 V kolinear, so spin polarizati problem, also in the non-colinear problem, means that if you have a system where the spin orbit is important, there are many materials where now the spin orbit is important from pero-skite or transition-medal, the calcogenide, other systems that are of interest nowadays, you can do it. Then I will show you some example of optical spectra calculated with Yambo. And then if I have time, I will go through the step-by-step solution of the Betesal Peter, so exactly what you will do this afternoon in the tutorial. So before moving to the Edine question, let me do some summary of what we have seen up to now about the response function. In the first day, Andrea spoke about the linear response and you remember that the chi is the crucial quantity that you need the microscopic response function to calculate the epsilon minus one. And so Andrea has shown that it's possible to derive a Dyson equation from the independent particle response function. You can obtain what is called RPA, including local field when you include this Coulomb interaction in this Dyson equation. And also, I don't remember if Andrea explained what probably he mentioned in his talk, you can obtain the response, the chi also in the time-dependent density functional theory if you include the exchange correlation term here that is the derivative of the exchange correlation functional potential with respect to the density. And so this is time-dependent Df30. But if you remember, this quantity is a microscopic quantity. What we need is the connection to the macroscopic quantity and you learned that in order to obtain the macroscopic quantity what you need is to do an inverse of this expression. So you have to do the macroscopic average and then make the inverse. And so at the end in practice what you did the first day is exactly to solve this equation. So we did in the optical limit q equal to zero because we were interested in the independent particle optical absorption spectrum, but exactly what you did is this one. Actually you didn't do this because you calculate only the zero zero without including local field. I don't remember if in the tutorial the local field were included or not, but the expression is exactly this one. But I have to say that in order to include this local field effect, so go from microscopic to macroscopic there is another way to do that has not been mentioned up to now that is to obtain an equation for a chi that is called chi bar. And the only difference is that in this equation the Coulomb potential must be cut at the long range part. So this is shown mathematically that if you do in this way the macroscopic electric function is directly related to the macroscopic average of this chi bar. So the way that you solve this equation eliminating the long range part that is exactly this definition you obtain directly the macroscopic electric function. So you can do in this way or in this other way. You will understand in a moment that in the beta-salt-peter equation is much more convenient to use this definition to obtain the macroscopic one. Let me do another statement that was mentioned by Andrea that a way to derive this equation or this one. So we understand that chi and chi bar are the same, the only differences the introduction of a local field effect here is that you can obtain this equation or this equation moving from an equation for the irreducible polarizability that goes from the independent one to the irreducible polarizability where you include here the exchange correlation effect in the response. Then you can use a Dyson equation that connects the irreducible polarizability that here I called chi tilde to the reducible one that is exactly this one and then doing this and combining with one of these two you obtain this one or this one. So now you understand why I had to include this chi tilde that you didn't... So it is another one quantity is useful for the derivation that I will do now. So here are the definition of the property in the complex space of the response function in linear response. Ok, so now I will do a step forward because I will start from the second day lesson. So the chi was introduced in the linear response theory if you noted in the many body approach in the Dyson equation Andrea and Daniela used the definition of pi that is the reducible polarizability clearly the two are quantities that are clearly connected we know exactly how to connect them but it is just a formal point that I want to make and if you note here I called the reducible polarizability not pi, but pi tilde. Ok, you understand in a moment why I do in this way. Ok, so if you remember this is the definition that you have seen in the two lessons of yesterday and in particular what I can say that the reducible polarizability is connected to the reducible polarizability in a similar way of the equation that I've shown before so you can go through this Dyson equation from irreducible quantity to reducible one both for the pi and also for the pi bar so you can define an equation where if you want to have a macroscopic quantity you define the Coulomb potential v bar eliminating the long range part and then you can move directly to the calculation of the macroscopic electric function through this equation so you don't need to invert but you use simply the pi bar and this is again a reducible quantity while in the edin equation we are speaking about irreducible quantities that are calculated diagrammatically ok so the goal now is to find an equation for pi tilde beyond the independent particle approach that is what we have seen up to now because we have calculated only pi zero let's say that is the independent one and then we want to calculate the equation essentially for pi bar because my goal is to calculate the macroscopic electric function and so we come back to the edin equation so here instead of calling p I call p tilde just to remember that this is the reducible quantity that I want to calculate in another cycle iteration in the edin equation and this is actually what I do so I start with the self energy that is what we learned yesterday that is g w and we stop here but if you do, if you remember gamma is defined in a compact way in this way so there is a functional derivative of the self energy with respect to the g and now if you start with the sigma that is g w you end exactly in this term e w and this part and this part is what Fulvio mentioned and also Davide before mentioned that is generally the first approximation that is done in the Betis-Alpeter equation so you assume that this term is negligible so you delay it completely and so the functional derivative of the sigma is simply e w and so at the end we arrive to an equation for the vertex that if you remember is a three point equation in space and time that is this one but my goal is to obtain an equation for pi tilde and so the idea is that from gamma I have to arrive to an equation for pi tilde and so what I can do I can multiply the gamma vertex obtained in the way that I shown by two g green function here and here and then doing an integration I obtain exactly an equation for this pi tilde three vertex equation that appear diagrammatically in this way so you have the delta delta let's say vertex g g that is this one and this part that is represented by this part here but this is not the end because to describe the propagation of the electron at whole and see their interaction we need a four point quantity so you can imagine to open also this point here three and generalize to a four point polarizability tilde and so at the end we obtain an equation for the irreducible polarizability from the equation doing another cycle with an approximation the electron hold interaction inside so this is the independent polarizability the rpa and this is what we have derived from that in a question doing a second iteration so these are the definition so doing in the same way that I mentioned before you can use this equation for the irreducible polarizability and the equation that connect the irreducible polarizability to the reducible one in particular pi bar because I want to the macroscopic quantity and then if you combine the two you arrive to the equation that is exactly similar to what Fulvio presented before with some difference that now I will try to explain so you see that here you have the p0 that is the free electron hold propagation and here you have this kernel bubar because I am interested in macroscopic quantity and w that as you can see is a negative quantity and the represent I will show you in a minute the electron hold attraction so this is what Fulvio called before there was an i of difference in the definition but I mean is inside the definition of p so they cancel out at the end so the question is this one diagrammatically you can say that the equation the four point equation of the polarizability is this you have the pi zero that are two independent electron and hold propagation and then here is the kernel that describe the interaction and again you have this term you see that if you go through the index I use the compact notation but if you use all the index they are done in this way so if this is 1528 the connection is 1281528 so they are not equivalent they are really four point you cannot have the same diagram you cannot in principle have two point from here so this part is what we call electron hold exchange interaction and this is something that is repulsive and that this actually from the derivation is an electron hold exchange but actually comes from the heart with M so is quite strange but we are speaking about electron hold exchange interaction and here this is the electron hold direct interaction that is attractive so this is negative so which are the approximation because up to now the W in principle is a time depend from time so here in principle you have something like T and T prime are different so the main approximation that we do actually and is done in the code is assume that this interaction is static is what we were discussing before so is exactly the W calculated at the RPA level and this is in this way you understand that this is the first approximation that we do and also we have neglected the change of the screening with respect to G the approximation that I have shown before and the other things is that clearly in this equation there is G0, G0 and here the normalization factor the dynamical factor Z and kappa that Andrea was discussing before in principle should be put there instead what we do is to assume that is 1 and this means to cancel out the dynamical effect in the definition of P0 and the dynamical effect here so was what Andrea was discussing before so now that everything is defined we can look at the connection with the macroscopic quantity and as you remember since I move to P bar I can simply do the average and the average in G space means to take the G, G prime equal to 0 term this is the expression and this is the macroscopic one clearly if you want to have the polarizability what you have to do is to contract the two index at the end of at the beginning at the end of the in this way so if you do that you can pass from the Fourier transform definition of the P bar this is this one and then as before you can move this is what is done in the code to the transition space so multiply for a couple of block state N1 and 2 and 3 and 4 and you have the definition to move from the real and time space omega space to the transition space so in this way we arrive exactly at what Fulvio has shown before if you remember before he used chi instead of pi bar but we are exactly the same and use k to index the transition here I explicitly put N1 and 2 to mean a transition from N1 to N2 that can be Nk if we have an extended system only valence conduction or valence valence or conduction conduction here is just a couple of index to another couple so what we do now is assume that we can do exactly in the same that Fulvio discussed so we can write this P bar through the inverse of an excitonic Hamiltonian that is already introduced by him and the excitonic Hamiltonian it is again in this approach you see immediately that the energy is the proper way to introduce the quasi-particle energy because we started from the calculation of the energy in the GW approach so here is very natural to include the quasi-particle energy calculated at the GW0 GW0 level and here this is the kernel these are the occupation between two states and if you go through a more explicit definition of this excitonic Hamiltonian you can see the excitonic Hamiltonian in principle is appearing this way and you have that this part we will see later but you see that in this non-diagonal part the independent part the independent particle part here is zero so you don't have this also here all these two parts are completely zero of the matrix due to the fact that both these and these ones are zero and then what we do is the fact that since we want to calculate the P bar you see here this term our occupation conduction this term cancel out this part of the matrix does not enter in the calculation of the P bar we are interested only in this part and this is exactly the part that full view actually shown probably only this one so this is we arriving to the final definition that you have seen before ok so we cancel out this in the definition of the P bar because we need only the transition from occupied to unoccupied state but as you can see the matrix is more general and you have a couple two parts that are ok I show you here are generally called the resonant part that are between a balance conduction prime and the opposite principle but you can show that one and the other one are exactly one the conjugate matrix of the other and then you have the part that is the coupling that is called K and you can show that the matrix this part are is a hermitian that is called resonant part this is the anti resonant part and this is what is called in the code also coupling and it can be shown that this matrix is symmetric so in principle what you have in the code is all these matrix but what we will do this afternoon for instance generally it can be show that the coupling part is small means that these two parts that represent the propagation of electron a whole let's say anti-electron and that whole can be the couplet so these two term if these two term are small these are the couplet and you hand in what in TDDFT approach is called the Tandankov approximation and this approach is generally good to describe the optical absorption spectra of all the material when the transition related to exit and plasma are very well separated so generally in semiconductor in traditional material standard material is a good approximation and actually you can see that since these two matrix are exactly exactly the same at the end you can completely neglect also this part what you will do this afternoon is calculate this matrix that is called the resonant part of the Hamiltonian and this is the approach that is good to calculate the optical absorption spectra of most of material remember that if you are interested in steady energy loss calculation it has mission that in general it is important to include these two term in the definition of the microscopic directive function ok so now what I want to do is to go more deep in the definition look at the matrix element of the excitonic Hamiltonian in particular C which is the role of spin because up to now we use all the index N1 and 2 without saying which is the spin of the electron and hole so we have essentially three cases we can say that we can have a system that we call non-colinear so non-spinpolarized where S2 and SZ are all good quantum number we can have collinear case so system determinatic where the spinpolarization is important and in general SZ is a good quantum number or we have what is called non-colinear case where nor S2 or SZ are good quantum number so this means that in principle the definition of the wave function that is in principle a spinor but one of the two components are completely zero so you have states that are up or down and in general in the non-colinear in the collinear case spinpolarize it will be a random combination of up and down wave function and then instead for the non-colinear case actually the wave function is described really by a spinor so this means that in the collinear case in principle you have that the wave function of the up and down channel are equal epsilon up and down channel are equal in this case they are not but in any case when you describe the propagation of the electron you can define the index n to indicate the state valence or k whatever you want and also sigma that means half or down so is a very good definition while when you want to describe diagrammatically an electron a hole that propagate you have to describe the state considering the fact that you have I call here and capital that embed the fact that you are describing a spinor so you have both the component up and down in this expression and the energy clearly depend from this index n so you have a double respect to the ampolarize case ok, so now we can go to look at the excitonic Hamiltonian and making explicit the index of the spin so the excitonic Hamiltonian is defined in this way so we start with the collinear case that is useful when you have a spin polarize system so we have this expression I called here the diagonal part that is what we have seen before and now you note clearly that there is a conservation of the spin sigma 3, sigma 3 prime sigma b, sigma b prime so it is completely diagonal and then we hand with the exchange part in this case if you remember the diagram is done in this way so means that you have a conservation of the spin here, of the spin component here and here clearly I assume that the Coulomb potential here does not depend from the spin that we are doing this approximation so in this way this means that if you go to write the matrix element of the exchange part that is what is called v of v bar depending from the definition you end with this matrix element so you have the same spin here and the same spin here in the w we have the diagram that is opposite exactly the same like v and so you have the conservation of the spin at this point and this point and so if you go through the definition you see that here you have the same index in two conduction c, c prime and the same index in w and w prime ok, so how looks like the matrix in the spin space if you go to see the matrix in the spin space you can see that the matrix appear in this way and you have a lot of zero that do not mix channel where you have a spin slip actually so you have two blocks and you can see that actually this block here means transition like this half down half that clearly I put this plus minus because this one is not equivalent to this one so you have the dependence of up and down in the matrix element through the energy and through the wave function so the elements are not all the same in the general case and then you have the other part that is this block here that describe instead the transition half up or down down and this clearly are the only transition that contribute to the optical absorption spectrum within the dipole approximation due to the fact that you have the spin selection rule to be applied ok but what happen when you have the spin polar the non-spin polarize case in that case clearly not all the s z that is a good quantum number but also a square so it is more convenient to rotate the matrix in the space of singlet triplet so what you can do is a combination of the couple up up down down in this way to obtain the singlet and triplet exciton and so if you do that you obtain a matrix that appear in this way so these are exactly the equation that we have seen before Fubio did not mention but here the two comes really from the fact that you are describing singlet exciton ok and this part here instead is what we call triplet exciton that are completely equivalent so actually if you want to calculate singlet exciton this is the way that is what we will do this afternoon if you want to calculate triplet exciton you have to turn off this part so what we learn from here is that the triplet for instance are always lower in energy with respect to the singlet and this is because this term is negative so you have the positive quantity here and the other things is that clearly singlet can be bright because they are spin alloyed then from other selection rule while triplet are dark because they are spin forbidden ok so what happen instead when we move to the full spinolier case the main difference is the following as mentioned before in this case each propagator has a sigma that is the component of the spin that goes with it and so you have that since the coulomb potential the interaction does not depend from spin there is a conservation so sigma sigma and what you have to do in the equation to obtain the matrix element is do an integral only over the spatial define it here instead when you have to calculate the matrix element at the end you have term where you are transporting a spinor that is i and j in this case what I called before n and so what we have to do is not only do an integral over r but also sum over the sigma component so you have to sum over the two channel of the spinor that you are transporting and so at the end that the beta-salpeter formally seems very similar to the previous one what I have seen in the very initial slide but here the index are the index of the spinor state so clearly this means that with respect to the non-spin polarized case the matrix is four time larger because you have the double of state so the four time transition so the matrix is much larger than before and actually if you look in the matrix element there is this sum in the two components of the spinor that I was mentioning before and so this is the end of the discussion but now just me show you how the matrix element appears so what is programmed in Yambo the exchange kernel if you transform for real transform is like you have seen yesterday w is appear like a single sum in g-space the Coulomb potential what we call the oscillator that was defined like rho probably by Daniele ok, while the correlation part that is this direct part as you see as a sum over 2g, g and g prime because there is this screened Coulomb potential w that enter here and so this is much more cumbersome to calculate like in g-w the exchange part is much less time consuming that the correlation part ok, so ok, so now how up to now we have seen all the definition of the matrix element so now we have to understand how to obtain the spectrum so we have to remember the relationship between the p-bar that we have assumed is related to the inverse of the excitonic matrix and the macroscopic epsilon and in Yambo you can do in two way you can invert the matrix and you can do with the Scalapac library but it's not very efficient and there is a way that is the recursive approach laxos-hydok approach you will do the tutorial this afternoon and it's very efficient and also if you are interested in looking really at the excitonic state for instance you want to plot the spatial distribution of the exciton you want to have the excitonic eigenvector so you have to use the spectral representation but as you can see here I put a more general one also because the matrix in principle is the full matrix that is not only the resonant part so the full matrix is pseudo and medium matrix that means that you have an hour left part here and so this can be done with the Scalapac in Yambo parallelized also but there are also new subroutine that are called slap that are appropriate to obtain the eigenvector of a large scale sparse matrix and in general is interesting if you have very big system so very large matrix to be analyzed but you are interested specifically only in few eigenvector or eigenvalues so there is a way to do it we didn't put in the tutorial but if you are interested you can ask and so the question that appear if you go through the previous definition that I mentioned before of the P bar and using the full spectral representation this one is exactly this one so you see that the expression is much a little bit more complicated clearly you see here that you have the real and the imaginary part of the full matrix if you go to calculate only the resonant part of the excitonic Hamiltonian that is what you will do this afternoon actually the eigenvectors become orthogonal so you have only the medium part of the matrix and you end up here so I took only the imaginary part of this expression simplified with this expression that is exactly what Fulvio has shown before probably what you can see here is that you can end up exactly with the definition of Fulvio if you use the definition of the velocity in this way you end up exactly in what Fulvio has written before so this is the dipole and the dipole is the excitonic dipole means that you are doing transition from the ground state to the excitonic state lambda and you want to calculate which is the probability to have this transition through this dipole that is the exactly excitonic corresponding dipole with respect to the independent particle that is this one so in the independent particle approach you can calculate this one in the beta salpeter you are mixing through the excitonic eigenvector the dipole of the independent particle approach ok, so this is to obtain the spectrum through excitonic eigenvalues and eigenvector instead to obtain the spectrum from the iterative solution you can start from the final definition and do some simple mathematics so rewrite the velocity through the definition that I have shown before and then you can explicate this modular square in the two element and use the sum over lambda lambda lambda that is equal to 1 so at the end you end up with this definition of the spectrum and this definition of the spectrum because you remember that if you have the spectrum in this way where this is the initial vector that is exactly what I have shown here this one you can define this matrix element so transform the matrix in a three diagonal way and then invert through this recursive approach so what you need is to calculate the coefficient i plus 1 and then use to this definition to calculate the spectrum so clearly in this way you don't have eigenvalues, eigenvector you have only the final spectrum and this is very efficient so you can use very big matrix when you have problem of memory this is the only way to do it beyond the approach that I mentioned before and it convert very fast generally less than 100 iteration so you can visualize and scale very well with NPA parallelization and note that you can do this not only for the resonant part of the Hamiltonian but there is a work by Mirta Groening and Andrea Marini where they extended this solver, iterative solver also for the full matrix so if you are interested in the calculation also consider in the coupling part of the Hamiltonian so the last part I am time is that I show some example that are quite old so probably not very interesting but okay this is the school so this is the very famous spectrum of silicon you can see this is again the RPA normal RPA, this is GW so you shift to up energy and then you have a completely renormalizacion of the intensity of this to peak E1, E2, only through the solution of the beta-valpeter and so you have a comparison with the experiment that is quite good it is not perfect because here the temperature effect in the solution of the beta-valpeter has not been taken into account probably you know that it is possible to include the thermal effect in the calculation of the beta-valpeter from Andrea Marini and recently people use frozen phonon approach to calculate the thermal average and so calculate the beta-valpeter spectra including the thermal effect okay and so this is the other example that has been shown before probably again this is a solid and if you calculate the RPA this is what you obtain if you calculate the GW what you obtain so the electronic gap is very large about 14 EV while if you include electron hold interaction you end up again with a very nice agreement with experimental curve and again this difference from 12 to 14 is what is called excitonic bonding energy that clearly in a material like argon and so here other example this just to show you that in the past what has been done by Andrea is used the beta-valpeter for three material that are here and so the agreement with experiment that are the blue curve with the red, with the black is quite good but in this paper if you go through this paper you realize that in this derivation what they were trying to do is trying to use the time-dependent approach to calculate the optical absorption spectra and I don't know if you are aware in time-dependent DFT generally the optical spectra are good only for finite system and this is due to the fact that the exchange correlation kernel is not exact in particular you don't have the long range term 1 over 2 squared is instead inside the W that is in the beta-valpeter so what they did in this work is start from the many body kernel so exactly used the equation for chi in time-dependent DFT and the equation for p in the many body approaches of beta-valpeter so they extract this many body kernel from the beta-valpeter and they obtain an FXG that is defined like many body kernel and in this way you see the red and black are perfectly equal so in principle is possible also to do calculation in time-dependent DFT using this kernel but from computational point of view up to now is not convenient in the sense that in any case you have to use the beta-valpeter extract the kernel and then do time-dependent DFT so it's not very useful for in this moment so these are other results this is a very old results from our group always this is the reflectance sinusotropy spectrum of a surface that is the diamond 100 the surface when you cut along this direction form dimer at the surface and this dimer break the isotropic optical response of the material and so if you use light polarizing in one direction you obtain a very strong signal that comes from the initial surface state and so you can calculate, sorry, measure this anisotropy and what we did here is to compare the measure that is this continuous curve with excitonic calculation and as you can see the agreement seems very nice I didn't present the RPA that is completely off so the nice thing is that it seems that there are excitons that are bound like 1EV that is quite strong for a normal semiconductor material this is another example where doing an average on molecular dynamic snapshot they were able to calculate the optical absorption spectrum of liquid water that seemed to reproduce this optical feature of the experiment probably there are more recent result from that but I'm not aware so now we can move to other results that are for instance nanowires here is another work where what we have shown is try to compare the excitonic gap with the optical gap measured in different nanowire of different orientation and different size what you can see that clearly doing quasi particle calculation only the gap is completely off from the experiment that are the yellow while when include for three different orientation the excitonic gap we move in the region of the experiment and also in this way at that time we were able to explain the optical absorption of porous silicon that is experimental is a combination random combination of different wire of different orientation and here again you can see that using the definition of singlet and triplet exciton you can see which is what is called the difference between triplet and singlet the exchange splitting and again for this material you can see a nice agreement between theory and experiment for different nanowires and more about low dimensional material this is another paper also this one quite old where you can look again at the dimensionality in fact what Fulvio discussed before here these are different nanotub this is the bulk of boron nitride and these are the absorption spectrum this is the boron nitride single sheet spectrum is more or less similar and these are different optical spectra calculated for different tubes of different dimensionality and you can note that reducing the dimension of the tube the bonding energy that is the difference between the first peak optical peak and the GWF that is indicated here by this arrow is very very big this is because more the system is smaller and more the bonding energy increases but another thing that you cannot here is this one that the GW this calculation clearly are obtained in a repeated cell approach and at that time it was not included the Coulomb potential the cutoff in the Coulomb potential and what you can note here is that in principle if you want to simulate an isolated material you have to increase the vacuum and you see clearly that the GW have increased the distance between the tubes in this case while the bonding energy increasing the distance increases too instead the position of this peak of the optical peak more or less is stable and so this is due to the fact that you have a compensation between the opening of the GW and the closing of the beta cell peter so the two effect more or less compensate but clearly you see a very strong effect due to the dimensionality if you want to do better what you can do and is actually implemented in Jambonau in order to simulate an isolated material include the cutoff in the Coulomb potential in order to eliminate the image-image interaction here you can see differently from before you see that the bonding energy is more stable with vacuum if you include the cutoff but the nasty things that you can learn from here that in case that you use cutoff if you are interested to do it you have to pay attention to the convergence in k-points because if you look at the macroscopic directory function the material you discover that in Q-space has this shape and this means that you must have a very, very dense k-grid to describe this part if not you are not able to reach convergence so if you are interested in cutoff you can do it and so you eliminate the problem to increase the vacuum size but pay attention because you have to pay attention in the convergence in k-grid so this is another example just to show you that it is also possible as mentioned before to do calculation in spin-polarized system this is a boron nitride again sheet where Claudio Andrea and other people did calculation for different vacancy and different effect in boron nitride and these are the difference between the level, the DFT level and quasi-particle level so you see that there is a very strong upshift of the unoccupied state and downshift of the occupied state and also from this paper what we learn is that all these defects can be responsible of the visible photoluminescence in this material that seems quite interesting for a single photo emission in this material photometer in this material the last example is this one this is another material that is quite interesting in this here in this paper what I shown is that it is possible to reproduce the experimental optical spectra of monolayer belayer and bulk only including spin orbit because these two peaks are due to the splitting of the valence band essentially due to the spin and they are what is called A and B exciton in this figure you see what is the plot of the excitonic wave function so if you want to know which is the distribution of the exciton in space you can do it you will do it this afternoon and this is actually the modulus square of the wave function ok the last example is some example that dankov approximation is not good so means that in some material you are obliged to include all the matrix in the description because this part is important I have to say probably I didn't mention before that this term is important all the time that you have a material that is quite inhomogeneous in the charge density so for instance in all the organic material is quite important in all the molecule is quite important to include all the matrix if not you do not end up in the proper description and here is the difference for instance for the dankov approximation that are here for these two chromophore and the experiment you see that are completely off from the experiment while if you include the full beta salpeter and this is another example shown by Mirta and Andrea in that paper where they have shown that also for nanotube if you are interested in impinging electron and measuring electron energy spectra at certain angle so you are considering also the perpendicular component of the microscopic electric function you end up in big difference in some peak dankov and non-tandankov approximation similar, this is an example in porfirin a quite interesting material in the gas phase we were able to reproduce with GW approach the experimental optical spectra that are here this is the small version of this matrix without this four part but you can note that again the agreement is good only you go beyond the dankov approximation so the peak reproduce the experiment all in that case and also for crystal of porfirin is the same situation this is the experimental spectrum done by a group in miran for two light polarization in one direction or in another this is a crystal of zinc porfirin the porfirin in the gas phase is completely isotropic what turned out in the experiment that you have a very anisotropic peak in this region and we were able to reproduce this feature only including the coupling part so we had to go beyond the dankov approximation because the position of the peak change completely if you do dankov you have a peak here if you don't do dankov you have a peak here so it's quite important and also you can see again there is a way that you can look at this afternoon when you will do the calculation in the 2D material there is a way to understand when the coupling part is important this is what we called amplitude that give you an idea if the exciton lambda is built up by electron hole or electron and hole that are called anti-electron and that hole so the anti-resonant part of the matrix you see that there is a big part of this amplitude of positive energy so normal transition valence conduction and a small but not negligible part from conduction to valence in the representation ok and so I can go very fast in the last part or no I will do this afternoon we can do this afternoon ok this is the practical part that we will see this afternoon ok