 Protožite vseh lečenih je vseh lečenji, da je dobro je z Paolo Moras, kaj je o prejmenjavnjem težavnem, in občasnih problemov in nekaj zelo, o arpecih, angolo rezultator, spetroskopija. Vseh počutite, da sem počutila, da je to zelo, da je izgovoril z materijnej perspektive. tako, zelo vse, ki očujemo naprejno izgledati spetroskopu kako je tudi izgleda, da se kako načinimo, in boh, kaj je, to je poživlja, nekaj, zrešanje spektrali, ta je celosti, da si se pogleda od grijenih, ta je ta početka, tako, vsaj ustajemo grijen, tudi, da se grijen se glasba, ki so se način ni dvevini očiniti in začiniti začiniti na volj tukaj do spetralj nekaj velik popravlj. In vse eto najbolji všeč se pričoje sei malo zrpustite spramo del know, kjer je zvonila, per ir moj MATOKS. V veli lahko bo ti notes打č prav you nekaj včelj, a pa je z polj. na pomečenja delana in dobro nape tudi, nekaj pogled je zelo, da je še delalo in da je nrgli, da je počela do sljeda, noj del je prosah zel, ali tudi je srednje dzivječ, tako tako ne ZhiW čeč, več gleda toga tukvaja. Znosimo z Arpesa. Zelo pa občalaughter. Zelo je tukvaja, tko vse jazam z pridine vse, skupajse tega v režim, skupajte vse, nekaj vse, kot tudi sem, pa je vse. Pridine, kako se prejde, tako začne, nekaj, tega nr. in tudi, nekaj, nekaj nekaj, nekaj, nekaj nekaj, nekaj nekaj, nekaj,ần Samspers遯zda elektronargega Respectroskopija. The are macro-classes of spectroscopies. The important point from a theoretical perspective is that some of these are charged spectroscopies which means that electrons are either added or withdrawn from the system. Direct photon maken that with a system having n minus one electron So now, inamo, you want to make a electron to the same system and this pa je neko druga, čärgieta. Zato so nekako nekako, da se je počukne, in nekako se počukne. Čoha je, da nekako. Nekako se je zelo. Čaho je, da se je zelo, da se je zelo. Stajte rečen, nekako se je nekako. To je zelo. Čaho je. Čaho je. Čaho je. Čaho je. Čaho je. a to v Čustku bila vse neč opera. Tukaj se naredite, ki je vse ta slavery, če lahko doelo, brz 110 in nekaj je zdravil pozut. Nadejte zelo na piekateru, da je tako drugaj, ko je zašavljena, v kratku, na kratku se polarizacije. Vespečenja je počkala kainetik energij in elektron. Idejno vse kompleti in komplementi informacije, kaj je vsega elektrona, in kako smo videli, je to vsega spina elektrona. Vsega spina rezovuje arpec in angol. Zprin, to je tudi tudi tukaj, kaj je izgleda izgleda izgledo vzajem. Tukaj je tudi tukaj, ki je však vzajem o svojih nog. Nekaj tudi tudi tudi tudi izgleda izgleda izgleda izgleda izgleda izgleda izgleda izgleda izgleda izgleda. According to, it may be better to be corrected. Ten is the best number for a model. There is a model, a cartoon. We see here a simple density of states. We have course levels, we have a valence levels with a dispersion. And this is a metal. We have a firm level that crosses the valence. leko bi zdeni, tako nekaj so našlih zelo za delanje. Ni, ni, ni, ni, ni, ni, ni, ni. Selo se iz toga, ni, ni, ni, ni, ni. Foti so našlih, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni, ni. Vse nekaj ne vseče, kaj je zelo, da 💚 neko sem da bo, da je to neko, ko se nekaj. Če so nisi, da smo tega, da smo na 2 režite z nisem, tako da nekaj nekaj nekaj nekaj, da smo na ektroni in nekaj smo na režite na režite za srečke. Srečke z nisem na stačnega, nekaj nekaj nekaj nekaj. To je najpravj kako je. To je tudi srča, o kaj se če se izvajte. Kabil smo videli, da smo vse sem vse klasnostili komputer elektronične kajče se vse več način na sistem, da smo vse se vse skupali, sredskega, da se zelo odlegašenje elektroni, to zelo, da smo vse zelo odlegašenje elektronične kajče se zelo odlegašenje elektronične in zelo, da smo da nekaj nekaj... zelo, da smo nekaj nekaj... da pa ne zelo vzelo vzelo, da je zelo vzelo. Vzelo, da je tega čas, da se je metalik, vseč je tukaj zelo, da je, da je tukaj, vseč se je, da se je tukaj, tukaj, da je tukaj, da je tukaj, da je tukaj, da je tukaj, If this is a cartoon, this is realistic data out of... I think these are obtained here in Trieste desynchrotron. This is a very nice arpe's picture for a system here is graphene absorbed on nickel. Here you see some bands connected to graphene, but graphene on nickel is interacting, ki bilo vzal šnega konega, bo je za to, kako je konega. Zaterga. Da je za to, dokom je zrta, ki je se za tudi. Zato, bo rather vzal. T side to je bilo vzal. Zato vzal je to, da je tukaj popular. Tukaj mora bi to se možeme zrčititi. Vo, mi to, ovo je tega. Here is a deeper look at this point here and so on. So, theoretical treatment, this is very well established. Here I listed a couple of reference papers. To me, and we can start using the Fermi Golden rule if we want to describe this process, so we want to describe an excitation. izglednja z izglednjem tajšnji foton, kaj smo v iniskej stvari, kaj je pravda stvari izglednja v systemu, in tajšnji stvari, kaj je kompozit, in elektron, kaj je, kaj je, izglednja izglednja z systemu, idej na stvari, kaj je stvari, v vakium, in v tem, tudi bila nge sprednja, ki je tako nekaj, kaj je sreč, kot je nekaj, da je nekaj, kot je zručna stet, ki je za moins 1, je zato nekaj, ki je za minus 1, ki je zručnja stet. Zato je tukaj, kot je zručnja stet. a zelo te Mustafa in se nač Memberil je, ki pristim tukaj, tudi, da se je začunila, in da je tukaj, ja ste počutngli, da jo drž 근데 svač ne ampak. Kajbe bi ste počutna, da so dobro, da je na zelo počutnju, zelo si začunila, da je načinil, da se tak, da je bilo, bilo,istas načinila. Zelo ga seč načinila, da se dobro zelo bo. okazovati nog načenog zelo vzelo. Pravno je, da ne vzeli da injem nastiramo, či že je n k TradeSkog, a začin, to je, da je dobro nog traj vzela, in je tati, da je tati, zelo nekaj zelo vzelo in to je n k fuel, ni momentist vzeli, nače ne pozve. Vežno vzelo to je vzelo, tudi v tem formu. Zelo počekaj smo počekaj s sestem v n-1 elektroni. Zelo smo počekaj 3 elektroni v vakuumi. Zelo smo počekaj tudi, da je tukaj skaterin teori, da počekaj se počekaj, da počekaj počekaj je bolj komplikato. poslednimo to tem sistemem, ki bi transformsiti diskom tez orega skopr. Zde nekaj, ki je vse visi prijevaj readset, da je je vse prijevajーんos na tukaj dosadno. In pa je in izgleda različno p outerješke www.marker.com Zelo se tako občatno, da je pravda, tako je čečen, mačno je in je to, da je tudi prije, zelo je to, da je vse. Kaj je očine vse prije, da je izvajno početno? Početno, je to, da je prije, da je boj, da je dobro dve pise. This is eventually called three step model. It is probably the simplest approach to the problem. First, think of an excitation from where ... your valence panel, or where your electron isしい to some available excited states in this system. In potem je tudi občasno zeločnje, da se počutite izvršenje. Vždyš, da se počutite, da se počutite in tudi občasno zeločnje, tako samo, da se počutite izvršenje spektrosniki. Tudi je elektronične počutite izvršenje, da ne zeločne energi, vsega namočenje v se obtravljenjom, a potem govite na vsega principu in zaprave na elektroni. S tem vsega je obžaljava, pravno, da je obžaljava separati v tretje vsega. To je obžaljava, je vsega režim. In mi se bo hvala, da stefano Baroni vsega bila vsega vsega. Zelo je ne početnico, da ne pomečajem, ki so se početne. To je zelo vzgleda, ali tudi je vzgleda namov. Prej našli tudi pristih koncezov, ki je vzgleda, naredijo. Prej se tudi neseljamo, da je najbolj nekaj. Proste pripljamo zeljak, da je zelo, da je nekaj se vzgleda, ki vseh zelo je več več nakončena, je več več nakončena v vakium, da pogledaj si posrednji posledo, če so da smo pošlibe taj vzelo na kratu, na skrivenju, na glasbo delu, zato, da je tudi skupnjen. Zato, da pa se je tudi skupnjen. Vseh zelo je nr 1 vsega particlea plus an extra electron that is in vacuum, so this is the approximated final state and if you do the math now, so if you compute the photomission current that is this, you obtain this expression here, this delta term here contains basically the matrix element with the perturbation that then will end up in square, but here importantly we find an object that is theoretically very clean that is the spectral function that is defined in this way here, so this is sum over all occupied states, in this case there are these brackets here that I'll show you in a second, basically play the role of single particle orbitals, these are going to be called Dyson freman orbitals, so this is pretty much psi star psi psi star and then delta and after we'll give a more formal description of this, we'll see that this spectral function is pretty much close to when taken diagonal to the local density of states, is the very many body definition of the density of states, so we'll come to this point later, but I think this is important, we are used to think that we have single particle cartoons of the electronic structure of the systems we study, in this those cartoons can be made mathematically korrekt, when we take into account the proper definitions, so basically orbital, single particle orbitals will become Dyson orbitals, single particle energies will have a proper definition in terms of excitations of the many body states and so on and so forth, but the point is this can be made mathematically exact in the many body problem. Okay, so somehow this completes the picture and from the experimental point of view and the main quantity that ends up in our theory of the photoemission is exactly something that is related to the density of states, that indeed is more or less what we expected because we end up with something that resembles a density of states, if you want a density of states that is k resolved, that are the bands and indeed here we end up with a mathematical quantity that is exactly related to that. So let me just conclude this first part about the theoretical picture of photoemission by taking a statement from this review model physics from Damachelian coworker that is talking about arpes and it kind of reverses the picture we may have, so it says that arpes has a large impact on the development of many body theories just because it provides information on the single particle Green's function, so somehow it's an interesting perspective to me, since typically we think that we have the Green's function and then that can help understanding photoemission experiments, if they say that indeed the advancements on photoemission can actually help understanding what are the features you should get out of a properly defined Green's function, but the main point is that the two are tightly connected. One more comment here that anticipates some of the content of next day's lecture is that charge excitations most of the time can be treated with the one particle Green's function. We'll see that neutral excitation instead in order to be properly treated in terms of even qualitative features would require some treatments, namely leading to excitons, that are going to be discussed in perhaps described in the Betty Salpeter, that instead need a Green's function that is the two particles Green's function or are tightly related to the two particle Green's function, so that discriminant between charge and neutral excitation is exactly because theoretically those lead to very different methods. Ok, so let's go further into the connection with the Green's function, so let's introduce the Green's function. We have seen this yesterday already, this is the textbook definition, is the ground state expectation value of a time order product of two field operators creation and annihilation field, time ordering means that we have basically two contributions for t1 larger than t2. First we create a particle in x2 and we annihilate it in x1, vice versa, if t2 is larger than t1, we first annihilate a particle or if you want to create a whole, evolve it in time and then take the scalar product or the amplitude at x2. And somehow this describes this physical process where we create this electron here, evolve it in time and then look at the amplitude. So how does this state here project on the state that is at x1 t1 with t1 later, than t2 and vice versa for the whole. So one more comment here is that this construction here may seem a bit hot at first because why should we have this time ordering that, ok, we kind of understand but why should we deal with this extra complication while physically we would like just to have direct evolutions in time, that is what we really want to know about. The main point is that mathematically time ordering is very crucial to use Vick's theorem that was introduced yesterday by Andrei Marini that eventually leads to Feynman diagrams. So the very, very reason by which we need time ordering is exactly to be able to cast the final expressions for the Green's function, the self energy in the formal Feynman diagrams. If we were using the other related quantities like the retarded or the advanced Green's function, that simply wouldn't be, so those are typically thought or said to be more physical, like the physical response functions typically are the retarded ones, but do not lead immediately to Feynman diagrams. So this just to clarify why this extra complication here. Ok, so that expression for the Green's function is interesting but is not effective or efficient to use. There is a very, out of the definition basically you can immediately cast the Green's function once Fourier transformed and in fact while here, when the time domain, here we are in the frequency domain, the Green's function can be cast just out of the definition in this form here. And this is also very interesting to me because this, if you are familiar with the Green's function, is pretty much the form of a Green's function of a non-interacting system. It's exactly the sum over states, product of the orbitals in x orbital star in x prime and then a denominator with the single particle eigenvalue. Indeed, this expression is exact, even in the many body formulation, if we pay the attention to defining the orbitals and the energies in this way. So energies are actually these differences between excitations at n-1 or n-plus-1 with the ground state at n. So this is the charge excitation at n-1 electrons, charge excitation at n-plus-1. And this is what is used for what we call occupied states, for states that are, let's say, with energies below the Fermi level. And this is the definition that we use for empty states. So, somehow, when we look at a spectrum, we have structures, structures that are below the Fermi level corresponds to involve many body excitations with n-1 electrons, structures that are above the Fermi level involve excitations that are with n-plus-1 electrons. By the way, if you want, this may have an experimental counterpart that is, so the quantities that we put together in this density of states spectrum basically are very different qualitatively. So involve excitations n-1 and n-plus-1. So probably it's not by chance that a single experiment like photoemission can access just half of it. So photoemission just access the occupied states. In verse photoemission, the empty states, it's true that there are experiments like STS, so Scanning-Tanning spectroscopy, that is the spectroscopy you do with STM tip, is local, you can sweep the bias from negative to positive, so you can actually access both occupied and empty states. So it's not impossible experimentally, but somehow, I mean, this fact here I think reflects in the fact that experimentally we may not have access to all the quantities together. And so this is for what concerns the energies, and these are the orbiters. The orbiters are just these amplitudes here, ground state, annihilation operator, and then we take the scalar product with excitation state at n-1, similarly for the empty states. These are called Freeman-Dyzon orbiters, or also Dyson amplitudes. Probably Dyson amplitude is more popular in the chemistry community, can be computed, so there are actual words where these are computed, those are pretty similar to the orbitals we are used to deal with, though, I mean, they carry the full many body information, importantly. So what are then the main differences between this expression here for an interacting system and for a non-interacting system? So already just from here we can see that these orbitals here are not orthogonal among themselves, among themselves, while they are in a non-interacting picture, so in a non-interacting picture those are just eigenvectors of a non-interacting single particle Hamiltonian. These are not normalized, second thing, except that the sum over all of them needs to integrate to the density or to the occupied ones, to the number of particles, so eventually we end up also with having many more states. This is one of the most important features, so if in a non-interacting system we have, roughly speaking, n electrons and we have n orbitals occupied, pretty much. Here we have many more orbitals, actually infinitely many more, and this is one of the main features of an interacting system, so if you want the spectral weight is not just n Dirac deltas for n electrons, that is what you would get out of Konešan Gefti or Artrifok, we have infinitely many peaks, not normalized to one, as if the spectral weight actually would just split in multiple structures, that is the effect of these complicated many body excitations. Actually, the number of excitations that we have in a many body system is much larger, if you want the size of the Hilbert space is much larger than a non-interacting counterparts. We will see, so this is math, we will see the physical connection, the physical information we can get out of it. I've said before that what we end up having here in the spectral current, sorry, in the photomission current is actually the spectral density that has this definition here, and now you can recognize that these guys here are really related to these guys here, just in the basis chains, but this is exactly product of those Dyson orbitals there. So, let's keep going, this is the same definition I've shown you before, how is the spectral function definition? This is defined, this is basically the imaginary part of the Green's function, could be defined this way, g minus g dagger, probably taking into account a change of sine in the time ordering, and again mathematically this is the object that can be used in a kramerskronik-like transform, this is also a spectral representation of the Green's function, if you want spectral function is the mathematical source of the Green's function via this kramerskronik. If you take this definition here, you plug it here, you end up with this definition here. So, if you want, this is a density matrix, is a frequency resolved or energy resolved density matrix. And if you take it local, so not x prime, but just x, you obtain this that should look very familiar, this is the typical definition we use for density of states, very basic, we see probably these very early classes in quantum mechanics, this is the proper many body definition of the density of states. Okay, so, this is a definition, how can we compute this quantity, so I'll assume you are familiar with this equation here, this is the Dyson equation for the Green's function, so somehow if you want to compute your Green's function, you can start from the knowledge of g0 that is non, in this case is the non-interacting the Green's function where we have simply dropped the interaction, but here in principle we could use any reference Green's function, paying attention then to get rid of the extra potential we use here, but so this is just a side comment, but this is an exact equation for the Green's function that can be computed if we know the self energy, this would be the third part of my talk, so just bear this in mind for the moment, formal solution to this equation is basically operatorially this inversion here, and if we keep in mind the Lehmann representation for the Green's function, we can arrive at an expression like this. So, first comment, this is true only for systems that have discrete states, so this is usually called quasi-particle equation, but this is true only when we have discrete states, I'll come back to this point later that is mathematically subtle, but for these discrete states we can, and the main point is that, what is the equation obeyed by the Dyson orbiters? Well, it's pretty much a Schrodinger equation, except that basically the self energy is here and plays the role of a potential, but the self energy is frequently dependent and basically each orbital fills a different potential, so fills the self energy is a different potential that is exactly its own eigenvalue, so somehow if you want this is also a nonlinear problem and so on. These are real because of discrete states, but this is in a sense the paradigmatic of the extra complexity you have to deal when dealing with quasi-particles or Dyson orbiters that is that you have an orbital dependent potential there and that is at variance with mean field cases. Also, if you want, this is the origin of the fact that these orbitals here are not orthogonal. If you think about basically each Dyson orbital obejz a different Schrodinger equation, so they are not the different eigenvectors of the same Hermitian operator. For discrete states the self energy is Hermitian, again not granted in general for when you have a continuous states, but each of them have a different belong to a different operator, so they are not orthogonal. Okay, so this was so far the Lehmann representation. We'll see now another different representation that can undergo the name of quasi particle, can be named as quasi particle representation. We just start again from the formal solution of the Dyson equation. Basically out of here we can just look this problem as an inversion problem with a parameter omega, so a parametrized inversion that we can cast in the form of a parametrized diagonalization problem. Omega here is a parameter. In this case the self energy is no longer Hermitian in general, so we need basically to when we have a non Hermitian diagonalization, we end up with left and right eigenvectors. Eigenvalues left, right are the same, but maybe complex. Everything is labeled with omega and if we take this inversion here out of this diagonalization, this eigen problem here, we end up with this expression that looks similar to the previous one, if you recall this, but it's not the same and the reason are first, here we have left and right eigenvectors that are not one, the complex conjugate of the other. This is more complex and actually involves, if you want the overlap matrix to the minus one in passing from one to the other, this is standard linear algebra. We have this omega dependence parametric and also the omega dependence is here in the denominator, while here we had real numbers and complex conjugate of the orbitals. So, in principle, relevant poles of this expression can be obtained just taking this denominator to zero, and so if you, let me try to be a bit clearer here. This is an exact expression. So now we are trying to look at the features of these expressions and the main feature we would like to locate the poles of this object here. The point is that being these non-termician problem, this can be complex, and so the fixed point of this denominator equal to zero, that is where the poles are, can be in the complex plane. So, this expression here, basically locally, can be described as a sum over discrete poles that are far from the real axis. Ok, while, if we look at the lemon, if you recall, this number here is differences of total energies. So, there's no way that is complex. This number here whatsoever, many body complicated system we have at hand, are real all the time. Total energy differences. Ok, here is the two representations are exact. It is the way how to reconcile them. So, this is the picture, if you want the density of states, spectral function integrated, that we have out of the lemon representation, as we said. A lot of states, ok, feature of the many body problem, not normalized one, a second feature of the many body problem, and these states kind of envelope and form structures, and these structures can locally also be described by, instead of multiple poles close to the real axis, as discrete poles far from the real axis. Ok, so if you want, these are two analytical continuation to the imaginary axis of the same object, and if we are dealing with discrete states, the two actually collapse one onto the other. That is why we can have a quasi-particle equation with the self-energy. So, every single time we have a representation of the Green's function that involves the self-energy that typically is related to this quasi-particle representation. If instead we just have the Dyson orbitals that is related to this lemon representation, the two basically are connected for discrete states. So, somehow if we are not in, when do we have discrete states, when we are not in the thermodynamic limit. So, here, that is also interesting mathematically, the thermodynamic limit makes a mess, as very often happens, that is basically this discrete states gets into a continuum, create a branch cut, and then our Green's function becomes a polydromic function, that is why basically we can have two different analytic continuation, otherwise the analytic continuation should be unique. Here we have two different ones, and the reason is that we have at least two branches of the Green's function. So, the math gets pretty much subtle and complicated. If you really want to dig into this, there's Benam Farid, that has been probably former Cambridge professor, has written a lot of math paper on the math here. If you want to dig into the complexity of the math here, just have a look at this works here. Okay, now, let's go back a bit to physics, so we've seen that, in principle, if we have a self-energy, we can compute the Green's function. Let's have a look at how it looks in an interacting system. So, this is the spectral function of a non-interacting system, just one peak, sharp, dirac delta-like, then we put a self-energy, and here we can simplify the exercise, assuming that the self-energy is diagonal on the basis of states that diagonalizes the Hamiltonian, so this becomes a scalar problem. We can have a Taylor expansion around this point here, epsilon i, so the non-interacting eigenvalue, that is this point here plus the real part of the self-energy. We do the Taylor expansion, and we end up with this form here of the Green's function. So, this is, first, a pole that has a main structure located at capital EI, so the pole is shifted, then we end up with the finite broadening of the pole, so here is infinitely sharp, here we have a finite broadening, and this is really related to the imaginary part of the self-energy. This is important, so the imaginary part of the self-energy has a role, has a meaning that is the broadening of the spectral features we get into the calculation, and then we have something that is also interesting, and natively many body, are normalization factors, so this is not one, this normalization factor deals with the derivative of the self-energy with respect to the frequency, so it's related to the frequency dependency of the self-energy, and so this basically says that the main quasi-particle peaks does not integrate to one, but integrate to zero seven. Where is the extra 0.3 weight? Here basically goes somewhere else, typically into a satellite structure here. So, take a message here, sharp features in the non-interacting case, broadened, shifted, and broadened feature in the interacting case, with also some spectral weight due to the renormalization factor. Okay, here is the nice cartoon again, probably from the Damashelli paper, so this is k resolved, so arpes from a non-interacting system, so k resolved density of states, this is how it looks like in an interacting system. So we have sharp peaks close to the Fermi level, then the farther we get from the Fermi level, the broader the peaks get, the more damp, and when we reduce the spectral weight, we also need to compensate, so we have extra structures that are the satellites. Good. All of these are intrinsic features that can get already out of the spectral functions, the spectral density, and are due to the dynamical, so frequency dependence, and non-ermician nature of the self-energy. So this extra complexity that we have in the self-energy is actually what mathematically gives the physical content of the many-body complexity. Okay, GW self-energy. So this point we have understood that the Green's function are interesting, that they actually carry a lot of physical information that can be used to interpret experiments, but we don't actually know how to compute them in practice, and here is the last part of this talk that is how do we get a self-energy that we can use to compute our Green's function, and a very popular approach to this comes from Eddins equation, so Eddins in 1965 introduced this equation, basically this perturbation here, and then used a linear response approach with respect to this perturbation. What I think Andrea, as they call the Schringer approach, it leads to this set of equations, five equations, self-consistently, so set of five closed equations. One is the Dyson equation that we already know for connecting g to sigma. There is also a very similar equation connecting w to p, that is the irreducible polarizability, and then we have an expression for the self-energy that involves g, w and gamma, that is the vertex. This object here that is pretty much related to the linear response of the Green's function itself. So this is a thing that g gamma g is the linear response of g itself, and this is probably the g minus one in the v, so it's the linear response of g minus one, so this is really linear response. If you're familiar with time-dependent DFT, it's the fxc, but here the kernel is way more complicated, the kernel is the self-energy, and indeed we have the derivative of the self-energy with respect to g. So this is the linear response input into the vertex, and this enters, of course, in the linear response cast into the form of polarizability. These are closed, very difficult to solve and connected in this form here, the so-called adding-spentagon, mathematical note, all these Dyson's equations can be cast in the form of geometric series. This form here that you can re-sum, and if you sum the series, it exactly ends up with the formal solution of these equations, importantly, I tried a couple of times to do this, and this works pretty well if your vp is closed. So the fact that this series converges within a radius of convergence of one is true, and so somehow you can use this if your vp is small. Otherwise, instead, the actual Dyson equations do not suffer of this problem here. So the step in going from this contracted expression to this basically involves extra assumptions that lead you to convergence issues. So adding this equation can be cast in the form of Feynman diagrams, and I think that these equations being this complicated here, diagrams are very useful just to describe and to look at the topology of the equation, so which Green's function vertex is connected to which other bear or screen interaction and so on. Now, out of these equations, basically we can immediately devise an approximation that is called GW if we drop this extra term here. So basically these equations, there's an internal hierarchy, need to be broken somewhere if you want to start guessing some approximations. So if we drop this and we approximate the vertex just as one point, that corresponds to a simple linear response for the Green's function as if the sigma were not depending on g. Basically we have then dirac deltas here, dirac deltas here, and this gives us immediately the polarizability written as gg that is probably cast in a different form, but it's exactly the polarizability you would get out of a non-interacting system whose Green's function is g. And the self-energy becomes g times w. Andrea yesterday introduced the GW approximation in a juristic way looking at the physics of the screening you get there. This is a mathematical way we can look at it. Here in terms of diagrams is the way it looks like, g times w and if we take into account the geometric series for w we have g times v then g, v, v and a single bubble that is a polarizability two bubbles, three bubbles summed up to infinity. This is so-called RPA screening. So, some messages. GW we've seen, we have dropped part of the adding equation accounts for dropping diagrams and there are a number of diagrams that are clearly not there and the first one that comes to mind is already the second order and is the second order exchange diagram. This is not there and yesterday we were talking about self-interaction. This is the main responsible for GW not being exact in one electron cases because this diagram is not there and it doesn't correct for the contribution out of this diagram here so basically we have self-screening in GW. Second thing is in this language of Green's function and Coulomb interaction we can write the Artery Fock as g, v and the diagram would be this. GW is just something that is similar to Artery Fock but takes into account screening not just static screening, dynamical screening the two have to come together in a sense so if you want is just one step pretty complicated one in GW there is much more complication but somehow is next step to the Artery Fock. Still in terms of resemblances GW, we saw this expansion here up to infinite order so this is all resummed we can see some similarities with the second born approximation that in terms of an energy is related to so second born is a self-energy but can be derived from the MP2 total energy of quantum chemistry without self-consistence then there are some fine details here if we take this MP2 total energy and the second born self energy we see that the first so this is the exchange diagram second order direct diagram is there and then in MP2 we just match the second order with its exchange counterpart so this is particle symmetry so this is exact in one electron cases instead GW keeps going with the infinite terms of polarizability so it's not exact GW on the other side has a very much better polarizability so for systems that are very polarizable GW weighs better than MP2 so somehow MP2 actually is used for small molecules in chemistry and so GW works by not a secret it works very well but as we said issues self interaction is there I've listed here a number of issues there are number of recent papers even more recent than these about how to better include vertex corrections so vertex corrections has been a mantra in the looking at the GW literature every now and then there is a suggestion for vertex corrections this is pretty much still an open field so satellites in GW we've seen that satellites are genuine features of interacting systems actually they are pretty bad in plain GW so if we do G0, W0 they tend to be at wrong energies if we take some self-consistency in there basically the satellites are dumped and you get this vanilla satellite that is spread all over the frequency so they are really dumped here the fear is difficult to read but you can really see that if you put self-consistency in satellites are gone here there's Tomaso Chiarotti's poster that has some more results on the in this case similar results were also recently obtained by I think this is diamond fully self-consistent GW that is solid line basically the dash satellite is just gone this is another very interesting work on satellites by the group of Lucia Reining that just compare recent photomission measurements in silicon in this case and this red here would be position of the GW satellites that clearly you see doesn't match the photomission position you have to include extra terms to GW and they do it by doing GW plus cumulant I think that nowadays that is a pretty common beyond GW approximation this was 2011 has been used since this days is very popular and when you add cumulant on top of GW you actually get decent positions of the satellites then in order to better compare with the photomission amplitudes you need to take proper into account the three-step model so not just the first step that is this spectral function but also secondary losses extrinsic losses and so on easily and you get some very decent comparison also here just other recent work thanks