 So, we started with the Slater determinant as a basis for the n electron systems because Slater determinants by definitions are anti-symmetric. So, we said that in the last class that Slater determinant are anti-symmetric by definition. So, they are valid wave function for the n electron problem. The question is whether they are exact or not. They may not be exact wave function but they are valid representation of at least an approximate wave function. And we discussed of course the case A where the Hamiltonian was non-interacting which means it is sum of just some one particle operator. In that case, a Slater determinant is an exact wave function and this is something that we discussed in the last class that the product if it is a non-interacting product of one electron operators is a valid Eigen function. However, the product is not an acceptable Eigen function so we made it determinant. Please remember that determinant is anti-symmetric. With anti-symmetry is the most important rule. The first rule is anti-symmetric. So, anti-symmetry has to be followed then all other things can come. So, if h is non-interacting of course Slater determinant is exact function. What we will discuss today is what happens when h is interacting and that is the case for our atoms and molecules and we have discussed it. Why? Because that Coulomb operator which is sum over 1 by Rij that cannot be written as a sum of one electron operators. So, we discussed that mathematically it is not possible to write. So, that is the reason. So, essentially interacting Hamiltonians are of the form which are of this form. So, we have a sum of one electron operators and sum of two electron operators for the Coulomb problem, for the atomic and molecular problem that we are discussing. This Hamiltonian contains of course kinetic energy of the electrons and all other external potentials which means electron nuclear attraction energy in particular for atoms and for molecules also electron with various nuclei not one nuclei for molecule. So, that there of course we invoke what is called the Born-Oppenheimer approximation. We will come back to that, but let us assume we have already done those we have already done Born-Oppenheimer we will know. So, for molecule of course we always discuss this under the Born-Oppenheimer approximation and we have the form of this Hamiltonian. So, then the question is what do you do? So, S-Later determinant is of course not an exact wave function there because I cannot write this as a product because of this curve. So, we have to find a way to write this. So, this is where I will again go back to a simple two electron wave function and show what I now call the form of the exact wave function and we will actually tell that an exact wave function form is certainly possible. However, it is in principle or in practice never implemented and that is the reason we say that approximate methods have to be developed, but there is a form of an exact wave function for interacting problem. So, we are talking of this case B, we are talking of the case B where Hamiltonian is interacting and we would like to show what would be the form of the exact wave function in that case. So, that is what we will try to do now. So, let us take a two electron problem it is always good to have a two electron problem. So psi is 1, 2 the Hamiltonian is H of 1 plus H of 2 plus 1 by R 1 a convenient example is a hydrogen molecule in Born-Penheimer approximation or helium atom. So, depending on what is your H, so external potential is only going to be different. So, this is my Hamiltonian and this is the wave function the wave function has of course coordinates which are I have repeated again not just space coordinates, but space and spin coordinates. So, I repeat that we have defined what is called the space and spin coordinate x 1 which is equal to R 1 and the spin coordinate x 2 which is R 2 and spin coordinate. So, omega 1 and omega 2 are spin coordinate and our spin functions are alpha omega and beta omega. So, I think it makes it very clear just like our space orbital functions are phi of R our spin functions are only 2 alpha and beta, but they also depend on coordinates those coordinates are what I am calling omega just to give a name. So, x is a composite coordinate of R and omega and I repeat that this is four dimensional because R is three dimensional. So, if you want to be more correct then you should write this as R vector and the x 1 and x 2 each of them is four dimensional. So, whenever I am writing 1 and 2 please remember they are actually x 1 and x 2. So, this is actually an eight dimensional function 4 for 1, 4 for 2. So, because the anti-symmetry must always include the spin. So, now we define what would be my psi of 1, 2 in principle without even solving this Schrodinger equation without even trying to solve the Schrodinger equation. We first note that any function for example, if it is a function of phi of x can always be written as a linear combination of a complete set of functions, this is a complete set of functions f of i. I can always write any one electron function as a linear combination of a complete set of one electron function. I hope this is a mathematics that everybody knows. For example, if you have sin x, cos x, exponential x, everything can be written in powers of x. Powers of x, x to the power 0, x to the power 1, x to the power 2 etc for a linearly independent complete basis. So, you can always write in terms of such a function. If you know this, then let us try to analyze this function first as a one electron function. How can I do that? It is a two electron function. It will be one electron function provided one of the coordinates is constant. So, I am only looking at it mathematically. Let us first look at x1 as a constant. For a given value of x1, the psi now depends only on x2. Then of course I will change x1. It is like writing an f of xy in curves as a function of y for different values of x. You have done that in chemistry or mathematics many times. I will try to do that. I actually say that for a given value of x1, it is only a function of 2. Let us assume that I have a basis of 2 which is the basis of chi i of x which is a basis of spin orbital. Of course, it must be a basis of spin orbital. So, this is my basis. So, do not forget this and this is the basis that we had introduced. What is this basis? This requires an orbital times alpha and beta omega. So, again I repeat the chi i of x would be some phi k of r times alpha omega or beta omega. So, that is my chi i of x because x is four dimensional function. I have a space part which is like hydrogenic atom maybe and alpha and beta part. I assume that I have a set of functions which is complete. I have a basis which is complete which is the basis of chi i. Then if I want to write psi of 1, 2, how will I write? I will write exactly as a function of 2 keeping x1 constant. So, this now becomes chi i of x2, multiplied by some coefficient. Let us assume that this is now a one electron function depending only on the coordinate of the electron 2 which means x1 is constant. So, for a given value of x1, this is no longer a function of x1, x2, it is only a function of x2. So, I am writing that as a linear combination of chi i of x2. However, if I want to write the full function, I have to change x1. Every time x1 changes, the form remains the same, the coefficients will only change. For another value of x1, it is again the same form. But what will happen? The coefficients will change because it is a different function, is it clear? Like sin x cos x, everything I can write in terms of x to the power n but the coefficients will be different. So, similarly for another value of x1, I will still have the same form but the coefficients will be different. Hence, I can say that I can write this as a chi i which is a function of x1 times chi i of x2 because now chi i will change as a x1. So, chi i will have a functional form of x1. I hope you understand. So, basically I am writing, so let us say I have a two dimensional function x1 and x2. So, this is my x2, this is my psi of x1, x2. I write this as a function for a given value of x1. Then another value of x11, let us say this is x10, x11 and so on, so functions keep on changing. Each of the function can be expanded in this form. So, in doing this, what is the difference? The difference will be only linear combination coefficient. When I expand two different functions in the same basis, only thing that will change is this. So, this is changing because x1 is changing. So eventually I can say that this chi is a function of x1. Since now chi is a function of x1, what I am going to do is to expand the chi of x1 again in the same basis. So, I am trying to write for a given chi of x1, I am going to write an expansion in the same basis, let us say dji chi j of x1. Note that for each i I am expanding, so I have to bring another dummy index which cannot be i, which has to be some other dummy index j, so this is the specific i and the dummy index is j. So, dji, actually it is dji chi j but dji depends on i, so I am keeping track of which i. So, I am writing dji of dji, so this becomes a two dimensional matrix and that is now a number. This is now a number. So if I put this here, then this will become sum over j sum over i dji chi j of x1 into chi i of x2, chi i of x2 is as it is there, the dji is now another constant. This constant depends on of course j is expansion of ci, so it is like writing c of x1, how would you write dji chi j, right? So instead of dji, since it is for a different i, I am keeping track of which i, so for every i dji will be different, ci is a number but it is a function of x1. I told you, number means what, for a given value of x1 it is a number. What is the number and a function? f of x for a given value of x is a number, correct? So I told you that I am expanding this for a first for a given value of x1, so then it was a number, but if I keep changing x1, this number keeps on changing, so that is a function. So that again I can expand like this by the same basis but now my chi j will depend on x1 because I am expanding ci of x1. So eventually my psi x1 x2 will be a double summation of j and i dji and a product of chi j and chi i. Is it clear? This is a very important part to understand because you can easily generalize this for n electron now, for n function. So if you understand this, then I can write x1, x2, etc, xn. How many now summation will be there? I have two summations here, n. Is it clear to everybody? Because first what I will do, I will expand in terms of xn keeping x1 to xn minus 1 free fix, then I will expand that in terms of xn minus 1 keeping up to xn minus 2 free and keep doing it recursively. If you see for every expansion, I will have one dummy index. So I have a dummy index which will be like i, j, k, l, etc, etc, n times and I will have a coefficient d, l, k, etc, j, i, whatever and then you can write this as chi, l, xn, chi, k, xn minus 1 and so on up to chi, i of x1. It depends on how you write it, does not matter. So I can keep expanding this as you can see the way I have written it here. If you want to keep the same i, then this should be i, j. So I can change this around. So this should be i, this should be j and so on, this should be whatever is the last one chi, l of x1. It does not matter. Those are all dummy indices. So it is a immaterial. The point that I am trying to say that for the n function, it should have product of these basis functions n times. There should be n number here and these coefficients will have, these are all numbers but this is now not a number, a number which depends on all these n, n indices just like here dji. So if you call these two dimensional matrix, this will be a much larger dimension matrix. So it will have all different numbers. So eventually what is important is that this is a product of n basis and if you remember one of this product was what I called an exact wave function for a non-interacting Hamiltonian. In fact, for a non-interacting Hamiltonian any one product would be exact function because the wave function is a product of the spin orbitals which are Eigen function of the H. I can take this as a basis for example. So it could be made as a exact wave function but for interacting Hamiltonian, I have to take all combinations of this product. So that is the main difference, not one product. Here one product was an exact wave function, here I have to take a combination of products. I hope you are under linear combination of products. So I have a basis functions chi i. So from this chi i, I continue to multiply. Remember number of basis function is much more than capital N. So I can keep on multiplying. It is not that I have the only one product. If of course you have n functions and n electron problem, you will ask me what can I do? Chi 1, chi 2, chi 3 etc chi n. But also that this wave function must be anti-symmetric. So what do I do? Every product I make a determinant just like here. If every product is a determinant from our discussion, please note that no two spin orbitals can be identical. So if I have n basis and n electrons, how many product I can write? 31. Is it clear? Because each of them has to be different and I have to write n product and I have only n basis functions. Let us say I have only chi 1, chi 2, chi 3 to chi n. I can only write a product which is chi 1 times chi 2 times chi 3 times chi n. And then any interchange does not make a new one because it is a determinant. It will only change sign. However, if I have a large number of basis which is capital M which is much greater than n, then how many products can I write? So let us say I write as a chi 1 into chi 2, one function, two electron problem. So chi 1, 1, chi 2, 2, forget about some plus, plus or minus, chi 2, 2, chi 2, 1, chi 3, 2 and so on. You can write but each of them is not a symmetry, how can I anti-symmetrize? You cannot anti-symmetrize between them because each product contains difference between orbitals. One case it has chi 1, chi 2, one case it has chi 2, chi 3, then it is not a separate function. Then it is the same function you are talking about. See, initially I have n product out of M. So this product can be any one of them. Each of them can go up to 1 to M. So it is m into m into m into m, so m to the power n. How many terms will be there? If each of them goes 1 to m, 1, 2 to m, 3 to m, it will be m to the power n but that many number is not allowed because that violates symmetry. So to make it anti-symmetric, we have to make determinant. So now I am asking you how many determinants you can make if you have a basis of m for an n electron problem, mcn, is it clear to everybody? Because all you have to take is a different sets of n from a basis of m. So you pick up any n out of m which are different. So none of them can be same because it is a determinant. Without determinant I will not proceed because I am repeating anti-symmetric governing principle. So my basis must be an anti-symmetric basis. So my basis is no longer just the product but a determinant. So I will have to write the wave function in terms of all possible determinants that I can generate, n electron determinants that I can generate out of an m basis which is mcn. So I will write them now. This is just the combination coefficient for each determinant and I am just keeping track it does not matter. So eventually let me write down if h is interacting and I have a basis of m spin orbitals and let us assume that it is a complete basis, then exact wave function for interacting Hamiltonian can also be written. I was always telling that it is not possible to write. It can be mathematically written as a linear combination of all n electron determinants that can be formed from the m basis. So this is the statement that we are making that I am given a basis of m spin orbitals. An exact wave function can be written as a linear combination just as I am writing as a linear combination of a determinant. Of course the combinations will not be all i, all j, all k, all l because I know that I cannot be equal to j. So you have to order them and there is a way to write how to order them. I will write for a specific 2 electron, 3 electron and so on how to write this running coefficients, running indices but that is very easy. In physics it means you have to find out all possible n electron determinants that you can form out of the m basis and that number, this number is mcn. The number of n electron determinants that you can form out of m basis is mcn and finally your wave function is just a linear combination of this mcn number of determinants. I will give examples to further elaborate this. So this is an exact form. However, before I go ahead with the example let me tell you that this is not really in principle or in practice it can be implemented simply because you will never get a complete basis first of all. A complete basis is usually an infinite dimensional basis. So the problem will simply become infinite by infinite problem. The number of determinants that you will require is infinite in number and it will never reach the exact wave function. So that is the real problem of writing this but mathematically a form exists which however is really useless but I think it is a good starting point to understand because now we can make approximations from here to cut down you know how to choose a basis and lots of questions will come up. If we choose a good basis maybe you will at least get closer to the exact very quickly. So let me just to elaborate this let me take example of another again a 2 electron problem and write a determinant. So let us say that I have 3 spin orbitals so I am making it as simple as possible. I have 3 spin orbitals which is my basis chi 1, chi 2, chi 3 obviously it is not a complete basis. But let us assume that how can I write the determinant chi 1, chi 2, chi 3. So to show the problem that you are generating. So what will be chi 1, chi 1, 2? Now chi 1, 2 will be determinant of chi 1, chi 2. I hope once I write this everybody can write this letter determinant. I have already shown this in last class row and column so it is a full form with a normalization constant but that is not really important. So let us call it some d of 1, 2 okay d of 1, 2 is just tracking this spin orbitals it is not important I can call it something else d 1 also. Then I have another one which is d 1, 3 chi 1, chi 3 right and one more will be there what is it d 2, 3 chi 2, chi 3 is it clear that this is all that I will have. If I have a 3 basis it is like taking 2 out of 3, any 2 out of 3 from my basis of chi 1, chi 2, chi 3. If I have chi 1, chi 2 I cannot have chi 2, chi 1 because it is negative of itself. I cannot have chi 1, chi 3, chi 1 I cannot have chi 3, chi 2 of course each one is not there. So instead of 9 if you had all summation 1, 2, 3 instead of 3 square 9 I have only 3. And that is your MCN 3C2. If you have 4 orbital you will have 6 so it is very easy to do this calculation and these are the coefficients which I am just keeping track in some way. So in general if you look at these coefficients however written I have written orbitals which are ordered chi first one is less than the second one or greater than one of the order you follow. So I can write this as I less than j same thing I can write the d ij chi i chi j so that is a very convenient way of writing. So this is now exact the same thing you understand the same thing I have now written at d ij chi i chi j but d ij this ij can be changed to some other index also. I can give an index for example I can call this 1, 2, 3 it does not matter right now I am just keeping track of the orbitals but later on I need not because all it all it just for a minute all it means these are numbers and eventually these numbers have to be found out. So whether you call it d 1, 2 or I call it d 1 it does not matter yeah all on it everything is later determinant. Slater determinant is the only basis yeah the point that I was trying to tell you if you have a non interacting Hamiltonian then each of them is an exact wave function one of them will be ground state one will be an excited state but each of them is an exact wave function because it is a product chi 1, chi 2, chi 3 if I choose chi 1, chi 2, chi 3 as an eigen function of the h and if my Hamiltonian is only h of i and this was not there then each of this would be a exact function. So I can trivially do an exact function by single determinant by ensuring however that the basis is an eigen function of this one electron operator that is important if the basis is something else then you will still not get then you still have to expand yes because each of them is anti symmetry this become minus it is not required it is not required no no 1 and 2 means you are you are so now what what you are doing that you make this chi 2, chi 3 this you make chi 1, chi 3 I did not understand no no no they have a they have an anti symmetry in the wave function in this in this d. So the coefficients will make sure when you do a full full mapping see eventually I have to write this into the Hamiltonian right so the coefficients will make sure that there is an anti symmetry built into the coefficients when I interchange them so that will make make it anti symmetric finally right now we are only looking at a at a basis which is anti symmetric the actual anti symmetry that you are talking of will come because of the coefficients the coefficients when they you interchange 1 3 and 2 3 they will all change sign so so I am so I am not I am still not understanding of problem you are saying psi 2 1 you are not in change talking of chi 1 to chi 2 right the spin orbitals and coordinates are different you are talking of changing of spin orbitals or coordinates which one you are talking about coordinates if you coordinate then everything will change chi 1 chi 3 is also what is chi 1 chi 3 let me write it down it is a coordinate of 1 and 2 so it is chi 1 of 1 chi 3 of 2 so I am not understanding what you are talking of no I was talking thinking of you are talking of entire determinant there is a symmetry of the determinant that will come out from the negative side if you are talking of this is trivial because everything is 1 and 2 coordinate I think you are again confused with spin orbital index and coordinate index because the entire wave function is a function of 2 electrons it is just that I am also using 1 2 3 for here I can use a b c I mean that would have been better probably to you are confusion I should use chi a chi b chi c the interchange is in the coordinates but there is a another anti symmetry that comes later that we will talk later when you do chi there is another anti symmetry I thought you are asking already that that is the how do you if you interchange inherently 2 levels spin orbital level then there is a relation between a set of coordinate coefficient to another set of coefficient so that is a much more complicated thing no this is a simplest example for a 2 electron problem if I take 2 spin orbital that is trivial because you will have only one determinant so that so this is the simplest example that I can think if you want complicated I can do it you can take n m it will be m c 2 I mean after that it is it is just trivial if you understand if you understand this is trivial right in general you can have 1000 you will have 1000 c 2 for 2 electron for 3 electron 1000 c 3 for n electron 1000 c n I mean this mathematics is trivial right you are only picking up that many out of this set so if you do for example 3 electron problem so let us do a 3 electron problem then I have to of course increase the basis set otherwise it becomes again trivial so let us say I take a simpler example chi 1 chi 2 chi 3 chi 4 then I can again write psi 1 2 3 so now I will have a how many determinant I will have 4 c 3 which is nothing but 4 right so what are those 4 determinant so let us say d 1 2 3 chi 1 chi 2 chi 3 plus this this number is not important d 2 3 4 chi 2 chi 3 chi 4 how I index them is not important d 1 3 4 chi 1 chi 3 chi 4 plus 1 more will be there 1 4 1 2 4 over and it is a trivial exercise anybody will I mean any mathematics people will do it can keep doing it I am I am just you know showing it is not really necessary to show once I made the point that it is a combination of all n electron determinant it means all m c n you write this coefficients in some way so I will again write exactly in the form that I wrote I less than j less than k d i j k and chi i chi j chi k so all i j k I am summing but I must be less than j must be less than k so I am so I am ordering them this ordering ensures that I get only m c 3 in fact if you look at this ordering this ordering ensures actually that I only get this those numbers that I want where I am never allowing any one of them to be to change interchange so I am always fixing the first index to be less than second index to be less than third index you can do the reverse the first can be greater than the second second can be greater than the third okay so this is just a way of writing and you can keep writing now 3 electron 4 electron so your determinant will be an n electron determinant that is all so that many or spin orbitals will be there okay so there is something else that will come up in the d that is what I thought he is asking but anyway that is a later part we will see later that is trivial because the coordinates are always 1 2 3 each one okay so each determinant is anti-symmetric that is the reason we call it a basis so these are anti-symmetric basis okay alright