 If we appreciate this, it is important for us to move from a physical vacuum to a vacuum which is Hartree-Fock itself. Because all our correlation problems start with the Hartree-Fock, so starting from Hartree-Fock we are exciting, so Hartree-Fock is our base reference. So every time if I have to expand Hartree-Fock in terms of vacuum and A1 dagger, A2 dagger etc., that becomes a complicated exercise. So is it possible to think of a new description in which Psi0 itself is a vacuum? Just like we are doing, so we will not expand Psi0. So what kind of description will make Psi0 as a vacuum? What is the property of a vacuum? Let us do this practice problem later, you will be able to do, if you do not do it on Thursday. But I think this was only to show an important topic to which I am going. So the topic is essentially that there is a simplification if I do not expand Psi0 in terms of the vacuum. So otherwise every time I do correlation problem, Psi-Hartree-Fock has to be expanded in terms of A1 dagger, A2 dagger etc., up to A1 dagger, A1 dagger on the vacuum. Can I avoid this? I can avoid this if my Psi-Hartree-Fock itself becomes a vacuum. This exercise was an exercise in that direction that do not expand Psi0 but make Psi0 as a vacuum use these killer conditions. Vacuum must have a killer condition that something annihilating you cannot annihilate. Clearly I have not really been able to do this. I am saying A1 dagger acting on Psi0 is 0, A1 dagger is not an annihilation operator. So how do I make sure that that is a vacuum by suitably defining my operators. So this brings to the concept of holes and particles, they are very very important for correlation problem. So instead of electrons, so far what we have done is the creation and the annihilation of an electron in an orbital. We are now going into a description where we create and annihilate what I call holes and particles. So holes and particles are my new description then you will see that this will automatically come. So let me first define what is a hole. Note that I have a reference determinant Hartree-Fock. This is particularly true for the electron correlation problem where Hartree-Fock is always my reference determinant. So I define hole as absence of electron or a vacancy, absence or vacancy of an electron in one of the Hartree-Fock spin orbitals which I call occupied spin orbitals. If there is a vacancy that means there is a hole, so you have to think little differently. What is a particle? Particle is the presence of an electron that is a normal thing, presence of an electron in an unoccupied orbital or a virtual orbital. So this is really normal, normal annihilation operator and creation operator but this is exactly opposite. If you look at now Hartree-Fock size 0, in terms of holes and particles, note that when I said physical vacuum is because it has no electrons. Now try to think of Hartree-Fock. Does it have a hole? Hartree-Fock reference determinant itself. Does it have a hole? No, because there is no vacancy. Does it have a particle? No. So then the Hartree-Fock which is actually an electron becomes a whole particle vacuum. It is not a physical vacuum, this is not a physical vacuum but it is a vacuum in the sense that it does not contain holes, it does not contain particles. So I should be able to define creation and annihilation operators not in terms of these but in terms of holes and particles. Then I need not have to write such an unphysical statement that something annihilation equal to 0 and I have to keep track. Something creation is equal to 0 and I have to keep track of what it is. I can write everything in terms of annihilation operator just as in the normal vacuum. Normal vacuum you remember any AI acting on the physical vacuum is 0. So I want a similar structure that any annihilation acting on the Hartree-Fock should be 0. So how do I do that? It has no holes and no particle. Let me first try to define creation operators for holes. So how do I define creation operator for holes? This is no longer creation operator for normal electrons. So how do I define creation operator for holes? From the Hartree-Fock I have to actually annihilate an electron in the Hartree-Fock orbitals. Let me define this creation operator as X just to distinguish from the small A. A was electron creation operator. This is whole particle creation operator. So I define a creation operator XA dagger as equal to AA when A is an occupied orbital. Because holes are always referenced occupied orbital. So this becomes my whole creation operator. So whole creation operator essentially means I annihilate an electron in the Hartree-Fock. So whenever AA will act on Hartree-Fock it will actually generate a hole. So I have actually created a hole. Is it clear? So obviously the adjoint I can define exactly the same way that the annihilation of a hole is nothing but creation of an electron in the occupied orbital. So if there is a Hartree-Fock of course you cannot create as I told you here A dagger psi naught is equal to 0 which means you cannot annihilate and that is the meaning of vacuum. Vacuum means you cannot annihilate because why cannot you annihilate because holes are not present. So I cannot do a whole annihilation operator. So this is for the holes. So I define creation and annihilation operator for the holes and A must be occupied orbital that is important because holes are only creation annihilation in the occupied orbital but exactly opposite. So normal creation of an occupied spin orbital is actually annihilation of the holes and vice versa. Let me now define particles. So for particles I am going to use virtual orbitals because this is the presence of an electron in the virtual orbital. So let me write this as xp dagger. I am again using a, b, c, d for occupied orbitals p, q, r, s for the virtual orbital. So xp dagger and this is nothing but a p dagger standard definition and xp is also equal to a p where p is now virtual orbital. So that becomes creation annihilation operators for the particle which are standard xp dagger is equal to a p dagger because I am just creating an electron. So I am creating a particle. Is it clear? The definitions? So now you can see I will come back to this problem or any other things. So let us see now that if I have a heart reform why this is a vacuum now if there is reference to the hole in particle. So let us assume that xa is acting on the psi heart reform. Now you see a is an occupied orbital so xa by my old definition is a dagger heart reform and we know that is 0 because I cannot create an electron which is already present or in other words now there was no hole so I cannot annihilate. Now I have to change the language. In terms of normal thing I am saying there is an electron present so I cannot create an electron again. In holes and particles I am saying there is no hole so I cannot annihilate holes. So what is important is of course look at this part because I am now defining hole and particle. So xa acting on psi heart refock is 0 and similarly xr or xp acting on psi heart refock is of course 0 because xp is nothing but ap particle is not present so I cannot annihilate anyway. So now I am able to define two sets of orbital a two sets of operators holes and particle whose annihilation on heart refock is 0 and hence I am arguing that the psi heart refock is a hole particle vacuum. It is not a vacuum in terms of normal electrons creation operator but it is a vacuum in terms of hole creation, hole annihilation or particle creation, particle annihilation. So from this vacuum I can only create, either I can create a hole or I can annihilate. So remember when I was doing this problem I had to physically remember that this is AA dagger so if I bring AA dagger here AA dagger psi heart would be equal to 0. I have to physically remember because A was occupied orbital. If I use this then I do not have to bother. This will be like bringing regular annihilation operators to the right, whatever it is and if I do that then automatically I will ensure that the results are correct. Now you can of course write the anti-commutation relationship between holes and particles because they are essentially related to the creation annihilation operators of spin orbitals. You can easily write by taking the adjoint or wherever it is not necessary like particles it is exactly identical. So you can actually see that they will follow exactly the same anti-commutation relationship between the two particles, between the two holes or so on or one particle one hole. You can actually use the same anti-commutation relationship so that is very easy to do. And then we can rewrite the entire thing in a very simple way. So for example if I come back here I can now rewrite this actually what I wanted to do psi 0 because that is my vacuum now heart refog is now vacuum. Now I will not use physical vacuum and then how will I write psi a r? How will I use psi a r? First one will create which one I will create first a or r all are creation now. I am going to write in terms of x, I am going to write in terms of x so why you are saying that I will do x r dagger, x a dagger psi 0. Now everything has to be creation because it is a vacuum now. From the vacuum you can only create unless of course this is equal to this but this can never be equal to this because one of them is whole one of them is particle. So between the holes and particles the holes cannot be equal to particle remember because they are occupied side particle is unoccupied side. So there is a condition by which they can never be equal. Now you can actually look at this as a vacuum which means nothing exists. So first I create a then I create r and then what I do is write psi a r as psi 0 x a x r just take the normal vacuum and then of course a i dagger a j x a dagger x, x r dagger x a dagger psi 0 and start expanding this in terms of the anti-commutation relations. Now again you have to be careful here because these are now ordinary creation annihilation operator these are holes particles so you cannot use an anti- commutation between x and a. So you have to actually write this in terms of creation annihilation by dividing this i j between holes and particles. So a subset of i can be whole i can be particle j can be whole and j can be particle and then write in terms of x so that everything will become in terms of x and then use anti-commutation relations is it looks very difficult but actually these are quite simple things to do of course there is a sum over i j and there is h i j but the important simplification that comes in is the fact that this psi 0 I don't have to expand and this is very important for a large as n tends to infinity in a many body problem where n becomes very very large this is a very important simplification because otherwise there will be too many too many determinants too many creation annihilation operators on both sides either sides this particular problem not it depends on what is the problem if I give you psi a r theta 1 psi a r of course there it will be so you can actually do this problem this problem is fairly simple as you can see so you can divide i and j in terms of holes and particles and then say that this is then apply the anti-commutation between holes particles yes yeah so you you you can call this when it is hole you can call this whole annihilation operator right this will then become whole creation operator when j is whole when this is particle they will have a normal this thing so you divide into four subsets i can be whole j can be whole i can be particle j can be particles of all four possibilities right so then it becomes easy to yeah just easy to break down and many of them will just become zero actually by very simple this thing you can actually make out what they are see note that aj is acting on psi not so what can be j see if if if aj acts on psi not it cannot become zero if it remains zero then the problem doesn't exist right so first of all all sum over aj aj has to be occupied j has to be occupied orbital which means i am creating a hole this should not be a whole annihilation this should not be annihilation operator in a whole particle time so j must be occupied and i must be virtual okay and then you can do the creation of that you will see there is a delta air condition basically this will be a this will be r so that is all that will survive aj so this will become r and this will become this will become a this will become r hij right so this is i this is j so j should be a i should be r that is all that will come because this is what is going to survive finally right so i know my first quantization result is rha so and you can actually see this that this j must be whole okay because then only it can act on psi so this will be actually xj dagger this will eventually become xj dagger okay or xc dagger or whatever you have to sum don't use a you can use xb dagger also sum over b and this will become r so sum will become b and r and obviously then you will see by the second quantization that only b has to be equal to a and s has to be equal to r so it's very easy to do this so do this exercise i mean it's very simple because most of the terms will become zero so only one term will survive when this is occupied this is virtual this is only term that will survive in all the four subsets and you should be able to get hra as a result okay so the point is that for two electron i gave you a first problem where it is a two electron you expand in terms of physical vacuum it was quite easy okay for two electron but for large electron system this is not tenable so that is why the hattrifog vacuum becomes much simpler so that you can expand everything in terms of whole creation whole annihilation operator so your theta one and theta two i can actually write in terms of whole creation and annihilation operator so let's do that and that is what i have actually used it so let's say i write theta one note that i have a ai dagger aj so i should write this as the following manner theta one as both whole ab so then this will become xa dagger xa xb dagger okay into hab so basically so i and j are both are whole they were ai dagger aj but because they are whole in terms of x operators they have changed it has become xa xb dagger correct plus ar so now i is a so ai dagger aj so this will become xa aj will also become now xr har plus sum over r a so now ai dagger will become xr dagger this will become xa dagger h sorry h a h ra plus rs all particles all particles what will happen again ai dagger aj so this will remain exactly as it is xr dagger xs hrs so the same operator i can define in terms of whole creation particle creation by taking ij both whole i aj particle i particle j whole both of them particle four subsets and i should be able to write so this is exactly what i am doing instead of applying the ai dagger aj i am actually applying this operator and then i am showing that only one of them survives in this case the rest will all become automatic in the zero i hope that is clear because what i have done is i have written theta 1 as ai dagger aj hij so this was all i all j i have broken it up so when i is whole ai dagger is nothing but xa so ai dagger as a creation operator in the occupied space which is basically whole destruction operator whole annihilation operator aj exactly opposite becomes xb dagger and both of them are holes so one of them whole one of them particle only this changes so this becomes xa this remains as xr okay one of them r and a then both of them become annihilation creation operator and then one creation one annihilation for the r and s r and s they will follow this okay because they are particle so you can see that all combinations are given creation annihilation creation creation annihilation but this is in the whole space this is in the particle space both annihilation both creation one of them whole one of them particle depending on which is whole which is particle one of because whenever there is a whole and particle i should be able to create both there should be a possibility of creating both but this now constitutes the entire form of the operator when i act on a size zero depending on size zero depending on what i am acting on one of these some of them will become zero okay their action will become zero but this is a general form of theta one i can write similarly theta two is more complicated but one can actually write a similar expression and then do the do the matrix element algebra the matrix element algebra tells you that if we get eventually in the first quantization what we got by Slater condon rule so that was important because Slater rules gives us a matrix element so exactly the same thing that we are getting alright i don't know if i will have time to do the two electron part yeah when it acts on size zero but this is a general form of operator yeah actually it will be good if you can do this problem of size zero theta two size zero just once if you can do it it's again there in the exercise either by using whole creation particle creation or by using normal creation annihilation operator but making a physical intuition that only a dagger will act on size zero is either way the same whichever you feel comfortable so please try to do once at least in the exercise it is given so that you are comfortable that you get this so size zero theta two size zero just show that it is equal to half of a b a b minus a b b by using the expression of theta two which does not have a remember which does not have a antisymmetric integral which is a regular integral but there will be two conditions will come which will bring in the antisymmetric integral okay so just expand this a and b are of course occupied orbitals in size zero theta two has a general expression but i think by simple argument in fact this is there for those who are interested should see this in the page 96 of Zabahaslu textbooks so please see this i am giving as a kind of an exercise so that you see if you have a problem on Thursday we can come back and discuss this so this is a one problem at least you should be able to do so that for the two electron also you are comfortable that this later rules are obeyed by this second quantized operators alright so let us we will go back now to the mp2 and try to write this in second quantized notation maybe in the next class we will see and the diagrams i told you the diagrams eventually simplify not that the diagrams are necessary but they simplify the notation and later on it is very easy to go e correlation at the mp2 level somebody can tell me the expression 1 by 4 a b r s a b r s r s a b i hope you remember why these two terms came this actually came because of the Hamiltonian matrix element between two electron w xr a determinant with heart reform and conjugate conjugate of that right so this was the actual that matrix element this is the conjugate and 1 by 4 came because originally number of w xr at configurations are actually a less than b or less than s anyway a cannot be equal to b neither can r can eb equal to s because of Pauli principle and we are we are talking in terms of spin orbitals so if you look at the determinant psi a b r s now can i write this determinant in terms of holes tell me all four so xr dagger xs dagger now you have to be careful xb dagger the way i have written x a dagger note that i am creating a hole now i will talk in terms of holes i am creating a hole so this of course acts on psi 0 which is my vacuum i am creating a hole a which is actually annihilating an electron in the spin orbital again remember then i am creating another hole b of course a and b have to be different and then on the b i am creating a particle x and a particle r on the a so my original notation was remember a r dagger a s dagger a b a sorry a b a that was my original notation acting on psi heart reform now with the new heart reform vacuum i am going to use everything which is holes and particles and everything is creation operator just like you do in a normal physical vacuum from physical vacuum you just start creating so this determinant is normally called two hole two particle determinant i hope the meaning is now clear so compared to my heart reform which is a vacuum this determinant contains two holes and two particles my heart reform did not have any holes and particles there that was a vacuum in terms of hole and particle so all the nomenclature in the correlation problem will actually include holes and particles so if i have a psi a r this is called one hole one particle state one hole one particle determinant so it has one hole a one particle r this is called two hole two particle determinant all right so this mp2 formula if you remember had come from a less than b r less than s psi a b r s so which is not to two hole two particle state v psi 0 which is a vacuum psi 0 v psi a b r or reverse it does not matter normally we would like it this first and this later because we want to show this from psi 0 to psi 0 but does not matter divided by the energy difference epsilon a plus epsilon b minus epsilon r minus epsilon so this includes this contains matrix element of the perturbation operator with the vacuum and two hole two particle state so this is now a two hole two particle state in my language so it is a matrix element of the perturbation operator with respect to psi 0 then remember what did i do i wrote the perturbation operator as 1 by r i j sum over 1 by r i j minus v heart refock i remember v heart refock i is very important because that must be subtracted in the perturbation operator because my h0 is sum of the fork operator right so my h0 was sum over h of i sorry sum over h of i plus sum over v heart refock i correct so when i write v it is sum over 1 by r i j minus sum over v heart refock i because this must cancel eventually my Hamiltonian is this plus this correct however the one particle operator did not bother us because this is between two determinants so it was 0 so you are actually only one by r i j which transformed into this and similarly this transformed into this or rather this transformed into this this transformed into this does not matter so and i get i got that expression if you apply second quantization now we know how to apply second quantization on this i should be able to get the same expression you know this is something that is a part of the exercise that you can keep doing so i have a i have a two whole two particles state i have a vacuum and i have one by r i j and v heart refock i v heart refock i will anyway give you 0 that you also you can check in fact this may be a good thing to check in second quantization also that if you do psi a b r s theta 1 psi 0 it is equal to 0 expand theta 1 in terms of holes and particles okay and just show that this is equal to 0 by second quantization okay so do not assume by the first quantization is 0 but second quantization you can show this so many of this practice problems you can do so the point that i am trying to show is that they can all be derived from the second quantization m p 2 whatever expression that we are showing can now be derived from the second quantization and the algebra of second quantization can be translated much easily in terms of diagrams there are diagrams by fine man goldstone hook and hole there are several diagrams are there we are going to follow mostly the hook and holes diagram which are easy to follow for the second order perturbation so what i will do next time is to give you the relevance of diagrams in terms of second quantization first and then show with an m p 2 algebra how can i transfer this into a diagram very simple diagram once i know the rules then every time i can draw a diagram i can write the algebra immediately diagram will be much simpler and then i will show in the perturbation each order how can i construct all the diagrams