 Welcome to lecture number 34 of the course quantum mechanics and molecular spectroscopy. Previously, we have been looking at the selection rules for rotational, vibrational and electronic transitions of diatomic molecules. In this lecture, I am going to talk about rotations of a polyurethane molecule and the consequent selection rules. If you look at the rotations of a diatomic molecule, let us consider diatomic molecule A and B and under rigid rotor approximation, we have the energy is given by B into J into J plus 1. And if this has centrifugal distortion, then this will be given by D into J square plus into J plus 1 square. So this is because of the, we talked about centrifugal distortion because when a rotating body what happens is that the length increases because of the centrifugal force. So that is covered by centrifugal force. For most of the molecules, D by B is equal to 4 times B divided by vibration whole square and it is just much less than 1. V vibrational is the vibrational frequency, for diatomic molecules such as NO, D by B is approximately equal to 3 into 10 power minus 6 at J equals to 1. But if I go at J is equal to 60, D by B will be approximately equal to 0.01. That means even for very large values of J, one can ignore D by B or the D by B value is very small. That means the value of this D is going to be really very, very small. Only when it goes to very large means let us say if J is equal to 200, then the centrifugal distortion constant will start making effect. So one has to go to very, very large values of J make the D that is here centrifugal distortion will make D prominent otherwise in general one can ignore. Now, this is about the diatomic molecules. What about molecules that are larger than diatomics, triatomics, polyatomics because in chemistry most of the time you are encountering with the molecules that are much larger in shape. Now, if you take a polyatomic molecule before we get to the selection rules for polyatomic molecules and how the rotation spectra of the polyatomic molecules is interpreted, let us look at how polyatomic molecules will generally behave ok. If you have a molecule in which you have let us suppose a molecule. So in general the polyatomic molecules are divided into three possibilities ok. So the three categories of polyatomic molecules one is called spherical rotors, second is called symmetric rotors and third one is called spherical symmetric and asymmetric rotor. Now one of the analogy that I can give between spherical rotor, symmetric rotor and symmetric rotors is like triangles. One of the things that I can tell about these rotors in terms of a triangle think of it in terms of a triangle ok there are three sides A, B, C correspondingly one could have three different ways one can. So there is one rotation axis around this I will call it as axis A and there is another rotation one can rotate along B and other one is along C. Now for an equilateral triangle you will see that A, B, C will be same. So spherical rotor is analogically is like a equilateral triangle ok. Now the other thing is a scale and isosceles triangle where A and B are same and C is different one could have like this or one could have one could have it like this. So you will see that the C axis and one you can see that A and B are equal in such case one can take A equals to B equals to C ok which is a consequence of A is equal to B is equal to C in the symmetric rotor is more like A is equal to B not equal to C which is also a consequence of A is equal to B not equal to C. Similarly if you have a scale and triangle then of course A all three sides are different. So A, B, C in that case the rotation along A the rotation along B and the rotation along C are different. So A is not equal to B not equal to C which is a consequence of A not equal to B not equal to C. So what is that I am trying to tell you here is that if A, B, C are the lengths of the sides of the triangle then depending on how they are placed with respect to each other ok. You can form either a equilateral triangle or an isosceles triangle or a scale and triangle. Similarly if you have a spherical rotor depending on the rotational constants A, B and C ok then the they can be categorized into three of them that is the spherical rotor, symmetric rotor and asymmetric rotors. Now how do I call this? For example if you take an object a cube ok this is just an example to example but of course when you have molecules they are three dimensional objects. So if you just think of a cube ok then one has one axis is this axis, second axis is this axis and third axis is this axis and the lengths are let us call it as A and I will call this as A axis this is B, this is B axis, this is C length is C and this is C axis ok. Now simply A axis does not mean it is like this but it is going to be rotation along this axis is A. Similarly we will see that if I want to rotate along this axis that is my B axis ok and the C axis will be perpendicular to A and B ok. So conventionally you can have three axis but if it is perfect cube for a perfect cube you will see that A must be equal to B must be equal to C. So whenever you have in such a possibility ok now you can think of a cube as a all the eight corners being on surface of a sphere ok. Similarly one can think of the molecule that is tetrahedral in shape like methane. Methane also is a cubic molecule because the carbon atom will be at the center of the cube and the hydrogens will be at the opposite vertices ok four opposite vertices. So then you will have a methane that is a tetrahedral. So anything that is more of cubic in nature or can be any cube can be fitted on a surface of a sphere. So if molecules have that kind of shape they are called spherical rotors ok. For spherical rotors in general for spherical rotors they should be at least two non coinciding at least two non coinciding axis axis of symmetry with n greater than or equal to 3. That means that will be at least two non coinciding axis that are C3 or more ok coinciding axis of symmetry Cn ok. Of course Sn will also do because Sn has a element of Cn in it improper axis of symmetry. Now first symmetric rotors one Cn axis with n greater than or equal to 3 ok. Otherwise for a symmetric rotor of course if you do not have any C3 axis molecules which do not have C3 axis C3 or Cn axis with n greater than or equal to 3 ok. Simplest thing that I can think of is for example if we take ammonia and we know there is a C3 axis. But if we take water there is only C2 axis. The maximum axis is C3 for ammonia while that is C2 for water molecule. So for water it becomes an asymmetric rotor and ammonia becomes a symmetric rotor ok. And you can think of many other situations. So for example if you have methane that is now you can think of you know this as one if you think of this as a plane and this is another plane ok. Each plane will have one C3 axis. So there will be one C3 axis here and this is another C3 axis ok. Since there are two C3 axis which do not you know coincide with respect to each other then methane becomes a say spherical rotor. So that is how you classify. One can take a look at the textbooks and they will be list out how various molecules will be classified as spherical rotors, symmetric rotors and asymmetric rotors. Now if you take a spherical rotor it has 3 axis ABC and there is a moment of inertia along each of them. So then you will have IA, IB and IC and since it is a spherical distribution you will see that IA must be equal to IB must be equal to IC. That means the moments of inertia are basically distributed similarly along all the 3 axis ok. But when you come to symmetric rotors ok there are two possibilities IA equals to IB which is greater than IC or other possibilities IA is greater than IB equals to IC ok. In this case two of the moments of inertia are equal and the third one could either be smaller than the other two or larger than other two. If such is the case it is called oblate rotor and this is called prolate rotor ok. Now one can think of oblate and prolate are just oblate and prolate are various shapes that you can think of in a symmetric rotor. For example benzene if something is like a disc then it is an oblate rotor. So this has a disc shape for example benzene. This is ammonia more of a ball shaped or egg shaped ok. So in the symmetric rotors you have two possibilities one is the prolate rotor and other is an oblate rotor while there is only one in the case of the spherical rotors. But in the asymmetric rotors it could either be towards these asymmetric rotors could either tend to a prolate. So what are prolate rotors or an oblate rotor? For example substituted benzene with no elements of symmetry will be more like a oblate rotor but water will be more like a prolate rotor ok. So it will depend on the shape ok. The energy which is function of J so that will be equal to B J into J plus 1 ok and if you have centrifugal distortion constant that will be D into J square to J plus 1 square. And we found that the of course one can ignore this let us not think about ignore for small j's. So what you get is B J into J plus 1 ok and when you look at the transitions delta E J to J plus 1 this will go from what will it go from? So this value will be B J into J plus 1 into B J plus 1 into J plus 2 ok. Now if I look at the transition then what will happen? So B J plus 1 J plus 2 minus B J to J plus 1. So this will be equal to when I calculate this will be equal to B into J plus 1 if I take common and this will be nothing but J into J plus 2 minus minus J so that will be equal to 2B J plus 1. So this transition will be equal to 2B J plus 1 ok. So delta E will be equal to 2B J plus 1 ok. Now it turns out that each transition of course the next one will be also J plus 1 whatever that value of J will be. So it will keep increment by one value all the time. It so turns out that this is also true for spherical rotors. Why? Because one cannot distinguish between the three axis of symmetry that is A, B and C or IA three moments of inertia are exactly the same. So one because the value of B will depend on depends on is inversely proportional to 1 over I is for inversely proportional to moments of inertia. So what happens is that since all of them are same so will not be able to distinguish between each direction that is A, B. So even though it is a three dimensional molecules you will not be able to separate out the A, B and C dimensions ok. Therefore the spectrum will look like a spectra of a diatomic molecule. So spherical rotors will have a very simple spectrum similar to diatomic molecule. However in the case of symmetric and estimate rotors the spectrum will get complicated which we will discuss in the next lecture. Thank you.