 ...zaj sem se počutil. Če so naparali, da smo učili o prejzivne prejzivne, kaj je, da ne boš v zelo, pravda ne boš počutil, ...zaj vse je, da se pričel ima nečične režite ...zaj načinovetje do kondakčnju. V sej, da se pričel ima, da se pričel ima ...zaj načinovetje in da si pričel se vse ...zaj načinovetje počutil, zato v trha there is no more than this when you do, for example, an optical absorption experiment. This ben gap, the quasi-particle ben gap, is the one that you will see. In general, you will see the optical gap that in semiconductor can be at a lower energy than the quasi-particle ben gap. This is due to the fact that the excited electron-hole pair, actually, the hole in the electron they interact with each other to the chromic interaction, and these lower, so the energy vseživaterstvega elektronijsta vseva, pa je vsevaThat's the independent particle one. The electron interaction is taken into account in the Bethe-Salpeter equation. That we will try to derive, not super rigorously, here, using a time-dependent approach. Basically we will concentrate on the response function of the electronic system after it has been excited about an external electric field, like an optical laser. In videlš, da je pačnja priča, če se vseh periodov, tako vseh priča, ki je to priča, ko je to priča, zelo pačnja. A pa če pačnja, ko je vseh priča, zelo pačnja vseh priča in vseh priča. Če je to, da se vseh priča, neč dela, da se počakajte, da se počakajte, da se počakajte, da se počakajte, da se počakajte, tudi za densitivne matriče in tudi za responsivne fungšenje, in tudi skupimo elektronične interakcije in exitonje. Sveč, kaj smo prišličili o exitivnosti, kaj smo prišličili, nekaj smo prišličili o elektroničnje sistem, nekaj smo prišličili o exitivnosti, in tukaj smo prišličili o exitivnosti na komputeritivne spektroskopi. Tako, da ide se, da sem kaj da pokopal o presteru prektupov,ONGS, to je formula, je samo si kaj đešno držal, da se pr anger zetva, co pogodila so v defa kod do tudi družt, za povedu o prektupi, posledaj se na ektronične energije, najverkove energije z vrkone, energijo, je občas, nekaj? Hahovutne je, da ne vsimimo vzelo in na razlih, da bi posturjamo začunja skupnike, pa novimo tako, da dobro se je povresima vzelo. DFT, prezent, pa tudi tudi, zelo s betrem zespektem. To je neko mushrooms in je vzelo kurs, kar bilo šej pravi srednjih capacitorje barjez. Tako včeles tega, ne ki dobro mili, neko bili zelo elektronovne rejtire. A kče pa je izgledan, skupaj se predikšnjeni, da se počutite vzvega prizvrga. Vzvega si nekaj pridlj, ki je vredne, ali priče, da smo se počutiti na nekaj zelo, tudi je zelo in je zelo vzvega. Vzvega je vzvega in je načo vzvega vzvega vzvega vzvega vzvega vzvega načo kredim vzvega. Zato je, da pa izgledana vzvega in Av folding in single particle excitation energy is any longer, but they are the exiton energies. So how do we find these red quantities is the topic of this lecture by solving the beta-sulpehter equation? If we apply this expression for the optical absorption, we obtain a good prediction of the experiment. Actually, electron interaction is extremely important if you are interested in layered or 2D systems, tk. monolajer, boronitride. If you look at the independent particle absorption, of course it started the quasi-particle bandgap, it's here. But if I add the BSE, it's completely different the optical absorption that I obtain. In particular absorption starts two electron volts below the quasi-particle bandgap. So you see that in low-dimensional systems it's crucial to add the excitons, to add the electron interaction. And you don't do it without, right? OK. So now we see how to do that. Ah, OK, sorry. I wanted also to say that why this happens. Why we have such a large exciton binding energy here. That is because in lower-dimensional systems the electron interaction is weakly screened. You see here the field lines between electron and hole are mostly in the vacuum space. There are no electrons more or less here. So much is screened more weakly than you would have in a bulk system. So this is generally the reason. OK. So to derive the BSE, in general, there are various approaches. For example, you can do the Schringer approach. You use four-point quantities, do some functional derivatives and then arrive at the result. If you have already arrived at the inspentagon, you can manipulate it, in particular starting with the vertex function. And then you also get the BSE. But today we will follow this other approach, which maybe is a bit simpler to follow, let's say. That is what I mentioned before, following the equation of motion in real time for the response function. So let's start with the description of the ground state electronic system. Here I wrote Artefoc. You see, this is the Arte function and this is the exchange, written in self-energy style. We don't need to start from this. We can start from Konshama Miltonian, Konshama plus GW. But this is just the simplest starting point for the electronic system for us. And so this system gives us the single particle energies and the single particle wave function. So in an extended system it will be a block function. Which, for example, if this is the Konshama Miltonian, we can compute in DFT, right? And the other thing that we have here is the equilibrium density. That is defined like that. And you see at equilibrium, basically, if we expand it in a single particle state, just give us the state occupations. So these F-occupation factors that you have also in the Yambo cheat sheet for a semiconductor at zero temperature, they are just zero or one. Nothing very complicated at equilibrium. However, now a laser is coming to the system and it's rearranging a bit this density. So the full time-dependent Hamiltonian in this case is like this. We have the original non-interacting Hamiltonian, like in this case I wrote Artifok. Then we have the external field. And then we have these corrections to the density functionals, because since the external field induces a change in the density, also the density functionals will change. So they will be corrected in the ground state once, the equilibrium ones. OK. And now we have a time-dependent density matrix that for same position, a Green's function, you can just find it directly from the Green's function. You expand it in single particle states and you obtain this time-dependent version. Perfect. We are not really interested in the density matrix per se. We are interested in the response function, in the linear response, cubo formula. Basically, this gives the variation of the density matrix with respect to the external field. So this is what we want to compute. This will contain the electron interaction. And this can also be expanded in the single particle basis set. It will depend on four single particle indices, as you can see here. And of course, even in this basis set, this functional relation is still valid. So we can write it like that. OK. Now, in order to proceed, we have to introduce the equation of motion for the density matrix. As you see here, it's time evolution. It's given by the commutator with the Hamiltonian. Now, you will hear much more about this tomorrow in the real-time lectures, where there will be a more general version of this equation, a time evolution of the Green's function, which is called the Kadanov bim equation. But for us, this reduced version is in linear response to arrive to the Betzelpeter. Why? Because if we now apply a functional derivative with respect to the incoming external field, then this becomes a equation of motion for the response function. And what we have... So our objective is to find an expression for this chi. What we have to do is to compute these commutators, compute this derivative, and then hopefully to compute absorption spectra. So, in order to make this a bit simpler, we start rewriting some parts of this Hamiltonian, in particular these density functionals here, we can expand them first in the external field. So these are some of the states and so on. And then we can use the chain rule to expand again, and then we transform these into two derivatives. One with respect to the density matrix and the other with respect to the external potential. This is useful because this is just a response function. So the quantity that we want. And then we are left with this quantity that we need to kind of assess what it is. We can do the same with the fork part or the self energy part. And then for the two of them, summed, we just obtain this expression. And the part in red will become the kernel of the Betzelpeter equation, the one that encodes the electron interaction. And now we will just compute these derivatives to get the simple form for this kernel. Let's start with the time dependent heart. As you know, it can be written like this and then expand in a single particle basis like this. You see here this is like fine one graph to kind of represent what happens. This line is the pulom interaction and this is the density. So this is a Green's function that starts and ends at the same position. We can expand this density into our single particle basis step. And we obtain this. So all the wave functions are here inside this integral. The time dependent part is outside. We call the big integral V and we are done. We are happy like that. An observation that I can make now is if we switch this graph to the momentum space, you notice that this pulom interaction cannot carry any momentum of itself because of momentum conservation. For example, here an electron is coming. If the pulom interaction will take away some of its momentum, then it would carry it here. But since this density must start and then with the same momentum, this is simply not possible for momentum conservation. So this heart term will be always at q equals 0. It will only carry any external momentum of the electron odd pair. And so finally we can write it like this. So it's quite simpler. Now we can take the derivative with respect to the density. As you see, it has become trivial and we obtain this expression for the hearty part. Now if we stop here, we obtain the RPA screening. It's what Andrei Amaini said on Monday. Time dependent heart gives you the RPA screening. But we have another term here, namely the self-energy, which now I have written it in a FOC case, pure exchange. So it's not GW, V. So the G can be rearranged as a density matrix that is not diagonally in the position basis, because you see it starts at R and then at R prime. But we can also expand this in a single particle basis set. And we obtain this expression in which now, of course, the N and L indices are exchanged with respect to the hearty case. However, here we encounter a problem. Because in general, we would like to start from a single particle description that is the most accurate possible. Namely, we would like to start from DFT plus GW, right? Very good. But here, if we take just the exchange self-energy, the interaction between electron and hole will be unscreened, will be bare. And this cannot be right now, because we are excluding the rest of the system. And then the binding between electron and hole will be much larger, much stronger than it should be in a real system with a charge density. So, what we can do at this point, we replace this V with a screened interaction. So, as you know from yesterday, the screened interaction can be computed with first to do the RPA screening, screens the Coulomb. Then we put it here. Now, this self-energy just becomes basically the exchange correlation self-energy of yesterday, in fact, it's called screened exchange, sorry. So, no correlation anymore, because the screening is static here. So, of course, what does it mean as screening is static? It means that when you create an excitation, an electron-hole pair, the rest of the system instantaneously adjusts to this excitation to screen it. In reality, there is a time delay for these adjustments. We are neglecting all the effects of the screened interaction, so, I mean, it's time dependent. But you will see this is needed to arrive at a workable form for the BSC. OK, so, now that we did this, we can, ah, sorry, I also wanted to say, at variance with the heart case, here the screened interaction can carry its own momentum. So, this will result in an additional integral over momenta in Fourier space. So, now we simplified and changed in this way. And we want to take the functional derivative with respect to the density matrix. The problem is now we, OK, we need to derive this part, but also this part, because now W itself depends on the function and on the density. But we just neglect this derivative here, because it will lead us to higher-order terms in W, so far we neglected. And then we also obtain simple form for the self-energy part. We go back now to the equation of motion with these simplifications that basically becomes like this. So, we have three commutators that we need to calculate, basically. And here you see you have these W minus 2V that appears. OK, so, now we need to compute them, but we can do it, for example, the first part, the A, we can just let the Hamiltonian, the equilibrium Hamiltonian act for the second commutator, we replace the time-dependent density matrix with the equilibrium one, because we want to stay at first order in the external interaction. Since R of t depends on the external interaction, we will not be at first order any longer, so we use the equilibrium one, and the equilibrium one, applied to the eigenstates, just gives a difference of occupation factors. Again, in a semiconductor at zero temperature, this is basically one or minus one. Not anything fancy. OK, and then the third term, the large one, actually goes exactly like this. So, it's not a difficult commutator. Analog is to be. And then we obtain this equation, after we also perform the derivative with respect to the external potential. Here, OK, I just emphasize that actually the response function depends only on the differences of the two times. So, the time when we have the probability of creating the electron pair and then the time when we have the probability of it to recombine back to the ground state. And here this k is the kernel. I just called w minus vk, just for simplicity. And now we can simplify it even more. You see, all these quantities depend, for example, the kernel depends on four single particle indices. Ah, sorry, yeah. Obviously, the w plays the role of an attractive interaction between electron and holes. So, it's the binding term. It's what creates the bound excitons. Bound excitons. Well, this is a repulsive contribution. OK. Now, we can simplify this equation even more. So, when we have an electronic transition, for example, we can imagine it is the n1 state is in the valence band, and then the n2 state is in the conduction band. It can be also the opposite, a transition from conduction to valence. In general, we can label a transition with a single index kappa, and we go in the transition basis. This actually simplifies a bit how this equation appears. And now it is clear that we can easily take the Fourier transform, because we have just the time derivative here and the chi to be Fourier transformed. We do it, and in Fourier space we have this simple expression. Now it's time to... Basically, we are almost at the end. Notice that if we set the kernel to zero, which means if we set the electron interaction to zero, then this equation reduces to the... must reduce to the independent particle response function, which indeed we get, and notice that it's diagonal in the transition basis. In the diagonal term, the one that mixes different electronic transitions is the interaction term, of course. OK, so now that we recognize the independent particle chi naught, we can just rearrange this equation into a Dyson-like equation for the chi. And this is like the pictorial representation. So this is the point of connection of this real-time approach with other approaches like a Dean, Schwinger, and so on. Because in those approaches we started at a higher level with time order, two particle gains function, and then you would arrive at a Dyson-like equation for that function, and then you would reduce it to a retarded response function which is what we have here. So this is the point of contact. Of course, this kernel we can divide it in this way. W, the attractive term, and V, the repulsive term. OK. But still we don't know what is chi. So we didn't really solve it, right? We just reworked it. But now we can solve it. We just bring everything to the left, like this. And then we recognize, if we isolate omega, that this really looks like the matrix element of an Hamiltonian. In particular, a matrix element of an effective two particle Hamiltonian that we just define it like that. And now we can invert it. In this form, we invert this equation and we obtain chi equals to something. You see, it is chi equals to 1 over omega minus a Hamiltonian. So this can have the form of a three particle propagator for the excitons provided that we are going to diagonalize this Hamiltonian, which we can do. And we obtain exactly the quantities that we will need later for absorption. That is the energies of the electron-hole bound pairs. And then, from the eigenstates, we obtain the external coefficients that act as weights in the linear combination of single particle transitions in the optical strength of the absorption spectrum. Now, of course, the response function can also be written in this basis, the excitonic basis, because we have diagonalized it, so we can use the spectral representation of this Hamiltonian and we obtain this. And then, if we want to express it in the transition basis, which is what we have, starting from a single particle picture, then it will just be like this. So remember that this just gives us, it is just the response function. It is the variation of the density matrix with respect to the external field. Now, here, actually, I took a shortcut, because I didn't say that this matrix is a median. And so there might be problems. In fact, here I am restricting the possible transitions to positive energy transitions that one can go from the valence band to the conduction band. This Hamiltonian is much more general than that. It can also account for negative energy transitions. In this case, it is not a linear median in general. More likely, it is a pseudoer median, which means that the eigenvalues are still real. But the eigenvectors might not be orthogonal, but this is a problem that can be accounted for with an overall mathematics. You will see a much more detailed description of this in the next lecture by Maurizija, which she will go into details about this thing. But for us this is already enough. What we did, basically, we rewrote the problem of this correlated propagation of electron and holes as effective to particle Hamiltonian to be diagonalized by the code. This particular Hamiltonian contains this off diagonal term in the transition basis, which is the kernel, and this kernel encodes the electron-hole interaction with a screen interaction term, which is attractive and responsible for the binding of electron and hole. And this comes from the self-energy in a single particle picture, in particular for the exchange correlation self-energy in a static approximation. And this term, which is repulsive, and this comes from the heart refunctional in a single particle picture. What do we need to diagonalize this Hamiltonian? What are the starting ingredients? We need some quasi-particle energies, the best that we can get, maybe. So DFT plus GW. Some single particle wave functions, so let's say DFT. And then, of course, for the W we need to compute the static electronic screenings, so static RPA. But these you did already in the tutorials with Yambo. So when we have these, we have these, so just to recap, we notice that the independent particle picture fails to reproduce optical absorption in many semiconductors due to the lack of electron interaction. But we can account for the electron interaction if you consider the dynamics of the excited system. We consider these dynamics using the equation of motion from the response function, which really work, we recast into an effective two particle Hamiltonian, and this finally is the correct quantity, the most accurate quantities for optical absorption. But now we have switched from the independent particle picture to the exiton picture. So that's it. There is no more. I just leave you with this painfully detailed graph representation of the Bessel-Peter equation. So here you see you have all the indices, the state index, k point index, you have the sums of the g vector for the plane wave and so on. So you can maybe go on the YAMBO wiki, take these and then maybe compare with the expression that you find in the YAMBO cheat sheet if you are interested in how maybe one can really use diagrams for doing calculations. Thank you.