 Na zvenim sešenju smo všdemo pravdelja o inrišne zrbnevo. Zdaj smo pošli do kompletiviralu. Zal wave. Do obnovu. Da, zelo stej v štive. Zelo se je. Tudi, ponili smo prič na to, tako bo poveda, da se zelo v radio? Vščeš, da se povedak prišla v radio, tako že ste več nek basiti. Tukaj si konče se zelo? Do dve. Tako da se poveda, da se zelo v grešenje. Vščeš, da se poveda? A beno je. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. Zdaj. tudi, da je uvo, da če sem jaz včesk, da jaz na nekaj elektroni nešta. In po vzorciju, svoj je zvuk, ko je vzostaj, svoj je injeračno, svoj, da je zvuk, bolj je zvuk, ko neštačili, sem považivati, včesk je zvuk. Včeš... ...eč, da je včesk, da je nekaj, da je zvuk, in se, da, in vseh nekaj je bolj, da se je poslutilo, zelo vseh je zrečil, in začeli se prv, kako se nekaj, nekaj nekaj, nekaj, zreči se je, nekaj nekaj nekal, nekaj nekaj nekaj. To je drugo, če se je zrečila, nekaj nekal, nekaj nekal, nekaj nekal, nekaj nekal, nekaj nekal, nekaj nekal, nekaj nekal, v velikovim regulacijstvo delim. The situation is much simpler. If you perturb a person in general, you can predict much more easily, which will be the reaction. This, mathematically, the result is that the reaction depends on the subject and not on the perturbation. You can take it out to the dependence on the perturbation. This function is exactly what we are going to compute in to je vsečen, da je inisijalne težkovala. Zato konšanje, zato konšanje, zato konšanje energij. Zato da me... ...redužim tvoj. Zato vsečen, da smo pričajati, sem zato, da se pričaj, zato, da se pričaj, izvah, ki so pričaj, da sem pričaj, da se pričaj, zato, da se pričaj, in, da sem pričaj, da se pričaj, da se pričaj, da se pričaj. In izgledam se, da je tudi mikroskopične teore, ki smo zelo izgledali. Zato, nekaj smo počutili neutoreksativnosti v systemu in počutili vseh rečenih, vseh rečenih rečenih in vseh rečenih mikro-mikrokonnečnih. In vseh rečenih mikroskopičnih teore vseh rečenih vseh rečenih vseh rečenih, z independent particle transition, to in interaction describing plasmon, the role of what are called local fields, polarizing fields, and everything which is beyond that, which goes under the umbrella of exchange and correlation effects. So, first the experiments again. So, these three experiments, which you can see in the title, the first one is is, which stands for electron energy loss. In je to šečen, da je elektron in, vseč je zelo srednje, in potem elektron in vseč, in detektor je detektor šečen v energiji in vseč je elektron in tega potreboj vseč je energija, in ki je vseč vseč je zelo vseč vseč je zelo vseč in vseč. In tudi se vseč se kaj je zelo vseč, da elektron je zelo vseč, in vseč je elektron, in to se potreba vse način, da jste prišli, da sem vzelo, kaj jo se poživajovati vse. Kako je nekaj način, kako je to vse način. Zelo tudi vlasti, aspečen rast, je samo način, kako jste vse vse začin, kako jste vse začin in vse začin vse začin. Način je samo, da vse začin je samo vse začin. In tudi vse začin je aspečen rast, kako jste vse začin vse začin, in photonout. Vtev, da je photon in. Mešl gather photon out, kaj vsef dotaršave se večka, s plusom, servecki vetrveniji. In je to, da naj reasono da je tuvis napahar bez vseened, s večrem. Odpečenim uvranjem in uvranjem vsezen in vsezed, ne so vsezne in vsezne. Tukaj, v sva dve. Vsezne ili se svezne, in počkaj sem tudi vzelo za moment, vzelo za valenci, pa je zelo v valenci in pravno nasljim v elektronite. In vzelo na vzelo za moment? Vzelo za vzelo, da vzelo za toga experimenta, napravte na dft-grana, dft-total energija, pa potrebujemo in system vzela v eno zestatov, vzelo v eno zestatina. V zelo v enu zestatine, To je,يف sestilo, nekaj nezapivno je vizit, ker na zdravu zdravu skupne, a je to na lease stavile, ki jaz ne sem maxill je zimno skupnju. Vseš je, nekaj nezapivne, na prejda in na povnešen del je kom brandsi, taj je zmah mu taj, skupnja se ta, ki je to pričel na povživna, skupnju stavile se skupnju. In z vzimno. A kupnju stavile, the conduction band and are not ejected. The energy range which you usually look into is the visible ultraviolet energy range, which is, say, the gap of most of the material depending on whether they are semiconductor insulators. And this class of experiments can be described by constructing, what is the macroscopic dielectric function. makroskopik-dilectric function for absorption, the imaginary part, and the imaginary part of one over the microscopic dilectric function is what you need for electron energy loss and in elastic x-ray scattering. So the question is how do we compute this microscopic dilectric function? Now let's have a look to the field equations in a material. This is just classical physics, as you know if you take an electric field, an external electric field. Sorry, an external electric field is the electric displacement, then the material will react with an induced polarization, and then you can sum the two to describe the total electric field. And in particular the polarization and the total electric field are related by a function which depends itself on the electric field, but if you go to the weak perturbation regime, you can expand it linearly and you have this alpha 1, which is the linear response of the system. And then you should take this alpha 1 e and you plug it back, so p equal alpha 1 e, you plug it back e r, you have a relation between d and e, which is exactly the definition of the dilectric function. So the dilectric function is 1 minus 4 pi alpha, which is basically, this is a relation between the linear response function and the dilectric function you are interested in. Now, the dilectric function is a tensor, it has different components, and depending on the momentum of the perturbation, so let's say that this is your electric field, it is a wave propagating along some direction, q along z, for example. And then you can split in longitudinal direction and in transverse direction. And so there will be a longitudinal component on the dilectric tensor, and there will be the transverse component and also the mixing components. In most of the materials, it is a very good approximation, if not exact approximation, to just consider the diagonal components. So just the longitudinal term and two transverse terms. So the transverse part is what you need to describe absorption. In particular we said absorption, it's an experiment where you always probe zero or transferred momentum, so you need the transverse tensor at zero momentum to describe absorption. So absorption captures transverse excitations. One instead, the electron energy loss captures longitudinal perturbations. So if you want to describe electron energy loss at any momentum, you need the longitudinal dilectric function at any momentum. Now, the point is that at q equals zero, these two quantities, so the longitudinal and the transverse response functions are identical, and so you can just care about the longitudinal one, and let's say write your theory in such a way that you just need the longitudinal fields. And this is a great advantage, because if I just need the longitudinal fields, this is the slide I had before, now the longitudinal fields can be expressed in terms just of a scalar potential v, and then I can write everything in terms of scalar potential, and I can forget for a moment about the fields. Everything is scalar. And in particular, the scalar potential in classical electromagnetism is defined by the density. So I have a framework or a theory where the density is enough, and I can work within density functional theory, time dependent density functional theory, for example. So in terms of the potential, again, there is a relation between the total and the external potential. It's the same one as before in between the fields. And in particular, the total is the external plus the induced, the induced depends on the induced density times the Coulomb potential. And then you can use this equation to write again, I mean, you can write the induced density in terms of a linear response function. And then you end up with a relation between the longitudinal dielectric function and the this chi response function. So the difference with respect to before is that before we had a tensor, epsilon, and alpha was also a tensor. Here we have just the longitudinal component of epsilon and the scalar response function, which is the density-density response function. OK, so if we do a simulation, let's say that we are able to compute our microscopic response function, then we need an averaging procedure to get the microscopic response function. So in your simulation, Janbo is going to do, to compute this, and then you need some averaging procedure. So you are going to get the microscopic dielectric screening, and then you need to take the average. So this is the meaning of these brackets. And now to do, when you do the average, you can recognize that the wavelength of the external field in the energy range, which we are looking at, it's much bigger than the unit cell of your material. So in the averaging procedure, you can take the external field and move it out of the average, because you are doing one average over the unit cell, and the field, you can consider it uniform in the unit cell. And so like that, you get that the average of the, so the microscopic dielectric function is nothing but the average of the inverse microscopic dielectric function. So let me stress that you have to do the average of the inverse. Yeah, this is what I was saying, you get the average. You have to do the average of the inverse, because if you write the relation like that instead, the total field will have the external field, which is still very slowly changing, plus the internal field. The internal field instead is the atomic field. It will change very quickly on the space scale of the unit cell, so you cannot take it out from the averaging procedure. So you cannot use this equation, because you cannot do this procedure for the dielectric function. So you have to stick on this one. OK, so we have a procedure to do the average. We have to take the average of the inverse. Let's just focus now on the microscopic part. Now the response function is the change of the density with respect to an external perturbation. And we need to figure out a way to compute it. So the easiest way is just to compute the independent particle response function, which is called kai not. And I mean, this is well known in the literatories. You have this, which is called Fermi-Golden rule expression, where basically you have the pulse of this object at the independent particle transition. So this delta epsilon means the energy of the conduction minus the energy of the valence, for example. So this is a very simple object to compute. It's just a post-processing of your ground state simulation. It's called independent particle approximations. But of course, it doesn't work very well in many cases. And then you need to go beyond that. So how do we go beyond that? And so for this lecture, I will use what is the framework of density functional theory, or better, time-dependent density functional theory. So the idea is that you have a system of interacting particle. You apply an external potential, and then the change of the density is very complex because of the interaction. So you map that into a system of non-interacting particle. And the price you have to pay is that now, when you apply an external potential, there will be a change also in the effective potential, which is the mean field, which the particle feels. Now the starting point of DFT is that your independent particle density describes exactly the interacting density to this effective potential. And then you can use that to derive an equation for the exact response function starting from the non-interacting one. So I try to do the derivation here, if you can follow good, otherwise not a big deal. So this is the effective equation for this non-interacting particle, which they move in effective fields. So the effective field is my external field plus the classical field or earthly field plus the exchange and correlation field. Now when I do a variation of the external potential, so in real life I have this equation, the total density of the interacting system will be chi, my linear response function, times the external field. In my non-interacting world is just the non-interacting response function times the change of this effective field, which is done by the change of all these components. Now all these components, in particular the arc and exchange correlation one, depend on the density. Since I mean the linear regime, I linear expand these two terms. So there is a term proportional to linear in the density. And I call these two objects, these two functional derivatives. I give to them a name, which is the, as I said, the arthritem v and the fxc term. So let me, so you have this change, this change, the change of the arthripotential with respect to the density is this v. And then here I have the change of the density. And I use again the definition of linear response, the change of the density is chi, the change of the external potential, so this equation. Then I plug all these together, and then they end up with an equation which relates the total change in the density, sorry, the total response function to the independent particle response function. We can take away the external fields. And this is what is the, it's called the Dyson equation, because it's a recursive equation. You start from the independent particle, and then you have a term, so you can plug this back one time, and then again and again. And here you have the effects beyond the independent particle approximation. So now the idea is to have a look to this, some example, and to see how they work. OK, before doing that, let me mention that you are doing a simulation in periodic boundary conditions, so the Jambo code works on top of quantum espresso. So your response function, which depends on r and r prime, would be, depends on a small r inside of the unicell, and the big r, which is the position of the unicell. You can go in reciprocal space, so you do a Fourier transform, and then you end up with something that depends on g, g prime, and the momentum. And then the average in procedure we were discussing before, so the average of the unicell is just the g equal g prime component of this response function in reciprocal space. So you have this Dyson equation. We can have a look to it. So it has these two terms. So one term, this Coulomb interaction, is the functional derivative of the artery potential. And it includes both the microscopic and the microscopic classical induced field. And this fxc contains, let's say, the exchange correlation of the quantum induced field. And it is common to split the classical term v into a macroscopic part, the classical macroscopic induced field, and the microscopic one. And again, this one is this black box, let's say, which has all the quantum information. OK, so let's have a look of how it works in practice. So what we can do is we take our full equation. As a first step, we remove everything, and we just take the independent particle approximation. And let's see how it performs. Let's say that I try to compute electron energy loss on bulk silicon. So I do this procedure with k0, and this is what I get. So is this good? And one way to check that is let's go to the next step. So this time we add the classical induced macroscopic field. We do the same procedure. And this is what we get. I think I'm missing the two. OK, no, sorry. This is what we get, and this is this black line. So the message here is, of course, the classical induced macroscopic field is super important. We cannot describe electron energy loss without, because we would be here. And indeed, with just this term, we have a very good description of electron energy loss experiment in bulk silicon. So this suggests that these two other terms are not very much important. So what if we move to absorption? So the blue line is by chance. I mean, there is a reason. If we try to compute absorption, which is the imaginary part of 1 over epsilon to the minus 1 chi with this level of approximation, it comes back exactly to where it was the electron energy loss evaluated with k0. So you see these and these, they are exactly the same. Now, let's say that this has a reason, and here, as a start, we can use this as an information to get absorption in an alternative way. So instead of solving this full equation, we can solve an equation for a modified response function, which is called chi bar, which is a solution of the same Dyson equation without the microscopically induced classical field. So if you use this chi bar, then you can directly get the dielectric function through this procedure. So you remove one piece, and also you change your average in procedure. So this is rather technical. You don't have to really understand it now. But the message is that when you will do calculation with Yambo, you will have some situation where we compute the full chi, some others where we compute chi bar, and this is the reason. So this is, again, summarized here. So we have our full Dyson equation for chi, and we use it to compute electron energy loss. And this term is super important. For absorption, instead, we use a modified Dyson equation for chi bar. We don't have the microscopic induced term. And, again, we can use absorption via an alternative average in procedure. So this is the, let's say, correctly derived one. This is the alternative one. And the reason why we can do this change is somehow due to the fact that we can either define the response function to the external field, or we can also define the macroscopic response function chi bar as a function of the total field. So we can suppose that we know the total field, and we just have to compute what remains. OK, so we have discussed a lot the G equals zero term. What about the other two terms? So this other term is the microscopic, classical, induced field. And if we do simulation in extended system, it does basically nothing. So this is in the inset. So the dash line is without this term, the red dash, the black is with this term included. And you see that in the beryllium solid, there are no changes. Instead, if you take an isolated atom, there is a big impact of this term. This is what is called in the literature, are called local fields effects. And they are important, the more the system is not homogeneous inside the unit cell. So you do the simulation in a box. If the system is changing a lot inside the box, then this term is going to be important. In a solid, the density is more or less smooth inside the box. And so this term is not important in an isolated system. The density, of course, changes a lot. It moves from zero to the maximum. And then this term is important. And I mean, if you move to some lower dimensional system, for example, 1D system, then you can already guess what happens. So if you look at the system along the direction of the material, then you see something which is more or less uniform. And this term is not going to be important. If you look at the material in the direction perpendicular, then the system is strongly non-uniform. And indeed, so this is the independent particle simulation, now is which on the field. And you see in the direction parallel almost nothing changes. In the direction perpendicular, there is a strong shift of intensity at higher energy. And again the same for, so these are nanowires, these are nanotubes, but the effect is the same. So this shift at higher energy due to the local fields is what is called the depolarizing effect. And I said that usually in extended systems is not important, but it is not always true. So this is the example of solid argon. You see the two lines are pretty different. And the reason is that some materials, despite they are bulk materials, they are pretty non-uniform in the unit cell. OK, then the last term is mysterious box, the exchange correlation kernel, which comes from the functional derivative of the exchange correlation potential, which is the functional derivative of the exchange correlation energy. So first of all, this term, we have no idea how to compute the exact one. We have to do approximations. And what we do usually, or at least the state of the first one is the local density approximation. So we pick up an expression of the total energy, which depends on the local density. Then we can do the function of the derivative for the potential, which is a simple derivative. And then we can obtain the kernel. And then we have to extend that to the time dependent domain. So we also do adiabatic approximation. So here you see it depends locally on position and time. And then we can get the time dependent potential. And from that the time dependent kernel. Now it's an approximation. I'm not going to discuss much these, but the main point is that due to this approximation, it's something which is local in time and also local in space. So it's a super local kernel. And well, in general, the effect of this thing, which is super local, are not so big. If you take an electron energy loss experiment like this one, so the RPA is the blue line, the black line is width on top, the correction due to the FXC. It's not big, but it improves the agreement with the experimental data, which has the red dots. And this is the same for inelastic x-ray scattering, is more or less the same as electron energy loss. Instead, if you move to the case of absorption, this is absorption of bulk silicon. So the experiment are the red dots. The RPA is this green dashed line. And in this, I mean, if you try this green dashed line, you would obtain that also without these two terms. You just start the local fields, nothing changes. You add this FXC term at the LDA level. And again, nothing changes. And you see we are a bit off instead of two peaks. We have this one peak and one shoulder. So here we are clearly missing something. And this is even more dramatic if you take solid argon, which is a wide gap insulator. So this is the adiabatic LDA solution, which is pretty similar to the RPA1. And this is the experiment. So here there is a clear peak, which is completely missed at this level of theory. OK, and this is one thing we will discuss in the next days. The reason of this failure is that we are missing the exit on the peak. OK, I think I start to stop now if there are questions. And then, depending on time, I have another presentation, or we can move directly. I don't know how we are doing. So let's see if first there are questions. Your response part is that actually is the core of everything. So you have shown the last figure that the imaginary part depends upon the interaction. I mean, you have to take the exatonic effects here. So if your imaginary part is not so good, so how do you, I mean, when you do the Kramerskonic relation, you get the real part. So even your real part is not so very good. But you assume that your real part is good, and you do the calculations. So how justifiable is to use the dielectric tensor at an RPA level? I'm not sure I got the question. I mean, of course, if my imaginary part of epsilon is not so good, also the real part will be not so good. I'm not sure what was the question. But we are using the one which is not good. I mean, at RPA, we are not getting the good dielectric tensor. But we are using the same things for every calculation. I mean, you start with an RPA computing dielectric tensor at an RPA level, and then you feed these things to GW or anything. OK, OK. So how do you justify using? OK, I see. Yeah, yeah, now I get it. So he's saying, so here for absorption, this thing is not a good description of the absorption. So we're saying it's not good. And why do we use it? Because in the next lectures, you will see we are going to use the RPA screening to build up the GW self-energy, for example. But the point is that here, we are looking at one thing, which is the absorption. And instead, when you look to the screening, what you are mostly interested in is that you have a good description of the plasmonic screening, which is somehow connected more to the electron energy loss. So I have shown you before, so here, that indeed, at the same level of approximation, the plasmonic peak is very well described. So it's not good for absorption. It doesn't mean it's bad in general. It means it's bad in describing the absorption peaks and in particular the excitonic peaks. But in any case, it has a lot of good physics. And this is why we can use it as a starting point for GW. Yeah, maybe. So there was a lady asking, wait, how do you do with the isolated systems? Is this applied only to periodic systems? Or can we handle the isolated systems? Nadia pa bae? Well, OK, in short, I mean, the code is designed for extended system. There is an implementation in G-space, which assumes that we have a replica of the unit cell. But you can also do an isolated system. Simply you take your simulation box, you make it very big and then you need enough vacuum around. And then there are also some other technical aspects, so you use, for example, a Coulomb cutoff for the interaction between the replica. And then you can also try to describe isolated systems. Then El Achari Tariq is asking, can we study disordered in the magnetic systems using Jumbo? So I would say that, in general, the answer is yes, you can try. But I mean, it depends a lot on the system. Of course, these are abinitio simulations. So they are, in general, quite demanding. If you take a uniform material, you have a small unit cell. Then if you have to include disorder, you have maybe a big unit cell with some defects, and then the computational requirements will increase quickly. So usually you compute directly the longitudinal response and this is like the standard procedure. So what can be like a hint that maybe you need to consider all the tensor? I think, in general, it depends on the experiment. So there are experiments where they look at the, I mean, they also need the transverse elements. And then I would say that you can easily extend all the theory. The only drawback is that here we have everything which depends on the density. If you start to include transverse fields, then you need maybe a current density functional theory or polarization density functional theory. But this is not a tissue at all. If you do many body perturbation theory, because you have a theory which is a function of the green function and it's enough to describe everything. I mean, I have a few extra slides on that, but I didn't have time to show. Thank you. Or maybe last, I don't know, but I think it's time to move. My simple question, if you have a defect system, do we need another physics to explain what is happening in the vicinity of defect, or how do you... Do we need the... Defects, yes. Do we need another physics to describe what is happening in the vicinity of defect, or how do you describe the system in this case? So I would say, one of the message, I mean, the final message of this lecture is that here we are capturing already a lot of physics, but we already missed something, which is we have to improve this LDA approximation for the response function, and this is true in an extended system, in a uniform system, and even more, I would say, in a system with defects. So... But again, it depends a lot on the experiment. So if you want to look at the absorption spectrum of a defect inside a bike, most likely you have to go beyond these to use the beta-salt-beta equation that you will see. If you, instead, are interested in looking into the plasmon, pick in a system with defect, I don't know, maybe you can try. I don't have any experience of plasmons in system with defects. Was I here? I mean, it depends that the tool is completely agnostic with respect to the system. So if in unit cell you can build up a unit cell with defects, disorder, or whatever you want, if you can handle it, you have a plane-wave representation. Plane-wave representation is pretty exact if you have a huge number of plane waves. So you can project the density in a specific region. You can do whatever you want. It is not a big deal. So you need to distinguish the material complexity from the theory complexity. So material complexity is something that can be handled as soon as you have computational resources. No problem. I mean, Janba is being used, for example, to describe families of nanotubes with hold inside to describe amorphous material. I mean, this was done on Chinega, running on Chinega. Yeah, that's it. It's a unit cell. Then the question is whether you can apply the theory that Davide just mentioned with FHC to describe that specific material. But that's an issue of the theory, not of the unit cell of the material, to distinguish very sharply the two different problems. So I guess we are going to move to this. Yes. So now we are, we will move there, and then we need to set up in order that we all run on the virtual machines. So, move there. Ate on, Ate on, Ate on, Ate on, Ate on, Ate on.