 A very tough but very important theory that you heard. I tried to be a little bit more easygoing, a little bit more semi-empirical, a little bit more descriptive, and to give you some examples of how enjoyable it is to use YAMBO and to play around with theoretical spectroscopy. So for this purpose, I chose this image here, which was done in my group. I will tell you in a moment what it is. But even if you don't understand what it is, I think you can enjoy the beauty. So the message of the talk is theoretical spectroscopy, electronic structure calculations in general is fun, it's colorful, and it's also useful, I hope to show you. So I slightly changed the title, which was excitons in low dimension. I generalized a little bit to theoretical spectroscopy, which of course involves excitons. And I mostly talk about two-dimensional materials. Well, a little bit of history on two-dimensional materials. You know, in 2004, there was a discovery of graphene. And few people know what happened in 2003. Well, that was the foundation of the University of Luxembourg. And that's a place where I'm working. And so the university is only 17 years old. And when I arrived, it kind of looked like this. So the reason of the university is that there was a steel industry in Luxembourg. And steel industry went down the drain as almost everywhere in Europe. So you see this huge abandoned site that belonged to Achsador-Mittal. Here you still see the high furnaces. So the first thing that Luxembourg built was a bank. That's this red building here. Second thing, you don't see it yet, it was a commercial center. And the third thing is the university, which now looks like this. So a few years later, the university is there. And it's very active and growing. And there are also opportunities to do physics in Luxembourg. Okay, so just a few words about the university. Now back to the topic. Two-dimensional materials. I also will not talk long about, I mean, many things can be said about two-dimensional materials. After the discovery of graphene, many other materials were exfoliated because since the materials are layered, you can kind of peel them off quite easily with a scotch tape method. And then a whole zoo of two-dimensional, different classes of two-dimensional materials have been synthesized up to now. And I like to cite this recent review from Castellanos Gomez, who ordered the different classes according to their band gap. And of course, that's important for optical applications. So my group is mostly working on these three materials, graphene with zero band gap, boronitride with an extremely large band gap. We heard already from Mauricio that, okay, it's important to talk about GW convergence to decide what exactly the band gap is. Anyway, it's much in the far ultraviolet. And then you have the transition metal like calcogenides somewhere in the middle. So I give another overview here. So graphene, you remember the famous linear crossing of the bands, which makes it very exciting for physics, but kind of not so exciting for semiconductor applications because there is no gap. Boronitride does have a gap. Well, for a long time it was called the wide band gap semiconductor. Well, it's rather an insulator. So 6.5 EV is rather difficult for transistor applications. It's possible, but not very efficient. And then some years ago, people started to talk intensively about molybdenum, disulfide, because that one has a gap of 1.9 EV. And so it's suitable for transistors and solar cell applications, et cetera. So I will discuss a little bit the properties of hexagonal boronitride and molybdenum sulfide. And in the afternoon, you will spend your own time calculating the properties. So I hope that this lecture will motivate and tell you why it's fun to work with this material, which, by the way, is rather easy. It has a boron with three valence electrons, nitrogen with five. So on average, this makes four valence electrons. So it's not a mystery why you can build the same hexagonal lattice as you do with carbon atoms. And also you can make boronitride nanotubes as you make carbon nanotubes. Now, before I really give the lecture and present some of the work that is being done in our group, I would like to show this slide because if I put it at the end, everybody is tired and doesn't pay attention. But really, as almost everywhere in the world, research in Luxembourg is being driven by very good PhD students and postdocs and some of the people here are in the audience. So Enric Miranda is not here. He is now, he joins the WASP team in Vienna. Fulgo, well, you know, Sven Reichardt finished his PhD two years ago and Thomas Galveny will finish this year. He's also in the audience and Alejandro as a co-organizer was a postdoc for five years in Luxembourg and now he's in Spain. Of course, I have this long-term collaboration with Andrea. It started when we were both postdocs in San Sebastian. So some 16 years, 17 years ago and also Davide with all the wonderful implementations that he has done has been a very important collaborator. Well, we are not only doing ab initio, many body perturbation theory but also what some people nowadays call second principles. Calculations meaning semi-empirical, tight binding calculations and this is in collaboration with the team of Honorar in Paris, François Ducastel and Akin Amara. Okay, but actually I would like to start with something very simple and you may wonder, well, why is this guy now talking about the hydrogen atom? Well, you will understand in a moment but I do like the hydrogen atom because I have to teach it every year when I teach quantum mechanics. We are not going to do the full derivation. Let me just remind you that we are solving the Schrodinger equation for the spherically symmetric Coulomb potential and we do this by separation of variables in polar coordinates. So we get a radial wave function that depends on R and then an angular wave function and that turns out to be the spherical harmonics. If we insert this ansatz and the solution for the spherical harmonics into the Schrodinger equation, we then obtain the radial Schrodinger equation which now only depends on R and due to the slightly complicated form of the Laplacian operator in spherical coordinates we have this one over R squared over the R squared term. Remember we also have the centrifugal potential L times L plus 1, it's kind of a repulsive potential that grows with the value of the angular momentum L and then we have the attractive Coulomb potential. And, well, we do solve then the... We can then solve the radial Schrodinger equation, we get this functional form, exponential suppression at long distance, the Laguerre polynomials and factor R to the L. Doesn't matter exactly what the solution is, what is important in the context of this lecture are the energy levels and you get the famous minus one half times the Rittberg constant and I write it out explicitly here, times one over N squared. So that is minus 13.6 electron volt times one over N squared. If N equal one, you are in the ground state and then if N equal two, you are traveling already a large distance and the higher you go, the closer the energy levels and then you are approaching the continuum. So if you go to positive energies, we switch from bound states to scattering states and we have a continuum of states. And of course, knowing the Laguerre polynomials and the spherical harmonics, we can visualize the different orbitals, one as two S and so on, as we are used to doing from chemistry and atomic physics and quantum mechanics. Okay, so that's a hydrogen atom, an electron and a proton. That's already quite complicated, but at the end, when you know how to do the math, it's quite easy. At least it's easy compared to solving the beta-salpeta equation as you will do this afternoon. So we can go to a similar problem in solid state physics to the hydrogenic problem in semiconductors. So assume we have an excited electron in the conduction band and a hole remaining in the valence band. There are charged particles. They will interact through a screened Coulomb potential and then basically you have the situation here. The electron is somewhere, the hole is somewhere else. The electron is attracted to the hole so it will kind of orbit around the hole and the screening now is no longer the vacuum screening epsilon zero, but it's epsilon zero times epsilon r, the relative dielectric function or dielectric constant. So at the end, you see it's the same problem as the hydrogen atom. Just we have a reduction in screening and we have a kind of different effective mass. So first of all, we have the effective mass of the electron and of the hole. So we have to calculate the reduced mass through the inverse, through the sum of the inverse masses. Remember, sometimes it's good to look back what does this mean, effective mass. Remember, when we calculate the band structure, valence band maximum, conduction band minimum, usually around the maximum and minimum, we can approximate the band structure by a parabolic dispersion. It's very good close to the maximum but not very good far away. Anyway, usually when we do semiconductor physics, we are in a region close to the extrema so we can just do a parabolic approximation, Taylor expansion to second order. The first order is zero because we are in the extremum and from that expansion, you get the expression for the effective electron and the effective hole mass. So now we take an electron from here, put it there, so then we have two almost freely traveling particles, an electron and a hole, but they are bound together by the Coulomb potential. So now we can just solve again the Schrödinger equation and we did it already for the hydrogen atom and now we just need to replace in the Rittberg constant the free electron mass by the reduced effective mass M star and in the denominator, we replace epsilon zero by epsilon zero times epsilon R and then we immediately obtain the result that we scale the hydrogen energies by one over epsilon R squared times the ratio of the effective reduced mass and the free electron mass. We can also look at the Bohr radius which is given by this formula here. So the Bohr radius will scale linearly with the epsilon R and inversely proportional to the effective reduced mass. So we have basically solved the problem. We just need an estimation for epsilon R and for the effective mass and if we look at some semiconductors, if the band cap is not too large, you have fairly large dielectric constants around 10 and usually the effective mass is smaller than the free electron mass, so let's assume it's 0.1 of the free electron mass and then you see here you obtain a factor one over 100 squared times 0.1 so you get a factor 1000, so instead of 30.6 electron volt binding energy, you get some tens of millielectron volt binding energy for typical excitons in bound excitons in semiconductors. At the same time, the Bohr radius becomes 100 times the Bohr radius of the hydrogen atom, so around 50 angstrom, assuming that the typical intratomic distance is two to three angstrom, then this is at least a factor of 10 larger than the lattice constant, so this means that the approximation of a freely traveling electron is a rather good one. So physics is consistent and this gives you already an intuitive understanding of excitons in many materials. And well, Mauricio already mentioned solid argon, well I never understood why people are excited about solid argon, it's not so common, but of course at low temperature you can produce it and make it solid and you can measure it and well the excitement comes of course from the effect that in the epsilon two, in the optical absorption spectra, experimentally you measure the sequence of peaks and also in theoretical calculations you get this kind of hydrogen-like, Rittberg-like series of bound states and of course that's at the end the reason why theorists are excited about solid argon because you get a beautiful result and it's rather simple to calculate. So the kind of ab initio picture, so this one is ab initio calculation with BSE, it's consistent not only with experiment, it's also consistent with this kind of intuitive understanding of electron hole pairs being bound to each other by the screened Coulomb potential. There is a slightly more formal way to do this connection because you can, I will not show it, but you can simplify the beta-sylpeta equation. First of all you throw away the exchange interaction which is usually not so strong, at least not at gamma and then you obtain what is called the vanier equation, it boils then down to one particle equation and R is the relative electron, the distance between the electron and the hole and then you see it really looks like the Schrodinger equation, V of R is the screened Coulomb potential and on the left side you have the energy of the exciton with respect to the energy of the gap. So this vanier equation then describes bound excitons and it's rather consistent with what we have seen on the results are consistent with what we have seen here and also with what is calculated in beta-sylpeta. Now this holds not for all materials, it holds first of all for three dimensional bulk materials and it holds under the condition that the binding energy of the excitons is not too strong so the electron and the hole must be separated by some lattice constant otherwise the effective mass approximation for example is breaking down. Anyway I chose this introduction to kind of remind you that when we talk about excitons we are following a fairly complicated many body formalism but at the end the physics that we are doing is rather simple and has been known for quite some decades. So my lecture is mainly about two dimensional materials so let's then go to the hydrogen atom in two dimensions as a first step towards an understanding. So again I try to do a full solution, actually Thomas Galvani might remember that we put it as an exercise for the students in quantum mechanics. I don't know if they, how they performed but I think our instructions were quite good. So anyway you start with the separation of variables in r and and phi. The eigenfunctions of phi are quite obvious e to the i l phi, l being the z component of the angular momentum but there is just one component because we are in 2D. And then when we put this into the Schrodinger equation we obtain the radial Schrodinger equation which kind of looks similar to what we had before for comparison. Let me just show you what we had in the 3D case. So one thing that changes is the centrifugal potential it's just an l squared and not an l times l plus one. And then due to the different shape of the Laplacian in 2D we just have one over r d dr, r times d over dr instead of the same thing with r squared. So looks very similar but due to the two dimension due to the choice of two dimensional spherical coordinates there are some subtle differences and those subtle differences give a different radial Schrodinger equation with different solutions. But as we do for the 3D hydrogen atom we can write down the spectrum analytically and you get the same Rittberg constant you get the same factor minus one half but here you get one over n minus one half in parenthesis squared. And if you now calculate the ground state n equal one then you see that for the ground state in 2D it's sorry it should be a factor four. So doing math at night is okay, difficult. So the two dimensional binding energy of the two dimensional hydrogen is four times the binding energy of three dimensional hydrogen. So instead of 13.6 electron volt you would be at more than 50 electron volts binding energy. So it's increased by a factor of four and of course in the real world two dimensional atoms do not exist but now when we go to two dimensional materials we are in the situation and this figure I took from Furbius lecture. So here you have the electron and the hole that can only move in a plane. So this means that in principle when we do the Schrodinger or the one year equation we have to do it for the two dimensional problem. Well, it's not exactly true because the material has a certain thickness even if it's just one atomic layer let's say graphene or boronite white there is a certain extension of the wave functions into the z direction. So when we talk about two dimensional materials they are not exactly two dimensional but let's say almost two dimensional. But anyway, we can already expect that excitonic effects will be much stronger due to the factor of four in the solution of the idealized two dimensional hydrogen atom. Furthermore, we have to talk about screening and now you see in the simple figure that between the electron and the hole you can now draw field lines and the direct field line goes through the material so that one might say is strongly screened but then many field lines and I didn't even show all of them are going out of the material so they go through the vacuum. So this means that we have a strongly reduced screening or almost unscreened interaction between the electron and the hole and remember that the screening goes like one over epsilon squared in the Rittberg constant so if we now reduce the screening again we will increase the binding energy between the electron and the hole so the excitonic binding energy. So for those two reasons two dimensional quasi two dimensionality and reduced screening we are experiencing much stronger excitonic effects in two dimensional materials than in three dimensional materials and that's why two dimensional materials are such a wonderful playground for the Yambo code. Well, to be precise and that's because we are also doing many semi empirical calculations in the group when we have now the potential in 2D it's no longer just a screened Coulomb potential it is what is called a Keldish potential. Okay, this is the Strover function and the modified Bessel function of first or second kind and it depends now on the ratio of the dielectric constant in the outer and in the inner region. If it's in the outer region, if it's vacuum it's one and it depends on the effective thickness of the material and just to show you if this is the unscreened Coulomb potential then the Keldish potential at smaller distances is strongly screened, so strongly reduced so you kind of narrow down the Coulomb potential to something that is more squeezed for small radii and that goes to the bare Coulomb potential at large distances. So we deviate from the ideal Coulomb potential so when we then solve the hydrogen hydrogenic problem for this situation we will no longer expect a pure Rittberg series that only holds for the one over R potential so the states will change with respect to the Rittberg series. Yeah, a small excursion to the one dimensional world so you know there are also one dimensional materials linear chains of carbon atoms of boronite white atoms, sulphur chains, a lot of different chains have been produced. So we could solve Schrodinger equation in one dimension. We don't even have to do a separation of variables. It's right, it just depends on one variable so in principle this should be easy so I tried to get you the solution and looked at the internet and then I stumbled on this article of 2016 by Rodney Loudon and it starts with the theory of the one dimensional hydrogen atom was initiated in a 1952 paper but after more than 60 years it remains a topic of debate and controversy so then I decided, okay, if it's so complicated I will not present you the solution. Well, the point is you can usually people then do a cut-off to kind of lift the singularity. If you do that, then you get a finite energy solution. If you don't do it, you get in principle an infinitely strongly bound solution so that's kind of an ill-defined problem but in condensed matter we don't have to deal with it because if we have a one dimensional chain then there is a certain extension of the wave function in the perpendicular directions so we will never deal with the purely one dimensional hydrogen problem but nevertheless, remember if you deal with one dimensional problem the binding energy might be even much stronger than in two dimensional problems. Might be because it really depends on the extension of the wave functions into the perpendicular direction to the axis. So that was just for completeness. I will come back to one day problems when I talk briefly about nanotubes but mostly I will remain with two dimensional problems. So then of course we want to use YAMBO and not just do some semi-empirical one-year hydrogen problem stuff so we really want to solve the exact problem so we do solve the Schrodinger, sorry, the beta-sile-better equation in its full beauty as it has been explained by Fulvio and Maurizier so I will not repeat this just saying, reminding you that we are solving for the optical excitation energies which can then be seen as a superposition of many different electron-hole pair transitions and they are mixed by the excitonic eigenvectors A, V, C, K and labeled with the exciton index S. So the first observation of quasi-two-dimensional excitons was in this paper by Brice-Arnault in 2006. I say quasi-two-dimensional excitons because it was done for bulk hexagonal boronitrite but layer-to-hexagonal boronitrite so and they found out that at the end if you look at the ground state exciton and if you look at the excitonic wave function from the side it looks pretty much localized on one layer. So the electron that you excite remains mostly on the same layer than the hole that was left behind in the valence band. So this is then a case where even though you are dealing with the bulk material you are seeing quasi-two-dimensional excitons. And that explains the rather strong binding energy of 0.72 EV, so 720 milli-electron volt which is rather huge for excitons in bulk materials. And again, the explanation was because excitons are confined to layers. So they looked at the different excitonic eigenstates with a very small broadening here and you see the optical intensity. Now, this was the first calculation for this at that time complicated material and it was a huge step forward but there were some problems. So, you know that we always have to look at symmetry and boron nitrite has a three-fold symmetry axis. So if you put an electron, a hole at the position of a nitrogen atom then the excitonic wave function should be symmetric around it, right? Or at least it's wave function squared. And that's what we are plotting. The probability density to find the electron should be symmetric. So that was not the case in their calculation. And the reason was they were using a k-point grid that was not centered at gamma. They did it for a good reason. They said, okay, if we have a gamma-centered k-point grid then we have many degenerate because they would be equivalent k-points and then we are wasting time because they have all the same eigenenergy. So it's better to shift the grid to get more different eigenvalues and eigenfunctions. But the problem was, and that's why you should never do that, that you break the symmetry because if you shift a little bit by a small epsilon in one direction, your k-point grid, then your k-point grid is no longer coherent with the symmetry of the material and then Yambo might give you results that are almost right but not exactly right. So at the time, it really made us think a long time about what is happening and we were fighting with this cake-relating wave functions. And at the end, the solution was that you get doubly degenerate excitonic states due to the symmetry of the, due to the high symmetry of the material. And then, if you have degenerate states like this one and this one, they both look as if they were violating the symmetry of the material. But if you sum up the two densities, then you obtain panel C and the sum of the intensities, it's perfectly coherent with the symmetry of the crystal. So that's an important message that you should follow when you calculate excitonic wave function that actually nowadays, Yambo is doing this automatically for you. So if the energetic difference between two eigenvalues is smaller than a certain tolerance value, then the two excitons are supposed are interpreted to be degenerate and then Yambo or the post-processing tool of Yambo automatically gives you the sum of the two excitons. And now if you compare our calculation to there, or by the way, we did two calculations at the time, one with Yambo and one with Wasp in collaboration with Georg Kressel and except for a small difference in the GW shift, the two codes at that time really gave the same result. And that's also important every now and then to check concerning implementation of different codes are still giving the same results. So let's look at the black dashed line. So now instead of getting in this region of 5.8 electron volt, instead of getting four different excitons, we only get two different excitons. They are both doubly degenerate and one is dark and the other one is bright. So you see, it's very important to look at symmetry and to get accurate results. But this arrow that they did, I mean, we wrote a comment and they replied, but at the end it pushed really the community forward because it really caught the attention to symmetry problems in the beta-salpeta equation, where symmetry is much more complicated because we are dealing with a two-body problem instead of a one-body problem. Okay, having settled this issue, we then in the recent years did additional calculations on hexagonal boron nitride monolayers, few layers and multi-layers, and one example is this one involving Thomas and Fulvio. So you see that we obtain kind of a Rittberg-like series of peaks with decreasing intensity. We will see in a moment why the intensity is decreasing and why the first peak is so utterly intense. So here you see the wave function for the monolayer looks like in bulk. We also did, but I will not talk much about the details, that's the work of Thomas, and there is a poster outside. We also did a tight binding approach to the beta-salpeta equation. Here you see the tight binding fit, which doesn't look very brilliant for the pi and pi star band, but since at the end we only need to fit the region close to the parabolic maximum and minimum of valence and conduction band, respectively, tight binding at the end does a good job. And you see that tight binding helps us not only to print the density squared, but also the wave function by itself. And you can also analyze symmetry and anti-symmetry of the wave function with the positive and negative lobes in the picture. So that's the wave function of the ground state. Now if we go to higher excited state, we get more fancy looking wave functions, and then we are back to the analogy with the hydrogen atom in 2D, or we are back to the analogy with atomic physics, because you know, or you should remember from atomic physics if you calculate radial wave functions, the higher you go in energy, the more delocalized your state becomes, because you are going from that region in the Coulomb potential closer to the threshold for ionization. So naturally, already classically, your electron will explore larger regions of the space around the nucleus. And you see, if you do it carefully, that beta-cylpeta gives you something that is very similar to atomic physics, meaning you get more and more extended wave functions. At the same time, this explains why the optical oscillator strength goes down, because then when you calculate the dipole matrix element, it turns out if the distance, average distance is small, it's very large, and then it goes down rapidly. One can say a lot about bright and dark excitons. You can really do now group theory on the excitons and understand which excitons are bright and which ones are dark. And you see here, E is always standing for doubly degenerate state in group theory, A1 and A2 are non-degenerate states. So now you can understand the figure that was on the title slide, because I don't know if you did it in atomic physics at the time, but you can solve the Schrodinger equation not only in real space, you can also solve it in Fourier space in momentum space. So then you get the wave function as a function of the momentum, and then there is an inverse relationship. The more extended your real space wave function is, the more confined your momentum space wave function is. It's like with wave packets, the more extended a wave packet is, the less Fourier components you have, because for localization, you need many Fourier components. And now you see that as we go up in energy, our momentum space wave function gets more and more confined around the hexagon, the corners, the K and K prime points of the first Fourier zone. You can also do some kind of symmetry analysis, but I invite you to see the poster of Thomas to learn more about this. Well, last but not least, we started some time ago with Eric and Exiton website. It's kind of not developing, but eventually we'll pick it up. We are in the age of high throughput computing, so one could put all the calculations that have been done. You put the spectrum, we did it for boronite right, you click on the different peaks, then you get the Exiton wave function, and you can take your mouse and turn it, zoom in, change the ISO levels, et cetera, et cetera. So it's under this address, it's still functional, and eventually it will evolve. It would be nice to be able to shift the whole position with the mouse, but okay, we all know that this means the recalculation with the jumbo post-processing tool, so that's not feasible for the time being. I think I can skip, well, I will just briefly go through it, but Mauricia told you already the importance. When we do two-dimensional materials, we are always, well, not always, but when we use plane waves, we are using a periodic supercell. So even if your vacuum is large, the vacuum distance, you may, you might put it here, but still it means that when you go far away, you are dealing with the bulk material. So the distance D between neighboring layers in your calculation is and remains an important issue. So at that time, we calculated the dependence of the binding energy on the distance, because we know that in bulk boron nitride, the binding energy is 0.7 EV, and if we go to a distance of 80 atomic units, so roughly 50, 45 angstrom, then the binding energy goes up considerably. This is because we are going more and more from a 3D situation to a pure 2D situation, and at the same time, we are reducing the screening. But you have to do the same calculation for the GW correction. So you see that in the same figure, the GW shift goes up, and that's not a big mystery, because at the end, we have the same terms for screening and electron-electron and electron-hole interaction that enter the equations. So then, if you add up the two effects, you get a rather stable position of the exciton, even if you change the vacuum distance between the layers. However, this is only valid for the ground state. If you really would like to calculate the whole Rittberg series, of course then, it does critically depend on the position of your GW gap, and then, nowadays, you should use the Coulomb cutoff, meaning there is a certain distance above which the Coulomb interaction between electron-hole and electron-electron is no longer taken into account in the perpendicular direction, and that allows you a much more rapid convergence as a function of the distance. Yet, pay attention, k-point convergence becomes then a very important issue. Okay, I will maybe just briefly, Maurizio already showed you. We did results on boron-nitride nanotubes, and we saw some shift of the excitonic peak, but mainly for very small nanotubes. If you go to this index, gives you the radius of the nanotube. If you go to larger indices, then the spectrum looks remarkably stable, and it looks remarkably like the spectrum of the single sheet of boron-nitride that you see in Penelbee. And the reason for that also became quickly clear, because here, in a paper by Stephen Louis that came out at the same moment as ours, they calculated the excitonic wave function for a very small radius tube, to say, here's the position of the hole. You see the probability of the electron. And now imagine if you go to larger tubes, then the exciton will be in a locally flat area. It will not even travel around the whole circumference of the nanotube. So for larger nanotubes, you will just have a quasi-two-dimensional exciton because the electron hole pair lives in a very slightly curved, but locally two-dimensional world. That is valid for boron-nitride nanotubes. It's different for carbon nanotubes, where the exciton is really localized around the whole circumference of the tube. So there you are maybe closer to the 1D situation, but even then it's not purely 1D because the exciton is really traveling around the circumference, and thus it's really not comparable to the 1D Coulomb problem that diverges. Yeah, another interesting material is MOS-2, or all the transition metal diacalcogenites. I don't know why this is cut off. Also Mauricia already presented this result more or less important is that we have the spin orbit splitting, and that why it was very important at the time that David Sangali just happened to implement the beta-cylpeta equation for full spinorial wave functions, and that enabled us to do this calculation and separating the A and the B excitonic peak, which is typically the split exciton is typical for transition metal diacalcogenites. But this did not prevent other people at the same time from doing the calculations without spin orbit splitting. So here you see just a doubly degenerate valence band maximum. And they also got some peaks, and they are great. There is the A and the B peak in the experiment. In my beta-cylpeta equation, I see more peaks, but I say two peaks, so great. I reproduce the experiment. This publication here. Well, the common is of course the splitting of excitonic peak is merely due to a very low k-point sampling. So if you have a low k-point sampling, then your excitons may not be converged. You artificially split peaks into several sub-peaks, because in the beta-cylpeta equation, there are not enough states to be merged into one state. So then you may get artificial coincidence with the experiment. And this also happens in so-called high prestige publications like in nature photonics. Again, no spin orbit coupling, 18 by 18 k-points. You kind of see an A and a B exciton. Here they discuss the dependence on strain. Okay, the effect that they see is probably true. The scaling with strain, but of course, such a calculation is underconverged and it's damaging our community if too many calculations of this kind are appearing in so-called high impact or high reputation publications. So we did, sorry, it's not really easy to see, but we did a calculation for different k-point samplings. The different colors are the different k-point samplings from six by six up to 30 by 30. And you see that only starting at 24 by 24, you are converged in the binding energy. If you go to lower values of sampling, you really strongly vary the k-point sampling. And only then you really merge into one peak. So this is with outspin orbit coupling and with the 12 by 12 sampling, you see that you get a splitting of the peaks even though there is just one valence band and one conduction band. Principle, it should be one peak. So only to go to appropriate k-point sets you really converge. And the reason is why you need so many k-points is if you look at the excitonic wave function in reciprocal space and then you see that it's really confined to a rather small region around the corner of the hexagonal Briemann zone. So you really have to have some different k-points in this region. If you just use a six by six k-point sampling, you have one point here, the next point is here, and so on. So then you have only one point that really falls into that region where the wave function is localized. So that's a pictorial way to understand why k-point convergence is so important. Okay, in the last five minutes or so, I will take it easy and give you just some ideas where calculating excitons in the 2D world is going. So the modern way is to do high throughput and that's what Christian Thyssen did in his group and they came out with this paper and a computational 2D materials database which I think has something like 500 different materials with fully or hopefully fully converged GW band gaps, beta-cyl-peta calculations and so on. So for example, I looked for tungsten diselenite and you find the GW band gaps, the GW band structure and also the beta-cyl-peta equation comparison with the RPA, both for X polarization and Z polarization. So that's actually a very useful tool. If you want to check your calculation, you have a database that you can compare with. It also makes it more difficult to publish papers nowadays because when at my time it was maybe enough for a PhD thesis, let's say for a paper to present a converged calculation of let's say MOS2, then okay, you could maybe publish a PRL as Stephen Louie did or a PRB as we did with the same results. But then, I mean nowadays, you can no longer publish a PRB just about the absorption spectrum of W82 because it's already in the database. So we have to do more smart things and here are some ideas, some activities that we are doing in the group. So what can we still do with the beta-cyl-peta equation even though there are already high throughput archives? But there are still many things to be done. Exit on dispersion, few layer systems, hetero bilayers, and then spectroscopy involving phonons, electron-phonon coupling. I guess this will become a topic tomorrow when you will talk about time-dependent spectroscopy. So there are many, many things still to be done with the beta-cyl-peta equation. We are really just at the beginning mastering the static beta-cyl-peta equation, reproducing calculations that have been done. It's an important first step for you. Going beyond is then something that you can do in your PhD and I will just show, for example here, it's a full dispersion of the excitonic dispersion in the monolayer of hexagonal boronitride, again comparison of Appinicio and tight binding calculations. Another example is now we look at the bilayer of boronitride. We see the splitting of the bands, we see the splitting into bright and dark excitons because the wave functions are different, even versus odd, and then we are back to the selection of atomic physics. You need to change the parity in order to have a dipole allowed transitions. And we did just showing you some pictures without explaining too much the physics. We did excitons in hetero bilayers, MOS2, WS2. You see you have a staggered gap, which means in principle that in semiconductor physics, you say, okay, holes will go to the higher gap to the higher valence band, electrons will go to the lower valence band on the left, so you get an electron hole separation and that is great for photovoltaics, et cetera. The problem is you also get an excitonic binding of electrons and holes between the different layers and that really changes the picture and needs to be understood in order to use hetero bilayers as a solar cell. We also did excitons in tri-layers with an electric field, so you kind of artificially induce a staggered gap by changing the position of the valence band maximum in the three layers and also there you see interesting effects as a function of the electric field. And then we started to, well, we did some work, but I will really stop here to look at lattice vibrations and then when you switch on phonons and more information on that will come in the next days, I guess, then you can do all kinds of theoretical spectroscopy in elastic light scattering, Fulvio has been doing this and Claudio in the audience, sorry, indirect phonon absorption and emission, that's what they did. We are doing Raman scattering in Luxembourg and the works that Alejandro did and that you will probably see tomorrow is the relaxation of excited carriers via electron phonon scattering. Anyway, I will just conclude here and come back to this kind of overview, motivating you to dig further into thinking how we can use the beta-cylpeta approach to do new things and exciting things in theoretical spectroscopy. And with that, I thank you for your attention. Thank you.