 Hello and welcome to this new segment of CD spectroscopy and Mossberg spectroscopy for chemistry. My name is Arnab Dutta and I am an associate professor in the department of chemistry. So in the previous day, we have looked into all the different symmetry that can be found in the nature and then we try to mathematically connect them to particular symmetry elements or operation. So we would like to recap that a bit and then go forward for our next segment. So let us begin. So previously we have discussed over here that we have 5 different symmetry operators or elements. So just to remind you the symmetry operators are the movements that we are doing around a molecule. On the other hand, symmetry elements is a geometric entity around which these operations are performed. And we found there are 5 of them. First one is a rotation around an axis. It is written as Cn where n is defined by how much angle I am rotating. This is the theta angle of rotation to achieve is superimposable and indistinguishable configuration of the molecule. Then comes the reflection around a plane. So over here, reflection is the operation, plane is the element. The rotation was the operation and axis or line was the geometric entity or the symmetry element. And this is given as sigma which can have 3 different forms, sigma H, sigma V and sigma D. So what is sigma H? Sigma H is a plane of reflection that lies perpendicularly to the principal axis. So what is principal axis? In a molecule, you can have multiple rotational axis and among them which one is actually the highest magnitude of n that means you are achieving a superimposable and indistinguishable configuration with minimal amount of the movement that is considered as the principal axis. And this principal axis and its relation with the sigma defines what kind of sigma plane you have or reflection plane you have. If the relation between the Cn or the principal axis and the plane of reflection is perpendicular to each other that will be sigma H where H stands for horizontal. If the sigma plane contains the Cn, then it is called sigma V where V stands for vertical where means the axis of rotation and the plane is actually on the same plane. But if it is perpendicular then it is horizontal. Then sigma D is a special class of sigma V. So sigma D and sigma V are same with respect to where the sigma plane stands with respect to the principal axis Cn. However the sigma D has one extra feature and that is this bisects to C2 which is typically present in this molecule. So other two C2 should be present and which will be bisecting by this sigma plane. So that is why it is known as the sigma D and D stands for dihedral. So these are the three different sigma planes we can also have. The next one comes reflection through a center or point and that is known also in the other way inversion center and given by this term I and this is typically in center which sits at the center of the molecule and we try to find the reflection is a symmetry operation and center or point is the symmetry element and if this is present in a molecule then this present almost at the center of the molecule. However the inversion center does not mean that it has to have an atom present at that inversion center. For an example like benzene at the center point we do not have any atom present but this molecule contains a inversion center. Then we go improper axis of rotation where we do two different operation first rotation around an axis that means it is nothing but a CN and the second operation is just next to it is a reflection through a plane which is perpendicular to CN that means we are doing nothing but a sigma horizontal operation. So over here we do two operations next to each other CN and sigma H and this is written in the form of SN where the N is coming from the CN again the angle of rotation we are talking about. So over here we are doing two operations rotation and a reflection and it does not mean that a molecule has to have both of them actually present. So if a molecule has a CN axis of rotation and also have a sigma H obviously this molecule is going to have a SN axis. However there is a possibility that the molecule does not possess a CN and a sigma H at the position we are considering. However after doing both these operation I am getting a configuration which is super imposable and indistinguishable with the original structure then we can say our molecule is having an improper axis of rotation. And we have given the example of methane one last day where we found that it actually contains an S4 axis however the methane does not contain a C4 or a sigma H plane but if we do this operation consecutively we found the methane is actually having an S4 axis. And this improper axis of rotation has a huge implication on the chirality of the molecule and that we will be covering in the next coming segments. And the last one we have the identity operator which is written as E where the E terms comes from this German term Einheit which means unity. And in this operation we do basically nothing but the leave the molecule as it is or in other way as we say we are doing a 360 degree rotation that means we are basically doing a C1 operation which means the leave the molecule as it is. So it is quite redundant typically that we are doing nothing in the molecule but we are taking this as an symmetry operator. That is required because all this thing if we put together Cn, sigma, H, V or D then we are taking the center of symmetry Sn together and we try to form a group only these four cannot form a group properly. To form a mathematical group we have to follow certain rules and one of them we have to have a identity operator. And that is why this E comes into the picture and with all these things together we can form a mathematical group and that we are going through because we want to define our molecule with this mathematical groups so that we can easily find what are the different symmetry element present in a molecule. So if we have a particular molecule containing certain symmetry elements we can actually divide them among different groups. So what are the possibilities? Let us take a look. So now we are going to do symmetry division by point groups. So what is this term point group means? Point group means is a mathematical group which is proposed and derived with respect to the different symmetry operations this particular molecule has. So let us take a look what are the different forms possible and we can actually divide them in four broad section. The first one we call them non-rotational groups. So quite obvious from its name it does not contain a CN axis it does not have N and that is why it is called the non-rotational group. And this particular molecule which does not have a CN it can belong to three different groups. The first one we call them the C1 that means this molecule has nothing but an E symmetry operation as we discussed just earlier. E is the identity operator which is nothing but leave the molecule as it is. So that is 360 rotation or C1. There is a possibility a molecule have no other symmetry other than E because that is the symmetry operator will be present in any molecule no matter what. And that is why this is the molecule with the lowest symmetry possible it is a C1 nothing else other than an identity operator that means it is totally a true asymmetric molecule. Next comes the CS point group which actually says you have a operator E because this will be present in any molecule and along with that you have a operation sigma a reflection plane where it is present in the molecule. Now the question is why I am not writing sigma a sigma v or sigma d and as we have discussed earlier this actually is defined by the presence of CN axis or any other C2 axis. As this particular point group does not have any CN axis. So that is why we are saying this as only sigma without differentiating as sigma a sigma v or sigma d. So this is the second point group possible. The third one possible is CI where the molecule is going to have an identity operator and along with that a center of inversion. So these are the two possibilities only these two operations and this molecule will be written as CI. C is the point group notation and this S1i the subscript is saying what are the other things present in there. So these are the three point groups possible when I do not have any rotational axis present. So rotational axis is absent in this particular molecule will come into some of the examples of such molecules later on. So after this we have another set of point group will come into it and over there we call them single axis group. So what do you mean by single axis group? That means it is going to have different CN where N will be 2, 3 and so on other integer even infinite. So these are the different single axis group possible where N is defined like that and I am coming into it little bit later what are the different possibilities. The first possibility is I can have a point group CN where I have an identity operator E then a CN operator at the same place I can have a CN 2 operator and so on and so forth by successive rotation I can achieve a CN N minus 1 operator. There is a possibility that there is a CN N operator that means I am doing the CN operation at the same position nth time for an example if I have a C4 axis I can do that C4 one time 90 degree second time another another 90 degree 180 degree. So it will be defined as a CN 2 I can do it 3 times 270 degree. So that will be 270 degree or in the opposite direction I am moving 90 degree. So it will be a C4 inverse and then I can do it 4 times at the same position 90 180 270 and 360 and that point of time the molecule is rotating total 360 degrees which is nothing but the operation E which is already present over there. So that is why at a particular position a rotational axis that I am talking about for a molecule if it is CN at the same position I can have CN 1 CN 2 so on and so forth up to CN N minus 1 because CN N is nothing but an identity operator and CN 1 we typically write as CN only. So that is the concept of successive rotations at the same place. So that means what I am saying this CN CN 2 CN N N minus 1 is present at the same position the axis is actually common for all of them. So that is one of the point group that is possible for this particular molecule. Then it is possible a point group CN v where in addition to this single axis rotation system we can have n number of sigma v's. So if you have sigma v in this molecule present or in this point group present it will have n number of sigma v present for an example if I am talking about a C 3 molecule for this kind of point group I will have 3 sigma v's I will have either all 3 of them or nothing. So if you have a sigma v present in the molecule you have to have n number of them and if it is present the point group is CN v then comes the next one it is called the CN H again CN minus 1 and along with that it also has a sigma H the horizontal plane of reflection and in a molecule because you have only one principal axis or CN and only one plane possible perpendicular to that sigma H you can have only one whereas sigma v which actually contains the principal axis you can have multiple numbers even infinite we will come into some of the examples later where even with a minimal rotation I am going to achieve a superimposable and indistinguishable structure mostly in the linear molecule and over here we have only one sigma H sigma v and we have any number depending on the symmetry of the molecule but sigma H irrespective of the symmetry if you have a sigma H in a molecule you can have only one of them. So with respect to that we move to the next one which is called S 2n point group what is S 2n point group where you have an E S 2n goes to S 2n 2n minus 1 we are writing 2n over here because this molecule has to be such that it is actually even number over here so whatever the number n is this 2 will ensure that it becomes an even number. So if a point group only have this symmetries other than identity operator if only improper axis of rotation is present nothing else it is only possible if you have a molecule with S axis such that this number is actually even number. The last one we have C infinite v which is nothing but an extension of C and v where you have E C infinity, C infinity 2 and so on and so forth and then you have infinite number of sigma v. So what do I mean by C infinity let us take an example let us take an example of molecule AB which is linear and now if I look through this particular axis how much angle I have to rotate to get the similar structure because A and B are such atoms which is present over here this is going to remain same no matter how minimal you rotate around this. So over there if you rotate any particular angle theta you are going to get a superimposable and indistinguishable structure of the original molecule so that is happening for a linear molecule and that is coming into the picture over here so theta is going to be very close to 0 degree because even with a very minimal rotation you are going to get a structure. So that is why over here n which is 360 divided by theta because theta tends to 0 this whole thing will turn up to be n tends to be infinity and that is why it is known as C infinite point group. So with respect to that we move to our next segment so previously we have looked into all this point group which is called a single axis rotation group because over here all the rotational axis C A and C N V, C N H, H 2 N or C N V, D V there is only one axis or one line where all the rotational axis are present. So that is why it is called a single axis group because we have only one line through which the molecule can rotate there is no other axis of rotation present in that particular molecule but there is a possibility that you can have a molecule which belongs to such a point group where you have multiple axis of rotation possible. So let us come to find out what are those and that brings to the third segment of this particular system which is known as the dihedral groups. Again in dihedral group we are going to use the number N and N is again 2, 3 or any integer up to infinite. So what are the point groups possible in dihedral group? The dihedral group is a molecule where you can have a point group of D N which is having E C N up to C N N minus 1 till now it looks like a very similar to a C N point group but the difference is over here you are going to have N number of C 2's also present other than the C N axis and those C 2's will be perpendicularly oriented to the principal axis C N. So that is an important addition to this particular point groups and due to this presence of this N number of C 2's perpendicular to C N we call them the dihedral groups. Then comes the next point group D N D which is again very similar to E C N C N N minus 1 we have actually N number of sigma D's also present and over there we are calling them sigma D's because we have N number of C 2 which is perpendicular to C N which is present over there and those C 2's will be bisecting the sigma so that is why it is called the sigma D. So you can see it is very similar to the C N V but over here we have again this extra group present over there and in D N D point groups other than this C N axis of rotation and N sigma D's and N C 2's we have also improper axis of rotation that will be also present. So that will be also present over here and over here you can see it is S 2 N so whatever the number of N is it will be the double number from that for example if I talking about a molecule of D 3 D so you are not going to have only C 3's it will also have S 6 so that is the meaning of D N D point group. Then we come to the next point group which is D N H so over here again we have the E C N C N minus 1 to N minus and we have 1 sigma H present so it is very much similar to C N H but again we have this extra N C 2 perpendicular to C N that is going to be present over here and along with that we will have N number of sigma V's also present over here. So this is what is actually happening in the point group of D N H which is again you can say an extension of molecular group of C N H point group of C N H but in D N H you have this extra factor coming into here you have N number of C 2's perpendicular to C N so that is actually differentiating between a single axis rotation group or diheter group. Then the last one is D infinite H where you have a E you have C infinity present you also have present infinite number of C 2 perpendicular to the C N along with that it is going to have a sigma H so that defines this D infinite and H nomenclature along with this they have infinite number of sigma V's present which was absent in the C infinity point group earlier the sigma H but over here you will have a sigma H and interestingly they are going to also have a center of symmetry and you are also going to have a S infinite. So that means you are going to have C infinite for obvious reason infinite number of C 2 perpendicular to C N which defines it is a dihedral group we are going to have a sigma H that is why it is D infinite H sigma V infinite number center of symmetry and improper access rotation at the similar place at the C infinity. So what this says this is nothing but also a linear molecule but over here both side of the linear molecule is same so that is we are actually finding over here so you can see there is the C infinity and over here at the middle point you have the I which also shows up the sigma H plane because you can see the sigma H is perpendicular to the C infinity and this is what is actually present over there you have this I you have the sigma H present over there and where are the infinite number of sigma V's I am drawing the molecule one more time say the C molecule A and A so you can have a sigma V over here you can have sigma V in the perpendicular or any angle in between them so all the possible angles you can think of that particular plane will contain these two molecules and the contains the C infinity so that means there is all sigma V's and you can have infinite possibilities of that and the question comes where is the C 2 position say again I am drawing the same molecule A so the C 2 one is present over here if you rotate 180 degree which is perpendicular to the C infinity so that is one of the perpendicular C 2 you can have a C 2 rotation on this particular axis which is perpendicular to the plane of the board I am drawing and that is also a C 2 and you can think about all the C 2's possible in between them all the infinite ones which will be all perpendicular and which are all in the plane of sigma H and those are all the infinite number of C 2's perpendicular to the C n present in this molecule which ensures this molecule end up to be a dihedral point group. So the difference between dihedral and single axis rotation group is again this presence of n number of C 2's perpendicular to C 2 that differentiates a single axis and dihedral group and the last one for this particular system is called the cubic groups. So these are special group and we have two of them that we commonly found one is a tetrahedral point group which is written as DD and tetrahedral point group is quite a high symmetric system a point group is known as a high symmetric system when you have a huge number of symmetry elements present over there more than number of symmetry elements present in a molecule or point group more symmetric it is. So tetrahedral is one such group so let us take a look what are the symmetry elements present over there in this molecule you have a E that is present in all the molecules you have 4 C 3 you have 4 C 3 2 you have 3 C 2's you have 3 S 4 axis that you have gone through also in the example of methane earlier you have 3 S 4 3 the similar position and 6 sigma D's. So all those things is present in a tetrahedral molecule and if you add them how many number of symmetry elements actually present with these numbers in the beginning. So those if you add it together we will find in together we have 24 symmetry elements present in this point group octahedral. So quite a highly symmetric point group and one goes a little bit higher than that which is known as octahedral point group and what are the point groups present it has a E it has 8 C 3's 6 C 2 it has 6 C 4 3 C 2 which is a little bit different than this one because it is present in the position of C 4 and if we are rotating it twice because C 4 means 90 degree if you rotate it twice you are going to get a 180 degree which is nothing but a C 2 that is what I am writing over here then you have center of symmetry I you have 6 S 4 8 S 6 3 sigma H plans present and 6 sigma D's present over there all together if you count it out you will find it has 48 total symmetry elements. So which is a huge number even compared to the tetrahedral point group so you can say octahedral point group is even more symmetric than tetrahedral. So these are the different groups present. So what we have found so far that you can have 4 different groups one is the non rotational group then you have the single axis rotation group then you have the dihedral groups then you can have the cubic groups. So non rotational groups are C 1, C s or C i single axis rotation groups are C n, C n h, C n v then we also have the other point groups like S 2 n or C infinite v dihedral point groups on the other hand we have D n, D n h, D n d and D infinite h the difference between C and D groups are whether you have n number of C 2 perpendicular to C n or not. So that is defining this two point group and cubic groups are the special groups like octahedral or tetrahedral which contains a lot of symmetry. Now the question is for a particular molecule if I have and try to find its point group do I need to figure it out all the symmetry elements in there I can very easily find out a rational way to figure it out much easier what will be the point group of a molecule and that we will be covering in the next segment. So for this particular segment we will stop over here and conclude that there are 4 different groups are possible starting from non rotational to single axis rotation to dihedral and cubic groups. And now the next segment will be how to find out which molecule belongs to which particular point group with that we like to conclude over here. Thank you.