 So, I think with this I will currently leave the CI and go to somewhat more technical issues. So, this is what is called second quantization, this is extremely elegant however, note that this word second quantization itself may puzzle you actually, what is second quantization? We have of course talked of one kind of quantization in quantum mechanics that you already know and that is essentially the first quantization if you want to say. So, what we are discussing is actually expressing quantum mechanics in another language. So, the second quantization will not necessarily bring in new physics, but it will actually tell that how these physics of anti-symmetry etc., etc., that we have learned can be written in a simpler way. So, that is what is second quantization. This is a language, but the language makes a simplification. In fact, if you look at the Hamiltonian, the Hamiltonian that we write, it is always written in terms of number of electrons. So, our Hamiltonian has a sum over h of i which is number of electrons, sum over 1 by i is a number of electrons. In the second quantization, when we will write the same Hamiltonian, we will not have explicit dependence on the number of electrons. In the similar way, when we write the wave function for fermions, we have to explicitly write Slater determinant. You will see that in the second quantization, even that is not explicitly required. It is automatic formation. So, these are some of the advantages. So, the two important points are of course in simplification of writing the wave function and expression of the operators. And then we will see how we can write matrix element of an operator between two wave functions. That is Slater rule. Two determinants on each side in the Hamiltonian matrix element. How does that translate in second quantization? The Slater condone rules. We have done this for type 1, 2, 3. Because wave function will have a different language, operator will have a different language. And so the expression of the operator and the wave function will look different from what we have learned so far. So, these two things, the expression of these both wave function and operators will change in the second quantization. And hence of course, Slater condone rules will also look different. And this will actually simplify in doing the algebra. And that is one of the reasons the second quantization is stopped. And we do not have time to re-express all the algebra. But please remember that all that we have done so far, Hartree-Fog, perturbation, CI can be rewritten in terms of second quantization. It will not bring in any new physics, but it will just simplify. And later on, when you go beyond MP2 for example, MP3, MP4, it becomes actually mandatory. Because to do that horrendous expression in the first quantization becomes extremely difficult as you go to higher order perturbation or even when we will do couple clusters. So it is better to use second quantization right from the beginning for many such cases. CI can also be expressed in terms of second quantization. But CI is somewhat easy because it is a linear expansion, it is only an eigenvalue problem. So it was somewhat easy. So that is the introduction to why second quantization is required to be learned. So what we will do is that we will start with the second quantization and as we start we will define two very important operators, two basic operators. One of them is called the creation operator. So let us say some a and this is the way to write this a subscript alpha, some number, alpha can be any number, anything that you want and superscript dagger. So that is called the creation operator. What does this mean? It means that it creates this operator, will create an electron in a spin orbital alpha. So this is the action of the operator. So let us assume that there is something called vacuum, there is a ket vector called vacuum. Vacuum means nothing exists. So it is an abstract concept, nothing exists. Then I allow a alpha dagger to act on the vacuum. What will happen? It will now create an electron in a spin orbital alpha and you will get a one electron state alpha. You will get a one electron state alpha or i, j, whatever is the subscript. This is i, you will get i, is it clear? Vacuum is an abstract concept that it is a ket vector where nothing is there. It is a zero state, null state. I just allow a alpha dagger to act. Then I will create a state alpha. Of course I can have multiple creations. So I can now have another state, another operator a beta dagger which acts on a alpha dagger vacuum. Now obviously this will create a two electron state. It is quite obvious and this will create a state which I now call beta alpha. Now what is important here is to notice that as soon as I create a two electron state I must make sure that this is anti-symmetric. I am talking of electrons. So obviously this beta alpha that it creates is automatically a determinant beta alpha. So I am creating a slater determinant which is automatically anti-symmetric just by having two operators acting one after another. So note that when this operator acts it is not simply creating another electron in beta or beta but at the same time anti-symmetrizing. So that is the job when it goes to many electron problems and so on. So for example if I have some AP dagger acting on already a slater determinant chi 1, chi 2, to chi n which can be a Hartree-Fock, n electron determinant. This will generate a n plus 1 electron determinant where the first orbital will be chi p and then chi 1, chi 2, to chi n. Note again the order of creation. The creation takes place immediately because the order is very important because there is a sign. It is a determinant. So creation takes place immediately. So the order is very important as I told you here when I put beta dagger the creation comes in beta. Quite clearly if I would have created a alpha dagger a beta dagger on a vacuum this would be alpha beta which should be negative of beta alpha by my definition. Please remember these are by definition. This is my definition so there is no question of showing it. So in principle if I have a alpha dagger a beta dagger acting on in general an n particle determinant chi 1, chi 2, to chi n this is negative of minus a beta dagger a alpha dagger acting on the same determinant. Is it clear? Because in one case beta will come first then alpha will come so alpha will become the first column then the beta in this case it will be opposite. So of course they have to have a negative sign change so if I interchange then they will be identical. Is it clear to everybody? Now we know that every wave function can be written as a linear combination of determinants which I call slated determinants SDs which is just slated determinants or any wave function can be written as a combination of slated determinants. Now look at a alpha dagger a beta dagger acting on a general wave function and a beta dagger a alpha dagger acting on a general wave function. Since they are linear combination of determinants and every determinant follows the rule then I can say that a alpha dagger a beta dagger acting on any wave function is negative of a beta dagger a alpha dagger acting on any wave function. Is it clear? Because any wave function is a linear combination of determinants. So in general I can write an equation that a alpha dagger it is an operator equation is equal to minus a beta dagger a alpha dagger operator. Now it does not matter with whom it is acting this I can write only because the operator acting on any wave any arbitrary wave function is going to be negative. So I can write the operators are equal correct. So this gives me a very important rule of anti commutation a alpha dagger a beta dagger is equal to 0 this is called anti commutator. I hope all of you know this that means a b plus b a equal to 0. So a b minus b a is commutator a b plus b a is anti commutator. So this is anti commutator so anti commutator of a b is defined as a b plus b a and of course you know how commutator is written bracket. So please be careful you know if this is the sign that is used it means it is anti commutator. If this is just this kind of parenthesis it is commutator. Sometimes some of the text books to make sure it writes plus just to make sure that you do not forget that it is anti commutator. So but anyway either way it is okay. The important point to note is that the alpha dagger a beta dagger anti commutator is 0 and this gives us a very nice understanding. Let us now which I have actually not talked about so far what is beta in relation to alpha. Let us assume that the beta is equal to alpha which I have not discussed so far I have assumed that beta is different from alpha but now I can discuss it means that a alpha dagger plus a alpha dagger a alpha dagger is 0 which means the operator a alpha dagger a alpha dagger is 0. It is a null operator in the operator language it is a null operator. What does it mean? It does not give back warp it is a null operator which means if it acts on any function if it acts on any function any determinant not just back warp any function it will give you 0. So the operator is a null operator it is a 0 operator and that is understandable why it is becoming 0 now physically think. If you have if you create one alpha here you cannot create another alpha again because the determinant becomes 0 two columns are 0 a two columns are identical which actually you could have mentioned here itself that the beta has to be different from alpha there cannot be a determinant by this definition alpha alpha correct. So and it is good to know that I did not assume that is good to know that the anticomputation is automatically giving you this you cannot create alpha twice that is the meaning of creation operator is fine but you cannot create a human being twice God has created and now you cannot create a once more right. So you cannot create what is already created by God see if alpha exists a alpha dagger cannot act because if alpha exists that means whatever whatever is there that function I can write as a alpha dagger something correct. So obviously I cannot allow another alpha dagger to act. So that is that is a physical interpretation and that that is a correct interpretation because that could have been actually argued right here I mean you should have seen right here the beta cannot be equal to alpha because if I have an electron in spin orbital alpha I cannot have another electron in spin orbital alpha right and that is the Pauli exclusion principle. So this is actually a statement of Pauli exclusion principle right or this statement of the symmetry itself that the wave function is anti-symmetric I mean it is one and the same thing. We will come back to both discussion of the creation operator but let me now go to the other operator I told you there are two important operators the other operator you will be called annihilation operator it is just opposite to creation operator annihilation operator which will be now written as just a alpha without the dagger what would we do it will annihilate so exactly opposite it annihilates an electron from a spin orbital alpha. So if an electron exists it will annihilate and I call it a alpha and I will tell you why I call it a alpha which is basically like an adjoint of the creation operator because dagger is like an adjoint right so dagger dagger is without the dagger. So let us try to analyze this operator we will come to the anti-symmetry anti-computation of this operator as well but let me first analyze this. So let us assume that I have a state chi j that means one electron state where chi j is there and I define a i dagger chi j where i is not equal to j so I get a state which is now called i j note that you know I am not writing chi all the time but that is understandable when I am writing i j it means chi j chi j the superscript is perfectly okay. Now I want to understand what is the adjoint of this state okay so the adjoint of this state is of course going to be chi j a i dagger dagger so let me take this state whatever is the adjoint of this state as a i dagger dagger so let's say I get this chi i now you can see that for this to act here first it has to destroy annihilate because they are already chi i exists if it annihilates of course this will have nothing will be there and I will immediately get this as a zero state null state but without annihilation this cannot act any two electron state will not be able to act. So if you take any state let's say k state so I am just trying to say why is the adjoint of this state called annihilation the annihilation is already defined as a i so I am trying to say that this must first annihilate chi i otherwise it just cannot act this matrix element cannot act. So basically if you look at an i jth state from there I am annihilating an i and I am getting a jth state I am doing the adjoint chi j chi j that is equal to one I will get this only if it can annihilate i so if I have k a equal to i j and I take an annihilation a i k I will get a state j now you know that j j equal to one so if I take an adjoint of this you will see that this is k a i dagger k a i dagger k is the original state and then I have a j which is sorry which is a i k so what is happening is that to get back my chi j chi j I must have to annihilate i then i has to be again recreated and then if I take the matrix element it will become one because this is already k so k is your i j so if you see here i j here you destroy i so I get j then I create i I get again back i j and k k becomes one and this is already equal to one so to get this I have to get this equal to one so why is it equal to one because whenever I am I am constructing either a one particle or a two particle state they are all normalized determinants so this so all the all the electron state one electron two electron all the states are normalized so that is the interpretation why the adjoint of a creation operator is called annihilation operator otherwise it will not even act then we will see the further inside when you look at the anticommutation. So let us now look at a alpha a beta acting on any any state chi one chi two etc so note that if there is a beta it will actually destroy so one of the important property of an annihilation operator is that a alpha acting on vacuum gives you zero just as a alpha dagger acting on a vacuum gives to your alpha state if a alpha acts on a vacuum this is going to give a zero because you cannot annihilate unless it exists so this is a what is called a killer condition it is a condition that we are actually using so how beta act only if there is a beta here if there is a chi beta exists then only otherwise it will become zero so let us assume that there is a chi beta so there are two possibility beta exists in the determinant or it does not exist if it does not exist in the case later a alpha a beta acting on this determinant is anyway zero because I do not have to do anything beta will first act and beta will make zero alpha has no role because as soon as a beta acting on the determinant is zero alpha has no role right so this is already zero if on the other hand beta exists then of course it has to annihilate beta how does it annihilate the rule of annihilation exactly like the rule of creation what was the rule of creation it creates right here so the annihilation can be only done if beta exists right here otherwise it cannot be done so what do you do beta can exist anywhere so you just do a permutation bring beta here by permuting each permutation will require minus one okay so let us say have k permutation required so you get minus one to the power k okay for the permutation and then you bring chi beta here chi one here and so on and then you allow a beta to act now it can actually sweep it off so you get the determinant without the beta chi one etc chi n with a sign which is minus one to the power k so we are now discussing a case where beta exists okay now let me discuss the case of a alpha of course again the same thing will happen a alpha will only act if alpha exists alpha also exists right otherwise again it is zero so let us assume that alpha also exists otherwise they have zero a alpha beta is zero so let us assume that alpha also exists if it exists then what will happen it will again destroy the alpha but to bring the alpha I have to bring it again back here sorry it will act here so let us say there is a chi alpha I have to bring it back here with again some amount of yeah minus one some amount of minus one the question is otherwise you can order even permutation okay so it is let us say minus one to the power k plus l or whatever then what do I do is to allow a beta a alpha to act on the same determinant chi one chi two so somewhere there is a chi beta somewhere there is a chi alpha okay so now first alpha will go then beta will go now what will happen is that in either of these cases there will be one x less sign so only a sign problem and you can actually see that if you take a simple two particle problem much more easily that let us take you can take a simple determinant alpha beta so do a alpha beta so what do you first do you write this as a minus a alpha beta beta alpha okay and then of course you knock out a alpha alpha and then get a zero state minus zero state but then this is not important I mean this could be you know if there are three electrons something will survive but for two electron let us say I have a gamma state alpha beta gamma then gamma will survive here if you do the reverse a beta a alpha then first I have to bring alpha here okay and I get a minus sign and then I again add this I will get again the same minus gamma state yes so one of them has to cross so there will be one odd sign because obviously you know when you do a alpha and a beta and a beta and alpha there will be one negative sign extra okay so the point is that will actually lead to the same commutation as a alpha a beta is zero this can be also derived by the acknowledgement that the a alpha is an adjoint of a alpha dagger because if if a alpha is an adjoint of a alpha dagger then obviously the the operator a alpha dagger a beta dagger is already zero so obviously a alpha a beta will be also zero so that could be another way of proving this but the important point is again to prove that the a alpha beta anti-commutative is also zero and hence if I go back to beta equal to alpha exactly the same way I will have a relation where a alpha a alpha is a null operator right so I will have a alpha a alpha as a null operator what does it mean if I have destroyed alpha once from somewhere I cannot destroy it again just as you cannot create second time you can't annihilate second time right it's a very important physical meaning that I have already destroyed so I can't destroy unless I recreate to recreate there must be a alpha dagger before I allow a alpha to act I am not defined I am saying that I first write this as adjoint then I show that this has to be annihilation otherwise I don't get jj see if this is not annihilation I can't I can't get jj because from the k is ij I destroy i but you are already considering this aik is equal to j I am just saying that the ai dagger dagger is equal to ai that I am not assuming because if you are assume that the dagger is adjoint then there is nothing to prove I am showing that the right that is defined what I am now saying is that the annihilation operator is the adjoint of the creation operator a dagger is a creation operator what is it a dagger a dagger is no I didn't understand yeah they are adjoint of each other it is a necessary condition to that to be an annihilation operator see if I have to define an annihilation operator let's say I would have defined in a different manner it would have I had to prove that this is still adjoint of the creation operator so that is the reason we use a simple symbol where the dagger is good because it is anyway adjoint then we know that a dagger dagger is already a so the symbol symbol is first defined to you then the justification comes with a symbol that I have used this a dagger I have used this a because one is a adjoint of the other now of course you could argue why not this dagger that doesn't matter that is a historical fact that the creation has been put as dagger and the annihilation is a but the point that we are trying to say and I will come back to this discussion tomorrow and I will start there is that this annihilation creation operator whatever symbol you give they have to be adjoint of each other that is the point here and once we prove this then of course the anti-computation is quite trivial from the anti-computation of the creation itself once I prove so either way it is okay but or you can prove it separately by assuming that it is annihilation operator and then also prove it without even thinking whether it is adjoint or not. So this actually gives us and I will come back because there are some people have left today that it gives us two anti-computation relationship which I let me write it down this is equal to 0 and alphabeta equal to 0 okay and I will elaborate on this as I start tomorrow's class okay there is of course a third anti-computation which is even more interesting that is anti-computation between one creation and one annihilation. So what happens to this so out of the two operators one of them is creation one of them is annihilation and this is little bit more complicated this will actually complete all possible anti-computation between creation and annihilation operators and then we will see how to use them in the quantum chemistry but of course it is very clear that in constructing wave function they are extremely powerful and we do not have to bother about anti-symmetrizing that is the most important thing because by definition when I am constructing this is already anti-symmetric by definition the only point is that where in the determinant where the column comes there is a fixed rule so column will be the first column of creation for annihilation the column must exist here so in the case of annihilation you have to do this but even in the case of creation you may have to do it that is not a problem I have just taken two electron states so it is very easy okay to show this that is minus beta alpha and minus alpha beta okay but if I have a long determinant I want to push it back that is a different matter but the creation takes place the creation takes place right in the beginning first okay annihilation also takes place right there so if I have to annihilate first I have to bring this spin orbital of the first column and then I can annihilate so then you require several minus one products of several minus yeah direct developer but what I am saying that it is not developed for quantum chemistry it is developed as a mathematical tool basically it has been used in physics extensively physics extensively it is used and field theory field theory is probably one where the most extensive use that I can recall and quantum chemistry came much later but you will see that it simplifies many things and whatever we have written can be simply written for example Hartree-Fock so the way the Hartree-Fock would be written is just following that by a creation operator so Hartree-Fock wave function determinant chi 1 chi 2 to chi n would be simply a1 dagger a2 dagger blah blah blah a n dagger vacuum over it is very simple okay I do not care and I know exactly what the columns are in the determinant the way I have written right so the so the last column is a n then n minus one the first column is chi 1 first column is chi 1 because it is coming like this so if I interchange any two it become negative and that is actually reflected by the anti-computation anti-computation automatically reflect so I do not have to bother over anti-symmetry as long as I know the anti-symmetry the algebra of anti-symmetry I am going to use I have ensured the anti-symmetry so anti-symmetry need not be explicitly talked about so that is the first thing that I said wave function how are the wave functions written and then later on we will see how the operators can be written operators also have a very nice expression so so here what is important is again explicitly number of the what are the spin orbitals what are the spin orbitals that are there the electrons are really underline okay so these these are again the just like insulated determinant you have spin orbitals so these are also creation of a of a in a particular spin of it but but this is going to become a alpha a beta dagger yeah plus a beta dagger a alpha not so if I take adjoint this is this will actually become a alpha dagger a beta so that is a little different thing you can re-express so the adjoint of this will be a alpha dagger a beta so I have taken alpha see of course you know this could have been beta this could have been alpha the point is that there are two orbitals they may not be different I have not said that there are two spin orbitals one of them alpha one of them beta I will discuss what happens in alpha is not equal to beta when alpha is equal to beta so that that will discuss this is a little bit more complicated it will require a discussion but I will first finish this part again revise this part I think the creation part is very clear very easy so the annihilation part I will discuss okay and then we will see how to represent the operators once you know how to represent the operators we can actually start to do quantum chemistry because you require wave function and operators in second quantization okay again I will do the second quantization in a very simple manner I will not go into much details we will actually tell come back to MP2 and show how to draw diagrams and the diagrams are actually evolution of the second quantization rules the matrix elements this letter condoned rule that we saw talked about so we will actually go back to the diagrams and re-express MP2 also tell you how to MP3 can be could have been done okay.