 I think this actually completed our one electron problems and we eventually moved to many electron problems and from where I will actually start, although we did a little bit of many electron problem, this course I will repeat many of the things that we did for many electron problems. So for the many electron problem, I first said that we need methods which are approximate because the problems cannot be solved and just as I told yesterday because of a term which is called 1 by R ij and I do not know if I mentioned yesterday, but the many electron problems let us say 2 electron H 1 on 2, if you write down the Hamiltonian, you have a kinetic energy of the electron 1, remember the helium problem that we did, the kinetic energy of the electron 2, again I am writing these in atomic units, these are all in atomic units. So I do not write h cross mass etc., of course it is a reduced mass actually but that is okay and then you have what is called the nucleus, nucleus so z divided by R 1 minus z divided by R 2, where R 1 and R 2 are distances plus 1 by modulus R 1 minus R 2 and we noted that so this is for the helium atom, so in particular for helium atom but in general we can write for any molecule or so, we noted that a part of this Hamiltonian can be written as a sum of 1 electron and 2 electron Hamiltonian uncoupled which means I can invoke the non-interacting theorem to solve that Hamiltonian, the wave function being product, energy being sum, but the problem really is this part, that this is a part which cannot be split as a function of 1 and function of 2, mathematically it is not possible. So you cannot write this term 1 by modulus R 1 minus R 2 as any function of R 1 plus any other function of R 2, let us say F 1 or F 2 in general, so this is not even possible, if it was possible mathematically then I would have put this R 1 function with 1, R 2 function with 2 and the entire problem would have been exactly solved, so the cracks of the problem is that this is not even mathematically possible, of course a lot of people have made attempts to write, can I write this as F of R 1 plus, some F of R 1 plus F 2 but it is not simply not possible, you can do a good job of it but that is an approximation, so this is where the problem of many electron theory starts that I cannot solve it exactly, so many of the solutions have to depend on approximate methods and two very important approximate methods that we did were actually variation method and perturbation method, and I remind you I have done this in great detail, so I do not know how whether I should again do in the same detail in this course but probably I will do this because that is I think an important part when I start I will at least repeat most of the things, these are two very important approximate methods which are actually used to solve the many particle problems, there are many other schemes but the helium atom is an example but you can have any many particle problem like hydrogen molecules, any other molecule which has more than one electron, you will end up with a problem where there is a term which is a non separable term which cannot be written in this manner and hence the problem is not exactly solvable, so I might mention that this is where the basic problem of solution of stationary state of the Hamiltonian starts, although I told you although we are interested in solving only the stationary states of the Hamiltonian, even that solution is not possible, so what we will be discussing further in this course is how to solve for many electron problem A psi equal to E psi and this has been the quantum chemistry for 70 to 80 years, how do you solve A psi equal to E psi for many particle problem where Hamiltonian is non separable, so far the problems that we have been doing were trivial problems, we learnt lot of quantum mechanics but the solution of A psi equal to E psi was a trivial problem some mathematical exercise of course I am not saying they are trivial but some mathematical exercise but doable exercise, when you scale particle in the box from 1D to 2D to 3D they were uncoupled, so there is no problem, when you scale harmonic oscillator to 1D, 2D, 3D they were uncoupled, for the hydrogen atom even for the hydrogen atom this solution could have been done and it was not very difficult to do this, the problem came when we actually want to solve the many particle problem where there is a genuine term which is non separable and hence the problem can actually not be solved. So we have to resort to approximate methods and the two approximate methods that we discussed were variation and perturbation and we have to see how to integrate each of these approximate methods to make solutions of this, so this is where we will start, we will also look at what kind of symmetry the wave function should have and this is something again we have said in the last class that they must be either quantum particles must be either bosons or fermions or electrons or fermions, the anti-symmetry is later determined, we have done all that up to that point but I think we will revise this very quickly, so we will see how to handle this problem because that is always, this heterogeneity is always there but we will try to see because then I could start but maybe I will lose another one or two classes to bring you up to that point, that is okay, I will do that in the next class. So anyway today we will close, all right.