 Welcome to this next segment of CD spectroscopy and MOSBIRS spectroscopy for the chemist. My name is Arnab Dutta and I am a professor in the department of chemistry at IIT Bombay. In the previous segment, we are discussing how we can define symmetry by mathematics. And when we talk about mathematical symmetry, there are two important parameters symmetry operations and symmetry elements. Symmetry operations are the movements that we do around a molecule or an object and we go to a totally newer configuration. If this newer configuration exactly matches the original one, we tell that there is a symmetry present over there and we define them as the term superimposable and indistinguishable. And this symmetry movement that we have done, it is known as the symmetry operation. Each of the symmetry operation typically performed around a geometric entity, either a plane or line or center, those are known as the symmetry elements. So far in the previous segment, we cover two of them rotational axis of symmetry where we are doing a symmetry operation rotation along with a line, the axis. And the next one, reflection through a plane of symmetry, we are doing the operation of a reflection through a plane of symmetry. Now, in the next segment, we are going to cover the third one, reflection through a center of symmetry. So that means over here, my symmetry operation is reflection, but over here, my symmetry element is not a plane as we have seen for reflections where plane of symmetry or the sigma, over here, the symmetry element is a point. That means just a center, a dot. And this is known as the reflection through a center of symmetry. There are another name of that called the inversion center. And that is why in short form, it is called I. So, as we have looked into all the other symmetry elements so far, rotational axis of symmetry is defined by Cn in subscript, then reflection through a plane of symmetry is defined by sigma. Similarly, reflection through a center of symmetry is defined with this I. Now, we will take some examples to define it. The first example I am taking is carbon dioxide. So, you can see there is a carbon dioxide, there is a carbon in the center and if I go through this center, I go to another oxygen. So, that means both sides are very similar. So, I can say there is a center of symmetry present in this molecule. If we take carbon monoxide on the other hand, you can see if you go through the center of the molecule which is the triple bond, if you go to the other side, there is a carbon instead of oxygen. So, we say there is no center of symmetry present, where in carbon dioxide it is. If you take nitrogen, dinitrogen molecule, again through the center of the molecule, you can see if you go through that you are going to go and take a look into another nitrogen atom. So, over there we will say center of symmetry is actually present. Now, take another molecule, xenon fluoride, xenon tetrafluoride and I am looking through the top view for this particular molecule and over there the center of the molecule. So, typically if you have a center of inversion or inversion center or reflection through a center of symmetry, it has to be in the center part of the molecule and the center part of the molecule does not always require to have an element present over there. So, over there xenon is present. So, if there is a center of symmetry, I should see a fluoride on the other side if I go through there I see go through there I see. So, that means I will say it actually has a center of symmetry. Now, if we take BF3, this is also the other molecule we have defined so far. So, if there is a center of symmetry it has to be present on the boron because there is center of molecule if I go through there on the opposite direction there is nothing similar for all these fluorines it is actually going through this imaginative center but it is not showing any other fluorine on the other side. So, if I reflect this fluorine through the center of the molecular boron I am not seeing the reflection on the other side where there is a fluorine already being present. So, it is not going to give me a superimposable and indistinguishable form if I do a reflection through the center. So, I will say there is no center of symmetry present in this particular molecule. Now, get to the next one say SF6 molecule which is octahedral in geometry. So, again the center of the molecule is a sulfur and if I go through that and reflect this fluorine these are reflected if I reflect through that reflected if I reflect through that they are all reflected that means the center of symmetry is present in this molecule. If I look into similar molecule of a metal metal typically try to have octahedral geometry say in this molecule and I have 6 similar ligands present over here exactly the same molecule. And over here I will say yes I have a center of symmetry because the ligands are all connected such a way that if I reflect through this plane through the center which is sitting in the middle of the molecule on the metal center it is having a center of symmetry. However, if I take the same molecule but say I am taking a bidented ligand say like NN bidented ligand is it going to have a center of symmetry. So, I am just writing NN is a bidented ligand bidented ligand means it actually binds with the metal with two different sides from the same molecule. And over here you can see if I reflect the nitrogen through the center I will see a nitrogen this nitrogen through the center I am seeing a nitrogen this nitrogen through the center I am seeing a nitrogen. But there is a connection there is some bond there are some atoms present over here if I reflect through them I am not seeing anything nothing not for this one. So, it is not only the atoms which is binding with the ligand and the metal is important but also the other entries everything has to be reflected properly through the center and this is not happening over here. So, you will say no there is no center of symmetry and the last one I would like to cover is the benzene molecule which we actually have covered earlier and over here you can see this is the benzene molecule and over here through the center of the molecule if I pass through this each of the segment is actually reflected to another portion which is matching superimposable and indistinguishable with original position that means benzene has a center of symmetry. However, if I reflect if I replace one of this hydrogen with an X this mono substituted benzene now does not contain the inversion center I anymore because this is fine happening this is fine happening but this one not happening. So, that is why the center of symmetry over here not present each and every portion has to match it is not like only a partial position is actually matching that means I can say I have a reflection through a center of symmetry we cannot say that we have to match each and every portion of it. So, that is known as a reflection through a center of symmetry. Now, we come to the fourth one improper axis of rotation. So, previously we have already gone through and rotational through an axis which is known as the proper axis of rotation because we are rotating it and along with a line and we are going to go to a indistinguishable and superimposable structure. However, in improper axis of rotation we are going to do two operations together consecutively. So, over here we have two symmetry operation the first one we are going to do is actually a rotation around an axis which is nothing but a CN rotation around an axis system we will do that but we will not stop it over here. Then, we are going to do a reflection around a plane that is perpendicular to that rotation we have just done. So, we are going to do two symmetry operation first is a rotation second is a reflection a relation between this rotation and reflection rotation is going to happen around an axis and reflection is going to happen around a plane and they will be perpendicularly oriented. So, obviously the symmetry elements we have over here also two of them one of them is a axis and the second one is a plane. So, we are going to do two of them together and if I am going to get after this two operations one is the rotation around an axis another is a reflection around a plane and if I am going to get to the similar looking molecule from the original that means a superimposable and indistinguishable structure only then we will say that we are having a improper axis of rotation and this is defined as SN very similar to CN but instead of C we are writing S and what is the N? N is defined again very similarly how much angle I am rotating over here during the CN operation. So, that is defined the N and this is the CN what I am rotating this is going to be the same as the SN. So, that is how it is defined in this particular system. Now, we are going to understand it better if we do it with an example. The example we are going to take is of methane. Methane molecule is actually having this tetrahedral structure in short form is a TD tetrahedral structure and this molecule we say we may have a S4 axis that means I have to first find out a C4 axis and a perpendicular plane that means nothing but a sigma H plane and if I do these two operations together I should get a similar looking molecule. We are going to do that operation and try to find if we are achieving a superimposable and indistinguishable structure or not. To do that I am drawing this molecule a little bit different orientation. So, I want to do the rotation around this axis and I want to bring that axis over here along with this where the CH bond is. So, I am going to rotate the full molecule and this is how it is going to look like. I am doing that because in this way we can understand the possible presence of the C4 axis better. So, now with this particular structure of this molecule if I rotate along this particular line 90 degree that means a C4 rotation. If I do a C4 rotation how the molecule will look like. Carbon is going to remain as it is. So, it is better to always do with each atom. These two hydrogen is in the plane of the paper and these two hydrogen is perpendicular to the plane of paper. So, when I do a 90 degree rotation this two hydrogen say like I am writing HA is going to come 90 degree to it because I am doing rotation of only C4 90 degree rotation. So, this particular molecule is going to rotate from this plane to this. So, this will be the HA molecules and what will happen to this say HP molecules. So, they are already perpendicular to that. So, if I rotate 90 degree they will now come to the plane of the paper and this is what is happening. So, now you can see I have done a C4 operation. This molecule where I start from and this is where I end. Are these super imposible and indistinguishable? No, not at this point with only C4 it is not. So, that is why methane molecule does not have a C4. So, it does not have a C4 axis because it is not going to give you a super imposible and indistinguishable structure. So, now I have to do a sigma plane a reflection during the improper action of rotation and this reflection has to happen perpendicular to the plane of the rotation. So, I have already rotate along with this axis. So, perpendicular to that that means this plane this is the sigma H I am going to do. So, if I do a sigma H operation over there what is going to happen? Carbon is going to stay as it is. This hydrogen A is what will happen there on the top in the beginning and there is my reflection plane. So, from the top they will come to the bottom in the reflection. So, they will come to the bottom these are the Hs. What will happen to this Hb which is actually lying below before the carbon? Now with this plane of reflection they will go to the top portion. So, this is I am going to get the Hb. So, this is the molecular structure I am going to get if I do a C4 followed by a perpendicular sigma or sigma H after that. And this molecule and this original molecule now you can see they are superimposable indistinguishable and we can say yes this molecule has a C4 and sigma H together if I run this operation that means I have a S4 axis of rotation. That means methane does not have a C4 but it contains an S4 axis. So, that is a very important conclusion because to have an Sn axis in a molecule if you already have a Cn, if you already have a sigma H that means you are going to have an Sn axis no problem with that. For an example when you talk about this Bf3 molecule it contains a C3 it contains a sigma H that means Bf3 will contain an S3 axis. So, if a molecule already contains a Cn and sigma H it will obviously going to have Sn. But it does not mean that if a molecule does not have Cn and does not have sigma H that it cannot have Sn. It still can have Sn and methane is one of the biggest example of it where I do not have a C4 and in this methane molecule I do not even have the sigma H the molecule does not have any sigma H plane somewhere anywhere else. So, no sigma H no C4 but it still have an S4 axis. So, that is why improper axis of rotation is very much important on this respect. And over here I want to mention one more thing that when you talking about Sn axis of rotation there is a possibility that you are going to have a S1 system what is a S1 system that means you are doing a C1 rotation followed by sigma H. Now, C1 is nothing but a rotation of 360 degree that means you are leaving the molecule as it is we are doing anything. So, that is why this is nothing but only sigma operation and because C1 360 rotation does not really mean anything we can simplify it to a sigma operation a plane of reflection. So, S1 is nothing but a sigma operation. On the other hand S2 can be equilibrate with a reflection through a center or inversion center. So, there are different explanation why it is so because it is a little bit out of the scope of this particular course we are not going into that but you can figure it out in any of the books or available materials where they have been proven that S2 that means a C2 followed by sigma H the operation you are going to do you end up as a same thing as a reflection through a center of inversion. So, if a molecule have S1 and S2 basically you are saying you have a sigma or a center of inversion. So, these are the two most simple S axis of symmetry you can imagine and you can also have higher order of SN where N is greater than 2 and over there as you have just discussed it does not always mean that you have to have a CN and sigma H present it may or may not be present and you can still have a new SN axis of rotation. So, that is where we are going to get to the next one. Now, the last operator is a identity operator which is nothing but a C1 operation the C1 operation that we just discussed a 360 degree rotation and over here we just basically leave the molecule as it is and this is known as the identity operator and this is defined by this term E. This term E is actually defined by this German term which is known as Einheit which means so you might be wondering like why do we need this particular operator it does not really mean anything I am just leaving the molecule as it is that is actually very much important because so far we have covered a few of the symmetry operations like axis of rotation CN reflection plane sigma which can have different variations sigma H sigma V and sigma D depending on their positioning with respect to the principle axis of rotation and also whether they are bisecting other perpendicular C2s or not. Then we have gone through center of inversion or I and we have also gone through the improper axis of rotation or SN where we are doing a rotation and reflection consecutively. And if we also put this operator E in the bunch all this five we can put together in a group which is known as a mathematical group with that it follows certain properties where they are going to have some interrelation between them. So, for a particular molecule I probably do not need to find each and every symmetry element out of them if we know a few of them which are actually finding and present in that molecule we can figure it out what is that mathematical group it belongs to and then from there we can connect to something called character table which is already available from mathematical description it is an equivalent of periodic table but instead of elements and their properties we look into mathematical operations the symmetry operations that we have just discussed all this five and from there we can define what are the different elements present over there and these are the symmetry elements that we are covering over here. So, now we would like to conclude this segment over here. So, in this particular segment we have covered the different symmetry elements that can be present in a molecule that can be rotational axis of symmetry C n that can be reflection symmetry through a plane sigma which can be sigma h sigma v or sigma d sigma h is a sigma plane which is perpendicular to the principal axis C n and what is the principal axis C n where I have the highest number of n or lowest number of angular rotation possible to get to a superimposable and indistinguishable structure. Sigma v is a sigma plane which is actually a plane which contains the principal axis and sigma d is a special kind of sigma v which is by definition is very similar to sigma v that means contains the principal axis but it bisects two C tools which are actually perpendicular to the principal axis C n. Then we have this center of symmetry I where it is nothing but reflection through the center of the molecule and it allows us to get a superimposable and indistinguishable form if it presents. The center of symmetry typically presents in the center of the molecule but it does not really needs to have an element present over there for an example benzene does not have it. Then comes this improper acceleration or Sn which actually covers two sets of symmetry operation first a rotation around an axis that means basically C n followed by a plane of reflection which is perpendicular to the acceleration we just did so it is doing C n and sigma h together. This Sn axis also defines that you can say the corollary that S1 is nothing but a sigma plane and S2 is nothing but a center of symmetry I. And we have also found that this Sn axis can be present even the C n or sigma h is not present on their own if they are present obviously you are going to have a Sn but if they are not present even then you can have an Sn and we have discussed that with the example of methane molecule which does not have a C4 which does not have a sigma h but it contains an S4 axis. And lastly we have this identity operator E which covers the full group of this symmetry so that we can create a mathematical group and connect that to character table to understand the properties of a molecule with respect to their symmetry. And over here we are thinking about okay now we know we are actually going to have this kind of symmetry elements but how to connect that with cd spectroscopy because cd spectroscopy connect that with chirality and over here one of the thing we are going to find out that a molecule which does not have an Sn axis it is going to be chiral but how to find out a molecule whether it has or has not any Sn axis that trick we will learn in the next segment. So over here we would like to conclude over here today. Thank you. Thank you very much.