 So you will see again the framework of the green function, it depends on what you need, but it will be focused more on the practical implementation of this method, so you will see exactly the equation that are implemented in the code. And okay, so you have one hour time. Okay, thank you Maurizija. So I'm Daniela Varsano from Italian Research Council and also of the European Max Center, which, I'm glad to say, is the main sponsor of this school. So today, I mean, the topic of this lecture is going a bit on the common approximation that enters in real GW calculations in the previous lectures, you saw, and I hope you appreciated the beauty of the glorious head integrations. But then and now we move from, let's say, the garden of Eden of the theory to the human suffering of real calculation in our computers. And that's important to know about implementation and about practical approximation that we need to run our calculations, because in this way we also know what to expect for the GW calculation in terms of in terms of accuracy and in terms also of what we can calculate and what we cannot expect to calculate. Sometimes we have a forum in the YAMDO code you can subscribe, you can access from the web page. And where we used to help users, we have problem, it wants clarification with the code that sometimes it writes questions from user that's something like, I'm trying to calculate the quasi-particle gap of strongly correlated, I don't know, metal oxides, nickel oxides. This doesn't match experiment, what I did wrong. This is the inputs and probably you did anything wrong just you cannot expect to match the experiment resorting on the approximation, on the GW approximation and the current implementation on the most common implementation of the GW approximation. So I will recall a bit some concept that have been already illustrated yesterday also today by Andrea to arrive to the equation we want to dealing with and we will see then extensively in the afternoon during the hands-on all the variables governing and to have to control to have a meaningful calculations. So we already seen that it's very useful to divide our excitation in our materials in neutral and charged approximations. The topic here GW deals with the charged approximation. Tomorrow we will see how to deal with the neutral excitation with absorption, essentially the excitonic effect. And now so we are going to look at what we want to calculate and give interpretation to director or director for the mission experiments and it's just important to see that essentially conservation of energy and conservation of momentum give access to our best experiments and the process in the systems is to pass from an electron system to n minus one electron systems and vice versa for the inverse for the mission we pass from an electron and electron system to n plus one. This is the conservation of the energy and from this we measure the density of unoccupied states in case of the inverse for the mission experiment. Let's have a look with the workhorse, first the workhorse of the electronic strato calculation nowadays. It is the density functional theory. These are the very famous consham equation of 65 that give the success of the theory because the DFT itself became very useful. I mean it's a theorem of quantum mechanics that became useful so once. We know how to solve this consham equation, all the many body effects, let's say, are contained in this exchange correlation potential and its glory comes also because of its moderate computational costs and we know that we have a very good prediction of ground state geometry and the electronic structures. So, what about band gap? Probably yesterday you already knows that the famous band gap problem and here in this plot from a paper of Markov-Schivskarder, here we compare experiments of a series of semiconductors which respect the gap calculated in the DFT, local density approximation in this case. As you can see that the estimated gap is one-half of the experimental one or even worse, the one-half, this is not a problem of LDA. This is an intrinsic problem of using DFT at least as difference of eigenvalues in estimating the gap of m materials. This can be also rationalized because if you look again here what you are looking at, I mean from a photomission experiment where we measure the kinetic energy of the coming electron and in coming photon energy, we see that these energy lever we are interested in are different of n minus one electron and so let's say from here you can just recognize that the ejection or removal of an electron is always a many-body process and in many-body process in the sense that when you extract an electron from your systems, you are going to extract also the electron hole around your electron so you have different phenomena of relaxation, screening, correlation, so it's hard to treat as one electron process. So here comes very useful that has been already commented today and also yesterday the concept of the quasi-particle, so the electron plus his screening cloud forming what we call the quasi-particle that will interact with a screened interaction. So anyway, we have just seen that we cannot calculate gaps from the DFT, but this is not exactly true just because the plot I showed before with a very bad agreement it's the usual evaluation of a gap in the DFT when we consider the homo and lume of a molecule or the minimum conduction band and the maximum valence band, but we can actually write an expression of the gap in this way performing different calculations over systems. We consider the system with an electron more, an electron and minus one and we can compute, I mean, DFT give access to total energy and we can evaluate the gap in this way. This is the expression of the gap indeed just here, it just has been rewritten, the energy in function of the sum of eigenvalues to the other components that form the total energy in density functional theory. This is the conchian gap, this is the one that is wrong, but if we consider all the other terms, here difference of eigenvalues of n plus one and then electrons, et cetera, the same for the exchange correlation potential and the Hartree potential. Actually, here this is a set of molecules and we can see that, okay, this is not perfect, but the agreement is not that bad. The problem now is so why we want to go to GW, many body when we could calculate gaps with density functional theory, which is most feasible is that for periodic solids adding an electron or a moving electron, the system is not feasible. I mean, because we are dealing with a bulk, we have a unit cell and repeated unit cell in the space, usually in play waves when we treat with bulk and the periodic solids, this operation that this kind of calculation is not feasible. So this is why as showed by Andrea one hour before, the green functions is the central variable that naturally contains as it spoils the excitation energy we are looking for, exactly the quasi particle excitation energy, exit on lifetimes, just to mention here, I will show some at the end of the lecture, some example, using a green function theory, we have also access to total energies and to any expectation value of one particle operator. So it's very powerful, this green, the many body to bashoon theory and the green function method. So this was already shown by Andrea, this is the definition of the green function, the propagation, I mean the creation of an electron at a time and position at 2, 2, 2, we propagate in the system for a time 2 minus t1 in an interact in Hamiltonian, we annihilate and this is the Lehmann expression and it can be easily shown that these Ej here, the poles, are exactly the many particle excitation energy we are looking for. Okay, now how to obtain the green function, how to calculate the green function? We are in many body perturbation theory, let's say that we start from something that we know, we know how to calculate, we approximate something that we manage to calculate and which we evaluate what is not known, hoping that the difference is smaller. So let's say we know a g-note, usually a non-interacting Hamiltonian, a g-note you can already imagine, I mean you also yesterday in the class which is our starting point, g-note is usually the green function of a density function of theory calculations, we add an interactions and this is the definition of the self energy, everything is unknown, or the interacting system is put in sigma, in the self energy. Okay, so from the equation, the motion of the green function, we can arrive to these equations and if we know the self energy, now we arrive to the definition, I mean to the approximation of the self energy, we arrive in the lemma representation to the quasi-particle equation, this is the equation we want to solve, this is the Dyson orbitals, this is the self energy, as you can see, I mean written in this way, naively, one can think about some similarity with today's equation, but important difference, conceptual difference is the self energy contains many body effects as the exchange correlation potential in the FTE, but here this is the potential effective system, this is the real potential felt by the added or removed electron from the system. The self energy is not Hermitian, is non-local, it is energy dependent, so this complicates a lot the solution of the equations and then as highlighted by Andrea, here the FS are not normal and the energy here are complex, so it's much harder than a density functional theory calculation. Okay, how to obtain now the self energy? This is the pentagon already shown by Andrea, this is the very famous edin equation I would say that it is quite impossible to solve self consistently, we can try to iterate and we start from g equal g note, so independent particle calculation, we know how to calculate the green function of this non-interacting Hamiltonian, we start just setting the self energy equal to zero, so this means then that the vertex is discarded, is neglected, equals to one and we arrive to the equation of the GW approximation, so sigma is exactly GW, here we have expression for depolarization, where we have neglected the vertex and, sorry, and here we have a Dyson equation for the green function and another Dyson equation for the screen electron potential. Okay, note, it's important to, when we talk about GW, so approximation is that the vertex is totally neglected, so it means that the GW cannot provide, there are cases that cannot provide the right solution. And here this is another important approximation that is usually done, not always, here as g, as green function of our systems, we consider the non-interacting green function, g note. Okay, so g is the green function of the non-interacting systems, this is the expression of our self energy. In time, by Fourier transform, we have here an expression in frequency domain and for what concern w, this is our Dyson equation for the screen electron potential and here the approximation g equal g note, we have also an expression for the polarizacion function that will be g note, g note, something that we can indeed put in our codes and calculate. So, first step is to solve an independent particle calculation, for example, an LDA, any other flavor of density functional theory and we can build in this way, this is the expression in real space of the polarizacion that is made of non-interacting electron and whose has showed here in this diagram. With this choice of the polarizacions, here we arrive, this is the expression of the screen potential that obeys a Dyson equation as was shown by Andrea yesterday. It can be seen also in a classical way so w will be the potential plus the potential fact here for a test charged to induced density by the perturbations. So, here we say that the induced density is such that the system responds independently as an independent systems to the total potential, not external potential, but the total potential, the external plus the induced and this is indeed the random phase approximation and this is an approximation. So, here we can see with this choice of p, this is the diagram, the infinite diagram that are included in the polarizacion, so the bubble approximation. We have p is p naught, g naught, g naught and all the bubble diagrams are included. So, we have seen now the sigma g w as a g naught and usually this is what is done in practice is to divide the self-energy in two pieces. We are just adding and subtracting the Bay column potential here and we arrive to a definition that what is called exchange self-energy and correlation self-energy. This is something we will see, we will calculate practically this afternoon. This is the expression in real space and this can be integrated in the frequency domain analytically and we arrive to the expression of the Hartree-Fox exchange term. This does not depend on the frequency anymore. This is a static term and here it is the expression for the correlation potential. The presence of w with this frequency determinant it made it computationally demanding and we will see it is the most consuming part of our calculations. So, practical implementation of these equations. There are many codes in the market. Most of them, many of them use the same approach in term of basis set. This is the reciprocal space and frequency domain. Just to mention, I mean, this is the YAMBO code, the implementation of the YAMBO code, but there are others famous code as Abinit, as GW Berkeley and probably many others I forget to mention, but it's not the only types and the only standard, the only implementation we have, there exist many codes, real space and real time implementation by Godby and Richard Nitz. Use of localized basis set that is very convenient when dealing with finite systems let's say molecule clusters because you use localized sets and these allow to be very performant in the calculation of integrants. Here there is the Rolfing code I don't know the name, but there is the code of Feliciano Giustino, the Fiesta code, where also Claudio that is here is involved. There is the code of Abien, Bruneval, MOLGW, they are very performant, but I mean they are meant for localized systems, for molecules for systems solids. Then other implementation paulomari make use of localized function. Surely I forgot many of other codes. And now we stuck in this reciprocal space frequency domain implementation. This is the expression that are coded in Yambo in in plane wave set. Just the expression before we have that in plane wave this is the exchange part of the self energy, this is the correlation part and you can see that we have to manage with the frequency integral here of the epsilon minus 1 with the frequency dependent. And so also we have seen, we have to arrive at the correlation, we have done already several approximations. So we are talking now to what it is called the G-note, W-note. So it means that the function is then G-note from the DFT calculation and W also the polarization as we calculate as G-note, G-note. So even at this at this level let's say the basic level the calculation is rather laborious. Here I start to mention some important ingredients that should be checked when doing this kind of calculations. For instance here we have an integration of the Breven zone that means that we need converged k-point sampling which is not the same convergencies that we achieve for the DFT calculation but actually for the calculation of this kind of integral. Here we have some over unoccupied states. Here it enters a closer relation so this summation in line of principle infinite and something that should be checked carefully in order to obtain meaningful results. I mean this is the basis of the best practice of GW calculation otherwise I think you can get whatever. And here the most complicated part maybe I would say is the integration in the energy domain. This is just the slide you have in your cheat sheet just what I already said here for the integration of the Breven zone some occupied bands here there are the name of the variables that governs the convergence of these expressions. Here the same about the correlation part of the self-energy. Now let's have a look to this integral in frequency domain. In principle you can calculate this numerically. This can be done in simple way you need to sample so you calculate your RPA the electric function matrix for a lot of frequencies then summing in order to have the integral. This can be done, actually YAMBO can do that. It's very cumbersome and usually what is done in many of this code and also YAMBO does is to use of approximations we model these matrix each element of the matrix can be modelled. Usually we have the imaginary part of epsilon minus one the macroscopic content is the plasma peak. So we can do in many cases the bold approximation that our function is a picket fraction around is plasma peak is plasma frequency. So we can model is this is the expression that you use to model. It is a one pole just one pole wave function. We have to calculate the residual and the pole of this of the epsilon minus gg prime once we are able to do that then this integral is analytic. How to evaluate these parameters here let's say there are many resipes in the literature. Here some reference, Ibertson-Lewitt, Godbinitz and Gelfarid. What YAMBO implements is the richer the Godbinitz resipes that means essentially to perform calculation for two frequency usually the static value omega equal zero and to a complex frequency which is an input of our codes of our calculation then this is two equations in two variables we get the two value and we can plug here in the analytic expression of the frequency integral. Here I want to show just the performance of different way, different resipes for the plasma pole approximation in particular this Ibertson-Lewitt Ibertson-Lewitt difference of the Godbinitz consider the static limit but then the coefficient evaluated through some sum rules of the epsilon minus one. Here you can see that this is the real part around the real axis of the epsilon minus one of the screening for a silicon diamond zinc oxide and metallic neon and in the case of silicon everything is similar here you can see some difference and here is the impact of different approximations Gn is the Godbinitz this is the Ibertson-Lewitt here are other models this is the reference this is the numerical calculation of the self-energy so calculating for a number of frequencies so let's say this is the reference and this is the experiment as you can see that beside the silicon everything works perfectly for other systems the difference arise let's look for instance to the zinc oxide here we have the Godbinitz that differs by 0.01 electron volt from the reference here the Ibertson-Lewitt seems to be much poorer in this sense but note that Ibertson-Lewitt that has been widely used somehow is more in many case is approach better the experiment but this is I would say an artifact in the sense that the reference here is the the reference here is the exact computation of the integral over there in the case of zinc oxide there is a nice work of Marty Stankowski where analyzed in details the performance and the convergences of different of different models so anyway this is the accuracy that we expect from the Plasmon-Poll approximation so remind, this is another approximation that enter in the calculation it's a bold approximation in many case well actually not so many but in some case the Plasmon-Poll model fails for instance interfaces where there is a work by Andrea on the electrons in copper where it is shown that the Plasmon-Poll approximation is not suitable and in this case full integration is needed and there are also other methods to perform the integral the control deformations alternative methods and here there is Dario there is a poster of Dario that actually is working in the function of the delectrin function and there are some preliminary results on that so we have a Plasmon-Poll you need calculation of the screening matrix for two frequencies a direct integration hundreds of frequencies so you can imagine the computational costs other methods are for two deformation 20-30 frequency I would say something like that now we are exploring an intermediate regime to deal with this integration ok this is again a slide taken in the cheat sheet that is what are the input variables governing the Plasmon-Poll so what do you have to care essentially you always calculate the static limit you calculate the delectrin matrix in this PPA energy fixed here and of course so you need a convergent calculation of your matrix for these two frequencies and these are the variables again the dimension of the matrix here gg prime this is the real stands for reciprocal lattice you will see this afternoon you can also give a cut-off in energy and here the number of bands in the submissions that you need to conversion in the electron-hole pairs in your polarization ok now once we know we see how to calculate g-note we see the w-note also it needs frequency behavior we want again solve this equation here we assume that our Dyson function are our coneshami integral this is another approximation that enters in the calculations so with this approximation now we can solve this equation the quasi-particle equation this is a non-linear equation what we can do is either develop around our coneshami again value and solve at first order this is what usually done in jambo there are other alternatives just interactive method to find the root of our expression of our function ok, so we have all the ingredients in place and we can arrive here to what we want to calculate the quasi-particle energy of our system I want to mention here an algorithm it implemented in jambo developed in the group of Xavier Gonza when calculated the self-energy here we have a summation of unoccupied bands this could be very painful I mean you have to calculate many many calculate include many many bands in your summation this means also that you have to perform a non-self consistent calculations with many many bands many many bands could be thousands of bands this algorithm that we named in the code terminator but this is our jargon consists simply in it's very simple to understand and let's truncate our our summation to a certain band this big NB and let's see the what is missing I mean what is missing bands we are not including and we do an approximation the extra polar approximation we assume that all the states above this this high number of bands of the same energy epsilon sigma making this approximation now we can exchange here the order of the summation and we calculate we can calculate here what is missing from the summations and so we can we can correct the value we obtain it summing up NB to these corrections and as you can see here these accelerate a lot the convergent with respect to the empty bands entering in our calculation this is the conduction let's bands minimum for bulk silicon this is a cluster I think and without this correction you need more than probably 1500 bands here they are already convergent with a half of this number of bands this is the valent maximum for a titanium dioxide nano wire many occupied bands this is the calculation without opting any correction here adopting different correction just playing with the value of the of the extra polar of the extra polar that we have introduced in order to collect all the transition above that energy and this can be also tuned in order to achieve here a speed up a very important speed up what happen in a GW calculation when we deal with non-periodic non-strict three-dimensional periodic system if we want to calculate nanostructure an isolated molecule a nano wire a wire, a polymer two systems so two systems they are non-periodic in some of the directions this is a plain wave code so our representation of the systems is always replicated in space we are always dealing with an array and the GW calculation I mean really feel the long range interaction and so the naive, the first thing that comes in your mind is a problem that you have also in a GW calculation if you want is to enlarge more and more so putting more and more in the non-periodic direction unfortunately the convergence with respect to the vacuum is very slow and the scaling of the code of the computational cost putting more and more vacuum is is very bad so here in just a representation this is a bulk in 2D this is a wire in 1D this is an isolated molecule where I repeat also here the bulk the supercell the supercell sites and there is a simple idea that don't treat, I mean to modify our column potential our interaction will be not 1 over r but will be 1 over r until a certain threshold and then will be 0 this setting properly the definition of this domain with respect also to our supercell sites permits to speed up the calculations with respect of the of the vacuum so this is our expression we are dealing in reciprocal space so we we Fourier transform this modified Coulomb potentials and for some definition of the domain of the region this is the case of a sphere for a molecule you can put in a sphere and this is you end up with this analytical expression this is more complicated in the case of a cylinder this is our best cell function that comes here this is the signal sites of of our of our supercell these are the reference where I will explain in detail take care as to be taken to the limit for q equal 0 g equal 0 of this potential here you don't have any problem but here you have divergences that are not anymore 1 over q square but 1 over q in this case it is logaritmi in this other case that you should take some caution when dealing with the limit with q equal 0 that is one you are interested in when calculate optical absorption ok, here is an example this is one this system this is an atomic chain here you have the inter chain distance here is the supercell volume as you can see here it shows a very very slow convergence with respect to the vacuum the modified column potential allow you to converge much faster I think you will see some application of this and tomorrow in the calculation of absorption spectra ok, let me summarize what is the workflow of a typical GW calculation this is something you have seen also yesterday the first steps is always a DFT calculation and this provides us consham again value and again function both are used to calculate our polarization our green function the polarization then is used to calculate our screen potential usually done in the plasma upon approximation even if it is not mandatory but if if it can be done it's very recommended we build up the self energy and then we can solve our quasi particle equation and extract the quasi particle energies ok so, as shown by Andrea before GW in many case semiconductors works or at least provide a very much better agreement with experiment with respect to the DFT calculation of the GW so we have a huge improvement with respect to the LDA in some case the agreement I would say it's perfect this is the case of diamond this matched experiment but for the wrong reason maybe Andrea it worked exactly on diamond effect of I mean so don't be scared if you don't match the experiment but don't be so happy even if you match because it could be the wrong reason the matching with experiments here other effect enters so there isn't no the story is not just the GW calculation that explains the gap of the diamond also with temperature metals we have talked before about semiconductor here metals also for metals with some care we have a very good matching with the experiment much better than LDA here it's important to note the case of metal and plasma this is a work by Andrea 2002 and here had to resort to full integration for obtaining the the self energy here if we can define here a local potential from the GW this is a metallic surface aluminium 111 and it can be seen that the GW is also able to catch image potentials image potentials this is a polarization effect and it is catch by the GW approximation as I told in the introduction in the first slides we can also calculate total energy using GW we have the green function we can have access to total energy here is the sample of three dimensional this is a work by Pablo García Gonzalez Rex Cobby where they showed here that the correlation energy here of the three dimensional electron gas at high density at smaller s the GW approximation but GW this is not GW I mean the self consistent one not the vertex the self consistent in terms of GW by the perfect agreement with GW calculation as far I heard this is not the final story I mean there are works there is Tomaso that is also working on that and maybe you don't agree totally on this perfect agreement we have also Mike here that is GW so probably this perfect match is not the final story ok here another work of Rex Cobby Pablo García Gonzalez the slab of Jellium they showed that GW is also able to match to represents the van der Waals interactions again on total energy here very briefly there are works that also calculate here lattice parameter for instance here we can see that lattice parameters in GW is quite good this is self consistent GW quite good means that I mean it's not better than GGA anyway but we know that the GGA is the workhorse for this kind of calculations so we have a good energy here but the same foot when you calculate the the electronic properties the electrospectrum we can see that the self consistent GW is worse the simple G not approximation so does not always the self consistency here is better the G not remember always that you forgot you have neglected the vertex so that could be important optical absorption if we this is the silicon I think will be shown tomorrow in many flavors this picture we have the experiment we have the RPA absorption if we just plug GW energy this does not the job I mean it's not here in this case it's just a redid shift so something is missing in the description of the absorption that will be the topic of the theoretical and the practical lecture of tomorrow so some important thing is missing and we will have the lecture by Maurizija and Fulvio addressing this this point so I write to some conclusive remarks GW many vicious parameter free method provides accurate results quasi particle energies also total energy also provides information lifetime of the excitations it is the starting point for optical absorption in many body perturbation theory as you will see that quasi particle energy are ingredients entering in the solution in the build up of the Becker-Salt-Petter Kernel and G not a tadium note but I will try to show you that even it's in it's simpler flavor it's a quite complicated calculation as a quite complicated workflow but I mean a virtue of this kind of calculation that the algorithm is suitable for HPC computations so they scale very well if well implemented in modern machine different architectures just to mention Jambo now works also on GPU accelerators so if your results don't match the experiments take a message before asking to the forum what I'm doing wrong check for your converger parameter I call this this boring part but we have a lecture just now addressing also how to make machine work for you for this part of the calculation it's just repeating or repeating local calculation increase in parameters even at G not W note again several converger parameter and approximation have to be checked here just the sum of the ingredients we discussed now that should be checked and also I didn't mention Jambo as many other plain way codes make use of sort of potentials so you neglect core electrons this have an impact also on double calculation and this is something that should be checked also in term of how many electrons enters effectively in your calculation so if your system to treat explicitly semi-core states this has an impact in the calculations of the quasi-particle energy so don't forget that GW is in approximations so it's not the final story this is already mentioned by Andrea before and something that I didn't mention but you can understand G not W note is not self-consistent so you always can have a dependence on your starting point and like a rule of thumb if your starting point your DFT starting point is more close to what you get then the GW connection will be more accurate so it's not independent of the DFT calculation behind the G not W note this is important for molecule and in Yambo now it's also possible to do a GW calculation on top of a hybrid calculations even in partial self-consistency partial self-consistency essentially we we mean again values only self-consistency calculate your quasi-particle energy and this quasi-particle energy are then used to build up again the screening and the green function so a new self-energy iterate, usually it converge very fast in this way the problem of the starting point is mitigated, is not resolved but it is mitigated but anyway about the wave function you always assume the initial wave function so here there is still reminitions of the of the starting calculation per pi level for the screening and the Plasmon-Poll model Plasmon-Poll model in particular can be critical for several materials so this is my final so I would say GW is successful interpretation in approachimation that can also also fail so let me would like to thank the sponsor that funded this school so the Max Center of Excellence the PCK network and the ECTP would like to leave here some reference, these are the seminal paper of the DINSE questions as seen before here some reviews put here just two very recent review that appear the last year and this year on GW it's kind of compendium that I found quite useful and these are the two paper on the practical implementation in the YAMBO code and I thank you for your attention