 Zdaj smo prišli svojo prišličnje spike, tako še je hodnje v sku, ne za vse, ali za prišličnje. Zdaj je Mateo Gatti, z EKOL Politeknika, v Franske. Kaj je prišličnje, nekaj nekaj detaj, nekaj nekaj potrebečnji teori, z vsej vsej, tako pa, Mateo. Čakaj, Davide. Dobro vse na RCTP. Det del je, da vse preddaveš, ali ja hamam tudi, kako prav tebe vse zej, a ki je tudi neko Lega, ti moj rebunje čudov, n prove svoje. Tako, nekaj možem tudi prijo, ki bomo neko, da bo ni bo, Or complain, you can use my e-mail here. So, the idea of Lodčilles Letcher was to make the bridge between what Pedro has just told you about made by Well agu N comptrionchuckles theory and the equations of rins mieg n shares theory and the applications. Any about spectroscopy, that you will start to static out from next lecture. Spotrofljenje pa z專adnjih vseh pavnoje obojevače je zaradi pobitno vsakro flaske, da se se udrije sprednje diskalitih. The last argument of these lectures is about what comes next, after you introduced all these formulas, what comes next. The answer, considering the point of view of the formula, Zato vzpoče, da je prejzupočno prejspoljeva, je to tudi tudi o petroskopu, še to je najbičaj zelo, vzpoče, da je začela, je to predikšnja, sej analizi, je to zvukovana spektra. Završenje spektra je zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo zelo. Tama tudi možno je nekaj materijali for solar energies, so new solar cells, new materials for memory devices, new materials for new colors for candies. So the applications of spectroscopies can be very, very different and these functions are made for calculation of electronic excitation spectra that are at the basis of spectroscopy. The first question we should ask whenever we start to use analytical formulas is why we do what we do. So this is the first question that I will try to address in the following. Why should we use these functions instead of, for instance, the may-body we function or the density. So what we are interested in when we want to do spectroscopy or in general when we solve quantum mechanics problem are observables. And we have learned from basic quantum mechanics that if you are at temperature zero and for a given number of particle n, is that observables are expectation values of some particular operators that are related to the specific observables. And expectation values are calculated as integrals with respect to the may-body we function of the given operator. If the system has n electrons, the final answer is maybe just a number, and at the end what we have to do is a very large integral with respect to many of the degrees of freedom in our system. So this is a very complicated object, but in principle, if you know the may-body we function, what is here is just a recipe that is telling you that if you give me the may-body we function, I can tell you which is the observable that you are interested in. You have just to perform this very complicated integral, but this is a functional that is in principle known. The only problem is that this is very complicated and you have to make a very large effort to calculate and to store this may-body we function, which in most cases of interest is impossible to do. And after all, after you have made this very huge effort, you are just interested in a number. So this is not a practical way to go. The alternative is density functional theory, and in density functional theory we have learned that instead of calculating observables using functionals for the may-body we function. We replace these functionals with other functionals. These functionals could be in principle functionals of the external potential that defines our system. And in density functional theory the idea is to calculate observables as functionals that in principle are different from the original functionals. And these functionals are functionals of the ground state density. And Oember and Kohn has told us that in principle we can calculate all the observables, not only the observables that can be explicitly calculated as observables in terms of these intervals of the density. So single particle operators, but we can calculate in principle any observable as expectation value in the ground state wave function in principle all the quantities that we are interested in. And this is in terms of functional of the density as I've just said. So instead of using the may-body we function as our descriptor of the system which is what very often we do in quantum chemistry when we do simulation for small molecules where the number of degrees of freedom is limited and this is called configuration interaction theory or in case of quantum Monte Carlo calculations where we make a stochastic estimation of this may function instead of using this as descriptor in density functional we use the density which is a more compact object as our descriptor. And density functional theory is telling us that we can calculate everything in principle as a function of the density. But the first question that we have to answer is where do we get the input that we have to insert in this functional which is the density. And the answer given by Konešan is the genius idea to build an auxiliary system which is the Konešan system that is just meant to give us the density that is again the input that we need to put in our functional to obtain the observable. It's an auxiliary system because as we have already discussed previously the Konešan system is just meant to give us the density but for instance the eigenvalues of these Konešan questions are not supposed to give us information about excitation energies. The only information that the Konešan system is supposed to give us is the density. And we know that we have to solve this system of equations that are the Konešan equations where we have the Hamiltonian that is an effective Hamiltonian where in addition to the external potential that describes our system we have the RT potential that is related to electrostatic interactions so classical electrostatic description of the system plus a correction that is the string correlation potential of Konešan that is a local potential, that is a function of the density that is formally defined as a functional derivative of the string correlation energy with respect to the density and this is the quantity that we have to approximate. So this is what Konešan gives us, it's the density but the second problem of DFT is that for most of the cases we do not know the functionals. So even if we have the density we are not able to express our quantity of interest as a function of the density. And the typical example is something that has to do with spectroscopy and excitation spectra. So this is what we are going to discuss in a moment. To summarize the exact Konešan system is designed as the good density. It's often used to calculate observables directly in the sense that we use the functionals that were in principle supposed to be used in terms of the many body wave function but we replace the many body wave function with the Konešan many body wave function and this is an approximation clearly and the problem is that for most of the cases we don't know the explicit function of the density. So the quantities of interest for us in principle are function of the density but we don't know them and if we use instead the original function in terms of the many body wave function but with the Konešan wave function then this is an approximation and it's not supposed to give us the good answer. This is because the Konešan system is an auxiliary system that is meant just to give us the density. And the first example where we discuss this is photomission spectroscopy. In photomission spectroscopy there is some light that is used to kick out one electron from the system. So typically you measure the kinetic energy of the photomitted electron and you can also measure if you do under-result photomission the angle of the emitted photoelectron. And if you combine this information related to the kinetic energy of the photoelectron in its direction, so its angle that you can associate to the k point inside the material what you can obtain is a collection of dots that you see on the right panel and for instance you can fix a given dot that corresponds to a pair of energy binding energy and k point that is associated to the kinetic energy and the angle of the photomitted electron and then you can scan the angles and the energies and what you obtain are these dots here that correspond to the non-structure of the material. This is what photomission spectroscopy is and if you combine direct photomission that measure the occupied states with inverse photomission that measure the unoccupied states and you can retrieve the structure of the material. So this is the experimental definition of a structure. Now in the independent particle picture what you would obtain in the description of this experiment is just the fact that you think that you are removing one electron from the material and this electron was occupying the single particle levels at different k points if you are in a solid and so the description is just a collection of very sharp peaks mathematically these are delta functions at the energies that were occupied by the electron before being emitted from the material. So this again is for direct photomission where you remove one electron from the system you go from n to n minus one and this would be for the unoccupied levels for inverse photomission where you put one electron to the system. So this is the independent particle picture just a collection of delta peaks at the energies of the single particle levels. And this is also what function DFT gives us and in particular this is a comparison between the LDA function, the structure on the left and the experimental results in the center right panel and you see that this is the band structure of bulk germanium and in particular you see here that in LDA you obtain a band structure that has a problem in particular around the band gap because LDA is giving a metallic one structure and this is the famous or the infamous Konsham band gap problem it's the fact that typically in Konsham LDA you get an underestimation of the band gap and this has two reasons first is that we are using LDA that is an approximation of the exact Konsham system and second and most importantly it's the fact that the Konsham system is an auxiliary system that again is meant to give us just the density but it's not meant to give us the band structure and also the band gap and this is also the case in if you have the exact exchange of correlation potential for instance you can also think that you want to calculate the band structure of the homogenous electron gas so in this case the LDA would be exact of course the homogenous electron gas is a metal so there is no band gap but you can still compare the band structure with more advanced theories like Green's Function Theory and you would see that even the exact Konsham band structure because LDA is exact so it's known is not equal to the exact band structure you don't get the exact band structure of the material in the case of the gas this can be proved directly because you have the exact approximation and you can have very good approximation for this potential in materials and you would obtain also in that case that the band structure is not equal and the reason is that again I want to stress the fact that we don't get the exact band structure because we don't know how to express the band structure in terms of the functional of the ground state density so what we have to do is we have to change the descriptor again instead of using the made by me function and density or the density we have to use a different descriptor and in our case the proper descriptor would be green's functions green's functions have been already introduced by Pedro these are describing the propagation of additional particles in the system and in particular I will first of all discuss about the one body green's function and it's linked to spectroscopy so this has been already introduced and the one body green's function is a function of two points in space and time and one frequency and it's a function that can be expressed in this way with this fraction representation that is called the Lehmann representation and in particular for what is interesting for us is the fact that this function has a function of frequency as poles that correspond to the zeros of the denominator and poles correspond to the addition that we measure in spectroscopy and these are those that describe the structure of the material so we use and we define the green's function indeed because the poles of this function will be the exact quantities that we are interested in contrary to and in particular a quantity that is of interest for us is the imaginary part of the green's function and this imaginary part of the green's function is expressed in the reciprocal space so you have just taken a free transform of this quantity and you go to reciprocal space, so in case space and then you can see that you can express this imaginary part of the green's function and this expression here where you have peaks at the additional removal energies that were the poles of the green's function that are weighted by these amplitudes that are at the numerator of the green's function that are called the Lehmann amplitude and this quantity is of interest for us because it's called the spectral function and it's the quantity that we can compare with photoemission and in this way you see that we obtain quantities that we can compare with photoemission as simple functionals of the green's function in particular in this case these are very simple linear functional of the green's function we have just to take the imaginary part of the green's function, the absolute value and obtain the spectrum now you have seen that the Gaussian system is auxiliary system where you have an effective potential that is meant to give us the density in the framework of green's function and one body green's function in particular we have to generalize this idea of auxiliary system and instead of having this local and real potential that is the change correlation potential we introduce a more complicated potential effective potential that takes into account all these change correlation effects beyond the art description of the system as in the case of this change correlation potential of Gaussian but it has more features because this new effective potential is now no local space so it acts as an internal operator with respect to the wave functions and moreover it's frequency dependent because the green's function is describing a propagation in time so it is no local in time and if you take the Fourier transform it gives us a frequency dependence so the self energy is not only an operator it is no local space but it is also no local in time and it means that all the quantities become frequency dependent including the wave functions the effective wave functions the quasi particle wave functions and the energies so this is just a new auxiliary system that is meant to reproduce the single body in green's function so a more complicated object and this means that this more complicated object requires a more complicated effective potential that is the self energy you have seen that green's function can be calculated as a solution of the Dyson equation and this Dyson equation is just equivalent of this effective equation that I've just written here and the solution of this Dyson equation can be formally obtained as inversion of these operators where you have the self energy and the single particle non-interacting green's function and easily expressed in terms of the single particle energies the non-interacting energies so this is a quantity that we are readily able to obtain to calculate and the self energy is introducing all the effects related to exchange and correlation and from the imaginary part of the green's function we can calculate the spectral function this is the expression that is explicitly written in terms of the real and imaginary part of the self energy the self energy is a complex object contrary to the consham potential that is real the self energy has a real and imaginary part so both contribute to the imaginary part of the green's function and to the spectral function and the description of the spectral function in terms of this more complicated object is much richer than in the case of the non-interacting picture so in addition to these single particle levels you have other features that are described by the contribution coming from exchange and correlation that is encoded in the self energy and in particular the real part of the self energy is renormalizing the energy of the peaks so it's moving the position of the peaks and the imaginary part of the self energy is inducing a broadening of these peaks so each of them is moved the imaginary part of the self energy is also adding new feature in the spectral function so all in all instead of having a single peak that is very sharp you have a peak that is more broad which is still a prominent peak very often and this prominent peak that has a different energy and it is a different broadening it's called quasi particle peak it is a dominant contribution in the spectral function this is what still defines the bond structure in an interacting picture but in addition to this you also have other features that are called side bands or satellites that are a pure effect of exchange and correlation this has something to do with the fact that when you remove one electron in the material and this positive charge is perturbing the system and it can create additional excitation in the material and you see these additional excitation as satellites for instance this additional positive charge can induce the collective excitation of the charge that are called plasmons in the material and you see these additional excitation as in the spectral function and this would be plasmon satellites so these satellites are due to the coupling of the hole of the positive charge in the material and additional neutral excitation that has been created by the fact that this positive charge is the perturbation in the electronic system again this is a pure effect of correlation and it cannot be captured in a single particle picture so the description in terms of the Green's function requires a more complicated object because it has to describe a richer physics which is related to a normalization related to broadening of the peaks and to the creation of these additional satellites now all this is in principle exact for the one body Green's function and in the case of the one body Green's function in this more complicated auxiliary system we have the same problem as in the conscious system we need to make approximations for this self energy, this effective potential the advantage of the Green's function framework is that we can define effective particles and we can have an intuition of the physics that we want to describe instead DFT is a made body theory of collective variables which is collective variable which is the density in DFT we have Cauchon electrons but no one has ever met a Cauchon electron so it's not something that we can measure we cannot use our intuition in physics this is something that we can exploit in terms of Green's functions there are different ways to approximate this self energy one way of approximating the self energy is very similar to LDA and it's developed in the framework of dynamical material theory where we express self energy as a functional of the local Green's function where local means that we just take a particular part of the Green's function in a particular space in the orbital space and we are interested just in this reduced part of the Green's function and we express make an approximation of the self energy as a functional of just this reduced Green's function but this is not what we typically do in many body perturbation theory and in what you will discuss in the rest of the school instead what we do in many body perturbation theory we just follow the propagation of this additional particle that is what the Green's function describes and you have already seen that you can follow these different stories that have to do with the effects that are related to the propagation of this additional particle design drawing final diagrams so whenever you have an error this corresponds to the propagation of one additional atom in the system and then you can think about what this is causing on the rest of the system and this can be expressed in terms of interactions so you have the coulomb interaction that is represented by the dash line and you can think that this additional particle is capturing with the density inside the material and this density in the language of final diagrams is represented by a circle like this so this corresponds to the artery potential the artery term in the Hamiltonian it's the scattering of the additional particle with the density you can have exchange diagram like this one that corresponds to the fork exchange in artery fork and you have more interesting diagrams where you have more interaction lines and in particular will be interested in this kind of diagram where the additional particle in the system is polarizing the medium and this polarization is represented here by this ring diagram also called the purple diagram where you have the propagation of one other particle and the propagation of a particle with the reverse sign because the error in this way has a reverse sign and this is corresponding to the propagation you can understand this diagram as the fact that this additional particle in the system is creating an additional electron in pair in the system and this is associated with the polarization of the material that is related to the fact is the consequence of the Coulomb interaction and it's the consequence of the fact that the system the medium is seeing the presence of an additional particle in the material and these polarization diagrams are those that are creating a screen a reaction of the material for the presence of this additional particle and all this is something that we have to take into account and these effects are all encoded in the self energy so the self energy is just the collection of all these possible stories that we have to keep track and there are consequences of the fact that we have an additional particle in the material and the spirit of making approximations in the Green's functional theory framework is just that we have to select the most important stories the most important physical effects that we have to take into account and in the physics in the GW approximation in particular we take into account in the self energy the exchange effects plus these polarization effects these screening effects and the GW approximation is called GW because the self energy turns out to be the product of the one body Green's function and the screen Coulomb interaction W and the screen Coulomb interaction W is screened by the inverse electric function and instead of adding this per Coulomb interaction P you have this screen Coulomb interaction W and the difference is given by this inverse electric function that is the scrubbing screening and this screen is related to this polarization this creation of additional electron pairs in the material so the GW approximation is describing the coupling of the additional electron or the additional Coulomb with electron pairs in the material that are screening the propagation of this additional particle in the material so you can understand we are reaching the question time so you have time for a few months ok, thank you so the GW approximation can be understood as the propagation of a boat on a surface on a sea, on a lake and this propagation of this boat is creating waves and these waves are those that are describing this polarization and this is in contrast to Artifox and Artifox can be understood as ice skating so you skate on ice and the ice is not reacting so it's completely frozen it's not able to polarize to react for this perturbation so in GW you take into account these additional waves in the propagation and GW is today the standard approximation to calculate one structure and it's in very good agreement with experiments so these lines we are calculating in the GW approximation by Rothing and Ljue and it's also correcting the Artifox estimation of the band gate now GW is not only able to calculate to give us the one structure but is also able to go beyond this quasi particle description and it's also able to give us satellites and this is a relatively fact that we can also calculate this frequency dependent object that is the self-range and here it's an example and then I will conclude with this it's the photo mission experiment of bulk aluminum in the middle panel and this is compared the calculation of the spectral function on the left that is calculated in GW plus cumulant, cumulant is just vertex correction to the GW approximation but let's say this is a calculation of the spectral function and you see that the main feature is this one that corresponds to the band structure of aluminum so you have just a parabolic band and in addition to this you have replicas that are satellites and if you measure the distance between the band and the satellites these are equal to the plasma energy of aluminum so we can understand them as plasma in satellites and you see that this GW plus cumulant calculation for the spectral function is able to capture the physics but it's not in quantitative agreement with the experiment if you want to be in quantitative agreement with the experiment you have to take into account all the phenomena that are related to the photo mission experiment in itself and if you do so then you can get a good agreement quantitative agreement with the experiment this is what we have done in this right panel and this photo mission spectroscopy will be the subject of the next lecture by Polina so you will get more details about this in the following I will skip the discussion about excitons we will have the occasion to discuss about excitons in the rest of the week and I will just go to the conclusion and the conclusions are the fact that we use Green's functions because of several we are interested in spectroscopies and the different spectroscopies are linear function of the Green's function so these are non functional of the Green's function if you calculate the Green's function and perturbation theory gives us a very powerful way to devise approximations because it's related to intuitive picture of propagation of additional particles in the system something we cannot do in Consum and in particular I have discussed the use of the one body Green's function for electron addition removal energy and you will see in the following we can use the two body Green's function for the calculation of electron excitations associated to absorption and other spectroscopies that are measuring neutral excitations like electron energy loss spectroscopy and elastic stress elastic stress scattering spectroscopy and the frequency dependence of the self energy and other kernels brings us information that is beyond the quasi particle picture and with this I would like to thank you for your attention if you want to know more information on our activities in particular on what Lucia is doing you can look at the website Lucia has also prepared an online kulsera mock on density fashion theory this is the address and there is also a book written by Lucia on body perturbation theory Green's functions and quantum of the car and this is the book and with this I would like to thank you for your attention and I wish you fun with Green's functions and the rest of the school so thank you very much Matteo and the session is open for questions if you have any or maybe Matteo I can start with one so of course in your group I would say the most expert group on satellites and you have shown this last example where there is this plasma satellite in the photo emission spectrum of bulk aluminium and as far as I understand you can have a plasma satellite because of correlation effects what you have described but you can also have some kind of plasma satellite because of all the effects which we do not account the fact that we compute the spectral function but then the electron is going out and exciting possibly other excitations so can you comment on that is there a way to disentangle when a satellite is correlation driven and when it's not? Ok, so first of all we are not the most expert group on this but we are one of the groups working on this aspect and the reason is that satellites are a qualitative feature of correlation so if there is no correlation you don't have satellites so this interesting because it's really a fingerprint of correlation Yes, then to answer your question, yes so in a calculation that is just a calculation of the spectral function there is no photon energy dependence so the spectral function doesn't depend on the photon energy instead experimental spectra do depend on the photon energy and there is always a contribution of the outgoing photomitted electron and this outgoing photomitted electron can also scatter on the way out from the material and it can also induce plasma satellites as the all that is left behind and then there is also interference effect between these two effects which contribute to plasma satellites and not only and this is a photon energy dependent so this is the step that we have to take into account when we go from panel A on the left to panel C on the right and this is very important if you want to reach quantitative agreement with the experiment indeed and very often we discuss about strong correlation effects so corrections to correlation because we find deficiencies in our description of photomission spectra but often I have to say that most important effects come from the fact that we are not calculating directly what is measured what is measured is the photomission spectrum and if we calculate the spectral function this is not exactly the same thing but there is a gap between the spectral function and the photomission experiment and indeed we have put a lot of effort to go beyond this spectral function description for photomission it's very important to try to work together with the experimentalist to bridge this kind of gap this is now clear maybe we can start with the questions online mukeš sing mukeš sing maybe he can do the question himself so can you try to unmute yourself because the name is mukeš sing if I spread it correctly ok maybe we can read the question so the question is sir how do we know that how do we know what term should one use for the calculating surfanage I guess that you are referring to this it's so in principle the self energy is describing all these change correlation effects in the electronic system so if you have it you have the exact answer for the one body Green's function so you calculate it exactly in practice we have to make approximations and the approximations are related to the physics that you need to describe and the advantage of Green's function theory is that you can keep track of the physics because you have these effective particles that are interacting in the material and you have to as always in life you have to make a choice and the choice is related to the most important physics of your material and in the case of the GW approximation it's a specific case we make the choice of identifying the most important physical aspect that is screening that is this polarization of the system for this propagation of this additional particle in the material it's not the only choice there are different choices that can be made in particular in the Green's function formalism we can associate approximations with couplings in the case of the GW approximation we describe this screening physics in terms of couplings between quasi particle and neutral excitation of the charge so electron pairs associated to the charge and for instance we completely miss the coupling with spinning excitation which are the other kind of bonds and in your material your system of interest as this specificity that this coupling is important you should choose other type so the choice is related to the physics Thank you Matteo and I read another question from a student online so it's actually a double question so the first one is what does it mean to speak about non-locality in time and then the second question is why the name self energy so why do we call it self energy non-locality in time it's due to the fact if we can understand it from the fact that the Green's function is the propagation of a particle from a time t to a time t prime so this is giving us already one reason why we have no locality in time the second reason indeed is more subtle in the sense that the mini body Hamiltonian is static in our case it's not time dependent so we have a static problem and when we introduce the self energy we make a choice which is we want to avoid to take into account explicitly all the mini body possible excitations in the system that are two particle three particle, four particle excitations and we fold all these mini body excitations in a frequency dependent effective potential that is the self energy and this is like an embedding strategy and indeed self energy it's called in this way because it's a way to describe the effect of the other excitation on the particle itself so this is one of the reason why it was called self energy Ok, so I think we have time for one last question if there is any from the audience here otherwise we can pick up Hello, sir. Hello So my second question was that let's say if we put the self energy and we can improve the results Will improve, sorry Will improve the result maybe energy and band structure Ok Again it depends on what you are interested in if you are interested in the density the Konsham system is enough because the Konsham system is meant to give us the density if you are interested in band structure you cannot use the Konsham system because the Konsham system is an auxiliary system so it's a system of effective particles that are auxiliary particles that are just meant to give us the density and you have to introduce a different framework and in particular the Krins function framework where you have this effective more complicated object is the self energy it is effective more complicated and the auxiliary system is meant to give us for instance the band structure then the quality of the results of this more complicated framework depends on the quality of the approximation for this exchange correlation send energy so if you make a good approximation for the send energy you get good results if you make a bad approximation you still get bad results ok sir so one more thing is there any other thing apart from the density of state which can be calculated from Konsham equation exactly not the density of states the density yes sorry density ok in principle you could calculate from the density and you know that in principle you could calculate all the observables as functionals of the density this is what DFT tells us yes but we don't know the operator you told the problem is that for most of the cases we are interested in we don't know this functional ok so in some cases we have good approximations to the functional in particular for the total energy we know that we can have good approximation to this functional which is also implicitly unknown but we have good approximations the LDA, BBE these are the widely used functional so for the total energy we can obtain good estimates for quantities that are related to spectroscopy this is much more difficult so in particular for bad structures we don't have these functional we don't have ok so maybe how do we know that the we move to the last question from a person here in presence and then we have to move to the next so you can put the question on the chat and we will try to address it ok ok thank you ok so how do you build this greens function so if from the equation you are showing we basically compute this from the Komsom Eigen states but like so when computing this self energy we do some Dyson expansion so we have like for example higher order greens function terms when we do this expansion so how do you compute these things because this is like an independent particle level at an independent particle level but somehow you have to include some interactions between these things for example if you are computing the dielectric tensor for example in the when you are doing this thing we have to compute the dielectric tensor so the imaginary part of the dielectric tensor is somehow related to the we have to consider the interaction and are these when doing this thing because when you are doing the GW I guess we don't consider these effects so how important are these effects in the bandgap collecting the bandgap and the bands ok, many questions but I try to be synthetic so you calculate in this Dyson equation representation you can calculate the greens function by any approximation of the self-energy ok, because this G0 is something that you can obtain directly from the single particle energies all these change correlation effects beyond the single particle description are encoded in the self-energy so anything that is related to interaction is in the self-energy any approximation to the self-energy can be used to calculate the greens function in this in practice you will see when you do for instance GW calculation in practice with Yambo how this is important the effect of the self-energy is represented here on the spectral function so you go from this single particle picture where you have sharp peaks to this richer description where you have peaks that are broad and where you have additional features how this is important for the band gap I would say not only for the band gap but for the band structure in general and for effects beyond the band structure that are satellites an example is here for germanium so instead of having a metallic material as in LDA or a very large gap as in RTFOC you are in much better agreement with the experiment and the difference between GW and RTFOC is in the screening so you have to calculate this in screening and you will do this in practice so don't forget to come also in the next days to the school OK, so thank you again Matteo