 because of the fact that a more powerful theory has emerged out of the synonyms idea of pairs and their product of pairs being important and that has subsumed the entire set of electron pair theory and that is known as the couple cluster theory. So that is essentially much more powerful method and that has subsumed the most of the discussion of the electron pair theory and also subsumes all of perturbation theory. Though perturbation is independently exist and and most of the electron pair this is a very debatable points of course because there are lot of people in Europe particularly in Germany who actually walk on electron pair approximation as you can see from the name Ulfeld Mayer from Germany and the many of the German scientists are very very attached to electron pair theory. The couple cluster was invented more in the United States of course then later on of course it has gone to US but it was originally done originally there are two Czech scientists who actually did it in atomic and molecular physics and one of them who is basically called the pioneer is Gini Čizek he was a Czech but I think during the war I mean those days you know that there is a lot of tough time in Eastern Europe he actually fled from Czech Prague to Canada and eventually got a job in Canada and then he actually was doing the couple cluster along with Joseph Puldous so both of them actually did a lot of initial work in couple on couple cluster and they were based in the North America I mean in a broad sense Canada and of course eventually it went to Rod Bartlett in the US in the Florida and and of course Europe has Europe become very active because you know they understood that it is in a way much more sophisticated version of the pair theories but I must say that the pair theories themselves were extremely important part of the the the couple cluster you know origin it started also from Sinanoglu actually so I must say that Sinanoglu gave the original idea the Nulfred Mayer developed so the Germans are still very very sometimes get very upset the old German scientist that the United States stole from Germany the couple cluster actually in some way so this is basically a North American start and as you know when the United States takes over they can advertise well so the couple cluster became so well advertised that it almost finished off the electron pair approximation so there there is a lot of discussion on that but if you look at the philosophy of the couple cluster in a way Wilfred Mayer was already doing couple cluster but he again could not give exact notations and how to do second quantization and so on couple cluster uses a lot of second quantization diagramatics which I already discussed for that for the perturbation very similar and eventually it could it was a very nice size extensive theory so it takes care of the dynamic correlation very well so normal standard couple cluster is size extensive which means it has correct independence so in if you combine with the reference space if the reference Hartree-Fock or whatever reference space fragments independently dissociates independently and this is a very important point which you know that was not followed in DCI the correct limit dissociation limit then the approximate couple cluster any approximate couple cluster also separates correctly note that these two points are little bit tricky first we are saying that the standard couple cluster is always size extensive which means for a given n electron problem the energy correlation energy goes as proportional to n however for dissociation or separation I need to ensure that the reference space must fragments independently if reference space does not fragment independently even if you do couple cluster standard couple cluster based on this reference and I will tell you how it is done this will not achieve size extensive consistency or size yeah this is called size consistency right we have already mentioned this is size consistency is basically dissociation so please remember there is a slight difference between size consistency and extensivity extensivity is merely correlation energy being proportional to n size extensivity a consistency is giving the correct dissociation but for that one important part is the reference function must dissociate correct so if I start with a RHS and I again want to repeat for Li 2 Li plus Li right it does not dissociate correct so whatever you do after this it will not help so even if I do couple cluster on this RHS although couple cluster is size extensive dissociation would not be achieved right on the other hand you look at CI the for the CI what happened the RHS was correct H4 going to H2 plus H2 was alright then I did a DCI then this separation could not be done this will never happen in couple cluster because the reference space if the reference space correctly separates couple cluster will guarantee the dissociation it will not spoil so it is very important to realize that the reference space dissociation is not necessarily retained by approximate theories when I do do any approximate theory perturbation does I actually mention that perturbation also is size consistent just like couple cluster I mentioned that yesterday that they have the right end dependence and hence it is size consistent so for size consistency theory must be size extensive there are two important points reference space must fragment correct and I had written down once that these are the two features for size consistency that this has to be there and this first part has to be there if both are there it will guarantee dissociation who is the couple cluster does who is the perturbation does but CI does not because CI the approximate CI does not because it it violates this first part of course approximate that full CI will do anyway there is no discussion on full CI full CI is exact function so I think that is essentially something that you have to remember about the feature of the couple cluster the couple cluster can take care of this DCI problem of H4 H2 plus H2 if you remember why the DCI problem was very difficult because when I did H4 DCI it had a doubles right which was only 11 bar to 1 a 1 a bar to 2 a 2 a bar and 1 b 1 b bar to 2 b 2 b bar right at the separation limit but when I take the product function the product must contain a quadruple which was 1 a 1 a bar I just look at the dimer for example 1 b 1 b bar 2 2 a 2 a bar 2 b 2 b bar right with the coefficients this product has coefficient it means coefficient as the product I think we have discussed this they not only require the quadruple excited but its coefficient cannot be a linear function it has to be a product and that time I mentioned that no linear answers can actually give it but an exponential answers can so this is exactly what the couple cluster do so I exactly what the couple cluster does is to start from a heart reform standard couple cluster I will not go into the details the more details in that the standard couple cluster which is basically a single reference based so you will understand the meaning single reference couple cluster CC so it starts from a psi heart reform and instead of writing 1 plus linear expansion it gives a expansion in terms of exponential of a creation operator this T now are just like CI it generates 1 particle excitation 2 particle excitation 3 particle excitation so in terms of holes and particles it is whole particle creation operator right if I consider the heart reform as a vacuum I have already mentioned that everything that you do is a whole particle creation either you hold create hole and create particles and you have just like in CI you have CI SD so you can have a CC SD depending on what is the T so T in general is T 1 plus T 2 etc up to T n so just like CI this is actually one body excited 2 body excited up to n body excited so you can actually generate several excited determinants by action of this T operator on heart reform just the T operator but since the couple cluster wave function the phi couple cluster is actually exponential T and not 1 plus T it guarantees this product okay and this is something I mentioned before I will not go into the details because exponential of this into exponential of this is exponential of 2 sum T sum a and b so it guarantees the exponential separation guarantees the product and this is why it does but otherwise the T 1 T 2 looks like exactly CI operators so can we write what is T 1 for example it is a one body excitation operator so in normal quantization what we will write like this a a a p some amplitude let me call it T a p and a p dagger a correct in operator form so what is it doing when it acts on heart refoc it creates it just annihilates an electron from or occupied orbital a and creates an electron in unoccupied orbital p so it generates a determinant CI a subscript p superscript correct with a coefficient which is T a p the coefficient is also important so when I take exponential of T 1 then you have the coefficient product T 1 square T 1 cube and those things will guarantee that product separation that I talked of similarly I can write T 2 as sum over a less than b this is all doubles p less than q I write T a b p q just like I used to write in CI C a b p q remember so I am writing T a b p q a p dagger a q dagger a b a a again note the order because I want b to be replaced by q so the order has to be changed in second quantization just like I wrote the two electron operator and I have T a b p q and I made it a less than b p less than q if you make this a all a all b all p all q you have to put a factor one by four these are all spin orbit so similarly I can generate T 3 T 4 etc each of these acting on psi heart refoc will actually generate determinant but it is not CI because the final wave function is not 1 plus T 1 plus T 2 plus T n that's the difference from CI it has an exponential T so the product will come and hence lots of nice things that which have to be taken care are taken care by couple cluster of course the calculation of these T amplitudes is much more complex I will not get into it in CI since it's a linear expansion one can show that this becomes a simple eigenvalue problem here it is not in fact the equations become much more complex and they they have a structure of a nonlinear equation algebraic nonlinear equation you can imagine why they would be nonlinear because you have this exponential T and that makes it rather difficult it's not difficult but it is more time consuming than CI I should say and today people know how to solve nonlinear equations by iteratively again iterative techniques are usually used and you can actually show a lot of correspondence in perturbation theory so some correspondences can be shown when you do the iteration just like you showed you need to CI iteration remember under some approximation we got PT 2 perturbation theory 2 here it is much more easy to show the orders of perturbation mp2 mp3 etc etc is much more easy to show and I will not get into this but I just thought as an overview I'll mention that the equations become more complicated which you should imagine I mean you can't have everything good right I mean so you have to compromise on something so the standard couple cluster of course you can you have a problem here if this reference space does not fragment I told you in that case you can actually have a reference space which is like mcscf or multi configuration then the exponential so you can do both good static and good dynamic correlation they become part of what are called the multi reference on cc mrcc again those are extremely advanced and probably the state of art correlation theory is today in the that when I take this as a reference space and then I put exponential t so they become the the reference things today I should just mention for one minute that all the discussion that we have done is based on calculation of a function right from the beginning if you remember was to solve a cycle t side so if you ask me what did we do in all these 38 classes and I have to say we tried to solve a cycle t side right and that equation that we are solving lest you forget you know it's just a cycle t side started many particle okay single particle of course you are great expert particle in a box harmonic oscillator that we did we went to many particle and then we said because of this 1 by r12 term it is not exactly solvable then we got into all these mess so we said let's make a simplification next let's make a model non-interactive problem a single determinant that gave rise to Hartree power because you already knew a lot about variation method how to do these and then we said not good enough because you are missing that 3% 344% of the total energy which is very very important for chemistry then we went to electron correlation theory we did ci further mess because what was good became bad in double ci what results are better of course let is lower than the Hartree fog then you went to perturbation theory how to handle complicated perturbation we taught diagrammatics and then we found that something more to do because higher order perturbations are very very expensive couple cluster can do better so we went to couple cluster later and then of course in between you talk to mcscf today so all these is really to solve s equal to s to get a structure of ci many particle ci there is another theory which is come which does not talk of wave function at all which directly talks only of density and I said the density is of course related to the wave function by very simple formulae which I will just give you that given the wave function and this is important to know are x2 sorry x x2 x3 xn ci x these are all vectors four dimensional quantities x2 xn integrate over dx2 to dxn completely and for the x part integrate only the spin so if you see it is a very complicated n minus 1 particle integration that I have performing one of them is left untouched x1 or whatever dummy variable you can call it for that you integrate the alpha and spin part then I get a quantity multiplied by n I got a quantity which depends only on a three dimensional variable are so what we are suggesting is that this is actually density if you do a simple heart refog or size simple heart refog this density was some of the orbital densities actually can show this but this is for a general definition of density so there is a different class of theory which which does not even talk of site which basically builds the energy directly as a functional of density potentially this is simpler because the density is just a three dimensional variable remember even if this is an any electron problem the density is just a three dimensional variable so it is very nice if I can write energy as a functional of density and not functional of site this was possible due to a very important theorems by Weinberg and Kohn we actually showed that this existence of energy grounds at energy I must say grounds at energy only as a functional of electron density is possible but nobody knows how to get this exact function so there have been lot of efforts to get exact functional and particular difficulty has been to get the kinetic energy exchange as well as correlation so all important things and lots of approximations have been done all these then you can if I have a functional I can get this energy to be optimum with respect to density just like variation method and I get what is called today DFT but it's a it's it's like a function there are so many things because you can keep on changing this functional there is no functional which people know which is correct the biggest problem with DFT is of course the advantage is simplicity but the problem is that there is no systematic way of improving this function we don't know what is better what is better and so on so that's a real problem I didn't discuss DFT in this course neither it was part of the scope of this course but I just thought I will mention about DFT that this has come out as a very very popular method and today if I say I am teaching to a layman if I say I am teaching advanced quantum chemistry they would assume that I have taught DFT and somebody told me oh you have taught so many classes of DFT I said no no I did not teach DFT at all so it's an embarrassment to actually to clarify that I am not teaching DFT and it is a part of quantum chemistry so but DFT has become such important thing for experimentalist so at least you should know that at least you should know that I have not taught DFT so first and but DFT is important so I thought I will mention this DFT connects to the wave function through this the density connects to the wave function through this formulae okay so I think with this I will close