 Welcome to this new segment of CD spectroscopy and Mossberg spectroscopy for KMS. My name is Arnab Datta and I am an associate professor in the department of chemistry at IIT Bombay. So, in the previous segments, we have discussed the significance of symmetry and chirality in nature and especially around the human life. And then we start defining how the symmetry is actually controlling the molecular properties of a molecule or of an object. So, now we are going to take a little bit of mathematical route to understand the symmetry in further details. So, when we talk about the symmetry, we have already defined symmetry from a literary perspective in the earlier segment. Today, we are going to learn what is symmetry from a mathematical perspective. So, let us begin. So, if I want to define symmetry in mathematical segment, there are two different comes into the picture. One is called a mathematical operation and another is a symmetry, one is the symmetry operation and other is and the other one is symmetry element. So, we will try to define each of them. First, what is a symmetry operation? Symmetry operation is a movement of a molecule such that after that after the movement has been executed, every point in the molecule, so every point in the molecule in its newer orientation coincides with an equivalent point of the molecule in its original position. It is a bit longer, but let us slowly break it down. So, we are going to do a movement. So, what are the movements are you talking about? The movement we are talking about is rotation, reflection, inversion. This kind of movement we are going to drew around this molecule and when you do this movement, the molecule is going to change its orientation. So, it is going to go to a totally new orientation and in this new orientation we are trying to find what happens to the different segment of the molecule and if we find the molecule is having a such orientation that it exactly matches each and every point with an equivalent point present in the molecule on its original position, then we say it is a symmetry operation. That means there is no change distinguishable happening in the molecule after this particular operation. Then we call this particular symmetry operation exist. So, this movement or the change you are doing that is known as the symmetry operation. What is the symmetry element? So, the symmetry element is a geometric entity. Symmetry element is a geometric entity such as line, plane or center. So, it can be a line, it can be a plane, it can be even a center with respect to which one of the symmetry operation is performed. So, that means you can say symmetry element is actually geometric entity. It can be a line, it can be a plane or it can be even a center. So, around which the symmetry operation is performed. Symmetry operation is this rotation, reflection and the inversion we talk about. So, you have to do a rotation around what? You have to do a reflection against which particular thing. So, those will be coming into this line, plane or this center. So, symmetry operation, symmetry element are different things by definition, but they are going to come together and combine together because we are going to do a symmetry operation against a symmetry element. You cannot have a symmetry operation without an element and vice versa. So, that is why both of them are very important. So, this is the definition of a symmetry with respect to mathematics. So, in mathematics it says you have to do an operation and have to element first so that we can combine them together. With that now we will go through the five very important symmetry operation or symmetry elements exist in the literature. So, the first one is the rotational axis of symmetry. So, for this what is the symmetry operation which is I am just writing as a short form. So, symmetry operation it is nothing but rotation and what is the symmetry element? The symmetry element over here is an axis and this is defined as Cn that is how it is defined a rotational axis of symmetry around an axis and this N is defined by how much degree I am rotating for a particular operation and if I divide C 60 degree by that angle of rotation I should get a integer number and that will be defined over here as Cn. So, it will be much easier to understand if we use an example. Let us do that. So, the example we are going to take is Bf3 molecule. So, this is how the molecule looks like it is a planar structure molecule like this and this is I have drawn such a way that one of the boron and fluorine bond is on the plane of the paper one of them is above one of them is down and if I look from the top that will look like the following. So, it is nothing but a 90 degree rotation of the Bf3 plane. So, now the Bf3 plane is sitting on the plane of the paper previously it was perpendicular to the plane of the paper. Now, over here where I can take a line and along with which I can rotate and see if my molecule looks exactly the same and you can already have an idea because it is nothing but a triangle in such. So, if I rotate it through the middle that means in this picture around this line and rotate it 120 degree I am going to get a very similar structure. So, for an example this fluorine over here will go to this one this fluorine will come to this this fluorine will come to this and boron is not even changing its position. So, what I am going to do after a 120 degree rotation I am going to get a superimposable and indistinguishable structure. So, that is what I am going to get and that says yes I have an rotational axis of symmetry over here and how I define it it is rotating 120 degrees. So, now using this particular system so 360 degree divided by 120 degree is 3. So, I can say I have a C3 over here and it is typically written like this a rotational sign followed by C3 and over there I can also show the C3 is over here. So, these two are the same molecule they are just 90 degree rotated along the plane. So, this is what is actually present over here BF3 I have a C3 are there any other axis of rotation present over there in this molecule and the answer is actually yes. So, let us find out where is the other axis of symmetry present in this molecule and that is again easier to understand if I take the view from the top and over here you can see along with this BF bond if I rotate 180 degree this boron and fluorine remain as it is in its own position because they falls on this particular axis of rotation whereas these two fluorines exchanges places. But even after 180 degree rotation this fluorine will go to this position and vice versa and I am going to get a again super imposable and indistinguishable structure and that means I would say I have a axis of rotation over here and what is the axis of rotation the N C60 divided by 180 degrees so 2 so I have a C2 over here. So, this is in this particular molecule is over here. So, how many C2's of such kind will exist and the answer is actually 3 because it can be present in either of this BF3. So, that is why I have 3 different C2's. So, I have now for this BF3 molecule 1 C3 and 3 C2's. So, now if you have a molecule such that there are multiple axis of rotation we have to find out which of them is going to dominate in the symmetry definition and their properties. So, over here we found that the symmetry operation which actually gives me a super imposable and indistinguishable structure with minimal movement that is going to be the higher symmetric operation. So, over there between C3 and C2's which is actually higher symmetry which is actually have to have less movement. So, in C3 I am moving 120 degree whereas in C2 I am moving 180 degree. So, 120 degree movement is lesser. So, that is why C3 will be my major principal axis which is known as principal axis of rotation. So, remember if you have multiple axis of rotation the higher value of n that means you are over here doing the least movement is going to be the principal axis. And this principal axis determination is very important because certain other symmetry element depends on it. So, that is why we want to have this principal axis understanding and this is typically the highest value of the n I should say highest. And over there between C3 and C2 C3 is the highest value. So, that is why we end up having the principal axis of rotation by C3. Now if we go to the next one we are going to the same molecules that we have followed in one of the earlier segment the benzene derivatives. So, this is one of the benzene where we have 6 different protons to it. Now over here if I ask you to find out where is the axis of rotation and you will see there is one in the middle if I rotate 60 degree I am going to find a new configuration which will be super imposable and indistinguishable. So, this super imposable and indistinguishable these terms are coming from the definition of symmetry from the mathematics where we define that this after this movement each and every part of the molecule has to match which is original configuration which is a simpler term super imposable and indistinguishable. And over there I am achieving it by rotating only 60 degree. So, I can say over here I have a C6 why 6 because 360 degree divided by 60 degree C6. Are there any other of them? Yes, you can say around this if I rotate 180 degree this hydrogen this hydrogen remain as it is these two changes these two changes. So, you have a C2 over here and how many such C2 I will be having 3 of them going through each of the opposing hydrogen. So, over here I have a C6 I have 3 C2 which will be the principal axis it will be the C6 because higher number of n or 6 and this is actually having a motion of only 60 degree compared to a 180 degree rotation for the C2. Now, let us look into the other derivatives of the benzene that we have drawn. So, over here we have say like this one the monosubstituted one. So, previously when you are looking into that we found that in benzene we have only one proton signal whereas in this one because of the presence of this monosubstituted X where X is not equal to H we found that we actually have 3 different proton signals and we said that the symmetry is going down and as well as the degeneracy of the hydrogen is actually breaking down it is becoming more and more asymmetric to each other. And over here if I try to find what are the different rotational axes is present in this molecule if I find all this only one going through this hydrogen and this X through this ring and there is a C2 present over there 180 degree rotation where these two exchange these two exchange these two remain on its own position and this is the only one it actually has and over there you can see from having C6 and 3 C2 now it is going to have only one C2. So, basically lower symmetry and that is why we can say the hydrogen degeneracy is actually breaking down. Now, take a look into the other benzene rings we actually have proposed. So, over there we can have this one the disubstituted one. So, over here obviously similar to monosubstituted one one C2 is still present over here and there are other C2 is also present because of this presence of this two X group over here I can have a C2 180 degree rotation over here. If I rotate 180 degree these two Xs changing position and this and this hydrogen change position and this and this hydrogen change position. So, I have one C2 over there and also I have a C2 over here very similar to where I have the C6 in the benzene if I rotate 180 degree only then this Xs are going to exchange. So, that is why I have to rotate 180 degree for that and when it is rotating that all the other urges are also going to this para position with this 180 rotation through this particular benzene ring perpendicular to the benzene rotation. So, over here you can see now you have more C2s 3 C2s present and because now you have 3 C2 we can say it is actually higher symmetric compared to the previous monosubstituted one and we expect the hydrogen will be very similar. So, that is why the hydrogen becomes symmetric one more time and the similar thing we can do for the ortho and meta position one say we are going to do having this position and the rest of them are all hydrogen and you have to just find out by where I can let down my axis it does not really maintain it should be going through always through the CH bond it can go in between bond because these are the imaginary ones and it does not have to follow where is a bond there can only be a symmetric axis or a rotational axis of rotation it is not. So, over there the C2 is present over here where these 2 exchange places similar to these 2 and these 2. So, over here you can see the number of C2s position is going to be changing depending on the structure of the molecule and that is how we are going to connect the structure and symmetry of the molecule together in the same place. So, that is what is rotational axis of symmetry important take home message you have to find out an axis around which the rotation will happen you try to find how much degree you have to rotate and because it is a 360 degree system you have to rotate such an angle by which if you divide 360 by that particular angle you are going to set an integer and whatever the integer is you will say that n integer defines my axis of rotation is Cn over there and you have found Bf3 as a C3 benzene itself and on social benzene as C6 whereas the substituted benzene ones are mostly lingering around the C2 axis of rotation. So, with that we are going to move the next the reflection through a plane of symmetry. So, by so far the first one we have defined rotational axis of symmetry where we have the operation is rotation and the symmetry element around the geometric entity around which I am doing the rotation is an axis a line. Now my symmetry operation is going to be a reflection and I am going to do this symmetry operation around a symmetric element and this element will be a plane. So, that is how I am going to differentiate it a symmetry operation and symmetry element a reflection around a plane and again this is defined by this term sigma. So, if you say sigma I am talking about a plane of symmetry. Now again how to find it out again we are going to take some examples starting from again Bf3. So, in this Bf3 molecule do I have a plane of symmetry and the answer is yes we have this molecular plane itself is a plane of symmetry. See if I draw that again with 90 degree rotation so I will get this particular molecule and this molecule is having this plane of the paper where this molecule is situating it is actually a plane of reflection where you actually put that molecule on the reflection field itself. So, what happens the fluorine reflects on itself boron reflects on itself same to for this fluorine. So, the reflection is exactly same as it is and that is why we call it is a plane of reflection and not only that we have some other symmetry planes also. For example, in this orientation when I draw in this particular orientation this particular plane of symmetry which is the plane of the paper in this particular orientation over here you can see it is containing this boron and fluorine and reflecting this one. So, this one is up this one is down. So, the up one is going to downer side the downer one is going to be the upper side and is going to have again a super imposable and indistinguishable structure and with that we can define there is a plane of symmetry present over there. Now, how many such plane of symmetry will be present? So, if I draw with this particular orientation where I am looking from the top view for this molecule this is the plane of symmetry I am talking about this one. Now, I can have 3 of them because it can go through each of the BF bond. So, you have 3 different sigma planes and there is a other one. So, total 4 sigma planes I have one is the plane of the molecule itself and 3 of them is perpendicular to the plane of molecule going through one of this BF bonds. Now, how to differentiate them? This differentiation of the sigma plane is actually found by the principal axis of rotation. So, now we try to figure it out what is the relation of the sigma plane and the principal axis of rotation is nothing but the Cn and previously we have found that the Cn at the principal axis is this one where you have a C3 in the BFC molecule. So, over here is this one C3 and now if you find your sigma plane is perpendicular to your principal axis. For an example, this is the C3 and you can see this particular plane is perpendicular to it. We defined it as sigma h. So, it is a sigma h plane and if your sigma is actually contains that means the parallel or contains Cn that means the main axis of rotation actually contains in the plane of the symmetry. For an example, this one over here the C3 is in the plane of the board I am drawing over here right now and it contains the sigma. If such is the case we say them as sigma V. So, this is a sigma V. All of these three are sigma V because each of them contains the C3 principal axis of rotation. So, this h defines horizontal and this V defines vertical. So, these are the different sigma planes I can actually have. Now let us take a look into some other molecule. Let us take the molecule that we are all familiar with water molecule. So, if I want to have a water molecule I have defined it a little bit asymmetrically. So, let us write symmetrically. This molecule first find the principal axis and you can see there is only one axis of rotation the C2. However, it can have two different sigma planes one is the plane of the paper and one is actually bisecting the plane of paper containing the C2. You can see each of the sigma planes there are two sigma planes actually contains the principal axis C2. So, that will there will be sigma V planes. So, in H2O the principal axis is a C2 and you have two sigma V planes and they are sigma V because they contains the C2 they are not perpendicular to it. Then takes another molecule XCF4. This will be the last example of this particular segment XCF4 and over there this is how the molecule looks like and again I am going to have two different views one from the top view one from the side view. So, this is the top view and this is the side view. So, the top view will get you a very nice example because all the fluorines are same you have to rotate just 90 degree to achieve the similar structure. So, it going to go through here you have to rotate 90 degree that means it has a C4 going through here. After having the C4 now I am going to look into what are the different sigma planes present. Obviously, this molecular plane is one of the sigma plane and you can see this sigma plane is perpendicular to this C4. So, that will be a sigma H. So, this is a sigma H plane. Are there any other sigma planes? Let us draw that. So, I am going to show in both side view and top view to having a better understanding. So, now you can see there are other sigma planes also present going through the opposing fluorines. So, one is this particular plane and other is this particular plane. You can see either of them contain each of them contain one fluorine, one gene and one fluorine and the other two fluorines will be reflected and vice versa. And these are the two different sigma planes. So, we will call them sigma V for now because they actually contains the principal axis C4. Both of them contains the C4 axis of rotation. So, that is what we have called them sigma V. But these are the not the only two ones. We have two other C2s and we have tried to find out exactly where those other sigma planes are and these are going through this one going to only xenon and this is reflecting these two fluorine to this one and for this reflection of this. So, there are other two sigma vis. So, this is nothing but the plane of this molecule and perpendicular to it very similar to this water I have drawn. So, these are the other sigma vis. So, there are four sigma vis and one sigma H plane which is the plane of the plane of the xenon fluorine. But over here we also found there are also C2s present. One set of C2 goes through the xenon if you rotate 180 degree you are going to get the same structure and the other two C2s going through the fluorine xenon fluorine bond. So, over here you can see the sigma V's over here are actually bisecting two C2s is bisecting two C2s and that is why they are going to name a little bit differently. They are going to be named as sigma D where D stands for dihedral and this is happened sigma D is nothing but sigma V but which bisects two C2 axis. So, all we learned there are three different sigma planes sigma H, sigma horizontal perpendicular to the principal axis, sigma V, sigma vertical which contains the principal axis and sigma D which is a special kind of sigma V but bisects two C2. So, that is what is a reflection through a plane of symmetry. These are the things we can have and also learned the rotational axis of symmetry, principal axis of symmetry. So, these are the two important symmetry operations along with their symmetry elements and over here we will conclude this section over here and the next section will cover the rest of the three symmetry elements present in a molecule. Thank you. Thank you very much.