 In our discussion of many electron atoms we have said already that here we have a situation where you cannot solve Schrodinger equation directly like we could do it for hydrogen atom or rigid rotor or harmonic oscillator or free particle or particle in a box. So that is why it is a cliche that the quantum chemists are often teased by other people saying that these people have only one equation and they do not even know how to solve it. Of course that does not make too much of a sense it is just a way of pulling each other's legs. But one thing that has come out very clearly is that we will not be able to do a very very rigorous solution of Schrodinger equation for systems here on. So we need approximation methods and we have already used one approximation that is orbital approximation and that is how we have sort of circumvented the problem that is there of electron-electron repulsion by considering effective nuclear charge and shielding constant we have been able to reach a situation where we can still use the same wave functions 1s 2s well not same but modified 1s modified 2s modified 2p so on and so forth that is we have been using. Now here just orbital approximation again is not enough we need to now indulge in we need to engage a rather detailed discussion of more thorough approximation methods of quantum mechanics. And you will see that when we do that this concept of shielding and all they come anyway. The only thing is that when we talk about wave functions we enter little more abstract domain we might be able to come back to 1s 2s etc later on but at least during the course of this discussion sometimes you might actually lose the memory of these good old orbitals that we have generated we will talk in terms of more general wave functions. So there are two kinds of approximation techniques that are very popular in quantum mechanics one is perturbation theory the other is variation theory these two theories are used to perform approximation and to start with we are going to talk about perturbation theory. So now you might be wondering in all this discussion where does this cartoon of a cat getting its cheek pulled by somebody comes in. Well we will see in the course of our discussion but before we get into it just so that we do not forget please have a look at the cat what has happened the cat this is the cat unperturbed cat and what the owner or whoever it is has done is that is tucked at the cheek. So only this cheek portion is distorted a little bit rest of the cat is what it was before its cheek was pulled please remember this then unless I forget all about this we will come back to this analogy later on but now we start slowly talking about perturbation theory we wish to introduce ourselves to this technique. So first thing I want to say is what is the scope of perturbation theory perturbation theory is an approximation that you cannot use anywhere and everywhere if it was then you would not need the more rigorous variation method later on. So in perturbation theory what happens is you start with a system for which you can have exact solution of Schrodinger equation for example as we said particle in a box or harmonic oscillator or hydrogen atom and these systems from where we start these systems for which exact solution of Schrodinger equation can be obtained these are called unperturbed systems. So what exactly is the formal definition of perturbation we will come to that shortly but we start with an unperturbed system for which we are able to solve Schrodinger equation without much hassle we can solve it exactly. Then the Hamiltonian for such an unperturbed system is called the zeroth order Hamiltonian what is the meaning of zero zero means there is no perturbation. Exact solution is there this is how we write it H hat for a Hamiltonian operator and we write a zero superscript in bracket which means zeroth most of the time I will try to say H zeroth by mistake I can say H zero please do not get confused about it what when I write this H zero or H zeroth what I mean is the unperturbed well zeroth order Hamiltonian of the perturbed system. Next we try to build a description of the actual system by considering a small deviation from this unperturbed system from which we just started and this word small is big in this context it is very important to remember that perturbation is a small disturbance well we call this a perturbed system but please do not forget that perturbation is a small change if the base value is 1 lakh then it is okay if the deviation is by 100 it is not okay if the deviation is by 20000 then perturbation theory would not work and you will see shortly why it would not work it is important that we never forget that the scope of perturbation theory is an extremely small well not say minus cube but small deviation from the unperturbed system please remember this perturbation is not very very big change and that is what brings us back to that cartoon of the cat remember the entire cat was exactly the same only a small portion of the cheek that was pinched by two fingers of this owner of the cat only that portion got distorted rest of the cat was the same so we might call that a distortion but if the entire cat was caught and elongated by some way that would definitely not be distortion and that would definitely not be perturbation that would be a very major distortion perturbation theory would not work in situations like those all right so once again let me emphasize in different colors perturbation is a small small change very very important not a small point at all great so now what we do is we write the Hamiltonian of the perturbed system as h 0th plus h first where h first this term is the first order correction to Hamiltonian psi of the perturbed system is written as psi 0 plus you can write psi first but generally we write delta psi but delta psi is a small change in wave function and E also is E 0th plus delta E where delta E is a small change in energy so if the so called perturbed system has an energy well if the perturbed system has an energy of 10 units and the so called perturbed system has an energy of say 50 units then perturbation theory will not work you cannot really work like this because delta E is going to become greater than E 0 or even comparable to E 0 or even a sizable portion fraction of E 0 then perturbation theory will not work so what we will do for the rest of this module is that we will try to build a an expression for this delta E and delta psi I am telling you right now and maybe when we talk about anharmonic oscillator later we will come back to this in a little more detail this delta psi or psi you can say is written as a linear combination of the psi 0s remember the wave functions the eigen functions of Hamiltonian of the unperturbed system constitute a complete set of orthonormal vectors I do not recall now whether we have formally discussed what a complete set is but let me tell you what it is let us think of real space the space we live in x, y, z 3 vectors unit vectors along x, y, z they are enough to describe the location of anything any point in this space right so x, y, z or i, j, k if you want to put it that way constitutes a complete set if I only take x, y then it is not a complete set for 3 dimensional space it is a complete set for x, y, space so what kind of a space you are choosing when I space I do not mean physical space I mean hyperspace that will decide what the dimensionality of this what is the required dimensionality for completeness no vector that is required to define anything in that function space as it is called is left out that is what completes complete set is all the vectors that are required to define the function space fully are there all right so that is a complete set so since these 0th order wave functions form a complete orthonormal set what we can do is we are talking about very small distortion right so we can say that even the perturbed wave functions are roughly going to be in that hyperspace that function space so whatever is the whatever is the wave function the perturbed wave function we should be able to write it as a linear combination of all the wave functions orthonormal wave functions in for the unperturbed systems so essentially you are going to write this psi as sum over i psi i 0th but we will worry about that later on we are not talking about wave functions just as yet right now for the rest of the module and maybe there will be a spill over to the next module we are talking about your delta E okay how do we find the small change in energy okay so let us proceed and by doing that we are going to again perform a formal discussion of a fundamental aspect of quantum mechanics that we had perhaps should have done earlier generally in classes it has done earlier but we wanted to get straight into the chemistry that is why we did not now we need it so we are going to discuss that very fundamental aspect of quantum chemistry properties of quantum mechanical operators also in the course of this discussion so that is a something additional something extra that we get to discuss here okay let us get ahead so what we do now is unfortunately I have gone a little far I do not know why so yeah this is what it is so this is what we have for the perturbed system we have the new Hamiltonian H hat new wave function psi new energy E and they all differ from the original unperturbed values by small small amounts okay so the next thing that we will do is we are going to simply write the Schrodinger equation for the perturbed system right Schrodinger equation for the perturbed system and for that we are going to write the Schrodinger equation for the perturbed system and for that we are going to work with this H hat psi and E okay so what is Schrodinger equation H psi equal to E psi simple so instead of H we are going to write H 0th plus H first like this instead of psi we are going to write psi 0th plus delta psi then right hand side is E psi instead of E we will write E 0 plus delta E and that will be multiplied by psi which is psi 0th plus delta psi simple as that so this here is our Schrodinger equation for the perturbed system great so now let us open it up a little bit let us open the brackets we are going to have 4 terms on the left 4 terms on the right and then as usually happens in physical chemistry many of these 8 terms are going to vanish thankfully and we will have a beautiful expression for delta E okay so now H 0th psi 0 plus H first psi 0 let us say these are the 2 terms I have taken these 2 terms in the first place well this I do not even have to say it like this everybody can do it this third and fourth terms will be H 0th delta psi plus H first delta psi simple right hand side similarly will be E 0th psi 0 plus delta E psi 0 plus E 0th delta psi plus delta E delta psi okay we have got our 8 terms now let us see how many of this are going to become 0 first it is very simple but let me once again emphasize the point that is not small the point that the changes are small delta E is small delta psi is small so what happens when you multiply a small quantity by another small quantity 10 to the power minus 3 is a small quantity 10 to the power minus 5 is another small quantity multiply you get 10 to the power minus 8 which is smaller than each of the small quantities whose product we have taken so delta E delta psi is going to be really really really very small and therefore we are going to neglect it remember we are approximating we are learning approximation methods here if we say that we will not neglect any terms then there is no point in doing this exercise okay so I am reminded of another cliche something that a very renowned quantum chemist had told a student the student did not want to do approximation so the professor said see I there is an ant crawling on the table I want you to work out the time that the ant will take to go from this side of the table to that so you have to consider that the table is stationary and then you have to work out the velocity of the ant and work it out now if you say the table is on the surface of the earth the earth is rotating about its axis and revolving around the sun the entire solar system is moving and the galaxy is moving as well so then you have so many terms in the equation then before you can even write it the ant would have crossed the table right so we do not want our ant to cross the table before we can write all the terms this is an absolutely justified approximation delta E and delta psi within the ambits of perturbation theory are very small so the product is even smaller so we are going to neglect this no problem with that if we cannot do it then we cannot use perturbation theory we need something else and there are situations that we are going to encounter where that is the case but then what it means is that you need something else you need some more rigorous theory you cannot use perturbation theory there you cannot use an easy theory and not invoke the approximations that you need to make for it to be applicable right so delta E delta psi is equal to 0 is anything else equal to 0 do you have any other quantity any other term where the quantities are both small H 0 psi 0 no H 1th psi 0 well H 1th is very small but not necessarily psi 0th H 0 delta psi again delta psi is small not necessarily well definitely not H 0th E 0th psi 0th if you neglect that then that means energy of the unperturbed system is 0 not a general case delta E psi 0th once again similar thing so wherever you have this blue terms multiplied by black ones you cannot neglect them but here what about this one blue into blue H first multiplied by delta psi delta psi is small and the perturbation is also small so the contribution of the perturbation to the Hamiltonian is definitely going to be small so we can neglect this H first delta psi as well so out of the eight terms that we have two of them are neglected because they are not they are very very small you do not have to worry about them okay let us erase them immediately that brings us to another interesting situation now we have six terms instead of eight can we get rid of more terms here let us see what is the first term H 0th operating on psi 0th what is the first term on the right hand side E 0th multiplied by psi 0th black and black black and black so H 0th operating on psi 0th and E 0th multiplied by psi 0th are they related to each other right these are Hamiltonian wave function and energy of the unperturbed system of course the Schrodinger equation right H 0th psi 0th is equal to E 0th psi 0th that is the Schrodinger equation for the unperturbed system so this term on the left hand side is the LHS of Schrodinger equation for the unperturbed system this term on the right hand side is equal to right hand side of Schrodinger equation for the unperturbed system they are equal to each other and so they cancel each other it is great starting from eight terms we now have four terms we have four terms starting from eight terms now we have first order Hamiltonian first order correction to Hamiltonian operating on 0th order wave function plus 0th order Hamiltonian operating on delta psi the change in wave function gives us delta E the change in energy multiplied by 0th order wave function plus 0th order energy multiplied by delta psi now our job is to try and make delta E just subject to formula and we have to see how we can simplify this further okay now one very simple technique ubiquitous technique we use every time in quantum mechanics whenever we have situations like this is to left multiply by an appropriate wave function and integrate overall space okay when you do that many of the integrals become either 0 or 1 by virtue of orthonormalization orthonormal by virtue of wave functions being orthonormal and you get many useful new integrals we are going to encounter many integrals of this kind later on exchange integral coulomb integral verus spectroscopy transition moment integral we have talked about transition moment integral very sketchily while discussing particle in a box so we are going to left multiply this but before that what we do is we will bring this terms in delta psi to one side h 0th minus E 0th delta psi plus first order correction to Hamiltonian operating on psi 0 gives us delta E multiplied by psi 0 why are we doing this because we want to make delta E the subject of formula now we will do what we were saying left multiply by the complex conjugate of the 0th order wave function and integrate over all space okay when we do that what do we get so the first term that we get is you multiply psi 0 star left multiply this thing you get psi 0 star multiplied by h 0 minus E 0 operating on delta psi integrated over all space now do you see that h 0th minus E 0th is actually an operator h 0 is the unperturbed Hamiltonian operator and E 0 is just a number right E 0th so an operator minus a number is also an operator so this h 0th minus E 0th is a quantum mechanical operator let us say a hat or something okay we are going to use this later so this is what we have the first term becomes integral psi 0th multiplied by h 0th minus E 0th operating on delta psi d tau looks quite formidable does not look like we can work this out but let us see whether things simplify a little bit second term of course will be from here what do we have here first order correction to Hamiltonian operating on psi 0th left multiply by psi 0 star integrate over all space you got the second term on the left hand side what about the right hand side delta E remember is a number so delta E comes outside the integral then we have integral psi 0th star multiplied by psi 0th over all space this is what we get okay integral psi 0th well delta E integral psi 0th star multiplied by psi 0th d tau and that is a happy situation because remember psi 0th is normalized okay if it is not normalized you can always normalize everything and it will cancel the normalization constant will cancel so integral of psi 0th psi 0 over all space naturally is equal to 1 okay so right hand side simply becomes delta E the question is what about the other 2 terms is there any way to find a simpler expression for the first term that we have here integral psi 0th star multiplied by h 0th minus E 0th operating on delta psi integrated over all space and to do this we are going to use a property of quantum mechanical operators remember quantum mechanical operators must have real eigenvalues that is where we are going to start from once again in case we lose track because of the sudden transition from perturbation theory to very basic quantum mechanics please do not forget h 0th minus E 0th is an operator it is operating on some delta psi okay so we are trying to find a way of simplifying this integral okay we want to know what this integral is because then finding an expression for delta E becomes simple so we might as well write this operator as a hat and whatever we have here this delta psi or psi we can write it as f f is a function so it has been said many many times in this course that we are going to get eigenvalue equations whenever we have a quantum mechanical operator it operates on the wave function if the wave function has the information then it gives the information about the physical observable in the form of the eigenvalue so a hat f equal to a f is the eigenvalue equation that we get all the time okay now what we will do is something similar because we want this kind of an integral integral some wave function star multiplied by operator operating on some wave function so we will just go through the motion and do it we are going to left multiply by f star so we get f star a hat f equal to a multiplied by f star f and then of course we are going to integrate when we integrate left hand side becomes integral f star a hat f d tau is equal to a integral f star f d tau okay a is a constant so it will go outside the integration sign as we have discussed already so we already know what integral f star f is overall space that is going to be 1 because f is normalized so right hand side simply becomes a now what we will do is since we have a complex conjugate of something here we are going to take complex conjugate of the terms on both the sides of Schrodinger equation that we have written in the general form so complex conjugate of a hat f on the left hand side should be equal to complex conjugate of a multiplied by f small a multiplied by f on the right hand side all right now what we had done earlier when we had a hat f we had left multiplied by f star now we have a hat f whole star so we are going to left multiply simply by f because complex conjugate is already there so you do that and do the integration you get what you get but before going there one thing that needs to be pointed out is right hand side is complex conjugate of a multiplied by f now remember quantum mechanical operators have to have real eigenvalues so a star essentially has to be equal to a it cannot be a complex quantity because the eigenvalue is supposed to be the value of a some physical observable it cannot be imaginary so this complex conjugate must be equal to itself so we can happily bring a out here so this right hand side becomes a f star so now we left multiply by f and integrate over all space this is what we get integral f a hat f whole star d tau is equal to a so we have two such expressions both are equal to a how do we proceed from here we can write equate the two left hand sides eliminate a and we get integral f star a hat f d tau is equal to integral f a hat f star d tau now we have something that is close to what we have here but not exactly the same because this expression if you see is something like f star a hat then g integrated over all space this one is integral f star a hat f integrated over all space they are not one and the same so we have to get from here to here we have to see whether we can get an expression where we have two different functions not the same one in this kind of an integral so that is what we are going to take up in the next module we stop here and we will continue right from here