 Ja, je bilo vzlušen. Tako. Zdaj, da se mi je vzlušen? Tako, da je. Vzlušen. Alberto zelo izgleda z VW, VW pravno. Vzlušen, da se vzlušen. Vzlušen, da se vzlušen. Tako, da se vzlušen. Zato sem bilo vzlušen, da se vzlušen. Zato sem bilo vzlušen. V prvi početnih, da sem bilo vzlušen. Zato sem bilo vzlušen. Z sentenced više Johnyp Nshop to je gled服č요. Lašš propos, da se vzlušen vzlušen. Zato sem bilo vzlušen, da sem bilo vzlušen. Zato sem bilo vzlušen, da sem bilo vzlušen. SERJAN GRUBIDGÅ Supreme adjustments. Dov deti f otroga, res ruins ja, a nrne enniko od d plungov. Zato sem bilo vzlušen. Vzlušeno od cm. In tukaj imamo tukaj dyagrami, kaj je korisponjena. Zdaj sem prišljena tukaj, ker je tukaj skupnjena. Tukaj je, da, kaj je tukaj pošličen, kaj je pošličen, tukaj je tukaj pošličen, kaj je korisponjena kaj je pošličen. But of course this is very difficult, and this is mainly due to the fourth equation, that is the equation for the vertex function, then the, what we can do is just to neglect vertex effects and this makes the self energy to be written as the famous GW and this is why this is called the GW approximation. While for the skinning interaction we have heh, a korrespondent grandpa approach & then we would solve self-consistently these equations here for the G that is also employed in polarizability and the self-energy, but what is usually done in practice is not to solve so consistently. The interacting greens function within these equations and these brings us to the genot-w-not approximation. this is because g naught is employed in the calculation of the polarizability, which brings to w naught, and also, this g here is computed with independent particle approximation. So what YAMBO implements, in practice is more of less, these five equations here, written in a more numerically explained picture, as we will see. Of course the fourth equation is treba famously solved. The main goal of this part would be to see which are the corresponding of these equations here, and these in order to deeply understand which are the main approximations involved, also in practice, and also which are the main converges, parameters that should be taken under control. But, first of all, it's important to comment also on the basic set representations, in tudi vseh operatorje se priče, da je pričočnje, način, na zelo. Vseh je zelo pričočnje, in pričočnje pričočnje. Jambbo je zelo dve zelo pričočnje. Prvo je zelo včočnje, zelo v Kvantume Espresso. Vseh je Konsham State, ki se je zelo v Kvantume Espresso. Vse več, da se vse zelo potrebe, je naredil v nekaj spasih. Zato, da bilo kulob interakcije, je vse zelo v zelo v pomečne vzene, kaj je vse zelo v 4 zelo v konšampstajske. In, da je vse vzelo v konšampstajske, is diagonal in the basis of conchium states, while it depends on gg prime and q in plane waves. Thus what Jan Bodaz is to solve the electric properties in plane waves, then to compute one element of the self-energy in conchium states, and this is an approximation as we will see in the next slide. Then there are the matrix elements to change from one base to the other, that are these ones here, and it's also important to notice that in practice we are going to cut off the basis function, thus two important parameters that should be converged are the number of plane waves, which are usually controlled by an energy cut off, as again in quantum espresso, and the finite number of conchium states, while the complete illberate space would be represented with an infinite number of conchium states. So how to solve the Dyson equation for g? Here we have the diagrammatic representation and also the Dyson equation. Ok, so the pointer is strange. And with the lemar representation for the interacting g, we can rewrite this Dyson equation in an Hamiltonian formulation. That is an eigenvalue equation for the quasi-particle energies, which are the eigenvalues, and the quasi-particle orbitals, which are the eigenvectors. This is kind of a difficult equation to solve, then the quasi-particle approximation is usually employed, and this corresponds to suppose that the quasi-particle orbitals can be conveniently approximated with the conchium orbitals. This is not always true, but this is usually implemented, thus one has to keep in mind whether this approximation is good or not. This brings the eigenvalue equation to just an equation for the eigenvalues, in which we have to a correction to the conchium eigenvalues given by the self-energy. Thus, in order to obtain the quasi-particle eigenvalues here, we just need to evaluate the self-energy at a certain energy. Then usually this equation is solved within the Newton method, in which the self-energy computed at the quasi-particle energy is still or expanded along the conchium energy. Then we need to evaluate the first-order coefficient that is usually computed with a finite differences method. This is thus the goal of a g0 W0 calculation, which requires to compute mainly the self-energy at two different frequencies. One is at the conchium eigenvalue, and one is on a slightly shifted frequency in order to compute the first-order coefficient of the Taylor expansion, thus these, which are called zeta coefficients. So this is the goal, and in order to reach the goal we need to compute the screening interaction. As we said, this is computed in g space, so W, and we start from the irreducible polarizability, which is a bubble diagram. This is the expression in g space. I'm not going to comment more on that. What it is important then is that we have to solve the Dyson equation for the irreducible polarizability, kaj not. Thus we are not including just a single bubble diagram, but all its geometric series. And as Andrea Marine shown, this is very important. In order to do this, and to solve this equation, I have put here just directly the solution, we have to perform a matrix inversion and a matrix multiplication. Once we have the reduced polarizability, we can compute the inverse electric matrix and the screened interaction in Fourier space just with matrix multiplications. It's important to note here that the calculation of the screening is often the most high-consuming part of the computation, and this is mainly because of the computation of the reduced polarizability, which requires an integral over the Brillouin zone, a sum over unoccupied states, and this for each g prime matrix element. Then we need also a matrix inversion, and all of this for each momentum transfer and for each frequency, we want to compute the screened interaction. So it's a lot of computation. Then it is also important to note here the fact that we need to tronkate the basis representation. As I said before, we need to compute these matrix elements on a finite matrix, so we need to cut off the number of plane waves, and also we need to tronkate the sum over unoccupied states. This means to have only a finite channel of excitation in this that is an electron hole excitation. We are not considering all the possible excitations, which are an infinite number, but we tronkate this channel up to a certain point. Then of course we have to converge this integral here with respect to the discretization of the Brillouin zone sampling. Once we have computed the screening, the next step is to compute the self-energy. Here we have the Feynman diagram represented in the Cauchon space. Thus we have an incoming Cauchon state labeled with a band index and a k-point index, which scatters, and then we have an interaction line, which is the screen interaction W, which brings some momentum and energy. Then we have also an intermediate state described with this genot, that is in a band index M, which brings momentum k-q due to momentum conservation and an energy omega. So due to the fact that in Feynman diagrams we have to sum over all possible paths, this is converted into a sum over all possible intermediate states, an integral over the momentum transfer and this integral over energy, which is the most difficult part. Then again we have to represent the screen interaction in this case in the Cauchon basis representation by changing the representation with the matrix element I've shown you before, while the genot is diagonal in the Cauchon space. So in order to proceed with this calculation and mainly in order to perform analytically the integral over the energy, the screen interaction can be divided into two parts. So the bare interaction and the correlation part of the screen interaction, which represent the response of the system, due to the fact that the self-energy is linear with respect to W, if we split W into two parts we have two contributions to the self-energy. So the first one is the exchange part and corresponds to the Artre Fox self-energy, in which the self-energy integration is performed trivially due to the fact that the bare interaction is not frequency dependent. While for the correlation part we can use, for example, the plasma and pole approximation as was described in the previous talk, in this way we can again have an analytic integration of the correlation part of the self-energy with respect to the energy. These are the final formulas once the integral over energy is performed. We can see here that, of course, the correlation part of the self-energy will depend on the plasma residue and on the energy of the plasma pole. And that, of course, due to the fact that we model each gg'q element of the screen interaction with the different pole, this residue and pole depends on gg'q. We can see that here we have some overplane waves, as we have seen before. Thus we have to converge with respect to this finite troncation and also we have some over m, which is the intermediate state of this propagator here. Then again to tronkate the number of Caucham states means that we are considering just a finite number of possible intermediate states for this g0. Then again we have to converge with respect to this discretization of the Brillouin zone, which means to discretize the possible momentum transfer here. And we are going to see that for 2D systems this convergence here is very important and not trivial to reach. So this is more or less the first recap. The G0W0 method in one slide so it's kind of a lot of equations, but not too much. Just to remember that we start from a DFT that brings us to the Green's function of DFT that is diagonal in Caucham space. Then we have to compute the electric properties which brings us to the screen interaction in G space then to evaluate the self-energy and to evaluate the self-energy at two frequencies gives us the quasi-particle correction then the quasi-particle energy within the Newton approximation. Thus to conclude the main approximations are that we are neglecting vertex effects and this brought us to the G. We are neglecting also the self-consistent solution of G and this brings us to the G0W0 then we are approximating the quasi-particle orbiters with the Caucham orbiters we have employed the Plasmon-Paul approximation and we solved the quasi-particle equation with the Newton method. These are the main approximations are the number of plane waves usually controlled with an energy cutoff as in quantum espresso and the number of Caucham states included in the calculation of the kainot and on the self-energy and the finite sampling of the Brillouin zone. There are then other minor parameters which should be under control but these are more technical and they also depend on the system you are considering. So this is the end of the first part in this second part I am going to consider how to perform a G0W0 calculation for a 2D material. We are going to see that there are some technicalities which should be taken into account in order to obtain meaningful results. First of all just the definition of what a two dimensional system is for us like for the case of graphine, graphine that is like the most famous two dimensional system we have that the two dimensional system is periodic in two directions for example x and y while it is nano sized in the third direction which is not periodic. In this presentation we are going to focus our self on monolayer materials but the same reasoning holds also for n-layered materials and also surfaces which are not periodic in one direction can be treated with the same scheme. Finally also extensions to the 1D case is more or less straightforward because in this case the only difference is that we have two dimensions that are not periodic and one that is periodic so more or less. So just a brief introduction about two dimensional monolayers as I said the most famous is probably graphine that is a semi-metal with direct cores that is composed of a monolayer of carbon atoms which are in an hexagonal lattice then if you have that the two atoms in the sublattice are different atoms you can have something like HBN which is a high gap insulator that is usually employed as a layer spacer in the systems which are compositions of 2D monolayer materials then we have transition metal decalcogenites that are composed on one plane of transition metal and two planes surrounding of decalcogenites and this can be both semiconductors or metals despite their chemical composition. Finally we have phosphorine which shows a peculiar structure and that it is studied because its electronic properties despite the number of monolayers which are interacting in your system. If we want to describe a monolayer material or a two dimensional material with a plane wave code like quantum espresso or Yambo that employs 3D periodic boundary conditions we can use the super cell approach the super cell approach is quite direct and says that you have a bike but you want to describe the electronic structure of the monolayer just increase the unit cell in the non-periodic direction that is L in this case and at a certain point the monolayers will not interact anymore so you add some space up to convergence with respect to L if you go to a convergence then you have the electronic properties of the monolayer but let's see how this convergence is for the case of the DFT with local or semi-local functionals everything is kind of easy for example just add two m strong of vacuum with respect to the bike gives the convergent gap with respect to the amount of vacuum and this is because the interaction between electrons with local or semi-local functionals is not long range for G0W0 calculations it is the opposite so we have a long range interaction and we can see here that the convergence with respect to the amount of vacuum is very painful due to several reasons first of all because this is interconnected with the convergence of the Brillouin zone then because we need really a lot of space between the layers here we have like 60 m strong and it is important to remember that the computational cost scales as L to the third power then you have probably to extrapolate the value for an infinite layer separation and finally we have also that vacuum states are deeply affected by the amount of vacuum then you have to keep in mind that all these lines require a convergence with respect to unoccupied states so due to these it is like evident that to remove the interaction between the replicas the strategy to add a lot of space between the monolayers is not a successful strategy then we need to explicitly cut off the interaction and this is done in this way so this is the bare qurom interaction that in reciprocal space is 1 over r simply while in Fourier space is 4 pi over q squared what we want to do is to trankate the qurom interaction between the replicas of the monolayers this is done in a real space with a heavy side theta and in this way each electron does not see any more other electrons which are plus or minus L-halves distant in the non-periodic direction so this electron here sees only other electrons in this place here thus we are removing the interaction between electrons that are near to different monolayers if we perform the Fourier transform of this expression here we found this expression here which is the one that should be used instead of the bare qurom interaction we can see here that in the long wavelength limit we have a different power low we have a different power low power low ok and this will change the electric properties so again if we look at the genot-w-not formulas in one slide we can see that to change the interaction change more or less all the formulas because of course the qurom interaction is one of the main ingredients of a genot-w-not calculation we focus on the the electric properties thus on the inverse of the electric function and here we can see for MOS-2 for the case of 20 m strong vacuum the difference of the head of the the electric function thus the g-g-prime equals zero component for the static the electric function with respect to the momentum transfer the difference between the case of a truncated interaction and untruncated interaction can be seen here that for medium to high wave vectors the response of the system is the same while in the long wavelength limit the two system the two responses are qualitatively different while for the case of the truncated interaction epsilon-1 goes to 1 and this means that there is no screening in the long wavelength limit if you do not cut off the interaction you end up with a finite value and this can be described with this cartoon here in which we imagine to have an electron and a hole and we consider the screening between them so long wavelength limit means that the two charges are very far away one to the other and in a three-dimensional material if you have that these two charges are very far away due to the fact that the material covers all the 3D space you still have macroscopic screening while for a 2D monolayer if you put the two charges very far away due to the fact that the material is nano-sized in one direction the vast majority of the field lines travels into vacuum and this means that in the long wavelength limit these field lines are all in the vacuum thus there is no screening thus we end up with the fact that if we do not include the truncated interaction we are changing qualitatively the description of the screening with respect to the truncated case that is the physical one but here we can see that we added kind of a very sharp behavior of the dielectric response in the long wavelength limit and we can imagine that this will so that it will be very difficult to converge these dielectric properties with respect to the Brillouin zone sampling and this we will see in the next part of the presentation so here we can see the convergence of the quasi-particle gap for three prototypical semiconductors with respect to the number of key points of the Brillouin zone and it can be seen here that very dense Brillouin zone samplings are required in order to obtain convergent gaps this is a general property of two-dimensional semiconductors this is a general properties of two-dimensional semiconductors like these three as we can see here this behavior originates from several reasons first of all to this sharp behavior of the gg prime equals zero element of the dielectric function of the inverse dielectric function then also those matrix elements which are orthogonal to the periodic directions are quite dispersive and here we can see the dielectric function obtained with a fine grid the 60 times 60 which are circles compared with squares that is obtained with a coarse grid this is a 6 times 6 and it can be seen that this dispersion cannot be followed with a coarse grid then we also have the problem that due to this behavior of the inverse dielectric function that goes to 1 in the long wavelength limit this means that the correlation part of the response of the system goes to zero while the divergence of the slab of the bare interaction then we have a multiplication of zero times infinity which can generate some numerical problems for this point here thus in order to accelerate the convergence of these quasi particle properties with respect to the Brillenzo sampling some attempts have been done in the literature how many time do we have? ok so ok, these are two possibilities but I'm gonna explain during the last part of the talk which is the one that is implemented in the YAMBO code and that we derived so here the scheme so our goal is to compute the quasi particle corrections and this means to compute the self energy so we can split the self energy into two parts the exchange correlation and for example we start from the exchange part that is easier and the integral over momentum transfer is usually transformed into a discrete sum so what is done in YAMBO is instead of using the numerical value of the interaction in the finite discretization is to use the average value of the around the discretization domain as can be seen here in which we have a discretized Brillenzo these integrals are performed with the Monte Carlo integration and this is possible to the fact that we know the analytic form of the bare interaction and this is usually done for a finite number of g's because for i-modules g-vectors to perform this average is useless due to the fact that this function becomes smooth so our idea to accelerate the convergence is to apply the same scheme also to the correlation part of the self energy that is now proportional to the correlation part of the discrete interaction the integral is again performed with the Monte Carlo method is applied on a sub matrix g-g prime up to a certain value g-lim but now we don't have the analytic expression which can be like computed 1 million of times in order to compute the integral with the Monte Carlo method thus the procedure is this one first of all we write the correlation part of the discrete interaction in this way in terms of an auxiliary function then we interpolate the auxiliary function between the numerical point and its nearest neighbors in reciprocal lattice units here we can see briefly the results of this interpolation on the top panes this auxiliary function and on the lower panes the discrete interaction for some selected matrix elements the first thing we can see is that we have that the discrete interaction changes its order of magnitudes while the auxiliary function do not and this is why it is better and more stable to interpolate the auxiliary function instead of the discrete interaction then we can see that the accuracy of the most important point that corresponds to the long wavelength limit is accurately described due to the fact that we have explicitly including the analytical trend of the auxiliary function thus we have used a particular expression for this case looking at the results for the gap we can see that for the case of MOS2 even the first so the core serigrid is sufficient to obtain convergent gaps and with convergent gaps here I consider the n error of 50 mV with respect to the extrapolated value at the infinite sampling of the Brillouin zone and it can be seen here that the extrapolation is justified also by the fact that these two methods converge to the same value to the same extrapolated value again also for phosphorine we can see that we can use a very coarser grid like the 8 times 20 times 1 instead of this 36 times 50 grid does saving a lot of computational time and the same is true also for HBN in which we can use the 6 times 6 instead of the 36 by 36 it's important to notice here that the computational cost scales as the fourth power of this number here because it scales at the squared of the total number of key points so the total number of key points in this case is 6 times 6 26 squared so to conclude there are three technicalities that should be taken into account in order to simulate a two dimensional system the first one is the supercell approach the second one is the cutoff of the potential and the third one is this W averaging method that speeds up the convergence with respect to the number of key points so with the combination of these techniques we are able to treat non-periodic systems with a plane wave basis set to use an amount of vacuum which is not too large like from 10 to 20 m strong for the case of monolayer and to converge the quasi particle properties with grids which are comparable with density functional theory calculations a brief note that these techniques ok this seems to like to speed up but in most of cases it makes the difference between a calculation that you can or you cannot be done thus these are more or less mandatory very soon after the paper of this method is published I'm gonna add a tutorial dedicated for the G0W0 of these two dimensional systems but unfortunately not in this school mainly because the paper is under the process of the submission thank you for your attention so, thank you Alberto just a couple of announcements before we open the session for questions after lunch we are going to have the poster session which will be a virtual poster session I think there are three of you here who prepared a poster so remember that you should, you are supposed to do a one minute presentation and you should have just received an email with instruction on how to handle the poster session and then due to the poster session we are not going by work to the canteen there will be a shuttle just outside the ICTP actually we are supposed to live now but we take a few more minutes to address the questions and then we go so questions thank you very much for the talk I see the question about the tronkation of this column interaction for the 2D system and for some 2D system like you have shown here which are periodic in just one dimension so it's very easy because the periodicity is very easy to handle but there are some systems that are very light in this dimension how do you handle the tronkation in this case or is it general so let's suppose to have several monolayer materials with a huge space is this true, right? so this is the case you were talking about well in that case this method is not so efficient because of course you need in your cutoff to include the interaction you want to include for example if you have several monolayers then this cutoff so this L can be still should be still quite large but in any case you restore the qualitative behavior of the dielectric response and this is really is still very helpful then of course the bigger is your system the more computational time you need and this is a little bit less efficient because you have to tronkate between different layers but do not tronkate within the layer and this ends up with a very like general rule but not so accurate that you need at least as a vacuum the thickness of your slab plus something like for the monolayer for MS2 it's like 10M strong if we have something that is like with the thickness of 10M strong so a very general rule is that maybe 20M strong can be sufficient but then of course you have to converge because in this case you have like an electron that is in one side of the monolayer that is so he needs to see more or less so the electron on the other side of the monolayer but he needs not to see the first electron of the replica ok, we have another question here so in the expressions when you are computing the W so you have some terms if I am not wrong they are the dipoles right and the very big ok, so the row they are the dipoles so I think here here ok, these are the generalizations of the dipoles so their expression is this one ok, so maybe I can point here so you can see so these are these expressions here which is the which are the matrix elements to change from the base of the Cauchon states to the base of plane waves and for q that tends to zero you can see that these are the dipoles because well this more or less becomes r it can be seen, r times times q ok, then the mathematics should be done in a proper way but this is the final result so for q that tends to zero and g equals zero these are the dipoles but then we need also these elements for q and g different from zero and yeah, this is the the expression that I think is one of the most unknown expressions within the jumbo code because it depends like on k n, m, q and g then the so these numbers here are this is one of the most demanding part computational part and it is used in many steps yeah but like the physics is trivial so just to change from one base to the other so I don't know I mean like could you please justify why your plasma I mean like for example if you take I mean this is why your like choice of your poles, plasma poles is independent of your thing so here you are taking somewhere I think if I remember correctly it is one heart ray or something you are using so why is it independent of the choice of this thing also like I don't know I mean like the poles of this w are basically the plasma energies right so which are the like but how do you compute those values I mean like how do you compute the poles of the inverse of the dielectric tensor ok so well in this lecture here it was mainly on the practical equations and approximations so here I just explained that with this plasma approximation which is the standard procedure and this is why I take the plasma pole approximation as an example because the main goal of this part of the lecture was like to show the equations of what is the standard and plasma pole is standard then you have that the approximation behind the plasma pole is that the if you consider the macroscopic case thus gg prime equals zero then you can see that this experiment of a lot of materials is more or less the you see more or less one plasma then you can see ok with gg prime different from zero that is the microscopic response you change the residues but the non interacting poles are the same then you can say ok we can approximate and say ok, there is just one pole also in the other matrix elements of the microscopic response this is of course an approximation justifies by the fact that the more these g modules are higher the less important is the response of the system ok, I think it's time to go now because there is a shuttle waiting for us and as yesterday you can leave your things here if you want we are going to lock the room ok, just as a I remark that if someone want to ask a question you can ask me in person yeah, during the tutorials