 Hello everyone, welcome to the module on intermolecular forces and potential energy surfaces. In the last few lectures we discussed about various intermolecular forces and what are the consequences of those forces on the properties of gas molecules. Now we will move on and start looking at what do we mean by potential energy surfaces and how do we try to understand them. But before we get into potential energy surfaces themselves let us just do a quick recap and try to see how intermolecular forces or potential energy surfaces are related with one another. So in the last few lectures we have been looking at role of intermolecular interactions in real gases. Here we had said that because of the induced dipole or what are called as London dispersion or van der Waals interaction real gases do experience an attractive force between them when they are at a close distance, the molecules are at a close distance and this and in addition to a repulsive forces which also come in when the molecules are really close to one another actually contribute to lot of the properties of real gases. And we had tried to capture these properties using the two formalisms or two kinds of methods. One is called as the Virial Equation of State and the second one was called as the van der Waals equation of state for real gases. In the van der Waals equation of state we had tried to construct the equation using the very phenomenological arguments or very intuitive arguments based on the interaction between molecules. And we also looked at some of the advantages and the limitations of van der Waals equation of state. And finally we had seen what are called as critical phenomena or critical points which are associated with it and some of the potential applications of these phenomena. For example in selective and benign extraction of chemicals from a given mixture and this can be achieved by varying the temperature and pressure. And we had taken the example of supercritical carbon dioxide to illustrate this. So having learned this now let us try and jump into what are called as potential energy surfaces. But just to give you sort of a brief distinction between the two what we looked at so far was interaction between molecules and by that I mean there was actually no change in the bonding which was taking place. So the bonding was intact that is I had a molecule A and molecule B which were actually floating around in solution and they were interacting with one another. And this is what we were primarily concerned with when we looked at intermolecular forces and the consequence of that on the properties of real gases. Now when we jump into potential energy surfaces we shall actually go away from looking at intermolecular interactions and we shall actually look at the bonding or how when two atoms or ions come together they actually bond to give rise to a covalent system. And that is what we would be primarily interested in and in this lecture I will try to introduce to you what are called as potential energy surfaces and in the subsequent lectures we shall look at some of the examples to concretize the ideas built up in this lecture. With this now let us begin by looking at intermolecular or interaction potential of diatomic molecules. Here let us say I have two atoms or ions which are interacting. For example I have an atom or ion A and similarly an atom or ion B which are actually now interacting and they would ultimately give rise to a covalent system that is AB. This is what I am trying to look at. How do A and B come together to form a AB covalent bond? So if you look at the interaction potential or how does the energy of the system vary as a function of this distance R then one would end up in a diagram like this. So here on the x axis what I have is R which is the distance between the two atoms or the ions that is A and B and on the y axis I have the interaction potential of the system V of R. And I guess you would already be reminded of something which we had looked at when we studied Van der Waals equation of state in the previous lecture. That also had a very similar nature of the curve where at a very large distances we had almost the zero energy which means the isolated energy of both the units A and B and as we come down along this line or as the constituents come closer to one another they do interact and they actually now form a covalent bond and this the length at which the energy is minimum is called as the equilibrium bond length or the equilibrium length and it is labeled as R0 or we can call it R equilibrium this point and one if we actually now go beyond this I mean if you try to even get the two units closer together even further then they would actually repel because you are getting them because the Pauli's exclusion principle and thus the energy would actually really go higher in energy. So this actually explains very nicely how the interaction energy or interaction potential varies when I bring in two atoms or ions such as A and B in this case correct and this as this potential is also called as Moore's potential. So the point to note is both Moore's potential and the Lenard-Jones potential might look very similar in the nature of the curve but they do have a very distinct and a different mathematical formula which describes them and another very key distinction is the Moore's potential is for a diatomic system where there is a covalent bonding takes place whereas the Van der Waals equation or the Lenard-Jones potential is for two atoms which are actually in a non-bonded interaction that is there is no bonding which is taking place between the two atoms or between the two molecules such as methane and ethane or two any atoms any molecules which are trying to look at. All right I hope that distinction is clear and so this might look very simple and easy how the interaction potential would look for a diatomic molecule. But now this is things become actually far more complicated as we go away from diatomic molecules because in reality we do not just have diatomic molecules such as Hf, H2, Ki and many different systems but we do have more different complex molecules to deal with in our life right like we deal with water we do have amide bonds or many different kinds of molecule entities which undergo reaction to form a covalent bond. So how would one capture the interaction potential of such a system which has more than two atoms in it. So let us go ahead and see how we can do this. So in order to do that we need to understand what is called as degrees of freedom and their association or their relationship with chemical reactions okay. So I am sure you all agree that if I take a system which has n different atoms or ions and if I now look at what are the different degrees of freedom I am sure you will all agree that the degrees of freedom is given by 3n right because 3 comes from the 3 axis along which the system can freely undergo any kind of rotation, translation or vibration. So you have x, y and z direction and each of the atom can go along these three accesses in the Cartesian coordinate or the similar accesses in a polar or spherical polar coordinates. So now that means that if I take any particular general system then I would have 3n degrees of freedom to that system where n being the number of atoms or ions in the constituting the system and these 3n can further be divided into what are called as rotational or translational and vibrational degrees of freedom. So that means the molecule has all these three ways in which it can actually move around. So if this is the case now let us go ahead and deconvolute or look at only the vibrational degrees of freedom and I will come to that in a minute why we are only interested in the vibration. So let us say if I take a molecule which is non-linear for example I have a molecule of water or ammonia or any other such system non-linear system and the total number of vibrational degrees of freedom of such a system are given by 3n minus 6. So I will tell you where this number 6 comes from. So this 6 comes because the molecule can undergo 3 degrees of motion of translation that is if you take this as a molecule it can go under it can translate along the one direction one axis other as well as the next axis right. So along all the 3 accesses that is x, y and z the molecule can undergo translation. So that gives me 3 degrees of freedom of translation and now the molecule can also undergo rotation along the 3 accesses that is x, y and z. So that gives me again 3 degrees of freedom which corresponds to rotation. So if I am interested in only the vibrational degrees of freedom then all I need to do is take the total degrees of freedom available to the molecule that is 3n in this case and from this I will subtract the contribution from the translation and the rotation. Then what I would be left with is the vibrational degrees of freedom for a given system which is non-linear. Similarly one can also do this exercise for a linear molecule and there is a slight difference and what you find is that the total vibrational degrees of freedom for a linear molecule is actually 3n minus 5 but not 3n minus 6. This is because you have the same 3 degrees of 3 translational degrees of freedom for the molecule along the x, y and z direction. However the degrees of freedom for rotation is now reduced by 1. This is because if you take a molecule which is actually linear and if you actually rotate along this axis itself then you do not see any change. So you only have rotation possible along the 2 orthogonal axis to this either this or this. If you rotate along the molecular axis you do not see any change. Thus for a linear molecule you only have 2 degrees of freedom for rotation and now if you do the same thing that is I take the total degrees of freedom for a particular system and subtract the translational and the rotational degrees for a linear molecule which is totally 5 and then I would end up with 3n minus 5 degrees of freedom for a given system. So this looks I think at least intuitive or easy to understand and you must be wondering why is it telling us this or what is the relevance of this to the potential energy surfaces because that is what we started out by discussing right. So if you actually look at the degrees of freedom and we are now currently only interested in the vibration because if you think of chemical reactions for a reaction to happen let us say if I am holding a bond like this for a reaction to happen that is either for this bond to break or for this bond to form between 2 atoms the molecule should be vibrating right. Only when the molecules are vibrating or when the molecules are actually undergoing some sort of torsion or distortion then you would have a chemical reaction take place and that would lead to a change in the potential energy. However, if you just have a molecule from one position going into another position or just rotating along an axis in the system then you will not have a significant contribution towards a chemical reaction. That is the reason why we are particularly interested about vibration because generally vibrational degrees of freedom are the ones which constitute or which which actually are the dominant degrees of freedom which lead to a chemical reaction when molecules actually come and collide with one another. So I hope this idea of vibrational degrees of freedom being connected to chemical reactions is clear that is if you can think of the atoms as actually like springs which are actually vibrating around or moving around and if the 2 units which are moving around come and hit one another then when because of this vibration the change in the bond order is what would lead you to your chemical reaction. So I hope this idea of vibrational degrees of freedom being connected to chemical reactions is clear. So with this we shall now go ahead and try and look at slightly more complex systems but if you want to actually understand the previous example which we learnt you can do a very simple exercise that is if you now go to this linear molecule case which I told you that is you take 3n-5 and now put n is equal to 2 in this then you would end up with 1 degree of freedom and that is exactly what we saw when we took 2 molecules which are actually linear in the previous slide when we talked about atom or ion A and atom or ion B there the only coordinate was there the distance between the 2 units. So I hope this gives you an idea about the vibrational degrees of freedom and their connection to chemical reactions. Now let us go ahead and look at the potential energy surface for a multi-atomic system or for a system which has more than 2 atoms in it. So here what is shown is a potential energy surface for a triatomic system that is particularly water and in this what you have is one of the accesses that is typically the vertical accesses energy and the other 2 accesses in this case the x and the y are in one case it is the length bond length of the OH and the other one is the bond angle which is again the 2 degrees vibrational degrees of freedom that is you have the bond length which can actually stretch or you have the if you think of this intersection as oxygen and the 2 hydrogen this bond angle can actually change right. So what one can do is you can take it every point or every change in the bond length and the bond angle and you can look at the energy of the system and if you now plot the energy of the system at each of the different angles and the bond lengths then you would end up in this so called hyper surface and in this hyper surface the point which corresponds to the minimum is the is corresponds to this angle of about 104.5 degrees and about 0.0958 nanometers as the bond length right. So this is how one goes about constructing the potential energy surface for a multi atomic system that is you keep one of the accesses as constant that is the energy of the system and then you take many different accesses which each of which corresponds to either a bond length or a bond angle or any different kind of vibrations and then you look at how with each change in either bond length bond angle or a vibration or bending the energy of the system varies and if you plot that you would get a surface which looks something like this which is a hyper surface and in this way once you get a surface like this you can actually extract many, many information out of this. So this still looks okay to understand for a tri atomic system but let us say if I go to multi atomic system which is even which would look even more complex. So here on the right hand side is shown a potential energy surface for a system with more than 3 atoms and where what you see is actually that please ignore these labels because they are finer details which we need not get into right now and what you see is that there are actually there are hills and there are different valleys in this right and this is what is called as a potential energy surface or a n dimensional hyper surface for a given system where each of the accesses would represent some of the one or the other change in the coordinate that is it could be again a change in the bond length or bond angle or any of the other vibrational degrees of freedom. And you are looking at again how does the energy of the system change when you change all these different parameters or vibrational degrees of freedom. So I can understand that this might look a bit daunting to understand or figure out so to make it easy let us take an example of a mountain range which we are all familiar with if you look at this image from a Google image what you see from a top view you see that there is a this is a beautiful mountain range and here you see lots of valleys and hills and various kinds of pathways which go through. This is exactly how a potential energy surface would look like for a real system. So you can imagine a mountain range as a potential energy surface and the various valleys and hills actually correspond to each of the states that is it could be intermediate, it could be a transient species or it could be a transition state or it could be many different kinds of species. So the best way to visualize a potential energy surface is to think of it as a mountain range where you have different kinds of let us say lanes going through one another and we will dive into this in a bit more detail and see what are the important informations one can understand or extract from such a potential energy surface. So now let us go ahead and look at three important ideas in this potential energy surface what are called as potential energy surface and reaction coordinate diagram and another term called as saddle points or transition state. So to understand that here I have shown you this rather simple looking potential energy surface from a paper in science where what you have is you have a energy minimum which is you can see it here I hope you can see this energy minimum wave which is a dip and then you actually come up in energy and then you actually again fall down into another valley where the energy is even lower and then on either side of this you have the energy which goes up. So if you take a 3 dimensional or n dimensional hypersurface like this that is what people typically call it as a potential energy surface. So I will just write it down here so that it is easier to remember. So let us say 3 or n dimensional surface is called a potential energy surface. So this is point number 1 and that is what we are having currently here this is what I am trying to show. And now what you can do is you would say that let us say the molecules are in this particular valley here somewhere and they are trying to come up through this and traverse this path and then go down the hill and end up in this particular valley. So then you would say that I am interested in only looking at this particular pathway and I am not because this is the one which is the energetically the least expensive for the molecules to undergo and I am not looking at actually what happens here or what happens here or let us say what happens here. So if that is the case then what I would do is I would actually take this hypersurface and I would actually make a cut or a slice of it along this plane. So I could actually make a slice along this plane and use a different ink. So if I just cut it along this plane then what I am doing is I am taking it after I had a 3 dimensional or n dimensional surface I have cut it into a 2 dimensional surface or in other words I have mapped the 3 dimensional surface into a 2 dimensional graph and in this what we typically look at is the following. So you will have something like this you will have the energy the potential energy V and this is what is called as a reaction coordinate and one would have something like this. You had a let us say A going to be via some T and with the important thing to notice this reaction coordinate is one of the element that is it could be either one or more than one or combination of one element that is let us say if I am looking at a molecule of water for example. So I can look at the change in this bond length R and I could also look at this theta which is the angle which is making angle between the H O and the H. So this reaction coordinate could be one or more than one of this bond lengths or bond angles which are changing which are now mapped from a 3 dimensional surface into a 2 dimensional surface. So that is the distinction between what is called as reaction coordinate diagram and potential energy surfaces. So this is what is called as RC diagrams the one which I have just drawn up. So I hope this distinction is clear between a potential energy surface and a reaction coordinate diagram. I will again repeat a potential energy surface is a 3 or n dimensional hypersurface in which you look at how does the energy of the system change as the function of the different vibrational degrees of freedom that is 3 n minus 5 or 3 n minus 6 based on the system. Whereas a reaction coordinate diagram is a 2 dimensional slice of a potential energy surface to represent the important steps which reaction in which you are interested in. So reaction coordinate diagram is typically a 2 dimensional slice of a n dimensional potential energy surface. So now we shall look at one important point which is called as a saddle point. So if I actually wait to look at this point here on the potential energy surface that is the atom or the system is trying to come from A to say B and it going through the point let us say T that is what we have represented here and if you actually look a bit carefully at T what you see is that at T there is a it has a very unique feature that is on 2 sides of T you have the energy which is decreasing that is towards A and towards B. Whereas if you go orthogonal to A and B on the other 2 sides the energy is increasing that is if you go along this path or if you come along this path the energy increases whereas if you go along this and this which is going back to the reactant or going to the product you have a lowering of the energy of the total system. So such points which actually resemble like a saddle on a horse if you ever sat on a horse I am sure you will realize that both of your legs would come down on either side and the place where you are sitting would look something like this where your legs are on either side here. So this is what is called as a saddle and this point is called as a saddle point and this invariably represents a transition state of a chemical reaction. So with this brief introduction to potential energy surfaces we shall stop here and in the next few lectures we shall take a few examples to understand the potential energy surfaces in a bit more detail. Thank you.