 Dario León, who will speak about frequency dependence in the GWO approximation, so we stress the role of the frequency part in the screening. Thank you very much for the introduction. Yes, my name is Dario León, I am a postdoc at CNR Modena, and as Dario said, I will speak about the frequency dependence in the GWO approximation. This is the outline of my talk. First I will introduce starting from the GWO equation. We will get to the frequency representation and the idea is to understand the computational complexity of the GWO calculation. Then I will introduce the most commonly used method for integrating the self-energy starting from the Plasmon-Paul approximation. I will explain briefly its origins, the generalization, the different implementation, and the achievement and limitation of this model. Then I will also explain another alternative full frequency approaches like numerical integration, the real axis, the contour deformation technique, and we want to understand the computational cost compared to the previous approach. In the end I will also introduce a new methodology that is called the multiple approximation that has been developed in Jambo. I will explain some of the main motivation, and we also want to compare its accuracy and computational cost with the previous methods. Then starting from the GWO side, this is the hidden pentagon of the equation, as already the previous speaker from yesterday explained very well. In the GWO approximation we neglect the vector correction to get this simplified cycle. The second point is usually a simple approach like DFT, so we start by Conexam G0, so the self-energy is null, and then the first iteration, we need to compute the polarizability at the independent particle level, and then the reducible one, and then the self-energy, that is this product of the beam function, and the screen of potential W. Then how to arrive to the frequency space? I recall this picture of the beam function that represents transition of electron or holes from ambient position at a given time to different coordinates. This notation is very compact. Here I am putting the coordinates one or two, represent these coordinates both in space and time. Then I will just use the time variable to arrive to the Fourier transform to arrive to the frequency space. Since we have a simple product of two operators, this results in a convolution in the frequency space, both for the polarizability, there is a convolution between the two beam functions evaluated at different frequencies, and the self-energy. So this is the problem, the integral we want to solve in the GWO approximation. Then as I already introduced yesterday, we can arrive to a lemma representation. So here this is an example I took from the book of Stefanucci. So for instance, we take the retired beam function. This is the function in time. We have a Hamiltonian. Then we just simply Fourier transform this beam function. We use this expression for the heavy side function. Then we arrive to this equation here. Then if we introduce a complete set of, I guess, states for the Hamiltonian, we get the lemma representation. So we have poles in the excitation energies of the system. Then from the lemma representation of the green function, we can compute that starting from a Conechamp basis set. So here again the poles are in the excitation energy. Then we have also the projector of the states. We can integrate the convolution I showed in the previous slide for the polarizability. And we arrive to a lemma representation for this operator where it's a very similar expression. Now we have poles instead of the excitation energy. We have poles in the transition energy. So this delta E corresponds to different state transition between a state k, i, and k minus q, j. So this is the full expression of this polarizability. In all the values we have used also the Fourier transform of the spatial coordinates in a crystal. So we have also symmetry. This result in g prime and also q coordinates. Then this object enters in the Dyson equation. There is a nonlinear equation. And we obtain the reducible polarizability. So we want to understand what happened in this Dyson equation. So we start by a representation that is... We have a lot of peaks. Delta light peaks are in the excitation energies. But then this somehow this picture is destroyed because what we want is... We want to describe collective excitation. We are switching on the iteration. So this picture is destroyed for the reducible polarizability. What we can do is to compute this subject at each frequency at the time. We can compute it numerically. So the computational cost of the evaluation of this polarizability can be found in this paper for instance. So we have a... Let's say we have an integration over the brilliant zone. We have summation over conduction and balance bands. And also... C is the number of plane waves of the basic set. Then there is also a crucial point that is the method we use to integrate the self-energy. This for instance will determine the number of frequencies we need to use for evaluating the polarizability. And there is also a further sum of bands in this integration. You can see the full expression in the Yamuchichi and also in the next presentation by Alberto Wandalini. So the origin of the Plasma Polar Policimation comes from the study of the electronic correlation effect in simple metal. That can be modeled by a simple electron gas. For instance, we can take an aluminum as an example. Here we realize that the relevant excitation for a small cube correspond to a strong peak in the magnetic part of the screen potential that have a very small width. So there is to model this excitation with a simple delta light peak as I am showing here. So we have simply this pole. This is the magnetic part. And the further application of this model has been used for the excretion of the main longitudinal Plasma Mode, observing a specific health experiment. Here I am showing a simple comparison of this Plasma in Hartree for a set of materials. material, we can see that it's in very good agreement with the experimental value. Then this model has been generalized to the full metrics. So we are extending the model also for G prime and Q coordinates. So this has been done by Harvesting and Louis. This model is less accurate for large Q, large G vectors. So we don't have any more, a single peak. We have many more peaks. So this is less accurate. But in general, W is dominated by the small value of Q plus G because it's multiplied by a Coulomb potential. It has an inverse relationship with this model squared. So this is the model generalized for each QG prime. We compute this single matrix element with this Plasmon pole. Here I am showing the analytic continuation. We have two parameters, the poles and the residue. And there are two main ways to receive it, to obtain the parameters. So we need to impose some condition. One possible option is the epsilon rule. I am showing here using the Johnson formalities. This is just a recipe for computing. Even if we don't know an analytical expression of chi, there is a recipe for computing this integral as a function of the electronic density. The other method is an interpolation of the parameters. For this, we just need two frequency points. So set in this case is equal to 0 and set equal i omega p. So we are computing a frequency along the binary axis. And then with these two parameters, we can interpolate the model. This recipe is by Godbein needs. In the end, I prefer in the end. So here I am showing the comparison. So this is the porous ability computed for three different materials. So the parameters on top correspond to the porous ability computing along the binary axis, where we get a smooth dependence. Because all the poles of the porous ability lies on the real axis. Then if we sample the porous ability along the binary axis, what we see is something like this, very smooth. We see the main structure correspond to all the coordinates. Then as we go far, we lose the information. But then, so I'll tell you, we can compute one frequency at a time, the reduced porous ability, then interpret the model for instance along the binary axis. But then, since we have an analytical model, we can evaluate the model on the real axis, where actually we want to compute the self-energy. And this is the comparison with the full frequency case. So here in this Dyson equation, what happens, all the poles of the k-node that are single particle are overlapped, and we get this overall structure. So we have this for silicon, it's very, very simple to see. So we have a plasmoon, and it's not always like this, no? We have more complex material, but in the end, we have an overall structure. So this is a table from this paper I am taking for comparing different plasmoon pole results of the vanguard in this set of materials. So here, beside the high-resilient Louis and Godwin's recipe, there are more refining of the plasmoon pole model by Linder-Horch and Engel Farid. They use all the same Johnson's rule, but the refinement lies in the spatial coordinates. So we also have understood this Johnson's rule that can be written in the form of this limit here. Then we can rewrite high-resilient Louis equation for the plasmoon pole in the similar way of the Godwin's version with an interpolation of two frequencies, and the second one is at infinity. So just to say that the sampling is a general procedure. So as I showed before, the plasmoon pole provides quasi-particle with energy, with a rational level of accuracy in many materials, but it has several limitations. The first one is that we are not able to describe the imaginary part of W. So this result in the representability issues. In the self-energy, this is the self-energy of aluminum. As a simple case, we can see that the self-energy has a very well-defined peak full frequency, but here, so in the self-energy, we have a summation over the state. That's resulting in the discretization of all these peaks in the plasmoon pole case, because W doesn't have the imaginary part well-described in the plasmoon pole model. So if we compute quasi-particle along the Fermi energy, we will get very accurate result compared to full frequency. But as we go far from the Fermi level, we will reach a region very noisy, and we will get some problem in the plasmoon pole model. So also, there are materials with the self-energy. It's more complex, more structured in frequency, and the plasmoon pole is expected to fail in these cases. So regarding the integration of the self-energy, we can see that we can use the plasmoon pole. So we have an analytical model for W that is simple to integrate the convolution in the self-energy. But there are also alternatives that are called full frequency. The first simple example is if we can integrate sigma in the convenient basis set or to use a spectrum representation. But these approaches are usually accompanied by some approximation. For instance, we can approximate the probability matrix as almost diagonal. So the integration is exact in this basis set, but there may be some approximation. So this also may be system dependent because, for instance, we use a localized system. We can use localized basis set in order to add the less number of the basis set to be as small as possible because the computational cost is plagued with the size of the system. There are also numerical integration methods. For instance, by using quadrature rules, for this we need to compute the integral numerically in a very fine frequency grid. This is the real-life scheme implemented in Jambo. Or we can use the full frequency control to the formation or the analytic continuation through with some models. For instance, an integral model like the Padre approximate. Then this list is not meant to be a comprehensive list. There are many more methods. For instance, we can use the Fourier transfer to imaginary time or some stochastic approach to compute the integral. But we also add our new development which is also analytical model that we call multiple approximation. So I will explain now the main approaches implemented in Jambo, so the Plasmon pole, the full frequency on the real axis. I will also explain the contour deformation because it's a very useful technique and then the multiple approach. So if we substitute the green function in the self-energy integration, we get this expression. We can reduce the limit of the integration. If we use the symmetric property of w, so that w is anti-symmetric. Then this integral can be computed in quadratum-fruits. I am listing some of the possibilities here that are compared in this paper here. So the version implemented in Jambo is called piecewise linear. So what we do is to approximate w linearly in each interval that we are computing in the very fine frequency grid. So this approximation is better as we use a finite mesh. So since then we approximate w linearly in each interval and since we have now an analytical expression of w, we can integrate the self-energy and we get this expression over here that is the one implemented in Jambo. Then in order to understand the computational course of this approach, we need to analyze the pole structure of the EW operator. So the green function has poles on the excitation energy. Connicham, if you use the W0, YW, so kind of have poles in the transition energy. Then this poles enters in the priceability but the lemma representation is general. So the priceability also has a lemma representation just that we don't know where are the poles. But W also have this pole structure in the complex frequency plane. Then so in order to represent this object, we need to use because we cannot put an infinitesimal imaginary part in a computer. We need to use a finite damping parameter and we can interpret this in two different ways but equivalent. So either we interpret the poles to have a finite imaginary part, this corresponds to broaden the poles so we don't have delta like peaks anymore. They have a small broadening. Or we can interpret that the imaginary part goes to the frequency in which we are evaluating the operators. So in this sense, we can think of as a time order real axis due to the sign of the imaginary part that I'm showing here. So in this sense, we can say that the poles lies actually on the real axis and this is why we have the name of full frequency on the real axis. But in the end, we need to use a finite damping in order to represent this object. So as if we use a very small number, damping, we need to use more frequency to represent this delta like peaks. It's a parameter also in the code. Then also in this world, it can be shown that. So if we go to the frequency plane, the complex plane in general, we close a contour, semicircular contour on the upper semi plane. We will englob the same pole as if I use a contour like this, the semicircular in the left side of the plot. So with this, we can transform an integral on the real axis in an integral along the main area axis. And this is one of the first idea of the contour deformation technique. So in the general situation, we have a convolution. Then the general situation, the frequency in which we want to compute the cell furniture is not zero. Then there is a chief of the poles of G with respect to W. So as you can see here in this plot, it's very complex, but this is a semiconductor because I forgot, but in general, so these black points are the poles of W while the crosses are the pole of G elevating omega minus omega prime. So in the case of this omega is zero, there are a line with respect to the main area axis. But in general, they are not. Then there are two different formalities to integrate. The self-energy using these kind of techniques, one is called a minority path axis integration that is used in the KELDI formalities in order to define this kind of contour. So it's the same idea. An integral on the main area axis can be transformed into an integral. So sorry, an integral on the real axis can be transformed in an integral along the main area axis. I am in the Dutch line represent the closure of the contour in order to include the same poles as I rotate the contour to get the integration on the real axis. Then there are also, we can also use the residue theorem and to perform a contour, like a landing scale that is represented here. But in the end, these two approaches are equivalent, just you will find different names in the literature. But in the end, we are including the same poles within the contour. So the final expression you use the residue theorem is that the integration on the real axis is equal to the integration along the main area axis plus the sum of residues entering in the contour. So in order to, so it may seem that we did nothing, but as I showed you before the dependence, the frequency dependence along the main area axis is much more simpler than on the real axis. Then in order to solve the integral along the main area axis, we can use for instance numerical evaluation using similar quadrature rules. But then the mesh will be, so the number of frequency that we need to use in order to convert this integral would be much less than if we do it on the real axis. So we could use also interpolation, feed models or the analytic continuation technique that I will explain in a second. So since this dependence is very simple, we can use a simple form to interpolate this. And then we can do two different things. We use this equation here, compute the integral analytically with one of these two analytical form, or we use directly the analytic continuation of the self-energy that is, we compute the self-energy. The complex plane in general with these two analytical form and then like in the Plasmon-Paul scenes, we have an analytical form, we can evaluate it on the real axis without the need of compute the residues. This is the ring of the two different names. Either we use the contour deformation, we have this much of the residues, this is more robust, or we use analytical model for the self-energy in order to just evaluate the self-energy on the real axis with this model. Then I will explain our approach we call multiple approximation. You can see as an extension of the multiple model. Here we model the polarizability with a finite number of poles. So again, this is inspired by the Lehmann representation, but here the difference is that we don't want to use a very large number of poles. So these poles in general are relatively complex in general. Then the idea is that we are passing from a single particle picture where we have a lot of transition like in the kind of depolarizability at the independent particle level to another picture in which we want to reduce the number of poles. So what we want to describe are collective ascitation of the system. Then there is the way to use, reduce the number of poles. By allowing the poles to have a finite and binary part. The way of obtaining the parameters by an interpolation procedure, in this case we need to use a number of frequencies that is twice the number of poles. And this leads to a non-linear system of equation that we can solve with two different approaches. One is a mapping to an equivalent linear system and the other is based on the Pader-Tilly et cetera procedure to solve Pader-Approxima because this multiple model can be written as a particular Pader-Approxima. Then the key point of our approach is to choose how to choose the frequency we want to use in the interpolation. We have to do this several combination and we have proposed this recipe that we called double parallel sampling. Here in this plot I am showing isolyte corresponding to a temporalizability that has 200 poles on the real axis. And the idea of this sampling that has two branches, one closest to the real axis and another one further away is that the blue line introduces a simplification because we are far from the real axis. What we see there is more smooth than what we get on the real axis. But then we also add points close to the region we actually want to compute the self-energy and somehow this combination is optimal. So the details can be found in our paper but with around 10 poles let's say is enough to get a very accurate description as I will show later. We also use a failure condition in order to avoid a physical position of the pole. This is already present in the Plasmon-Poll model in the Godwin implementation. If this condition is satisfied we usually fix the pole to a constant value of one heart tree. We have generalized and improved this condition in the multiple case. Then we have, we proposed two quantities to monitor the interpolation, one is the number of failures in the whole metrics and the relative standard deviation of the model with respect to the sampling points. We can see here these two quantities is the average in the matrix as a function of the number of poles so compared with the Plasmon-Poll case for these three materials. As you can see there may be a very large number of failures already in the Plasmon-Poll that is close to 40% in the case of the I02. The multipole with one pole even while fixing the same number of poles the error is lower because we have changed the condition. But in general both quantities go to zero as we increase the number of poles. Then this is how looks the polarizability one random matrix element the case of HBN computed with different number of poles and compared to the full frequency case and we can see that eight poles is already very good. So here I show different matrix element of HBN so we are using the same number of poles for all of them we can describe real and in minority part of the priceability at the same time and work for the owner and non-the owner matrix element in general. This is a kind of silicone that is more simple so the first element that is the most important one is looks like a single pole then there is also the I02 that is more complex but the multipole with a small number of poles managed to get the overall description. So some elements are more complex but as I mentioned before for larger G this element is less important so you can see on top the scale so compared to the first element. So in the end we get a very good description. So then we can compare the result of the quasi-particle with these three different approach plasma pole for frequency and multipole. I am using the linearized quasi-particle equation as I show here and this is the comparison this is the difference between the full frequency gap computed with full frequency and the multipole case and also compared to the plasma pole for the three materials. We can see with around 10 poles we get very accurate result with comparing to full frequency results. Then the multipole model also provides an analytical form for the self-energy. So simply this is the integral this is an extension of the plasma pole case. So we have the green function we have a multiple analytical model for W and then we can integrate the self-energy and we get this expression that we can test against full frequency. This is the comparison for the silicon case. So here we have two quasi-particle corresponded to the gap of silicon computed with three different approaches plasma pole in blue and gray full frequency in red and orange and the multipole with eight poles in black and gray. So we can see that the multipole provides a very good description of the whole energy range relevant for both the self-energy and the spectrum function. So as a summary, so I show you there are several methodologies addressing the full frequency dependency in the EW approximation. We need to be careful to decide which is better for the system we are studying. There are three main approaches in the Jambo code. One is the plasma pole that uses two frequencies. So we have one pole. We need two frequencies. The number of frequencies provisional to the computational cost. In the real axis approach, we need at least 100 frequencies depending on the material for 30 to 40 instant I have used 1500 frequencies and this can be compared to a simple multipole model where we use 10, around 10 poles that correspond to 20 frequencies. So in the end, the multipole, I have shown that the multipole with an ultimate sampling provides a very accurate description compared to full frequency method and much lower cost. So this is the bibliography I have used for the present presentation. So the first book is just for introduction on DFT. I like it, I use it for my thesis. The same books that have been shown before. These two are the Jambo papers. And if you want to take a follow of my presentation, you can go to the paper. I already posted several times or to my PSD thesis. With that, thank you very much for your attention. So the session is open for questions. If you have any. It's just a curiosity. You said that in order to improve the plasmapole approximation, the first simple thing you can do is to start to considering the reciprocal lattice structure of the plasmas. So you start putting plasmas with Q, G, and G prime. Yes. I was wondering, what's the meaning of plasmon poles with G different with G prime? So the energy of the plasmon disperses with Q. Yeah, okay, yeah. Now, but if there are reciprocal lattice vectors different, G different from G prime. So yes, you have a model. So you see this picture. Let me take silicon. So you have a whole matrix, no? Yeah, yeah, no, but in the multipolar, I understand, but okay. So what, the structure of the porous ability is like the head of the porous ability looks like a single peak for many materials. Then this, the amplitude diminished, but then we have more structure correspond to, because we have all the possible transition, no? And we have less overlapping in this matrix element, but the amplitude is diminishing. So the idea of the generalized plasmon pole is to approximate all the matrix element. So not only G, G prime, this is for Q, it was zero, sorry, for different Qs, we use the same model. So we have two parameter for each matrix element, G, G prime, Q, and it's less accurate for all the matrix element non-diagonal, et cetera, but also the amplitude diminished. The idea is to use the same model for all the matrix elements. Just I try to reformulate. I think his question is, if I got it right, why do we have a pole which depends on G? So I think he is, no? On G and G prime. So you're asking, why do we have omega G, G prime, Q? This is your question? Yeah, only for the plasmon, yeah, the plasmon pole, okay. So maybe Davio, you can comment on that. So yes, because if you think you need to take into account non-homogeneous materials. I think the pole is not dependent on G and G prime. No, no, no, the poles, no. Sorry, this is the expression of the porous ability, the full expression. So you can see here that the poles do not depend on G, G prime. It's the same pole, just the residuals which are changing and then you see the in the frequency plot that. The residuals are here in the numerator depend on G, G prime. But then the idea of the generalized plasmon pole is to allow, since we will use one single pole, less allows to be different for all the matrix elements. So you use a model, but it's somehow flexible, no? And the multiple approaches use the same idea. So we have 10 poles that are different for each matrix element. Okay, so another question from here. Sorry. Sorry, I go very basic. I think you went a lot in the specifics, but maybe you can argue like the reason behind why you have this structure on the screen interaction. So I guess is that the region of the plasmon pole approximation, so why do you have these divergences? So can you give like a physical intuition behind it? And is it valid only like in molecules or it's only valid on solids or? Okay, okay, so in general, so this kind of is at the level of non-interacting particles plus Andrea Marine, it was explained in the end. If you consider the screen, we have collective excitation. The main contribution to this screen potential are plasmon. So our collective excitation of the material, no? The response, how the material responds to a perturbation that is removing or adding an electron. And this is a collective excitation. You don't have a single particle picture anymore. So mathematically, this is reflecting the Dyson equation. So this is a non-linear equation. You start with this kind of representation for each matrix element. So here you have third matrix element. You have a price ability that have delta light peaks in all the possible transition, single particle transition. But then when this subject entering the Dyson equation, you have a superposition of all these poles that changing, let's say the residue. So in some half, you have an envelope of all the transitions. And you have the overall behavior that looks like one peak, very big peak that corresponds to a plasmon. So if I can comment on that. So I mean, what you're asking is why do we have different structures in between the macroscopic, the head of the response function and the microscopic term. So I think that in general, the common knowledge is we know that the macroscopic screening at this big plasmon pole, and this what is used as an input for the plasmon pole. But we don't know what happens to the microscopic screening. And what is shown is that at the microscopic level, the response can be different. So you can have the poles which are shifted and even multiples. I will say that, so the microscopic screening is a bit different from the macroscopic one. And then if you think to a molecule, for example, a big molecule, the macroscopic screening tends to go to zero because you are in vacuum, but there is a microscopic screening, which is important. Okay, yes, I forgot to answer that part of the question. So you can use the plasmon pole model for any kind of materials. But let's say it's better for, doesn't depend on the dimensionality. So there are some cases like a marine half-student with a strong screening interaction. This means very complex frequency structure that you cannot approximate with one single pole. So the deviation would be larger in this kind. The example are metals with low plasmon energies. So we have this main structure very close to the origin of coordinates. This will result in poles. So let me try to explain. This object entering the self-energy and you will get the poles will be entered here. So you have in the priceability a pole close to the origin of coordinates. This will result in a pole close to the Fermi energy. And then the speech I made before that the plasmon pole is good. If you are in the tail of the priceability, so in the tail of the self-energy, it's not valid anymore because we have structure close to the Fermi level. So whenever you find a material that have structure in the priceability close to the origin of coordinates, it could be a candidate for not using the plasmon pole. Okay, I think it's time to move to the next speaker. We thank Dario again.