBC and if we take these two as the variables one can construct a three dimensional potential energy surface where two accesses are these two coordinates and the third access is the potential energy and that would look like what is shown here in the on the slide. I hope this is clear or at least understandable and if that is the case now let us go ahead and look at how does one actually take this and map it on to a two dimensional picture so that it is easier to understand these potential energy surfaces for further analysis. So again we start with to map this potential energy surfaces from a three dimensional to two dimensional picture let us just start by looking at this 3D picture which we have of a H3 system which is what we just looked at in the previous slide and just another way of looking at it is that you can now forget about the pink parts and only look at the gray area if you just concentrate on the gray area all I have done is I have just scooped out the pink portions and I have just left with the gray area and if you just now look at the gray area what you would see is something like this and the most important point to note in this is that here you have the reactant which is this particular part and then you go to the product which is this on this side that is the this coordinate but what is important is while going from the reactant to the product you go through a slightly higher energy position which is actually hidden if you look at this three dimensional picture you do not see this higher energy point when you look at it when you look at a three dimensional picture so that is the reason why we have scooped out the pink portions and then hopefully this is a bit more clear that while going from the reactant to the product you have a slide you have a barrier which you cross and then fall into the product regime. So now the question is how do I map this into two dimensional picture so to do this what people typically do is what are called as slicing or a counter maps and that is what you see here on the right hand side this is what is called as a 2D counter map of H3 system and the way to obtain this is first what we need to realize is that if you actually look at this picture the three dimensional picture here you have the you have the RBC and the RAB actually both pointing at one particular direction and whereas if you look at the now the 2D picture they are actually not in the in the exactly same orientation that is this two dimensional picture is obtained or at least it is shown by just tilting it at a particular angle. So please realize that this diagram is actually not in exactly the same way as this three dimensional potential energy surface but it is just slightly tilted so that the 2D map becomes a bit more clearer. So just to make that clear that point so if you now take this okay so if you now look at this picture what you see is that RAB and RBC are pointing towards the top right hand corner that is this whereas if you now look at the three dimensional potential energy surface you have these two pointing at the down so you will have to actually take this three dimensional object or three dimensional potential energy surface and rotate in such a way that this the bottom corner actually turns to the top right corner so then you will be able to translate or at least you will be able to imagine the how one can go from a three dimensional surface to two dimensional contour maps. So this is point number one and the second point is what typically people do is once I have done that rotation that just in plane rotation then what people typically do is you take a slice or in other words you choose a particular potential energy for example I will let us say choose this I am just drawing a plane here so I have chosen a particular potential energy let us say I call it P1 so then at this point I will look at the at for both the coordinates how does the potential energy surface look like and I see that would typically correspond to this something like this because you have this shape which is going here this particular line or this particular curve is what one would get as this when you cut a when you cut through it you would get this particular line right so that is what I am trying to show here and to some extent you would also cut this that is here somewhere you would cut if I pass a plane through it and these are the two lines you would get for a given particular potential energy now I will come down a bit and I will again put again slice it or I will again go to a different potential edges let us say I call it P2 so then I can actually keep doing like this I can just keep slicing this or taking a slice at different potential energy surfaces and look at how does the 2D map look like so as you keep doing this till you come to this point or till you come to this point you only cut these two sides that is this side and on the other side is this right once you hit this particular point somewhere as you keep going down the potential energy you will hit this point at that point you are mostly at this this juncture okay and now if you actually go down further then what you see is that you actually see a this is this can be imagined like a small hill here so if you have a small hill then if you are cutting through this then you would get a you get these kind of let's say curved lines this is what one would get as you go below the top of this hill this is the top this is the hill top if you go below that then you would start getting curvatures like this and you can actually as you go as you go even down further down you would this we would become actually the the radius of this or the the radius would become lower and you would end up at the minimum of the particular bond that is either the reactant or the product so this is what typically is done to obtain a 2 dimensional contour from a 3 or n dimensional potential energy surface. So I will again repeat a few of the silent features which are used to attain a 2 dimensional contour that is first you take the 3 dimensional surface and you slice or you take a cut at different energies or different points and then look at how does the 2d plot look like and that is all that you are trying to plot here on the right hand side and I hope this is you can visualize this to at least some extent that the way we are able to construct and the most important point is that if you take a look at this the bottom left picture then you will realize that there is a small hill with a barrier which is actually hidden in this 3 dimensional complete 3 dimensional potential energy surface and only below the barrier you start seeing the corresponding contour maps which are which correspond to either the reactant or the product. Alright so I hope this gives you a feel of what are these 2d contour maps so having had some feel for this now let us go ahead and try to see the H3 system in a little more detail. So here again what what is done is the same picture which I showed you on the previous slide that is the RBC and the RAB and what I would want you to realize or appreciate is the following that if you now look at I am just going to draw a dotted line here so this is the equilibrium distance for AB which is the product and similarly I can draw something along this line and this is the R equilibrium mon length for the reactant that is B and C right and if I were to this is where the they would all the energies would go toward the infinity or toward the isolated atoms. So if I have this particular picture so then if let us say I have a hydrogen A which is coming towards HB so how should it come on the potential energy surface right so there are 3 possibilities and we shall look at all of them in little detail. So the possibility number 1 is or let us call it A because that is the notation we are using here. So I have HA and HBHC so this is coming approaching this so what I am trying to show is that the HA is approaching the HBHC progressively and at that moment the HBHC is kept constant or the equilibrium bond length of HBHC is kept constant it does not move and at beyond a certain point now after this what happens is that HA and HB bond and then you have HC which is left out okay. So this is one particular scenario where I have the HA which is approaching the HA is approaching the HBC progressively and in all this time the HB and the HC bond length or the equilibrium bond length is kept constant it does not change. This is one particular scenario you can think of if that is the case let us see how does the potential energies or the trajectory would look like on a 2D contour. So we said that RBC is the equilibrium distance for the BC so that is what we have kept constant here if you now look at the trajectory which is marked with A that is this particular line if you start taking look at this particular line the R equilibrium is more or less constant it does not change pretty much till you till you hit this line till you hit this curve right and and all through all of this what is changing is the hydrogen atom which was at the which was far away is actually slowly coming in slowly coming in and trying to go towards the HBHC bond that is what we have shown here on the right hand side. Once it is at almost this particular juncture let us let us call this this dotted point at this point what happens is then suddenly the HBHC bond would break apart and then the HA HB would bond would form and you have significant lowering of energy and that would lead to formation of the R equilibrium AB which is the product here which is what you would see right. So this is one particular trajectory the reaction can take place but if you already notice what you see is that this is along a slightly higher energy path because you are looking at it from a this path almost coincides with an outermost line which means the reaction is taking place along a higher energy surface right. So now the other option or the other possibility is what is called as B let us take a look at the B. So the HA is here and let us say HBHC is to begin with okay. So in this particular case what is happening is HA is approaching the HBHC but in this case the HBHC bond is actually getting loosened up far more quickly it is actually going from the equilibrium bond length the HBHC is slowly opening up or it is HBHC are dissociating much more rapidly than the approach of the HA this is another scenario right. So you can think of these two are the most classical ends of the spectrum right. So if this is what is happening then you would go along this path which is shown here B that is at even a small approach of HA which is along this direction you see that there is a significant lowering of the R equilibrium that is this angle this distance is actually now going is increasing and by the time it comes to the equilibrium it has increased so much and then suddenly little then it will the HBHC will break and it will form a HA HB. Then one can write HA HB and HC is lost right. So this is another example where the dissociation is much more pronounced compared to the approach of the incoming hydrogen atom. However the another possibility is that both these processes that is the approach of the hydrogen atom and the weakening of the HBHC bond can occur to the same extent or to a similar extent. So that is what we will call it as a KC. So in this case the approach and the breaking these two processes are occurring at the same distance or occurring at the occurring to a same extent. So in that case what would happen is you would take this particular path which is the C you go along the path C and then you ultimately end up in the product here right. So this path where you get the where you go through this particular pathway is when you actually cross that small hill which I told you in the previous slide where we had scooped out the pink part and only looked at the gray part. I hope you remember that there is a small hill and along the path C actually you cross the you cross the top of that hill which is the C and it turns out that C point C is the or the C dagger which is shown here is the least energy path among the ABC. So the reaction would invariably would like to go along C rather than A and B because A and B are higher energy or this is the low energy path the C. So this point or the C dagger is what is also called as a saddle point or a transition state and the geometries in and around that are what are called as activated complexes. So if a reaction is taking place along this particular trajectory or path then there is more likelihood of the reaction falling into the products rather than when it goes along A and B okay. So I hope this has given you at least some idea on how to look at the trajectories on a potential energy surfaces taking H3 as an example with this we shall stop now in the subsequent classes we shall look at a bit more of the final details. Thank you.