 to the lecture number 30 of the course quantum mechanics and molecular spectroscopy. In the previous lecture we were looking at the rotational transitions and the associated selection rules ok. So, we had when a E rotation or E rotor was given by H BE J into J plus 1 ok and when we had this is without centrifugal distortion and when we had centrifugal distortion E rotation was given by H BE J into J plus 1 minus H DE J square into J plus 1 whole square ok. So, and we figured out that this is with centrifugal distortion. Now, we figured out that when there is no centrifugal distortion the rotational lines will be evenly spaced ok. So, this is going from J is equal to 0 to 1, 1 to 2, 2 to 3 and 3 to 4 this will be 2 H BE 4 H BE 6 H BE 8 H BE and the difference is given by 2 H. So, we will have equally spaced lines 2 H BE and we know that H BE is equal to H bar square by 2 ok. So, from BE is proportional to 1 over RE square. So, one can get the where RE is the and RE is the equilibrium distance or equilibrium geometry. So, by measuring the rotational spectrum one can get the bond distance or in general for polyatomic molecule you can get geometrical parameters ok. Now, it turns out that the rotational spectroscopy is the only spectroscopic method by which one can measure geometrical parameters or bond lengths in particular ok. There is no other spectroscopic technique that will allow this. Now, what happens when you have the centrifugal distortion when the centrifugal distortion is there. So, it kind of affects the because of this term it affects the larger J transitions more than the lower J. So, what happens is this lines become more and more packed ok. So, the distance or the energy difference between the subsequent lines keeps decreasing ok. So, delta E 0 1 delta E 1 2 this is delta E 2 3. So, if we so delta E 0 1 will be will be greater than delta E 1 2 will be greater than delta E 2 3. Of course, you know by looking at the pattern one can also figure out the centrifugal distortion and which is generally. So, DE is approximately equal to 10 power minus 4 times BE. Now, when we go back to the molecular Hamiltonian ok. It had many terms. So, H was equal to sum over alpha H power square by 2 m alpha del square alpha negative of this minus sum over i H power square by 2 m E del square i minus 1 by 4 pi epsilon naught sigma over alpha sigma over i z alpha E square by r alpha i plus 1 4 pi epsilon naught sigma over i sigma over j greater than i E square by r i j plus 1 by 4 pi epsilon naught sum over alpha sum over beta greater than alpha z alpha z beta E square by r alpha beta. Now this term corresponds to where alpha is the index of nuclei and i is index of electrons ok. Now, first term we know that this is nothing but kinetic energy of nuclei. This one is kinetic energy of electrons. This is PE of electron and nucleus. This is PE of electron and electron and this is nothing but PE of nucleus and nucleus ok. Now, it turns out that this whole Hamiltonian is written as H is equal to H nuclear plus H electron ok. So, this will be nothing but KE of nucleus ok. This will correspond to H nucleus and all the rest of the terms kinetic energy of the electron, kinetic energy of the sorry kinetic energy of the electron, the potential energy between the electrons between the nuclei and the electrons and the nuclei will constitute to be H electron ok. So, this has so, this one will only be this term and this will be rest of the how many 1, 2, 3, 4 terms ok. Now, one can think of this within the that is within the Born-Oppenheimer approximation. So, one can think of it. So, your total Hamiltonian H is H nuclear plus H electronic. Now, if you solve this. So, one can think of this as H is equal to H nuclear plus when you solve this, this will give some energy that is what I will call it as U or U electronic. So, this is the energy of the electronic part of the. Now, one can think of the total the Hamiltonian will be nothing but the nuclear Hamiltonian. Nuclear Hamiltonian here is only the kinetic energy of the nuclear, kinetic energy of the nuclear. Now, one of the things that we have to see is that the U electronic has also nucleus in it in 2 ways. One is the PE of electron and nucleus and the second one is PE of nucleus and nucleus. Thus, this U electron or this U electron will constitute a potential in which a nuclei will move. So, this is nothing but potential energy for the nucleus to move or nuclei to move. So, this is the potential. So, think of it. So, the your total H Hamiltonian is like kinetic energy operator H plus the potential energy after which I will call it as V electron or U electron. So, this is the KE and this is the PE. Now, so the electronic part of the Hamiltonian gives you the potential in which the nuclei will move. As I told you the kinetic energy part or the H nuclei will be nothing but for a diatomic molecule AB will be nothing but minus h power square by 2 ma del square A minus h bar square by 2 MB del square B. So, this we said that we could do in terms of center of mass separation when you do center of mass transformation. What you will get is minus h bar square by 2 mass of mass total mass del square center of mass minus h bar square by 2 mu del square internal ok, where capital M is given by MA plus MB and mu is given by MA MB divided by MA plus MB. So, this is nothing but total mass, this is nothing but reduced mass. Now, this is motion of center of mass which is nothing but the motion of entire molecule. Now, think of it like this ok, if there is a hydrogen atom and hydrogen atom has internal structure of several orbitals like 1 as 2 as 2 but those are independent of whether hydrogen atom is going to be moving or is at stationary. So, this particular quantity will not affect, will not govern the internal structure of the molecule, basically is a free particle Hamiltonian. Now, what we are left with only this? So, what I have is h internal is equal to minus h bar square by 2 mu del square and this is exactly what we had when we had the rotational motion. However, when we are considering the vibrational motion this will also be added up with u because for nucleate vibrate you know a potential energy for them should be provided by the electrons ok. So, the electrons provide potential energy, electrons in the sense very loosely the electronic energy which will also consist of the electron nuclear repulsion, electron nuclear attraction and the nuclear nuclear repulsion ok. So, this is your h internal. So, we have the Hamiltonian. So, for a vibrational problem your Hamiltonian h is nothing but minus h bar square by 2 mu del square internal plus let us call it v electronic and how do you get v electronic? v electronic is nothing but h electronic psi electronic will give you v electronic. So, this v is here. So, that is the potential. So, there is the this is the kinetic energy operator and this is the potential energy. So, we have to somehow solve this Hamiltonian. So, h is equal to minus h bar square by 2 mu del square internal plus v electronic. So, this is the Hamiltonian. So, the corresponding Schrodinger equation would be that is what we need to solve ok. Now, if I were to plot the function of r the v electronic ok. So, that will come out to be something like this ok. So, we all know. So, this is nothing but your r or equilibrium geometry and in the rigid rotor case we looked at this r e as the fixed distance between a and b but that is not the case because the molecule a b will move in this potential or vibrate in this potential. So, what we have is the curve something looks like this ok. So, we have a potential energy curve that looks like this ok. So, this is my potential energy and this is my distance and this distance is r e ok. Now, when you have that one can approximate this as a harmonic potential in that case the harmonic potential will look something like that at the bottom of the well one can approximate the real potential or the potential in the molecule of a and b as a harmonic potential. So, this is your real potential. Now, how do I get to this harmonic potential? Imagine there is a potential energy v ok and at the bottom of the well ok around r e I want to expand it as a Taylor series. So, my v ok is now given as some value v naught plus d by dr of v evaluated at r e. So, r plus one half of d square v by dr square evaluated at r e r square plus 1 over 3 factorial d square v by negative sign 3 factorial cube by dr cube at r e. So, this is my potential r cube plus etcetera. So, one can write an expansion ok. Now, what is a v 0? v 0 is this bottom of the potential. So, if you have potential this is v 0 ok. Now, there is two things that we can think we can energy is always measured relatively. So, I can always measure the energy can be measured relative to. So, if I measure energy relative to v naught then my v this term kind of a ignored can be ignored ok. Now, what you have is the far. Now, since we are look at the bottom of the potential when you know at the bottom of the potential the first derivative is 0 ok. So, d v by dr this is equal to 0. So, at the bottom of the potential d v by dr. So, I can somehow ignore the two terms. So, what I am left with is my potential v is equal to first term is v 0 I told you that you know we can measure energies with respect to v 0. So, we will ignore it second term is d v by dr and since we are at the bottom of the potential we will ignore the term because the first derivative will go to 0 at the bottom of the potential. So, what I have is half d square v by dr square to r square minus 1 over 3 factorial d square v by dr sorry d cube v by dr cube into r e into r cube and if I want to include the next term it will be 1 over 4 factorial it is power of 4 v by dr power of 4 evaluated r e r to the power of 4 less etcetera ok. Now, if I ignore higher order terms beyond 2 beyond the power of 2 then I have. So, which means I will ignore these then what I have v is equal to half d square v by dr square r e that is my potential v ok. Now, I can call my this as if I equate something called k is equal to d square v by dr square evaluated r e ok. So, that is the second derivative of the potential with respect to r ok and this is the force constant. So, which means your v is written as half k r square and that is called harmonic potential. So, therefore, your Hamiltonian h will now become minus h bar square by 2 mu del square internal plus v harmonic ok. So, this is the harmonic. Hamiltonian for the oscillator ok. So, h will be minus h bar square by 2 mu del square internal plus. Now, if I want to if I have a ok if I have a diatomic molecule a b and this is my direction ok. Now, if this is my direction then what am I doing it I this is my r this is e r e and this is the direction r, but r is just a simply a choice of variable. So, I could in fact write it as a x. So, one can write it as so, this is my x e and this direction I call it as x because variables are dummy one can always interchange variables. So, what will happen my h will now become minus h bar square by 2 mu ok this is my direction. So, when I have del square internal will become d square by d x square where x is equal to x a minus a ok plus half k x square. So, that is the Hamiltonian for which I need a solution and that will give me the solutions of the harmonic oscillator ok, but one must always remember that harmonic oscillator is an approximation ok. The actual molecule is not an harmonic oscillator because for harmonic oscillator potential something like this, harmonic oscillator some potential is something like this and it will. So, this is your what I will call it as x e it will never break ok. So, harmonic oscillator potential does not break at all it can go up to infinity ok this is your potential energy. However, real molecules break. So, when you go to some distance or some energy the AB bond will break ok. So, then this is the real molecule. So, the harmonic oscillator approximation is only trying to represent the bottom of the potential. So, if you go above ok somewhat higher in energy then the harmonic oscillator approximation breaks down rather drastically. So, when you are looking at properties of a molecule at the bottom of the potential or when it is sitting in this well really at the bottom of the well then one can approximate it to be a harmonic oscillator even then it is only an approximation and it is not a real thing ok. However, this approximation works reasonably well for most of the most of the molecules and this approximation can be used to measure the evaluate the wave functions and the associated selection rules for vibrational spectroscopy which we will continue in the next lecture and we will stop it here for now. Thank you.