 Let me tell you what happens when 2 becomes n. I think all of you can set up the CI matrix once again. I hope all of you can set up the CI matrix. Let us now look at not just dimer but not interacting n more. I have n hydrogen molecules, each of them again in minimal basis set. So I will have now let us say 1A, 2A, 1B, I hope you understand 2B, 1C, 2C etc. A, B, C, D etc. They are non-interacting which means each of them is far apart from the other. Once again if I do the exact calculation, the energy of this system will be n times the energy of the monomer. If I do exact calculation, full CI. So what will happen in DCI? So when I do DCI, of course for each of them it is exact. Once again I am talking of only symmetry adopted and if I do DCI for the monomer, for this n-mar, then you can see it is hopelessly wrong because lots of product terms which are excitation between these, these all are missing. n such terms not just quadratic, quadruple, hexatopole excited or all that will be n such term. So if you look at wave function, it will have 1, 1 bar, 1A, 1A bar plus C2A2A bar, then again 1B1B bar plus C2B2B bar and so on into 1C1C bar plus C2C2C bar and so on. So all these n terms will be there. So you can see first term is of course Hartree Fork, no problem. Then you have only WXI and you can chalk out, but there will be terms like this, these, these, these which are hexatopole excited. So there will be lot of terms which will be missing. So quite clearly the result is even more hopeless. So what we actually get is what I want to tell you as a correlation energy. So if you set up the matrix however, the matrix would look pretty much the similar matrix that we had. So 0, k12, that Hartree Fork to WXI it will still be k12 just like monomer I did that. k12, so all, all Hartree Fork to all doubles. So I have only WXI configuration. So all of them will become k12. Note that I am doing a WXI for the nmar. So I do not have anything else. So I do not have Hartree Fork to quadruples. Please remember. So here only doubles are there. Product has, but this is not a product thing. This is a C I for the nmar. So I will have similarly k12, k12, etcetera. And then you have 2 delta and then you have one doubles to another double, 0, 0, everything, 2 delta, everything else is 0 and so on. And these diagonals will all become 2 delta except the first one which will be 0. And then you have 1 by symmetry. I can keep writing 1, C, C, C, C, etcetera. Again, exactly identical thing equal to e correlation 1, C, C, C, just expand this, whatever I did. Just expand this from dimer to nmar and I can solve this problem in the same manner and the result that you will get is e correlation is delta minus delta square plus n times k12 square to the power half. Just as you would expect, there you got 2. Instead of 2, you will get n. If it is a trivial exercise, just set up the quadratic equation, solve it, solve the quadratic equation and you will get this. So as n becomes large, it becomes even worse because when it is 2, this value is slightly different from 2 times this value. So you remember the e correlation, this is n e correlation. This is for the monomer 1 delta minus delta square plus k12 square to the power half. So now we can write in a very general manner. If it is 1, only 1 comes. If it is 2, 2 comes, n, n comes here. That is all. So generally you can write this formula delta minus delta square to the plus k1 where delta is nothing but half of this matrix element. Half of this matrix element between expectation value of the Hamiltonian minus E hf between a W x added configuration, whichever you take, they are identical. So 2a to a bar, same W x added, 2a to a bar. So it is very simple. When n equal to 2, this is mathematics again. This result is pretty close to 2 times this result. Pretty close. n equal to 3, it is worse. n equal to 4, it is worse. You can do the mathematics. Take any value of delta and k12 and just plug it in and see how far this is from n times this as n changes. I hope you understand what I mean. So as n increases, it becomes progressively worse. So that is the point that we are trying to say. As n increases, e-correlation for the of n mar, again progressively gets worse. And as I said, unless you do full CI, exact CI, you will never get it right. Any approximate CI will not do, not just DCI. Although we did this exercise with DCI, it will never do. So in fact, it is said that for all DCI wave function, the energy and this you can see from here, the dominant part of the energy, the dominant part of DCI energy or DCI correlation energy is proportional to n to the power half as you can see here in terms of n dependence. Because you can see that this is, if I expand this, the first time will be n to the power half. Of course, I can do a leading expansion in power series, but the dominant part is actually n to the power half. And this itself is wrong obviously. That is the reason the problem is coming up for n mar. It should have been actually proportional to n as you would guess. So in exact correlation energy, again dominant part is proportional to n. So that does not come out. It becomes square root n and this is essentially the problem. So actually, this is somewhat different because if I am looking at the dimers or trimers or n mar at an infinite separation, the exact correlation energy is exactly n times this. As I told you, why did I write the dominant part? I am writing this because this is a general theme that occurs even when they are interacting. So even when they are not non-interacting, when they are coming closer, then of course it will not be exactly n times, but the dominant part should still be n times or proportional to n. So that is the reason we normally call this in the interacting regime. So this we can actually say even in the interacting regime. So even when they are start to interact, the dominant part remains proportional to n. So when you split them apart, they will be exactly n times. This particular property, the second property that we said is actually known as another name and this is a very recent understanding and it is actually known as size extensivity. So in a way, you can say that size consistency is a material manifestation of size extensivity because this is in the interacting regime. If it is non-interacting, it automatically becomes exactly equal to n times, but they are two different things. So while size extensive theory may lead to size consistency, what is interesting to note that the size consistency has still another level of rigor which I will not discuss right now, that rigor is that the original Hartree form itself must be separate. So this is what is the point here in H4. Original Hartree form was H2 into H2. If I do for example Li2, Li plus Li, the original Hartree form itself will not be separable. So that will lead to another round of problem for size consistency which I have actually avoided. So in a way, size consistency is a much more difficult property to understand and I may summarize this without explanation what would be size consistent if you first understand size extensivity. So size extensive theory essentially says that the exact correlation energy for a given system in the same external potential is always proportional to n when they are interacting, even the interacting system. The external potential must be same. In this case, external potential is same. So this is a very important part to understand that it is proportional, this is called size extensivity. On the other hand, size consistency requires two things. One is of course size extensivity. So what are the requirements of size consistency? One we now understand that if the theory is proportional to n then automatically it would have become n times because it is square root n, this is not giving you size consistency. So one is of course size extensivity, that is important. So if the energy even in the interacting regime is proportional to n, it will automatically lead to size consistency. However, there is a second requirement that is the Hartree form itself must separate or itself must be size consistent, which is the case in this example. So we did not bother about the second part. But if the Hartree form itself is not separable then even if you do a size extensive theory, the result is not size consistent. So this is the little loaded statement. So let us say that I have a theory which satisfies size extensivity. But the theory starts from a Hartree form which is not size consistent. Then the final result will not be size consistent because I have satisfied one but not two. So this is a little loaded. I will not explain much about it but please understand that you can have a size extensive theory. Of course now I know that DCI is not a size extensive theory but I can have a size extensive theory like MP2. MP2 is size extensive I told you. I can have couple cluster later but even then the final result may not be size consistent because two is not followed, two may not be followed. So I have to first ensure that the Hartree form is size consistent. So this question when is Hartree form size consistent I actually discussed during Hartree form itself that if I have a closed shell system if it breaks into close plus close the point two will be satisfied which is the case here for the H2 dimer that I discussed. But if it is Li2 Li plus Li I first have problem here then no amount of size extensive theory will lead to size consistency. I hope I will not go into any more details it is very loaded but that is the reason when you do a potential energy surface you have to be very very careful. You have to first look at the Hartree form then you have to see what theory are you using and both must be right. So this is the correlation part, this is the reference part from where you are starting. So if the reference does not separate properly even if you do the correlation correctly you would not get the result. So even couple cluster will not work in such a case. So how do you make then work? So that is where what is called the multi configuration theories come. This is a totally outside the syllabus of the course but just for reference I am telling you today I will stop now. So a multi configuration Hartree form you do first to make sure that it is size consistent and then top of that you do size extensive theory. If you remember you have MCACF, you have MRCI, you have multi reference couple cluster they are all basically to achieve this problem. So the theory is become much more complicated. So these are all names for you but obviously one has to do it. So these are all understandings which have come in last 20-25 years and people are still working on it. How do I do it? In fact I will make little bit remark on the multi configuration SCF in the next class which is just like CI but little bit different and then I will completely leave this topic and probably go back to perturbation theory and introduce second quantization. So second quantization is very important tool. I think many of the equations that algebraic equations that you write would become actually simplified in second quantization. First one is theory dependent because you must have a theory which is proportional to n. The second one depends on the problem. If the problem is good then you do not worry because second one is about the reference, the Hartree-Fock. So if the Hartree-Fock itself separate then do not do multi configuration, Hartree-Fock. You can directly do whatever is size extensive theory that is a couple cluster for example. I mean we will have a slight overview of couple cluster towards the end but I am just mentioning that couple cluster is a size extensive theory but if you do this even if two is satisfied the theory is bad. So that is what I said. What we are now discussing is a point one that H2 dimer Hartree-Fock was size consistent but because this is not size extensive the result is not size consistent. So the example that we had given DCI, so for DCI the example was exactly opposite. It is not size consistent but this was okay for the dimer. This was not okay so that T can cross changes. So that is the reason it is not size consistent. In the case of DCI, what we discussed DCI of H2, H2n, this is the problem. It is not size consistent not because the Hartree-Fock was bad, Hartree-Fock was actually okay but because the theory of DCI is no longer size extensive as we just now saw. It is actually square root n. So the square root n always comes here. n should have come outside that is not coming and that is very critical to understand because of this CI structure of the correlation energy that it was dependent on the correlation energy itself remember till everything it was okay till you actually solve the quadratic equation okay or iterative equations okay thank you.