Yes, thank you, Nicola for introducing me, what's that, now it's okay. So, yes, I'm probably famous for developing one of the main solid state codes in the field of density functional theory, despite this, these talks I will present here are not at all about density functional theory, they really concentrate on methods beyond density functional theory, and in particular, on methods that are suitable for weakly correlated systems. I've looked at the program, you've heard a lot about strongly correlated systems, as far as I can tell, weakly correlated systems have not been particularly on the forefront. And actually the first talk is the most difficult talk, the other ones are kind of more easy going, they will just give you a kind of easy introduction, but the first talk is a tough talk, and I think part of you, maybe half of you probably know all things I present here, but so then this is kind of a recapture of what you might know, and I guess the other half might not know about it, and I can only tell you that I can give you only a teaser that hopefully will give an incentive for you to look into this series, into this really very basic series, and actually explore them and read more carefully about them. I think these are really the very basics, essentially what I'm talking about in the first talk is quantum field theory, quantum field methods, and many-bodied perturbation theory. So it's really important here, this word many-bodied perturbation theory implies that we actually work in the weakly correlated regime, so we don't deal with strongly correlated electrons. So first and outline, in the solid state community, first principle simulations are usually done with density function theory, and I will try to explain why not only density function theory should be used, or why we have to look into methods that go beyond density function theory, and essentially my favorite, or what I have been doing the last 10 years is really many-bodied perturbation theory, and I will try to give you essentially an intro to this many-bodied perturbation theory, and this is really, I think if you want to do model Hamiltonian strong correlation, you probably have learned about this, but I think it's really important to understand the basics before moving on the strong correlation, so there's never, ever any strong correlation here. So what I will talk about is essentially the particle whole picture, so I will talk about second quantization, I'll give you an intro to that. I will then talk about the Gelman-Low theorem, the Wick theorem, and finally about the link cluster theorem, this is the very basic, and I will try to be pedagogical, but it's the first time to do this, at least in one and a half hour, and I know it's almost impossible to do it in one and a half hour, so again, it can only give you a kind of a gear. I will then come to Goldstone diagrams and introduce those as a way to actually present the many-bodied perturbations here, and you will see in the later talks that I will almost exclusively use Goldstone diagrams to talk about what we do in solid states nowadays. Very briefly also about Feynman diagrams, but I might skip that. If you find these subjects interesting, there are a lot of quite good books to read. They're usually in quantum chemistry. Sabo Ostens is the famous book in quantum chemistry that you might want to read. Shavit and Bartlett, many body methods in chemistry and physics, and this is a more quantum field theoretically implementation, quantum field theory for the gifted, and I must say I haven't read those, these are the books that my students read. Also I must say that these slides have been mostly prepared by Felix Hummer who just finishes his PhD work. I simply had no time to do it myself. So let's go to the very basics. So I'm interested in first principle simulations. I want to do simulations that don't involve any model Hamiltonians, but that really set out from the many-electron Schroding equation, and this here is the many-electron Schroding equation written down. So I will just briefly summarize it because I'm pretty sure you have seen this before. We have here a sum i that goes over all i, over all electrons. Yes, i is the sum over all electrons from 1 to n. This is the number of electrons we have in our system. This is the kinetic energy operator, the Laplacian operating on the coordinate r, i. This second term, well this is actually kinetic energy. The second term describes the interaction between the nuclei and the electrons, and again it involves the sum over all electronic coordinates. And now the nuclear electron potential is to be evaluated at the position of each electron. And the final term here, and this is the most relevant one because it's the culprit, it causes all the troubles we have, is the electrostatic interactions between electron i and electron j. The sum is usually restricted to all j indices being larger than the index i, and the index i runs again from 1 to the number of electrons. So if you solve this equation you have to determine this quantity here. This is the many electron wave function, yeah? And the number of coordinates you have here is proportional to the number of electrons, and this should be a capital N to agree with the slide above. And essentially it tells you that this is going to be really a hard problem. Actually the solution of the many electron choice integration is an exponentially hard problem, so the compute time that you need to calculate that quantity scales exponentially with the particle number. Okay, that's already obvious if you just inspect these many electron wave functions. Imagine that you have one electron, a single electron, and you create a grid to represent the density or the wave function of this electron. So you introduce it here, it's a two-dimensional grid, you discretize essentially the electrons in space. And we introduce a three-dimensional grid with 10 by 10 by 10 grid points in each direction. And obviously this quantity will involve something like 10 by 10 by 10, yeah, something like 16 kilobyte. 1000 words, you multiply about, yeah, why 16 here? Well, just take it about 16 kilobyte. Why 16 kilobyte? What's that? Probably more like 4 or 8 kilobyte. Well, it's complex. If you use complex data in double precision, it's indeed 16 kilobyte. So this can be done fairly easily, that's almost trivial. But let's imagine we have three electrons, then we have one, two, three, coordinate, R1, R2, R3, and each one is xyz, so x1, x2, x3, y1, y2, 1, y3, and so on, and so on. And so that means you have a nine-dimensional grid to represent the quantity. And imagine, again, we have complex double precision, then we need something like 16 kilobytes. That's still manageable, so that's still what we can do on quite good computers. But let's move to something like five electrons, then we need a 15-dimensional grid, and that's already 16,000 terabytes. So that's not what you can store in even on a more than high performance computer. So it means to store this quantity with many electron-ray function is precisely impossible. So Walter Cohen had a nice idea to slim the problem down, and that's density functional theory that's quite routinely used in solid state physics. Essentially the idea is that for five electrons you don't use one 15-dimensional grid, but instead you try to represent these five electrons by one electron wave functions by so-called orbiters. So you have only five sets, and each of these sets is just three-dimensional. And so the first one, the lowest one, represents the lowest electron, the second orbital represents the second electron, the fourth orbital represents the fourth electron, and so on. And essentially each of these sets is on the three-dimension. That makes it of course much easier to work with the equation than with the original equation. It's no longer exponential, it's essentially scaling, usually cubically, the system size. Just a brief history of time, he was actually born in Vienna, but he left Austria when the Nazis moved to Austria, and he was lucky to be saved, and he was able, he was actually rescued to Canada by so-called Kindertransport, and actually he declined and said he's not in Austria and he's really just been educated in the US and so he's in America. He got, as you might know, the Nobel Prize in 1998 for this kind of theory. So this is the equation that you then saw, if it's a one-electron part, one-electron equation. So this guy here is essentially only dependent on a single coordinate, no longer on these many coordinates we had before. This often quantity, but it only depends on a single electronic coordinate, and that makes it much more easy. Complexity is essentially n cubed. So I want to give you first a few why density functional theory is in principle a great theory, in practice it's hard to use. So what you do is actually, or the problem is density functional theory is that you have something that is called exchange correlation potential, and this exchange correlation potential captures all the many electron effects that you have in your material. So here you have again a kinetic energy term that now operates only on the single coordinate. Here's the nuclear electron potential, and that you describe exactly. There's no trouble with that. You have a half a potential that describes the electrostatic interactions between the electrons, and finally, one term that embeds all the complicated many electron interactions. But that necessarily is an approximation, and the approximation most people use is some variant of the local density approximation. So you're discretizing space, and then you look at the electron density at one particular grid point. So you look at the electron density at this grid point, and you check a table what would be the correlation energy, the exchange correlation energy for the homogeneous electron gas at this density. So all you need is a table that maps the local density to the local exchange correlation energy density. That's all you need as an input. That's a pretty awkward approximation, and a very odd one indeed. So you map actually local properties onto the local exchange correlation energy, and you do this via the homogeneous electron gas. Nowadays, functionals not only depend on the density, but also on the gradient of the density S, or the kinetic energy density. But think about this kind of approximation. What does it really imply? Imagine that you have two electrons, one electron here on the left side of the nucleus, then the other electron is on the other side of the nucleus. You know that that's kind of what electrons do. They want to avoid each other. That's the effect of Pauli repulsion. It's also the effect of Coulomb repulsion, right? Electrons repel each other because of the Coulomb repulsion. Because they are fermions, they will never be in the same orbital. And if one electron is to the left, the other one is to the right. And they're kind of moving in a concerted fashion. So if this electron moves over, the other electron moves over there. But how can you capture something like that with the local density approximation where you look only at the density at one space point? That's kind of impossible, right? But it's approximately possible. And this kind of correlated, I always call it a correlated tense, or entanglement as people call it in quantum information series. It's the very hard of correlation. So electrons are correlated when one electron is to the left. The other one will try to avoid this region and move over to the right. More precisely, it's more likely to find it on the other side than to find it at the same place. This is intrinsically non-local and also, in principle, one could do it with DFT. It's very difficult to obtain this information from the density alone. Okay, this sounds esoteric, but let's give you some examples where similar things do happen. Fundavals interactions. We have two nuclei now, one nucleus here and another nucleus here, and let's say we have two electrons. So one electron always sits around this nucleus and the other electron always sits around that nucleus. Now what happens is that these two electrons again move in a concerted fashion. So if this electron is on top here, so it is more likely to be found up here, above the nucleus, then the other electron interacting will be usually preferentially also above here, above the nucleus, and they will move in a concerted fashion. So if this guy moves over, so if it kind of moves over to the other side of the nucleus here, then this electron will, in a concerted and tangent fashion, also move down. That's what you know. It's actually nothing but van der Waals interaction. So this is exactly van der Waals interaction, that these two electrons really move in this concerted fashion around the nucleus. And that lowers the energy because you have a dipole here and another dipole there, and in some, yeah, kind of a positive dipole interaction, this lowers the energy. It's another case that is really difficult to describe with density function theory. There are a lot of fixes now where you inspect the density of two points in space, but anyway, in a ballpark, this is a property that is pretty much difficult to describe this density function theory. I want to be a little bit more precise. What we can also look at is the wave function as a function of the difference between two coordinates. We have this many electron wave function of two coordinates, and now we look at the wave function as a function of the difference between two coordinates. So these are two electronic coordinates, r1 minus r2, and we look at the wave function as a function of the difference of two coordinates. And there are two important features in this wave function. One is this cast pier. Well, let's look at electrons with equal spin, then you know the wave function must be anti-symmetric. If the electrons have the same spin, two electrons with the same spin, the wave function is anti-symmetric, so there's zero probability to find second electron in this place. But there's another correlation effect, and that's essentially a combination of Coulomb and Kinetic, and that's called the cast. So even if electrons have a different spin, if they have a different spin, you don't need to consider anti-symmetry in space, because you have a spin coordinate. But even then there is some peculiar property at r is equal zero, and that's called cast. It kind of expels the electrons essentially by a combination of Kinetic and Coulomb repulsion. Here, close to the nucleus. Or sorry, not close to the nucleus, but close to the coalescence point where r1 becomes r2. It's called cast. It's very famous in quantum chemistry. Another thing is that if one electron sits here, there is a greater likelihood to find the other electron at certain regions in space, and less likelihood to find electrons at another reason, at another state point. This is exactly what I told you about here. If one electron sits here, it's more likely to find that you find one electron here. This is this area here, and it's less likely to find an electron over there. It's very likely to find it here. This is this place. Less likely to find it there. So all these effects you would like to describe somehow, and that's really, really tough to essentially function as zero. So all this is clearly very non-local, you see. All these effects, these kind of correlation effect, these two handles, these short range effects pretty well, but the large range, large distance effects are almost impossible to treat. So let's do something else. And this was my motivation, and now I will try to derive a more rigorous framework how to deal with electronic correlation. It's not density function as zero. It's essentially a many body pair to patient theory, and I will try to step through the basic series and later in the other talks I will come back to this and use the kind of algebra I've developed today, the algebra I've developed today, and apply it actually to different levels of theory, how to actually use it in practice. So we want to solve, we want to go back to our original equation. We don't want to use this D, we don't want to use this equation because it has its limitation. We want to use this equation here, which is really the exact many electron Schrodinger equation. So no DFT from now on, and here's again our many electron Schrodinger equation. I want to do materials, so I don't deal with model Hamiltonians. I want to do what is exactly written down by Schrodinger and Heisenberg. So I want to use essentially this exact equation and apply it to materials. So this is the kinetic energy operator again, sum of all electrons and that not dropped in this electron index. So this sum runs from i to the number of electrons. The Laplacian operates on each coordinate. This is again the nuclear electron interaction. This here is now the position coordinate. Sorry for this mix-up, so there should be x and r should be identical, so this should be also an x here. Great. So I've worked carefully through all sides, but now that I've presented, I still find mistakes. So this should be an x here. This here again is the description of the Coulomb repulsion between two electrons and that's the interaction between the nuclear and the electron. So the simplest way to deal with this exact equation is to use an ansatz for the main electron wave function, and the simplest ansatz you can do is the Hartley-Fock approximation. So in the Hartley-Fock approximation, you make an approximation for the main electron wave function, which is the single-slator determinant. So you insert, you plug into this here in ansatz for this wave function, and that's the single-slator determinant, and the single-slator determinant is essentially, yeah, it's a product ansatz, it's a very simplified form for the wave function. So this later determinant is actually written here, this is an anti-symmetrization operator, and we have now only index one to the number of electrons orbital, and each orbital depends on the single coordinate. So your simple ansatz is essentially this, and you plug this simple ansatz into this main electron Schrodinger equation and then solve it, and it will not go through the algebra here, this yields the standard Hartley-Fock approximation, which you should have heard before, but I will give you an idea what you have to do. So this equation is the equation that pops up, so the kinetic energy operator, and now this guy it can operate only on a single electron. This guy is an orbital, so it's a one electron wave function, it doesn't have all the main coordinates, it has a single coordinate only. This is the nuclear electron interaction, and this here is now an effective potential, and that's opposed to dT. This potential is in Hartley-Fock's theory a little bit different, and essentially I will use diagrams to represent this interaction. I will later come back to these diagrams and we'll try to explain how they come about, but the first term that comes up is actually some of our old particles J, from J is equal to one to the number of electrons, and if you look at this, this is just phi j, no this is psi right? Help me, psi right? I'm really bad with this, so psi j complex conjugated at the position x, psi j at the composition x prime, this is obviously charge density, and the charge density we can represent using goldstone diagrams by a circle, and I will explain a little bit more in detail later how this comes about. So this means here actually we have an outgoing line which will present this orbital here, and we have an incoming line that represents this orbital, and this is nothing but the charge density at the position x prime, then this here is the Coulomb operator indicated by the blue with the urine, and the Coulomb operator creates out of this density times r minus r prime, it creates the Coulomb potential, so this is exactly the electrostatic potential created by the electron, so this guy here is this here, this here is the density of the closed circle, and it creates the Coulomb potential, and then this Coulomb potential acts on another sum over states on the density created by the other electrons, so this is the operation of the effective potential onto and over to at the position x. This is the first term, this is exactly the Hartree interaction, you will see or try to recall this diagram if you have never seen this, this is essentially the Hartree interaction, maybe I will chop it down just that we don't forget this, because we will come across this today and next time, it's essentially the Hartree interaction. Now the second term and I will come again to this term and repeat it actually in the next lectures, is the exchange interaction, so here we have now again this orbital, but now at the position x prime, as well as the position x, and here we have the same quantity represented by this green error, the difference before is now that we don't close it at this point, but we leave it open and contracted differently in space, essentially we connect this position to this orbital and that position here to the other orbital, so we have this kind of diagram here and here's again the Coulomb interaction r minus r nine. This here is the exchange interaction and as you know this has no classical analog on in electrostatic, so this is essentially the electrostatic interaction or Hartree interaction, this is the exchange interaction that is related to the amplitude of the man-electron wave function, now it works. Hartree-Fock is a good theory, it's quite simple theory, actually this theory is variational because we've made a simple answer for the man- electron wave function and because it's variational, you kind of variational optimize it, you get an upper bound for the true, for the true energy of the man-electron system. We have made this very simple answer for the man-electron wave function, so it's for sure the case that this here is our Hartree-Fock energy and it's necessarily about the real ground state energy of the true man-electron system. In DFT we have a similar effective potential, actually the potential also contains the Hartree term but this term here is usually not present, the second term the exchange term and it's local and replaced by function of the density only. This quantity here, the difference between the Hartree-Fock energy and the true ground state energy will be called correlation energy. The idea now in many body perturbation theories that we set out from this simple solution, from the Hartree-Fock solution, and go from the Hartree-Fock solution to the true man-electron solution. So, we start from the original Hartree-Fock orbitals and try to actually describe what happens if we switch on the man-electron Hamiltonian. So, we try to see what happens if we start from the Hartree-Fock Hamiltonian and make perturbation theory to the true man-electron Hamiltonian. Essentially, we switch from one guide to the other, the abatity is slowly by slowly switching it on and we will try to actually determine what is the magnitude of the correlation energy if we slowly switch from the Hartree-Fock Hamiltonian to the true man-electron Hamiltonian. Now, this method, you must think clearly, this method cannot work always either, right? This method will only work if there is some connection between the Hartree-Fock ground state and the true man-electron ground state. So, otherwise perturbation theory might fail and diverge or yield intrinsically results. So, there must be some way to switch from the Hartree-Fock to the true man-electron system or these two must be in some way related. We will come back to this later. This, by the way, is usually the case if the system is weakly correlated. So, these talks, all the talks I have are really constrained to systems that are nowadays called intermediately correlated. This is not about strongly correlated systems, so this gas of perturbation series will always fail if you have a strongly correlated man-electron system. So, then you have to apply methods like dynamically mean-filled theory or DMRG density matrix renormalization group theory. In quantum chemistry, you then have to use multi-reference methods and essentially the difference is in quantum chemistry that you don't start from a single Hartree-Fock determinant from, but from multiple Hartree-Fock determinants and then try to do something like perturbation theory, which turns out to be really, really difficult to correct. So, how do quantum chemists approach the problem? I told you before, what you do first is you calculate the Hartree-Fock ground state and later we'll come back to this. You can also use the Konchem ground state for the starting point of your perturbation theory. So, you calculate the Hartree-Fock ground state and this diagram is very easy to understand. So, essentially what you have here are one, two, three, four, five, six, seven, eight states with eight electrons. So, you occupy, you put your electrons in the ground state into these lowest line orbitals of the Hartree-Fock Hamiltonian and the residual orbitals, the other ones are essentially empty to start with. Eight electrons, these are the Hartree-Fock orbitals, the lowest Hartree-Fock orbitals. In many cases if you do two systems you have a band gap between this here and that state, so there's truly most cases a band gap. So, if you do a molecule but even if you do a solid there is a true band gap and now I can say a little bit more about this perturbation theory. Actually this perturbation theory usually works if there is a band gap setting out from the Hartree-Fock Hamiltonian moving to the two-man electron Hamiltonian if this band prevails throughout the literature and then usually the many perturbation theory works. So, in many perturbation theory what you now do is you actually expand your solution into the ground state Hartree-Fock determinant. I call it ground state, it's not the two-man electron ground state, it's the Hartree-Fock determinant, this is your society. Now what's this symbol? Phi, thank you, phi zero. Actually my slides are not entirely consistent so I switch from phi to psi in several places, unfortunately. I realized that too late and since Felix Homer did many of the slides he used an inconsistent notation. So, this is the, well, I had different notation in him so and it was too late to change it here and he didn't want to to be honest okay he thought just you need to be more flexible, okay. So, this is our phi zero, this is our this is our single-slate determinant where we have put the electrons into these occupied orbitals and these are unoccupied. Now in quantum chemistry but also in quantum field theory what you do is you expand the true solution, the true many electron wave function into single excitations and you probably have seen this maybe this slide maybe also in the talk of Ali Alavi this is probably my slide originally. So, you expand it into single excitations and what does it mean? It means that you remove an electron from this orbital and place the electron into the orbital into a previously unoccupied orbital. So, you remove an electron kick it out here and move it up into this previously unoccupied orbital. This is the single excitation here is a double excitation where you actually take out two electrons from the ground state and put it into these orbitals A and B. In perturbations theory what you then need to do is you just need to determine these coefficients here in front of these single excited states double excited states triple excited states and so on and so on. Here we have eight electrons so you can have actually up to eight excitations from the ground state orbitals into previously unoccupied orbitals that's the maximum you can have. Actually, does this help us? This expansion of the many electron wave function in principle is as complicated as our original problem we haven't gained a lot we have just rephrased the problem a little bit. Does this still work? Yeah. So, we have just rephrased the problem a little bit. What we have done is we now start on the how to talk determinant and apply a vibration theory essentially to determine those coefficients. Just a little bit of a rephrasing because the point here is how many coefficients will we need to use when it turns out the number of coefficients is gigantic actually the number of coefficients if we have 32 of these orbitals and if we have only eight electrons even then it's almost unsolvable because the number of coefficients we will have is 32 over eight that's 10 to the 26. Didn't help us at all previously we had like 10,000 terabytes this is even larger in practice so we haven't really gained a lot well we are now with eight electrons and no longer with five electrons but this is we are still having this exponential or rather combinatorial wall that we had already originally but the trick now is maybe we can determine these coefficients using some clever perturbation theory and truncate the perturbation theory at some time at order if we can achieve that and have a computationally feasible scheme I don't care by the way about schemes that are not implementable on a computer the scheme that I like needs to be implementable on a computer and needs to run reasonably fast so that we can do true modeling of materials so if we can determine those coefficients kind of perturbation a little bit kind of done that might be feasible to implement on a computer program okay how does one do this perturbation theory there are infinite many papers on this probably and in the 30s 50s well between the 30s and the 50s there have been many different ways to phrase this perturbation theory it's always about the same it's always about how you determine those coefficients yeah this is essentially the point but there are many different ways to derive the equations and from my point of view the most elegant one is via second quantization gelman-Low theorem Wicks theorem and the link cluster theorem these are essentially the four theorems that you need for a very compact way to write down this perturbation theory again my from my point of view these are the very basics if you have got those basics you can move on to strong correlation but first you need to completely be firm with what you do in these four theorems again I mean this will be very tough because I have only one hour left maybe I will take 10 minutes more yeah I will try so let's talk about second quantization and I've heard you have already had some talks about it some introductions to it I anyway want to give for those few that have not heard about second quantization give you an idea what it does second quantization is not something new it's just a clever way to introduce an algebra to solve the many electron-sharing of equation so it's nothing there's nothing new in second quantization it's just a clever replacement of the many-body wave function of the many-body Schrodinger equation so essentially it's just a clever algebra to solve this equation here yeah I don't know why this can anyone explain to me who is wiser than I why is it called second quantization anyone has a clue yes yes that kind of makes sense but anyway it's still the same quantization I thought actually it's a quantization of fields yeah okay I will not answer the question now anyway it's not relevant it's nothing new so we have the many electron-sharing equation all we want to do is to solve it and that's why we introduce this algebra and the algebra is nice because it captures all the intricacies all the complications of fermions in it so in bodies all the complications of fermionic properties of fermionic guys so I will introduce this in a quite I try to introduce it in a quite similar simplistic way okay so we we actually introduce in this fox space so in this particular space we introduce operators and we introduce the operator c plus k and this operator essentially creates an electron in one of those orbitals so it essentially creates an operator in the healthy fox space and puts an electron in one of these orbitals for the time being I don't make a difference between the originally unoccupied orbitals and the originally occupied orbitals I just write k as a kind of general index yeah so this can be either occupied or unoccupied so this creation operator if you apply this creation operator to the so-called vacuum state this is a state that doesn't have any electrons yet you create an orbital in the state k this here is essentially the slater determinant so this sign here is essentially calculating the determinant so this operator c dagger k creates an electron in the orbital in the singular electron orbital k now imagine that we create a second orbital or create a second orbital electron in an orbital j so it's c dagger j in this case we create we add actually to the slater determinant to the very left another orbital at the position x1 and x2 so we add to the left and down here essentially a second state so this is one electron state one electron slater determinant this is a two electron slater determinant and we can do this repeatedly so we can add another electron c i plus and this will create another column in our slater determinant so another rule we have is acting so this is essentially a rule how to apply the creation operator it's a strict rule it tells you what to do in terms of the slater determinant when you add an electron another rule is that when you act twice with the creation operator on the same orbital so c dagger j c dagger j so if you create two electrons in one particular state it will yield zero makes sense because these are fermions so we can't have two electrons in one orbital so c dagger j c dagger j that means going back here we create an orbital in A and then we try to create another electron in this and this is not allowed by fermions rule so this is just a mathematical rule if we apply this creation operator twice onto the ground state orbital we get always zero this takes care that we do not generate two particles in one orbital another thing and that follows from the algebra of our determinants and slater determinants interchanging the order changes the sign if you change j move j here and i you move the i over there and write down your slater determinant you will see that it has exactly the opposite sign than the original slater determinant you can see this already up here if you exchange j and k so if you move k over here and j over there and write down your slater determinant and you only need rules for determinants you see it will change sign what does that mean that means that actually our operators have to observe this rule c plus j c i plus so this creates an orbital in j and this creates an orbital in i is the same as first creating the orbital in i and then the orbital in j so this follows again simply from the rules of slater determinants and there's nothing really particular sophisticated about it anyway we already see here that we are dealing for all these people who know this very well many bodies here we see immediately that this actually impacts essentially the typical rules for fermions so how do we introduce the annihilation operator the annihilation operator removes the leftmost state from the slater determinant so if you actually use this operator c j it actually takes an electron out from the orbital j and just removes it let's imagine you want to apply this to this slater determinant and then before you're allowed to remove the electron you have to swap the columns here and swapping the columns changes the sign so you bring that to the front and this slater determinant has has the opposite sign of this guy here and this guy then destroys this column and collects this to a two by two determinant so if required the columns need to be brought to the leftmost side and the sign needs to be changed accordingly so this follows from the rule that swapping two columns in this slater determinant changes this sign we just had this on the previous slide so again interchanging the order of the annihilation operators changes the sign so and that allows us if you take the two previous slides and walk through them we come up with this well known algebra for fermions well known algebra for creation and annihilation operators so this here c dagger q anti commutes with c dagger p in other words c dagger p c dagger q plus anti commutation operator c dagger q c dagger p is equal to zero this operation also this also applies to the to the annihilation operators and the most important rule is this one if you first destroy an electron and then create an electron in the orbit of p it observes this particular algebra here and that's very well known all this is nothing but the usual algebra that you have in many body language for the creation and annihilation operators of a fermion electronic system again all that follows from the basic rules of slater determinants and from my original introduction of the slater determinants how they work in the slater determinant space so these rules, these very compact rules are strictly following from everything I had on my previous slides you can do it if you have not done it you should do it yourself at home and figure out that this is indeed true so in classical many body physics your ground state is usually described by the vacuum ground state so in quantum field theory for classical fields you usually act with all your operators on the vacuum state and in many body perturbation theory for solids there is one small difference that we set out from the original artifact determinant so our ground state is not the vacuum ground state with zero electrons but our ground state originally has already n electrons in the lowest occupied orbitals okay, yes so our ground state we replace the vacuum ground state that you have in usual many body perturbation theory or it's often written like that or like that we replace that by the Hartree-Fock determinant that makes actually things slightly more difficult up to this point I've talked about these creation operators c dagger a and c dagger i and now we create a new set or we introduce a new set of creation and annihilation operators that are closely linked to these original operators but they are slightly different, subtly different so for the unoccupied states for the originally unoccupied states I will from now use on the index a and I will write this again on the blackboard because it's an important thing always to keep in your mind and you might have seen this in the talks of Ali Alavi for instance a means always an originally unoccupied state this is a convention from quantum chemistry and i let's be more precise a b c i j k is an originally occupied state now for the unoccupied states for those states that were originally unoccupied we introduce or we leave the creation and annihilation operators as they were originally so we define new operators a and a dagger that are exactly equivalent to original operators c dagger a and c a but for the previously occupied state we swap them we interchange the creation and annihilation operators so we actually introduce i as an unrelation operator so we define i as c i dagger and we define i dagger as c i now this is a little bit strange and you will oops finished yeah this is usually what happens to me so I just hit the wrong key go to the bottom of the presentation you see I'm already getting tired so let's look at this why do we do that so for the previously unoccupied state our creation operator remains as it is so a dagger means create an additional electron in the previously unoccupied state but i dagger i dagger actually is the same as c i so it actually creates a hole so it takes out an electron from the system so c i that's the annihilation operator and i dagger then is nothing but a whole creation operator so i dagger creates a hole c i creates a hole right so it's the annihilation operator and i dagger is actually now identified to be the whole creation operator and that makes a lot of sense and from now on we have to be extremely careful and on purpose I use two different operators I use either c i plus which creates an electron in the previous local right state or i plus and i plus creates a hole and that makes a lot of sense because we set out from a later determinant where we have occupied certain number of orbitals we have setting out from the heart defect determinant where these states are originally unoccupied so our creation operators will create a hole in this subspace that's a sensible definition so i plus creates essentially a hole a plus creates essentially a particle so let me write this down here so i plus creates a hole in the occupied manifold and it's therefore equal to c i and a plus creates an electron in the unoccupied space so this is c a plus this is a useful convention and you should apply it for quantum chemistry you should also apply it if you deal with solid state systems so this slide really summarizes everything we had up to this point it kind yeah well we deal I mean in classical quantum many body theory you deal with the vacuum state so you deal with fluctuations in the vacuum state here we have a certain number of electrons let's say eight electrons in our systems and so it's useful more useful to deal with the system where we occupy initially so this is our ground state the orbitals the lowest eight orbitals by electrons and leave the other ones unoccupied that's our reference concept yes it makes the algebra far more compact and it's also I mean the difference again the difference is that in classical quantum field theory for fields you start from a vacuum ground state here our ground state our original ground state is the ground state of an n electron system the heart defect ground state of an electron system so our perturbation theory and that's what that we written down here sets out from the artificial ground state where you have eight electrons yeah originally already in your system this is different from quantum field theory classical field theory for fields yeah where you have really the vacuum state and want to calculate the fluctuation here you have really eight electrons to set out so your ground state your I shouldn't say ground state your zero order ground state okay is a state where you have eight electrons that occupy the lowest orbital and then how you change that well by introducing this creation operator that creation operator now needs to annihilate an electron here yeah so it takes out an electron I plus takes out an electron out of this orbital that's a useful thing to do and that's therefore the whole creation operator so this operator I plus creates a hole in the originally occupied in the zero order ground state occupied manifold this creation operator creates an electron in the originally unoccupied manifold so there we leave it as it used to be and for the unoccupied to change it okay makes sense right but it's a good no it's a good question and that's why probably I cannot explain it in in one and a half hour actually Nikola asked me whether I want to talk of four times one and a half hour and originally it's always impossible to feel so many lectures and now I think it wasn't mistaken I should have said more now this is really important I mean this is really important to understand this is our ground state in classical quantum field theory they would be all black bars there are no electrons in originally here the difference is that we set out from this state here where we have eight electrons already in the system to start with and then we introduce we kind of have this identification and I plus then means create a hole here because it annihilates an electron in the ground in the helpful concept yeah so here are the anti-commutator relations and this one I could write down both for i as well as for j and a so the anti-commutator relation a b is 0 a plus b plus is 0 and a plus b is equal to delta a b so again all these all these operators do is to embed all the intricacies of fermions so we have taken care of all the complications of fermions for instance these rules mean if we apply the creation operators twice to the same orbital if we apply the creation operators twice to the same orbital we get 0 what I've written down here c plus a creation operator on the orbital on the same orbital in succession is 0 right using the creation operator twice on the same orbital gives 0 that makes sense you cannot have two electrons in one orbital using the annihilation operator twice on the same orbital yields 0 the very few cases that yield anything and the only case that yields anything is this one here so you first annihilate an electron and then create one that's delta a b that's one of the few cases that can yield something and only if those two operators operate on the same orbital so what can we do with these operators well now we can rewrite the original Hartree-Fock Hamiltonian the original many electron Hamiltonian I had before we can now rewrite using those operators that's the trick so we really essentially rewrite the previous many body and Schrodinger equation and we will use those operators and here are some operators that are useful this operator annihilates an electron in the orbital p and then puts it back in so this annihilates an electron and then puts it back in when will this yield any values look at this here when will it yield values well it will yield a value if you first create a hole here and then put back the electron into the same place right it will yield no value 0 essentially if you apply to an occupied state so this operator is kind of accounting it's an operator that allows you to count the electrons right so it will only yield a value if this orbital was already occupied originally and then you put back in the orbital so you don't do anything to your state so it essentially kind of does nothing but if there was already an electron it will take it out and then put it back in if this is used for a previously unoccupied orbital you will get the vacuum state so if you if you operate with this operator onto a previously unoccupied orbital you will get the vacuum state 0 you will get not the vacuum state sorry I correct myself you will get 0 so this is a useful operator it just counts the number of electrons in the orbital p yeah it determines the number of electrons in the orbital p so actually from this we can immediately construct an operator that counts the total number of electrons this is this operator we simply sum over all orbitals yeah so this is essentially something like a density operator determines the number of electrons in a particular orbital p so it's something like a density operator and this here is the density matrix operator actually this creates this creates a hole in orbital p or takes out an electron of orbital p annihilation operator for orbital p and this is a creation operator in orbital q and this is called the excitation operator or the density matrix operator is essentially allows you to calculate density matrix between two orbitals p and q if you have the density matrix you can actually translate potentials local potentials or non-local potentials into the correct equation that you would use in second quantization so a one electron operator can be also read always written as the potential expressed in this page in this basis so we have orbitals yp so you have actually an v and what I write down here is v q p is equal to q this here is a one electron operator that has a single coordinate and this v q p is nothing but essentially the original operator in this particular basis of orbitals these here are the one electron orbitals from happy folk and all you need to do is you need to evaluate these one electron orbitals in this particular basis and multiply it with the corresponding density operator so this is a means to convert a one electron operator into this language of second quantization is very quick again for instance for a local potential just if you look at this you can actually rewrite this guy here by basis set transformation into the local potential times the density at the particular state point or the number density operator at the particular state point this is just the basis set transformation if you set out from here to a basis set transformation then for a local potential this becomes essentially this equation so this is just v at the position r the potential at the position r times the expectation value for the number of electrons so what about the two electron operator in our many electron Schrodinger equation there's only one two electron operator that's the interaction term between two electrons one over r minus r prime the Coulomb operator this is essentially this guy this is an operator that involves two densities at two space points yes it gives the interaction energy between two electrons and for this purpose we need the two particle density matrix which analogous to the previous slide the one particle density matrix can be written like this so this creates a hole or annihilates an electron in state s annihilates an electron in state r and then puts back in electrons in state p and this should be another mistake on my side a creation on the orbital cube this one i can correct on the fly because this one i have written well analogous to the one particle density matrix this operator is used for the two particle density matrix and this here is now the Coulomb potential in this particular basis i will give you a definition of this in a minute here on the blackboard but this is exactly the Coulomb operator in this basis of the artifoke orbitals and this here measures kind of how likely it is to find an electron in orbital s and r and then you put in the orbitals back into the orbitals p and q well to give you some feeling what it is i've run through the algebra here so you this here is essentially the main electron wave function here you put in your operators your two particle density matrix operators then you can commute those two using the rules we have had before the anti-commutation rules so you exchange those two here that changes the sign here and then this here comes from this delta function we have before so all i have done here to go from the left to the right is inserting the basic matrix algebra i've introduced before that the anti-commutation operator between two such states is that the idea or respectively i did already on this slide i've only used essentially this equation to convert this guy here into that here and the second step is that i identify i do a basis set transformation from this artifoke basis to the real space basis and here do the same thing this is exactly the same operator we had before this is the particle number operator at two different positions and essentially if you now do a basis set transformation of this guy and this guy and go to the real space basis just a basis set transformation i can't do this here on the blackboard it takes probably half an hour to do well takes 20 minutes to do it properly if you do a basis set transformation you can believe me that this essentially comes up here and the same thing you do here this is now real space so we go from the orbital to the real space representation and then you see immediately what this describes this here measures the density at the position r prime and this here the second operator measures the density at the position r and this here is the coulomb interaction so this is our original term in the many-body Hamiltonian and this is our rewrite in the language of second quantization so no miraculous things have gone on all we have done we have written our original equations that we had in the Schrödinger picture into the language of second quantization density operator is equal annulation operator iteration operator r prime here's another density operator which is essentially the complex conjugated of the first but at the position r then you do a basis set transformation you come back to these particular operators so this essentially allows us now to rewrite our original Schrödinger equation into the language of second quantization this was kind of the first part of the lecture and it is taking me one hour and it should have taken 10 minutes not quite so bad so this is our original many-body Schrödinger equation and essentially here is a rewrite of our many-body Schrödinger equation in the language of second quantization for all of those that have already seen this many times there's no nothing special about this we have essentially only rewritten this so let's look at the individual terms this is our original amplitude Hamiltonian kinetic energy nuclear electron interaction and some effective potential and this term can be written in this form here annulation operator in orbital p and putting back the electron into orbital p and this is the eigen energy in the one electron basis these are essentially our Hartree-Fock one electron energy this here is essentially our amper term particle Hamiltonian and the rest is the our perturbation and our perturbation describes the move from the exact many electron Hamiltonian this is this term here this here is our Hartree-Fock Hamiltonian so we switch off the Hartree-Fock Hamiltonian and switch on the exact many body electron Hamiltonian so this is our amper term Hamiltonian this is the Hartree-Fock Hamiltonian and now in many-body perturbation series we switch from the Hartree-Fock Hamiltonian to the exact many body Hamiltonian and this can be written as in second quantization we have essentially only the electron all electron interaction which is written down here this here essentially measures kind of the number of holes at one place and this is measuring the number of electrons at another place and this here is the effective interaction one electron potential that we switch off is the term I've talked about before or under two slides that were presented here usually one chooses the reference Hamiltonian to be diagonal in its own eigen functions what does that mean we usually set out from the Hartree-Fock Hamiltonian and the orbitals we choose for the second quantization at least in quantum chemistry are the orbitals that diagonalize the Hartree-Fock Hamiltonian so usually the reference is Hartree-Fock but as you will see later we can also use DFT so now the rest I have to do probably quite quickly but these are pretty standard procedures essentially we now have to what we have now tick this actually second quantization I will briefly now touch on Gelman-Low theorem then have the Wicks theorem and then the link cluster theorem so we have essentially done the first part and we have still three things to do Gelman-Low theorem yes I'm getting desperate and you're probably getting more desperate I guess because this is really complicated and you realize that only when you start talking about it you can look at your slides and you don't have a clue how it will work when you present it so this is not so difficult actually what we will do now is we will switch we will switch now from this Amperturbed Hartree-Fock Hamiltonian to the many body so this is a point for those people that have stopped thinking this is maybe a point to try to come back okay yes okay everyone has stopped thinking so we now switch from the from the from the Hartree-Fock Hamiltonian to the exact many body Hamiltonian the other thing to consider here that we are working in the language of second quantization so our Hartree-Fock Hamiltonian we use Hartree-Fock orbitals to diagonalize the Hartree-Fock orbital in second quantization this looks like this and this you have seen I'm sure in other talks here right and this here is the electron electron interaction so if you haven't understand the steps how we got there don't worry you can just accept it that this is our Hamiltonian in second quantization this is which is on the electron electron interaction and this here actually switches off the Hartree-Fock Hamiltonian so it switches on the potential energy terms in the Hartree-Fock Hamiltonian switches off the Hartree-Fock so both the Hartree potential as well as the Fock potential and we switch on instead the exact many body Hamiltonian so this is pretty standard actually we now make a few tricks this is our perturbation H1 and what we first do is we go to the interaction picture for time dependent perturbation theory essentially we rephrase our operator our time independent operator as a time dependent operator so H1 is in principle non-time dependent perturbation right and we introduce here time dependent perturbation theory the time is only here to switch later the time will be only used to switch from the from the from the unperturbed Hamiltonian to the perturbed Hamiltonian the interaction picture implies that we actually in the interaction picture define actually the interacting Hamiltonian as e to the power of i h0 t times the perturbation and here we multiply with the complex conjugated then states involve according to this Schrodinger slightly modified Schrodinger equation i t psi dt of i interaction picture is given by this Hamiltonian here in the interaction picture times this time dependent state first step is to derive a time evolution operator that describes how a state at time t0 involves to a state at time t this is our time evolution operator and you can obtain the time evolution operator by essentially plucking those definitions into the many body Schrodinger equation and running through the derivation the time evolution operator will allow us to propagate the state at time t0 to a state at time t by the elegant compact notation and here is our final result and this looks very much a little bit like a Dyson equation well not quite so we have here the time evolution operator is one minus i times our Hamiltonian in the interaction picture times and that's our problem it's obviously recursively defined because this time evolution operator comes up again again this is pretty straightforward basic mass you can do it in a few lines you just insert the defining equations into this into the many electronic equations to actually derive the time evolution operator okay this is obviously not a closed solution to obtain a closed solution you essentially start here with the delta function and obtain the first order term and here then higher order terms obtained from this equation essentially you start with u i is equal to one so you throw away this term here to obtain one here so then you insert the one here and then you iterate this equation to obtain a closed equation for the time evolution operator the good news is essentially here in this lecture will not go beyond this first order term so I will never actually I will only use essentially those two terms here in the time evolution operator and this is pretty much standard so it's one minus i and then the integral from t0 is the starting time to the final time t times the Hamiltonian in the direction picture at time t prime essentially I will use only this here and the Gellman law idea was now to slowly switch on this perturbation so we actually switch from the original Hartree-Fock Hamiltonian to the many body electron Hamiltonian this is here again we switch from Hartree-Fock to the many body Hamiltonian by slowly switching on our perturbation so we multiply this time dependent Hamiltonian in the direction picture by kind of damping term now let's look if actually tau this sorry this is eta right again weakness in this equation what actually I'm joking a little bit actually I know if I concentrate I know it but if I give a talk I always get it so eta is larger than zero that means if you evaluate this quantity at tau at the time t minus infinity eta is positive if you evaluate this at minus infinity obviously zero right so at t minus infinity we have zero here we have no perturbation and we essentially slowly slowly switch only perturbation and the t is equal to zero eta time zero this is essentially one at t is equal to zero we have then actually our exact many body Hamiltonian so we switch from the Hartree-Fock Hamiltonian slowly slowly slowly ideopathically to the many electron Hamiltonian okay well then essentially our many electron wave function is given as the time evolution operator from minus infinity this is the starting point of our transformation this is the end point where we have to pull many body Hamiltonian so this is our time evolution operator that propagates states from minus infinity to time zero if our perturbation series converge and then we can actually write our many body wave function as this time evolution operator from minus infinity where we have the Hartree-Fock Hamiltonian times times this Hartree-Fock wave function so this is the one single slater determinant this is the one determinant that has only a single slater determinant this is our starting point this is the Hartree-Fock Hamiltonian we switch ideopathically from this Hartree-Fock wave function to the true many body wave function yeah and all we need to do then is to propagate our Hartree-Fock wave function using the time evolution operator to the many body wave function that's nice because we have already closed expression for our time evolution operator worked out so essentially if we can take it at second order this is our time evolution operator so we have that and we have a means to propagate our Hartree-Fock wave function to the true many body wave function so we only plug this guy here in from the previous slide as simple as that so we plug this guy here and we are done okay can be evaluated the ground state energy of the many body system the ground state energy we now assume at t is equal to zero our Hamiltonian is given by the initial Hartree-Fock Hamiltonian plus the many body Hamiltonian essentially this sum here is again our true many body Hamiltonian this is essentially done here so we have our true many body Hamiltonian h0 is our Hartree-Fock and this is a many body minus Hartree-Fock potential terms and if we are lucky and this adiabatic switching works if the perturbation theory converges we can write actually this expectation value should be equal to zero plus some term delta e times the many body wave function so if the perturbation theory works this is our true many body wave function this guy is our true many body wave function and we can sort in so it really solves this is our true many body Hamiltonian this is truly an eigenstate of this true many body wave function so we plug that in and that will give us some energy e0 this is the Hartree-Fock energy plus a correction term this is this correlation energy so by means of this this is our e0 our Hartree-Fock energy by means of this we obtain the true many body energy uh Well this looks a little bit early still we cannot use that in practice and that's why we multiply from the left side with the Hartree-Fock wave function so all we do is we multiply from the left side with the Hartree-Fock wave function so you multiply this guy from the left with the Hartree-Fock wave function you multiply this guy from the left with the Hartree-Fock wave function there will kill me if i touch this right I make probably every plot on the white board, right? OK, so we multiply from the left with the harte foc wave function. And obviously this yields, and now one should do this on the blackboard. We get, we then bring this on the other side and divide through. So this is basic algebra. So you multiply from the left with this guy, and then divide by, I should do it. It's so primitive. And then you come up with these equations here. So this here is multiplied from the left with the vacuum ground state. This gives this term here. This gives that term here. And then we divide by E0 plus 10. Then we divide by the harte foc terms. The many electron wave function divide by this. I'm up with this equation. It's a very basic algebra. So this is essentially the true ground state energy of the many electron wave function. And now the only thing we have to do is for this guy, this is the many electron wave function. We insert the harte foc wave function times the time propagation operator from minus infinity to 0. So this is the true many body wave function. And we just plug that in and we are done. And we have now, for the first time, a compact equation for the total correlation energy. And all it involves are harte foc ground state wave functions, time evolution operators, and the amplitude of the Hamiltonian as well as the perturbation, as simple as this. So this is, again, exactly the same equation copied from the previous slide. We have now an equation that gives you the harte foc energy plus the correlation energy, which is the true many body energy. And it's given by this compact equation. The only sketch to this is that we have still an infinite sum in this time evolution operator. And in perturbation theory, we now go order by order through these terms. The other thing I want to mention, all operators are in the interaction picture. But from now on, I will remove that I here. So it's implicitly assumed that all of the operators we have here are in the interaction picture. So this is, in principle, nice, because we have a really nice compact equation for the correlation energy. However, we still need a rest. Well, actually, even better, these are really vacuum expected. These are really ground state expectation values. These are the harte foc wave functions. So this really, everything is, in principle, known. We only need now a recipe to calculate vacuum expectation values because these guys here are the harte foc determinant. In many body perturbation theory, traditionally, this would be the vacuum ground state. I've told you before, we have actually chosen a different ground state, namely this ground state where we have previously initially occupied eight states. It's the only difference. But this is equivalent to the vacuum ground state of many body perturbation theory. So we need a recipe to calculate these vacuum expectation values. And that's where the second theory comes in. So this was OK. The Gellman-Low theory worked pretty, pretty fast. So let's look at the Vick's theorem. The Vick's theorem is actually the backbone of all this. Who knows the Vick's theorem? Excellent. No, it's not great because actually, again, who knows the Vick's theorem? Who has never actually studied this?  No, no, no, no, no. Who knows the Vick's theorem? Who has heard about it but doesn't know it? And who doesn't know it? The sum is not the audience. Don't feel embarrassed. OK, here is something embarrassing. I had no clue about the Vick's theorem 10 years ago. And then I was already full professor. This is not embarrassing to know nothing about the Vick's theorem because in many classes, this is no longer taught. Many body perturbation theories slip from the radar. OK, now I need to entertain you at least a little bit. This theory came up in the 50s, so they are done. So they are kind of totally boring. They are totally, actually they were considered to be largely useless because you can't use them for evaluating real properties. So that's why they dropped from teaching, rather. But these are extremely compact theories. Of course, I can't teach it in one and a half hour.  You need a complete course to teach them. You need something like 10 hours, I guess, to really teach them thoughtfully. But people don't do it anymore because it's not relevant, right? But that's not true because the big point is now we can do them in solid state. We can apply these theories for the first time to solid state systems. Unfortunately, the paper, you might read. They don't mention these theorems anymore. They might cumbersomely revive the formula somehow. And I will do that later, actually. Don't worry. I will come back to very simple things later in my other two lectures. I will not involve anything that is coming up here. But we can do it. Normally, the papers in the 70s and 18s assumed that everyone knew the big theorem. And then they are very compact. And they write some equations down. And most people used the theories written down in the 70s, like Hedin's equation. And they have kind of forgotten the basics. So they're using Hedin's equations. They're using even very sophisticated matrix renormalization group theory. And all that rests on people that, of course, knew the theorem and assumed that everyone knows it. Now we are in a new generation that doesn't learn the theories anymore. Big theorem, all this perturbation theory is gone. They only learn this more sophisticated series. But that's a tragedy, because you have no clue what you're talking about. Because your basic is not there. So you have to learn this. I think there is absolutely no way out. But you have to learn these basic theories before you jump into something more sophisticated. You have seen full CI-QMC. The basis of that is, of course, second quantization. I've seen you at talks about DMHG. But you cannot understand those if you don't know what actually big theorem is. At least that's my feeling. OK. Now what does big theorem? Big theorem is actually, now I've got it, actually. It's right foot. Big theorem is a very powerful algebraic theorem. There's really, again, nothing too fancy about it once you understand it. Actually, it reorders the operators. And now be careful. This big theorem, the way I applied here in Hartree-Fock, is that this applies to these operators here. To the operators that create a hole or an electron. So my normal ordering operators work on the level not of these operators, of these fundamental creation and annihilation operators, but on the level of these guys here that create a hole, respectively, an electron. And I define the normal ordering to be that order that puts the annihilation operator first and then the creation operator. So normal ordering is nothing but to place the normal order of two operators is exactly given by placing the annihilation operators to the right and the creation operators to the left. Why do we do this? What happens if you apply this operator to the ground state? OK, think about it. What happens if we apply this operator to the ground state wave function, to our Hartree-Fock wave function? OK, we will do this just for fun. So what happens if we apply the operator I or the operator A to the ground state Hartree-Fock Hamiltonian? Let's go back to our slide that I had a few slides before. So what happens if we apply an annihilation operator? Let's do this first. And this A is equal to an annihilation operator. So essentially, it takes out an electron of a previously unoccupied state. It removes an electron from a previously unoccupied state. So what happens if we apply this operator to the vacuum state? So if you apply this operator and I really need more blackboards here, not good. So what happens if we apply the annihilation operator A to our Hartree-Fock determinant? So if we apply this is our Hartree-Fock determinant, this is our vacuum ground state, our redefined vacuum ground state. If you apply an annihilation operator for a previously unoccupied orbital, what happens? It's 0. What happens if we apply this operator here to the ground state? So and I now, I is now equal to CI plus. That was the definition. And that's why I introduced this. Maybe this answer is now better answered why we redefined our operators. Somebody asked me, why do I redefine the operators? Here it is. I creates a hole. Or actually, it annihilates a hole. It's equal to CI plus. CI plus, but these orbitals are occupied in the ground state in the Hartree-Fock determinant. So CI plus actually creates also a 0 if you apply it to the ground state. So that's why this normal order is so super useful because by normal ordering all the operators, if you normal order the operators, you get something. If you actually apply it to the vacuum state, you will get over 0. Or if you apply it to the Hartree-Fock ground state, you will get always exactly 0. That's why we introduced normal ordering. That's also why we actually rearranged and redefined our operators compared to those that are classically used in quantum field theory. So the normal order is the one where we have this i or a guys to the left and the i or a dagger. Sorry, we have the i or a to the right. And the i or a dagger to the left. Well, because we are dealing with fermions, the normal ordering also has to count the number of permutations that we need to do this normal order. And the sign is actually given by the number for required permutations, p. And it is just minus 1 to the power of p. So the great thing about this is the vacuum expectation value of a normal operator always vanishes. This is what I have just gone through here. So the great thing is the vacuum expectation of any normal order operator vanishes. So if we order the operator normal, then the vacuum expectation value vanishes. This makes life reasonably easy and is exceedingly helpful. Unfortunately, we are not in close yet. The second thing to define is a code. Well, you see there are a lot of steps in these derivations. And if you haven't heard this before, it's really going to be a little bit difficult. I'm repeating myself and all. The second step is that we define something that is called a contraction. The contraction is strictly defined as the difference between the original order of the operators in any arbitrary sequence minus the normal order operators. It turns out, and this is very easy to follow if you just play with the mask, the contraction is always a scalar and even nicer, very often it's 0. It turns out to be 0. So for instance, the contraction, and this is, you just have to use the basic definitions. The contraction of two annihilation operators is always 0. The contraction of two creation operators is always 0. The contraction of annihilation and, OK, why is this so? Maybe I should explain why this is so. But it's very easy to follow. So we just defined before that the normal order is the one where we have the, this guy is already in the normal order. So the normal order is the one where we have the annihilation operators, where we have the annihilation operators to the left and to the right. Sorry, I have a problem with left and right. Can you imagine me being in the military service that you had to do this left, right? That was really no fun. So this already is in the normal order. And the contraction is defined as the difference between the operators in their order as they are given here in the arbitrary order minus the normal order. And this here, this contraction is obviously 0 because it's already in the appropriate order. So the difference between a b minus a b is obviously 0. This guy is also already in the normal order. This guy is also already in the normal order. The only one that we have to define is this one here. So the contraction is the difference between the order I've written down here, arbitrary order minus the normal order. This is this one. And this here is our basic anti-commodator rule. This gives delta PQ. So a contraction only yields a final value if the annihilation operator is to the left and the creation operator to the right. Only then can a contraction yield something. And furthermore, this is a scalar. This is no longer a complicated weight function object. It's just a scalar quantity. It's just a delta function. So it's either 0 or it's 1. So it's a value. It's a number. It makes life even easier. This is not so important. The only other thing I want to mention here, in this context, the normal order is defined for the quasi-particle operators. So the normal order, and this is already what I alluded to here, the normal order is defined for these operators, I and A, and not for CI or CA dagger. Good. Now, weak theorem essentially tells, and I will not go through this. I've decided to make a shortcut here, weak theorem tells, and this is the power of it for a vacuum expectation value. So if you want, if you are evaluating the vacuum expectation value of a series of creation and annihilation operator, only the fully contracted term survives. This is an extremely powerful theorem that really helps a lot in doing perturbation theory. I know you can find paper by, I don't know, Presta, where it's cumbersomely derived third-order perturbation theory, second-order perturbation theory. All this is not necessary if you understand those theorems. You can actually write down third-order terms, fourth-order terms, probably beyond that, it's getting really tough, but in principle, these theorems allow you to do any arbitrary order of perturbation theory on the back of an envelope. Imagine, I mean, third-order equations are really, really difficult already to derive. But again, these theorems, in principle, allow you to get total energy changes in any order on the back of an envelope in a ballback. Here is again the important equation for a vacuum expectation value only fully contracted term survives. What does it mean? So if you have here any order of creation and annihilation operators, all you need to draw is any possible contractions, pairs of contractions, and that's it. Contractions are scalar, so even the wave function is kind of gone, kind of, all right? So it's gone because these are scalars. So essentially, there's no longer the need for the ground state wave function. Of course, this applies only if you need to evaluate the expectation value for a ground state wave function and for the Hartree-Fock wave function. Let's go back to our Gell-Mann law theory. The way I've written it here, these are vacuum ground states. This is a vacuum ground state. This is the Hartree-Fock determinant. So these are vacuum ground states. And this here are vacuum expectation values. So the combination of Wicks theorem and the Gell-Mann law theorem allows you to calculate the correlation energy or essentially write it down comparatively. This here is the ground state wave function or the Hartree-Fock wave function, really. This is the ground state wave function or the Hartree-Fock wave function. This is exactly a vacuum expectation value in the common way it's defined in quantum antibody theory. Only we have redefined the things a little bit. OK, quickly, we have done a few things here. For the normal order, we have done, well, it's not me that has done it. Actually, you can find it in many books. We have only changed a little bit the reference. The reference is now a Hartree-Fock reference. And we have redefined a little bit how the normal order is. The only thing I have essentially done out now I know where I'm going in the wrong direction, dreadfully. So good. Let's be practical and just show you what you can do with this theorem. This here is our many-body Hamiltonian in second quantization. It's essentially the perturbative term that we have. That's the perturbation with Wicks theorem. This guy, so the combination of Gell-Mann law theorem, so this is our perturbation. And we just do lowest order perturbation theory. Actually, this is first order perturbation theory. So we apply it. We just look at this value. And we assume this U of eta is just 1. So we set the U to 1. Then we just need to evaluate Hartree-Fock determinant H1, 0. And that's the other Hartree-Fock determinant. We just need to evaluate essentially this and we cancel and drop this particular term. This is H1. This is our perturbation. And what I've told you, and that's Wicks theorem, if you want to evaluate the vacuum expectation value, this is an vacuum expectation value. This is the Hartree-Fock determinant. Hartree-Fock is our vacuum state. All we need to evaluate are all possible contractions. For instance, A, B, C, D, A, C, contraction between those two, B, D, contraction between those two. A, D, B, C. These are all possible permutations that we have. There are no more. Now, many of these contractions are 0. For instance, if there are two occupied orbitals or there are two annihilation operators, it will be 0. This will be 0. So essentially, a lot of terms will drop out. And it turns out that the only non-vanishing term is annihilation to the left and the same creation to the right. This is this here. This is the only contraction that will give non-zero terms. So it's an annihilation to the left and the same creation to the right. This is the delta function, same creation to the right, right and left. And the only non-vanishing term is annihilation to the left. It's the same creation to the right. These operators are, however, ordered a little bit. Well, these are the standard quantum field creation and annihilation operators. Be careful. These are these guys here, not these transformed ones. And it turns out that only the contribution that we will have is if the right operator is a whole creation operator. So if this here, C i, is equal i dagger plus, only then we will have a contribution. So these are the only terms that survive in first order. So it's actually this C s must be, essentially, this C s must be a i dagger operator. Now this looks strange. Yeah, essentially, this C s must be l dagger operator and this C r must be a k dagger operator. And only if i is equal to k, you get a contribution. So only if i is equal to k, then you get a contribution. Only if l dagger is equal to j, you get a contribution. So only if these two guys are the same and these two guys are the same, you get a contribution and this is written down here. In my next lecture, I will use a much simpler reasoning to derive the same equation. You will see that later that this can be done really easier in first order perturbation theory. This is a little bit complex. There's an easier way to do this and I will come back to this. So don't worry too much if you don't understand it. But this here is exactly the half refock energy for the ground state determinant. Good. Now we can do this in higher order. We can do it in first order, for instance. So essentially, what we do is, here we have, again, the vacuum expectation value as we obtain it from Gellman-Lowes theorem. And now we take the forced order term for this. So we just take the first order term for the Hamiltonian H1 or for the time propagation operator. So this is our time propagation operator. And remember, we propagate from minus infinity to time t hero. Minus infinity was half refock. Minus infinity was half refock. And the t is equal to 0 if we get over to the true ground state. And essentially, in the second order perturbation theory, this involves just H, the Hamiltonian H1. This here is our interacting Hamiltonian. This is the adiabatic switching that we use to switch on the Hamiltonian. This here comes from the interaction picture. We call, I've told you that in the interaction picture, we have to left multiply and right multiply with the independent zero order Hamiltonian. And essentially, for this here, we use a spectral representation. And I'm not going through that. Essentially, this is our full Hamiltonian, our full time dependent Hamiltonian. Here is the slow switching. The only thing I want to say here is that here are now our creation operators. And any creation operator that comes up also involves a time propagation operator. Again, I will come back to this and show it in a more intuitive way later what this does. But look at this. What this essentially does is, if you create a particle, then here you have, in the interaction picture, you have actually, and this here is in the imaginary time, but you don't need to worry about the imaginary time. This is just how a particle would propagate in shooting a picture. In shooting a picture, a particle gets e to the i epsilon tt. If you propagate a particle, this is the kind of phase factor it requires and it propagates in time. So any creation operator will involve this here. Similarly, so this here is an annihilation operator, and that involves essentially propagation of a whole and therefore the negative sign. So any annihilation operator that will later come up will have these exponents essentially in there. This is just rewriting the original perturbation into this interaction picture. Nothing has happened here, but we have rewritten our Hamiltonian H1, our perturbation, into this interaction picture. That's all I have essentially done. So now we need to do, again, weak serum. And very quickly, it becomes cumbersome. Now we have eight creation, sorry, four creation and four annihilation operators, and it gets really like complete mess. So this is done here, and I mean, it doesn't really make fun. And now you might say, why do we have to then use these crazy serums, right? I mean, I've told you before, this is very powerful, but now you have all these possible contractions. Are you stressful, right? So it doesn't look like a lot of help, and that's where diagrams come in. Diagrams are an exact one-to-one correspondence between complicated algebraic expressions and graphical representations. They are very clever means, and if you want to, if you haven't understood anything, that's still OK now. You can still come back at this point, OK? The diagrams are lovely, OK? Because they are exceedingly powerful, and they just, maybe, forget about everything I've told you. I've tried to give you the background of all this, but you don't need it. That's a good point. From now on, you just need to laugh goldstone diagrams because they allow you to forget about the complicated derivations and do everything on the back of an envelope. Actually, I learned goldstone diagrams first and only then looked after series one. Maybe what I want to show you is you should read the series once, but you can also instead just use the rules that I will give you now for goldstone diagrams and construct the derivation series of an arbitrary order. So here are my diagrams, and I've tried to capture how they work. Here we have a Coulomb interaction, and here we have the creation and annihilation operators. Any creation operator has an outgoing line. Any annihilation operator has an ingoing line. So here we have an ingoing line, and here we have an outbound line. Goldstone diagrams are time-ordered, so there is always a time that propagates in these slides inconsistency between me and my later slides. There are Felix slides in my later slides. Time here propagates only from up, from the bottom here up. So time propagates upwards, and my later slides will always propagate downwards. So essentially, let's look what this means. This means an electron comes in in an orbital s. This means an electron comes in in an orbital r, and this here means that you annihilate those electrons at this point here, and then create at that point here an electron p and an electron q, and they continue their propagation in time. This here is the Coulomb interaction, and the way you connect it is p and s are at the same space point, and r and q are at the same space point. So essentially, what we do here is we convert the algebraic expression here into this diagram here. These are the two particle operators, the Coulomb interaction. These are one particle operators. So we have an electron coming in here and an electron moving out here. So essentially, coming in here with an electron, we annihilate it at the position q, and we create a new electron in the channel p, and this here is our Coulomb interaction, ppq. So this is essentially a graphically rule how to convert these second quantization terms into graphics. So the one electron potential in this talk at least will be replaced by a dashed plot that's usually done in quantum chemistry with one incoming and one outgoing vertex. And the contraction operators, I don't know what that means. Forget it. So here is one example. This is the Hartree-Fock term. We had just a few slides before. This here is the Hartree-Fock term. We had just a few slides before. This is this here. And actually, we can use diagrams to represent this. So we have here a Coulomb line and a particle i propagating here to this point. And here it's a particle j propagating backwards. And here is the Coulomb line. This is an exact conversion of this diagram here. We had a few slides before. Just look back. And if you apply those rules I've given you here, you can immediately convert the algebraic expression that we had in first order perturbation theory to this nice little diagram here. Now, believe me that this is much easier to remember than this every guy. Actually, a lot of physicists, including very smart ones like U.G. Reining, they don't like diagrams for some reason. And then they took 10 pages of derivation. And they looked one minute and tell them these indices need to be swapped. This is just by the means of diagrams, which are extremely powerful and extremely compact to represent the equation. This here is the Hartree interaction I've drawn before. So this here is the diagram that we had three slides before. This is how I wrote it down. And now you use these rules here. Here you have a closed circle because i is coming in twice here as i and i dagger. And the way it's connected and the rules I've given here means that they run back into the same vertex here. And here J runs also into the same vertex here using just these rules. So what happens is essentially that this vertex connects back to that vertex by the virtue that p and s are the same. So if p and s are the same, you reconnect the vertex. If q and r are the same, if these two are the same, q and r are the same. This is here. These two are the same. They need to be reconnected, essentially, graphically. Another rule I need to say here is if they are closed loops, you only need to sum over the occupied states. That comes also from big theorem. And it essentially means any closed loops. And also if the loops are at equal times, it's always only a sum of occupied states. So I will not walk through this. I will not walk through that. I will leave. This would be. Well, these are derivations that are a little bit complicated, but you don't need to know that. So essentially, at the end of the day, all you need to do is you need to draw all possible diagrams that you can figure out yourself, all possible permutations where the diagrams need to be completely closed. And for instance, in second order, you have at most two Coulomb lines. And try yourself a Coulomb line. I have really too little time. It's really difficult. OK, but we will come back to it. It's good news. So we will have later in my later lectures, I will actually only use the goals from diagrams. And you will see a lot of examples. And I will give you again an introduction to it. Here is, for instance, a second order diagram, a Coulomb line, a Coulomb line here. This here is an electron line. And since time here in these slides moves upwards, this is a whole line. This is an electron line. So whenever the arrows go in my slides when they go down, in those slides, if they go down, it's a hole. If they go up, it's essentially an electron. And this is one of the few lines. Now let's try to do some exercises. Imagine that you have two Coulomb lines, wiggly lines, and try to draw all diagrams where you observe the previous rules. So actually, you have two Coulomb lines. And from each of these Coulomb lines, you can have arrows going out, two arrows, and they need to be fully connected. There are not so many possible choices. You can connect those two. And then either you can connect those two, or you can cross those two. This is about all you can do. And this is shown on those diagrams. Here are really all possible permutations of two Coulomb lines. And these are the electron hole lines. So you can connect those like this. You can go from here to here. Well, you can swap i and j. So you can swap those two. But that's the same topologically. It's the same diagram as above. This is another topologically identical diagram to this. And this is, again, the same diagram. So obviously, the many swaps you can have yield topologically over the same diagram. The only other one I haven't drawn here is this cross diagram. This is about the only other diagram you can have. So Goldstone diagrams, essentially, are extremely compact representation of these complicated algebraic equations. Here is a third order diagram, for instance. Essentially, there are many more. But you can draw all by drawing three Coulomb lines, one, two, three, three Coulomb lines, and then connecting the diagrams according to the rules I have given. And now the second point is we can immediately translate this Goldstone diagram to an algebraic expression. How do you do that? It's very simple. So for any arrow pointing up, you introduce A. A means an unoccupied state. For any arrow that points down, you improduce j or i. This is one pointing up. It's B. This is C. So it's a previously unoccupied state. And this is an occupied state, k. Now you have to insert the Coulomb integrals. Coulomb integrals are V, A, j, P, i. These are the Coulomb integrals. They come essentially from the vertices. So each of these vertices represents for each of these Coulomb lines represent the Coulomb interaction. And now the denominators are very easy to determine. You just put the branch cut here.  So here's epsilon A, epsilon j, epsilon i, epsilon B. This is, unfortunately, again wrong. Here's epsilon A, epsilon i, epsilon C, epsilon D. So there's something wrong in here, unfortunately. So these diagrams are not correct. So it's very easy, A, j, i, B. This is the first denominator. The second terminator is A, i, C, k. These would be the right denominators that you should have. OK, there are also rules how to determine the sign. The sign can be determined from the topology. And it's given by minus 1 to the power of L plus H, where L is the number of closed loops. Closed loops, what does that mean? Here we have a closed loop. Where you can move around in this loop. And here is another closed loop, where you can move around this loop. So this is a so-called closed fermionic loop. So you can move in this loop without interruption. And that's a closed fermionic loop. So all you need to do is you need to count the number of fermionic loops. And you need to count the number of whole lines. And that gives you the sign. The number of whole lines are essentially, in this case, those that point upwards, A, B, and C. So the number of lines that go upwards determine the number of whole lines. And the number of, sorry, the number of whole lines are actually the one, yes, I, J, K. Yes, I have to count it correctly. And the number of closed fermionic loops, in this case, is 2. So this gives you the sign. Jesus. OK, next time I take double four hours, that's here now. I'm really sorry for this mess. OK, I will skip the link cluster theorem. The link cluster theorem is very important. But I will not do this. Essentially, it allows you to determine and use only those diagrams that are completely linked. So those that are unlinked. So you could, for instance, draw diagrams that are unlinked. And there would be many more diagrams that come up. And those don't count. So in this perturbation theory, you only need to include those diagrams that completely link. So do we need this? Yes, you want food, right? Next time. Yes, I'm almost finished. Seriously, I'm not joking. I'm almost finished. I will finish after this slide. I know I have 10 more slides in principle, but I will not show it. So maybe I will come back to it, indeed, next time. Probably second lecture. So these are all second order diagrams, all second order Goldstone diagrams that you can possibly draw in second order. I want to show this because I will show it again later. So what's the property? Second order diagrams are those that have one Coulomb line. This counts. So two Coulomb lines, or one effective interaction, and one Coulomb line, or two infective interactions. These are all second order terms. These are the one electron interactions that we have switched off. So we have switched off Hartley-Folk. These here are Coulomb sets. So this is second order, two of these blobs. Second order, second order, second order, second order, second order, second order, second order, second order, and second order. These are all possible diagrams we can draw in second order. There are not any more. And we can actually transverse or convert those diagrams into a compact derivation equation. Exactly according to the rules I've given you before. You just draw a line and then you label the states. You put in the Coulomb integrals and you put in the denominator. The sign can also be determined. Same here, same here, same here, same here, same here. If you start from the Hartley-Folk reference, and that's a nice thing why many people do that, if you start from the Hartley-Folk reference, all these diagrams drop out. So it turns out if you start from Hartley-Folk, these diagrams by some trick that's called Brillouin theorem become exactly zero. I will show you this a little bit later in a different way in a slightly different language. All these terms exactly to about the only one that survived in second order are those here. And these are called Merlopresset perturbation theory terms. These are exactly the terms in lowest order perturbation theory. If you start from DFT, these diagrams need to be included. Yes, I know. I will skip this because I will talk about propagators later. And I will probably skip the Feynman diagrams anyway. Yes. And with that, I thank you for your attention and I'm really deeply sorry that this was far too much. Typically for the first case it's presented, it's always too much. Anyway, thank you. We will come back to many things.