 So, let me go back to the Hartree-Fock. Now, I go back for the general Hartree-Fock. Do not talk of closed-sale, open-sale. So, I go back to spin orbitals. So, I write again, I write the problem minimize energy. Now, I write minimize E equal to what is E? I now write it chi i tilde. I write deliberately chi i tilde now because they are trial function plus half of ij chi i chi j anti-symmetrized again. I have written in a short form now anti-symmetrized matrix elements of the Coulomb and exchange are stuck together. Minimize this subject to chi i tilde chi j tilde equal to 10 times. So, I have now stated the problem mathematically. Minimize this, of course, E subject to this with respect to n chi i tilde. So, this is my parameters and I told you, I am looking for n 1 electron functions. It is just that I do not know the basis. So, I am looking for n 1 electron function such that that determinant form out of that n 1 electron function gives you the minimum energy with respect to all other set of n functions. You can choose any n functions. So, that is the reason I am varying with respect to chi i tilde. I hope this statement is clear. So, this is my Hartree-Fock problem. So, how do I do this? So, remember it is a conditional variation because I have this chi i tilde chi j tilde minus delta ij equal to 0. So, it is a conditional thing. So, how do I do a conditional variation? I construct a Lagrangian. I hope all of you know this. It is a subject to this variation. So, my Lagrangian then becomes sum over i chi i tilde h chi i tilde plus half of ij. I am going to come back in the next class and expand this anti-symmetrize. Eventually, I have to expand this, of course. Right now, let me write it in a simple form. Minus sum over ij lambda ij chi i tilde chi j tilde minus delta ij, correct? That becomes my Lagrangian. Remember, the Lagrangians are always written that whatever is the function I am optimizing plus a undetermined multiplier times the condition such that the condition is equal to 0. So, what is my condition here? That this minus delta ij equal to 0 for all ij. So, how many conditions are there? n square, correct? All ij, remember. So, I will have for each of these one parameter. So, that is why I have n square parameter. I am summing over all of them. Is it clear this Lagrangian function? So, this Lagrangian is something that I am going to optimize with respect to what this chi i tilde is? As well as lambda ij. The lambda ij is something that I do not know. It is very clear that the variation with respect to lambda ij will merely give me this equation equal to 0. So, automatically the condition is satisfied and that is the basic idea of the Lagrange multiplier. That if I vary this with respect to lambda ij, what will be the first order variation? First order derivative will be just this because lambda does not appear here. It is a partial derivative. So, this should be equal to 0 which means my condition is automatically fulfilled. So, if I vary this Lagrange yarn with respect to chi i tilde and lambda ij, I am assured that in the process of variation, my condition is fulfilled. So, that is that was the main objective. So, for those who do not know the Lagrange variation, this is basically the reason the Lagrange variation is used. So, if I have to minimize a function f of x subject to another function g of x equal to 0, then what do you do? You consider l of x which is f of x minus or plus lambda times g of x and vary f of x with respect to x and lambda. If you vary with respect to lambda, g of x automatically goes to 0. So, in the process of variation, I am constraining my equation, the way I want. So, that is the basic idea of the Lagrange variation. That is a very important mathematics which is actually going to be used. It is being used just because I have knocked up the denominator. Let us not forget. I have written only the numerator here. I have knocked up the denominator. That is why it is being used. Let us do forget why this condition has come. So, this is where we are going to start now. So, this is my function which I am going to minimize with respect to kaii till days and lambda ij. Thank you.