 I hope you have practiced some of the second quantization that I gave for the wave function. Then we wrote the operator, please see the derivation of the anti-commutation rules, the definition of the annihilation operator in relation to the creation operator. Why is it adjoint? And of course, the important thing is a i a j dagger that is the only difficult part. We see the derivation of the anti-commutation. Look at the exercise, because we are not going to do lots of problems, but I can give from the Zabou and Osloon a few exercise for practice, again you should do that. So for example, a1 a2 dagger plus a2 dagger a1, this is something that we can just practice. You can actually use the anti-commutation rule on any determinant K is 0, so you can take determinants like K equal to chi1, chi2, chi1, chi3, any determinant we can take and so on, two electron determinant. Just check that this is 0, obviously it has to be 0 because of the anti-commutation and if you do let us say a1 a1 dagger plus a1 dagger a1 on any of this ket, it should be equal to the same ket. So that is very important to realize. Any ket, so the ket may not have chi1, so it can be chi2 chi3, so do not get confused, chi2 chi4 anything because that is very important because you may be confused then how a1 a1 dagger when K does not have chi1 is still why it is not 0. I hope you understand that if you have let us say chi2 chi3, I first create here and then I destroy, so I get back the same ket. First I create an n plus 1 electron state although chi2 chi1 was not included here, I first create a chi1, chi2 chi3 then I destroy, so I get back chi2 chi3 and of course this is 0. So one of them will survive depending on if that particular orbital is absent or present. So there is sometimes people get confused that if the determinant does not have these orbitals how can this act? You may think this is always going to be 0 because you have a a1 a1 dagger, if determinant does not contain a1 it will first create and then annihilate. So you get back the same determinant, this will not act of course. So remember this is in the anticommutation, so just practice this so that you will understand the anticommutation properly what this anticommutation means. So that we gave the form for the two operators, so one is theta1, one is theta2, operator form we are not going to derive it but let me write down the operator form. So one was the theta1 which is sum over, in first quantization it is sum over h of i, i equal to 1 to n. So this is the first quantized operator, in second quantization this theta1 will be written in terms of basis alone, so that is important to realize that you will now have only basis by which it will be written. So sum over all basis ij, iaj and you have ai dagger aj. So the standard definition is that one which is on the right side is created, one which is on the left side of the matrix element is annihilated, that is the standard definition. So please follow this, otherwise everything will not come, first quantized formula will not come. So in the same way if I write theta2 which is sum over 1 by rij, it is a two electron operator in first quantization, so these are all in first quantization. So what I am writing is second quantization, note that in this case ij's are now spin orbitals, they are not coordinates of the electrons, quite obviously because I am using this is ai dagger aj etc. So they are no longer coordinates, they are actually the spin orbital basis. So that is the beauty of the second quantization, that number of electrons as I told you before that this actually vanishes. So the theta2 then becomes here half of ijkl and then there is an integral which is the regular integral ijkl and then you have operators which create ij annihilate kl and the specific order is ai dagger aj dagger al ak, that is the specific order. So first k is annihilated, then l is annihilated, in the place of l j is created because that is what it means, in the place of l j is created, remember this is a direct notation, so this means i star 1, j star 2, 1 by r1 2, k1 l2, deta1 deta2. So obviously you can see that wherever the l was there that must be replaced by j, so that is the reason immediate annihilation, immediate creation and then wherever k was annihilated I will be created. So please do not write ai dagger aj dagger ak al, that is what I am just, there has to be a switch here, al ak because of the, so this is the second quantized form of theta1 theta2 in which these are all basis, these are all basis, exchange is not there. So second quantized notation has only the 1 by r1 2, so that is what we will show how the exchange term comes. So first thing to do is let us say theta1 in first quantization f h of 1 plus h of 2, so again just a simple 2 electron problem and you have a given determinant, so let us say my psi0 in first quantization is chi1 chi2, note again these 1 and 2 are spin orbital, these 1 and 2 are coordinates and again repeating, do not confuse with these 1 and 2, this could have been chi3 chi4 anything, so these are spin orbitals, these are coordinates, so this can be written as a1 dagger a2 dagger vac1 in second quantization. Now we know that this psi0 theta1 psi0, what is the result of this, yeah I have just given you, it is a 2 electron problem, do not have to sum, yes, so h11 plus h22, so this is something the result that we know, again 1 and 2 here are not coordinates of electron, they are spin orbitals because coordinate electron becomes now dummy that is going to be integrated. So now let us apply the second quantization to show this, so what do you have to do, you have to write this as a vacuum, first you write the left psi0 vector, so it is vacuum a2 a1, so this is a specific index remember a2 a1 is a specific index 1 and 2 which are actually present there and then you have to write theta1, so what is the expression of theta1 is sum over ij, so you have a sum over ij now dummy index, then you have a a i dagger aj all a ij, a i dagger aj and then again a1 dagger a2 dagger vacuum multiplied by the matrix element h ij, so h ij I define as i hj, so a1 a2 are occupied orbitals, they are very specific, ij of course contains these occupied orbitals but they contains everything else because it says sum over expansion of the basis set. Now practice this again how to manipulate these quantities, first of all you have to remember the whole idea of manipulation that you must try to bring the annihilation operator to the right because moment it acts on vacuum that will become 0 but to do that you can see that first you have to anti-commute this aj a1 dagger, bring aj here then anti-commute aj a2 dagger bring aj on the right, each of this anti-commutation will act for example what will be the anti-commutation of aj h i dagger, this will become delta j1 minus a1 dagger aj, so I bring aj to the right, now delta j1 would automatically ensure that now my summation j must be equal to 1 otherwise this term drops out, so when you now manipulate make sure that your j is equal to 1, otherwise this term will drop out it will become 0 and then only a1 dagger aj remains then you manipulate aj a2 dagger another delta condition will come and just show that eventually what you get is simply h11 plus h22, so basically eventually i must be equal to j and they must be either 1 or 2 with the delta aj condition, so this requires a little but I just thought I will give you some practice problem so that you can do this, the other way sometimes simplification is that do not expand psi not, keep psi not as it is only expand theta 1 in terms of the basis and then let the orbitals act directly on psi not, so you can get an expression like some psi not a a dagger a b psi not you can anti-commute and you will get this as delta a b psi 0 psi 0 minus psi 0 a b a dagger psi 0 and then since a dagger is a occupied orbital it will act on this psi not to give you 0, so if you write this in terms of occupied orbital but I think right now I will introduce this particular vacuum which will make it simpler you can right now try with this full expansion for the theta 1 at least, theta 2 I will not ask you to do but he had a question how the anti-symmetric integral comes that is because when I start again anti-commutating two terms will come, so that is going to give you the anti-symmetric integral finally, so this is only for you to test that this is self consistent your first quantization because we are not proving it ab initio we are giving you the expression and then you can check that they are self consistent at least for one term you should be able to do this. You can also practice similarly problems like yes the following problems on singly excited determinant note that many of these problems are given in the text in the Zaboslu so please read this. So for example if you have a psi a r theta 1 psi 0 so that it is equal to r aj exactly in the same manner you can show that it is equal to r aj many of these you can actually show by keeping psi 0 as it is that is without even expanding psi 0 so this particular problem is not a two electron problem it can be anything so you should be able to write how will you write psi a r as a a a dagger a on psi 0 right can I write it otherwise around why why what is the difference because the other one was a yes what will happen if I create first and then annihilate I know but what happens to the determinant think in terms of determinant if I first create if I first create the r would come here right and then the rest will remain then I annihilate a so chi a is somewhere here chi a will be somewhere here so how do I annihilate I have to first bring it here so I have to make this minus a chi a then chi r will go here and the rest will there so then you get minus the same determinant with chi r so you can see I started with the determinant where chi a was here I am getting chi r but with this minus sign right so I am just trying to show why it is not I mean your point is right it is also expression of anti-computation actually okay so the but on the other hand if you do a r dagger a okay if you do a r dagger a then you first bring it here so there is a minus sign bring it to chi 1 there is a minus sign then then create a r then you have to push it back there again so that is why these two minus signs will cancel so I will get so one way to write this is a r dagger a a acting on chi 0 okay so please make sure these are all small small problems please make sure that you are able to handle and this is actually clear from the anti-computation that both cannot give the same wave function because they anti-computed okay so if you start from chi not without expanding without expanding if you start from chi not remember a is an occupied orbital r is a virtual orbital that is all you need to remember so if you finally get a a dagger acting on chi 0 that will become 0 note that when you are starting from a general set of orbitals what is our strategy when you are expanding in terms of vacuum our strategy is to get an annihilation operator to the right the only difference here is that instead of vacuum if you keep this chi 0 your strategy will be to bring a a dagger to the right a a dagger occupation because then automatically this is 0 I hope that is clear that is a killer condition okay so you can start with chi 0 instead of expanding from vacuum because if you expand from vacuum you have to unnecessarily write lots of terms like for example this 2 electron determinant you have to write a 2 a 1 here a 1 dagger a 2 dagger here so while you first first try this then eventually start to think in terms of chi 0 directly then how do I write this expression in terms of chi 0 directly so this expression in second quantization then will become chi 0 a dagger a r I will not write the whole vacuum whatever is there because it is too complicated then I write theta 1 which is of course a i dagger a j okay and then chi 0 and you have a h i j sum over all i j now you have to be little bit careful a is an occupied orbital you have to remember r is a virtual orbital i n j can be any index i n j however can be any index because that is coming from the expression of theta 1 expression of theta 1 which is any index now you must start to manipulate so our idea would be to bring a a dagger here if you can bring a a dagger to the right that is 0 okay so start try to do this and reach the result h r a so basically r dagger will come here and a a dagger will come here a r a a dagger will come here so obviously when you are trying to quantize this for example this itself you will have a delta r i so I must be equal to r otherwise that part is 0 and you will have a a i dagger a r and so on then a or you can start with here bring a a dagger here a a dagger i dagger and then eventually it is okay so start doing the manipulation so these are some of the algebraic manipulation that you should be able to show I have simplified this because I am not expanding psi 0 so if you do not expand psi 0 your killer condition becomes this a a dagger psi 0 equal to 0 where a is occupied orbital quite clearly this is much simpler simpler simply because I do not have to expand this here it was okay you can do this for 2 electron but if it is n electron then there will be n sets of orbitals here n sets of orbital here so this becomes a horrendous task to keep manipulating so that eventually all the annihilation operators one by one come to the right here all you have to do is to bring all the creation of the occupied orbitals to the right okay and of course if a r also comes to the right it is okay because that is also 0 so either you have to bring a r to the right or a dagger to the right eventually okay so start doing this manipulation because each of them is 0 a dagger a r psi 0 is also 0 if r is virtual orbital so this is the 2 killer condition that we will use and this is simpler because I am not expanding psi 0 okay please practice this if you have problem on the final day we will discuss some problems okay on Thursday because I want to finish the course because now I am going to come to an important topic.