 Okay, we have a session about electron phonon interaction. So just a brief introduction. With Yambo, we have several basic topics like GW, because of Peter, but in addition, there are several models of the code that allowed out in the years of produce different models of teaching models. So there is electron phonon or linear optics and also magnetic properties. So this year we decided to concentrate on electron phonon. And electron phonon is actually a theory that as GW is based on a series of pre-calculations. I mean, as a, you know, lego type theory. And the very basics, there is density function perturbation theory. And this is what Elena Kanucha will talk about today. So please, Elena, the stage is yours. So thanks, Andrea. Good morning, everybody. I have the pleasure to open the this morning section on the electron phonon coupling. And at the end of the day, you will live with, let's say, with the take home message in the sense that you will realize that the electron phonon, in order to calculate the electron phonon coupling, what you need, you need basically two main ingredients that are the phonon modes and the electron phonon matrix element. The electron phonon matrix elements are, let's say it's the matrix elements in the sense that they give you the probability that an electron in the state i scatters through a phonon in a state j. Both terms, phonon modes and electron phonon matrix element can be calculated in the framework of density function perturbation theory that I'm going to talk about with this lecture. So let's start from what is a response function. Imagine that you have a system and you want like a material or molecule, let's say, and you are interested in calculating certain property that can be expressed as the variation of a variable with respect to the strength. What are variables, what are strengths? The variable is, let's say it's a quantity which depends on the atomic coordinates of your system and it may depends also on an external perturbation that you may let act on your system. And the strength is the strength of this external perturbation. So then calculating the variation of this variable with respect to the strength gives you a certain property that you want to study. For example, by calculating the variation the double system with respect to the electric field you get the polarizability as a response function. Or if you calculate the variation of the stress or the pressure with respect to the you get the elastic constant. And if you calculate the variation of the force acting on an atom high with respect to the displacement J, you get the interatomic force constant like here. Or moreover, if you calculate the variation of the dipole system with respect to the displacement, atomic displacement, you get the born effective charges. The first two response functions act at microscopic level. So you can have some microscopic response functions. And the third and the fourth that I wrote in these slides are microscopic response functions. So depending on the system that you are studying and the physical properties that you want to address you can define a different response function. So the fundamental theoretical ingredient of the response theory is the Elman-Pheumam theorem, which has been formulated in 30s by independently by Elman and Pheumam. This theorem states the following. Let's start from a time-dependent Schrodinger equation that you can see here. This time-dependent Schrodinger equation so depends on an Hamiltonian, depends on Hamiltonian, which depends on a parameter lambda. And as a consequence, the wave function and the eigenvalues depends on lambda too. This parameter lambda can be for example, the strength of the external perturbation that I was mentioning before. Or in the case of the Bernoulli-Penamina-Miltonian, lambda is the set, the collection of the atomic position. So lambda can be also a set of parameters. Anyway, this lambda parameter, this lambda is just a parameter does not corresponds to an internal dynamical variable. And in the sense that the theoretician can set lambda at his or her will. And an experimentalist can tune it in the laboratory. The Elman-Pheumam theorem allows to answer to a specific question, which is the following. Imagine that you want to know how the eigenvalue e-lander, the variation of the, you want to calculate the variation of this eigenvalue e-lander with respect to the parameter lambda. So what you do is to differentiate the expectation value of c-lander, h-lander, c-lander. And by applying the chain rule, first you derive the bra, then you derive the operator. And then in the end, so the bra, the operator and the ket. But by the fact that the wave function is normalized to one, the derivative with respect to lambda is equal to zero. So you end up just with this term. We end up just with this term, meaning that the variation of the eigenvalue e with respect to the parameter lambda is calculated just by differentiating the operator h with respect to lambda. So you don't need to calculate the variation of the wave function with respect to lambda. And this is a powerful and non-trivial results because not only, I mean, the amount depends on lambda but in principle, so the wave function. So the first take home message from this lecture is that you know that, for example, you know the calculate force system, your system, you don't need to calculate the derivative of the wave functions. So let's see how this theorem applies in the context of DFT. So we are building, we want to answer to this question and we go step by step. First of all, the Elman-Fiemann theory also, when you can set your formulate, the problems in terms of the variational principle. And variational principles is written here. You have an energy e lambda, which is the minimum overall, the possible wave function of this expectation value. And so now let's imagine that we have a function generic function g lambda, which is the minimum of a function g, which depends on internal variable e x and the parameter lambda. And g lambda is the minimum of this function. So here x plays the role of psi. And the question is, which is this derivative, the g prime lambda? How can we express dg over d lambda? So this is our question. Well, since capital G depends on x and lambda, calculating the minimum means that you want, you have to, with respect to x, you calculate the derivative of g with respect to x and you calculate this derivative at the specific point that this is the minimum, that this is equal to zero. Basically means that for each parameter lambda, you have a different function, capital G, and for each one you have a minimum. So the function, the g lambda here is equal to the capital G function, calculated at the minimum, and for a given parameter lambda. And if we differentiate this function, we calculate then g prime lambda, this is equal to the derivative of g with respect to x, calculated at the minimum, plus the derivative of g with respect to lambda. But because of the fact that the derivative of g with respect to x at this particular point is equal to zero, we don't have to calculate, we don't need to calculate this term. So g prime lambda do not depend on this term, but just on the derivative of the capital G function with respect to lambda, where g has been calculated at the minimum. Okay, so these are the main ingredients that we need in order to see how these theorems can be applied to the DFT. So we go on, we go on, and we observe that we do another step forward, and we consider that we have a physical perturbation, A. A is a physical perturbation, operator A. And B is an observer that we want to measure. So A is the physical perturbation, so it's added to them, to them perturbed Hamiltonian, and alpha plays the role of lambda, the lambda that we saw before. The question again is, which is the variation of the unobservable that we want to measure with respect to the strength of this external perturbation? So the derivative of expectation value of B with respect to alpha. So we use a trick here. We say that B is the external perturbation of an Hamiltonian. And because of Elma Feynman's theorem, we can say that the derivative of the energy beta with respect to beta is equal to the expectation value of the derivative of the operator H beta with respect to beta, which then, because of that, is equal to the expectation value of beta, or B. And if we add the two, but if you have an Hamiltonian with the two external perturbation the same time alpha A plus beta B, the derivative, the second derivative of the energy alpha beta with respect to the two perturbation is equal to the derivative of B with respect to alpha, which is the response function that we wanted to, that we were interested in. So let's generalize a little bit. And we consider an Hamiltonian, which is given by an Ampertabes Hamiltonian plus a linear combination of perturbation where lambda i are the strength of the perturbation B i. And the energy, we can think of the energy of the system in terms of Taylor's function. So we have this term here, which is Ampertabes Hamiltonian energy. And then we have the first and the second order term. The first, let's leave a name to these terms. The first one is the derivative of the energy with respect to lambda i, calculated lambda is equal to zero is the forces acting on the system, which are used to get the structure optimization and molecular dynamics. And the molecular dynamics. Then the second order, we have the second derivative of the energy with respect to lambda i and lambda g calculated at lambda equal to zero, that are, as we saw in the previous slides, this is the second derivatives corresponded to static response functions, elastic constants or the electric, electric tensor. These are the response functions that I gave you as an example at the very beginning. And by the second derivative of the energy, this second order term, let's say, is also related to vibrational modes in the robotic approximation, teratomic force constants, and burn effective charges. So here you have the very same equation that we had in the previous slide. And so the forces, what is the forces is equal to minus, the expectation value of the perturbation v i, since I'm calculating the forces, the f i, let's say, the derivative of the energy with respect to the perturbation i. This is equal to the minus expectation value of the perturbation v i, calculated on the per-target wave function. And if the v i is a local perturbation, this can be written in this term, the integral v i, r, and zero, which is the charge-density distribution on per-target charge-density distribution. You see that, which is the power of the Elman Feynman team that in order to get, for example, the forces active on the system, you don't need to calculate the derivative of the wave functions. And going to the second order, the second order, the second derivative of e with respect to lambda i and lambda j, this is equal to two times the sum over n, overall the possible states, and the product of these two matrix elements divided by e zero minus e n. And in green, you can recognize the first order perturbation theory wave function. And so that allows to write that can be, so rearranged in this way. So the second order derivative of e is equal to two times the expectation value of the perturbation v i between c zero and c prime g, which is the first order perturbation theory of wave function with respect to the perturbation j, which is equal to the integral of v i r and the first order derivative of the density with respect to the perturbation j. In the same way, we can write, it's perfectly equivalent, equivalent writing that this h i j, these terms are equal to two times c prime i v i c zero, and the integral v i r and prime i r d r. So this is the integral. So then let's see how definitely, so now we see how to apply all we have seen before in the framework of condensate functional theory. So we take a potential v which depends on lambda, which as before, the parameters like for example, the strengths of this term perturbation expressed in terms of an unperturbed potential, plus the combination of external perturbation. So in DFT, what we know is that the energy or our system is the minimum of all the possible densities of the function, which is given by universal functional f plus the integral of this v lambda times the charge density distribution with the constraint that the integral of the density is equal to n to the number of electrons. So this is what we know from DFT. What we know from the Elman-Fiemann theorem is that the derivative of the energy with respect to the external perturbation lambda i is equal then to the derivative of this function here in parentheses that plays the role of the capital G that we have before, sorry. So the derivative of that function in parentheses with respect to the perturbation lambda i calculated at the minimum of this function. So we have the integral of n lambda r pi r dr. Okay, and then for the second order, sorry, the second order derivative of the energy with respect to lambda i lambda j is the integral of the derivative of lambda with respect to the perturbation j v i r dr. So what we do in density function perturbation theory is not to calculate in this system the second derivative in this way by doing the derivative but calculating this integral. And in order to do that, we see that we need to express the variation of the density with respect to the perturbation j. So let's say we have to calculate the response of the density. So the density in density functional theory is expressed in terms of the Kohn-Eschamm orbitals which are, so then it's the sum of the occupied states of phi v r square modulus. And the derivative of the density with respect to the perturbation is then given by this expression. So here we have the first order derivative of the Kohn-Eschamm orbital. But this is the first order derivative in first order perturbation theory, just from quantum mechanics is given by this expression is the sum of all the possible state different from the state that you want to perturb of this expression. Actually, the sum would split in two terms, in two sums. The first is the sum over all the states v different from v prime different from v from that one I want to correct plus the sum over the empty states. Okay, but when we put this expression for phi prime v in the expression for the n prime so the variation of the density, we observe that by summing over v this sum is equal to zero, so that this sum does not contribute. In the end, phi prime v is just the sum over c over the empty states of this part of this expression. But we, okay, and then we have to put this expression for phi prime here and calculate the density, the n prime r. The problem is that as you see, we express phi prime v as a sum over the empty states which can be, that actually is, but in principle you can have a lot of empty states so this is not the way that we are going to follow. But what we observe is that this phi prime v, we can actually demonstrate that this phi prime v is the solution of this inhomogeneous shedding equation, which is, so here on the left, we have h0, the perturbed Hamiltonian, the eigenvalue, minus the eigenvalue times the phi prime v equal to minus pc, which is the projector on the empty states, v prime, which is the perturbation, phi v is the connexia orbital, but we observe, so the way we calculate phi prime is not this, the first line, but the second one. And we also observe also that the projector, pc is equal to one minus p, the projector of the occupied states. So in the end, you can replace p, there's a projector p over the empty states by one minus the projector of occupied states. So in the end, you don't need to calculate any empty states. And this equation is solved iteratively with the same algorithms that are used to solve the connexia equation. So we established the parallelism between DFT and the density functional perturbation theory. On the left, you have DFT, you can establish these binuclear correspondence between the density and the external perturbation with zero, here. Then by solving the connexia equation, we get the orbital, then we calculate the density and the potential, and so on. And this scheme is solved iteratively. In the very same way, on the right, we have this correspondence between v prime and prime r. And so you solve more or less the same scheme in the sense that here you see v prime and prime and v prime. Okay, so now let's see how we can use density functional perturbation theory in order to simulate the atomic vibrations. On the left, we have the crystal with the atom in the equilibrium position. And on the right, this place, for example, the atom in the position r, of the quantity u r, or the atom in the position r prime of the quantity u r prime. And we want to, we ask ourselves which is the response of the system to this kind of perturbation. So we have, we have our potential is equal to them per target one, which corresponds to the left side of the slide, plus the linear combination of the external perturbation given by the displacement of the atoms. And the energy would be the zero, plus here we don't have the first order, the derivative, because of the term is equal to zero, since the, we started from, we supposed to start from the atoms in the equilibrium position, the forces acting on the atoms are zero. Since the forces are the derivative of the energy with respect to the position of the atom. And then we end up with the second order term for the energy. But what we know from, from before that that is the second derivative of the energy is equal to the derivative of the forces with respect to new to the displacement of the atom. These are, these are, this is one of the response function that I showed you before. These are the interatomic force constants. The interatomic force constant which are the forces acting on atom r with respect to the, the atom displaced in position r prime. And these interatomic force constants are principle very big matrices of the size of these matrices 3n by 3n, n is the number of atoms and 3 are the Le Cartesian coordinates. And the, the, the, the, the, the, the, the, and so these are the, and the value of these matrices are the omega square. So the, the, the final modes, the final frequencies squared. And I insist on the fact that we don't have to calculate the second derivative of the energy in this way, but by using them on Feynman theory, we calculate the density function of perturbation to use the perturbation. As I showed in the previous slide. So these are the, the diagonalization of these, these matrices gives you the final modes, omega square. If the calculation has been properly in principle, these final frequencies are positive. But if sometimes they can, they can be negative because of some numerical problem or in the case, you are in presence of a real dynamical instability of the system and in this case, you can address, you can, that may deserve to be studied. So now, so then when you calculate phonons in principle, you, you have to use a supercells, supercells because in order to accommodate, let's say the perturbation, which may have a wavelength, an arbitrarily large wavelength. So, but so the supercells, big supercells means a size of very, very big size of these inter atomic force constant matrices. So means that the calculation can be heavy. But in principle, the phonon, since the phonon ones can be classified by their wave number. So the symmetry helps us and so they each perturbation is as given a wave number as I show here, you have a perturbation v prime, which has a certain wavelength, which is given by 2 pi over q. And by using this factor, you have that in the right and the right hand side, the question of the that I showed you before, this is the right hand side. Here we have the perturbation and the function, for example, and you see that the right hand side, the total wave number of the right hand side is k plus q. And the left hand side must have the same wave number k plus q. In the end, it's possible to demonstrate, but I don't do that. The perturbation as a given q vector leads to the variation of the density which depends on just on that q vector and the self consistent potential also depends on that q vector, meaning that for each q vector you that each q vector is independent from the other, and the calculation can be done separately. And in the case of polar materials, things become a little bit complicated, more complicated because of the fact that these displacement generate microscopic electric field. So let me write the energy as a function of the displacement and the electric field in this way. And then if we calculate f, the forces acting on the system like the derivative of the energy with respect to the displacement and the electric induction like the derivative of the energy with respect to the electric field by just differentiating this expression we get the f and the special for f and the special for d in terms of the displacement in terms of the electric field. This the carl of the electric field in C++ is given by q vector between q and e, and if we have the displacement perpendicular to the q vector, this means that e must be equal to 0. This is true for the transverse phonons meaning that if e equals to 0, the force acting on the atom is just this expression so the frequency is equal to omega 0. The frequency of the transverse phonon is equal to omega 0. But if we have longitudinal phonon longitudinal phonon means that the electric induction is equal to 0 and this is the case of the longitudinal phonon meaning that this implies that the frequency has an extra term which is given by the and the electric constant so that's why in order to treat polar materials in when you calculate phonons in polar materials, you have to to calculate these two quantities by for example, this is true for GW you have to for quantum espresso, sorry you have to switch on this plug epsilon equal to 2 and that allows you to take into account the calculated electric constant and the benefit in order to properly calculate this extra term which is added to the to the frequency. This means that here you have the splitting between LO and TO and the frequency of the longitudinal phonon is larger than the frequency of the transverse phonon. This is not true for the two-dimensional materials as Tiboso showed in 2017 because in this case the longitudinal phonons acquire a finite derivative here as you see so there is not this splitting at the gamma, the splitting is not true anymore so these two phonons are degenerated gamma and this is true due to the fact that if you have two-dimensional materials it's a question of dimensionality going from 3D case to 2D case and actually if you have a two-dimensional material you increase the number of layers and calculate phonons you see that the breakdown of the splitting is removed by increasing the number of layers meaning that the 3D case is recovered so in order to treat properly the two-dimensional case I copied here the description of quantum espresso PW input you have to be careful to switch on this flag assume isolated and put it equal to 2D which allows you to treat properly the two-dimensional case with this you finished so I want to thank you for your attention and part of these slides have been inspired from the Stefano Baroni lectures from a previous quantum espresso school and see and that's it, that's all for me okay, so we are now open to questions and comments everything please Tom unmute yourself and make your questions yes thank you very much for that I mean I've never been brave enough to look at this perturbation theory derivation every time I see this projection operator it scares me but thank you it's very clear and well laid out okay early on you make the point that to express the coefficients of the Taylor expansion of the energy in terms of the density you have to assume that each of these coefficients whatever they are if it's first or the second order etc they have to be local with respect to the position so you can move the bra through it and then take the density on the left right here yes here yeah and the slides before yes we've been hearing a lot this school write about how in the electron case the screening is non-local and this is very important so is this just because in phonons you can assume that generally speaking things behave like structuralist dialectics and you can always move the screening through or I mean if there are vacancies and there is more structure and so forth you can't do anymore no okay let's see let's say that when you move from here to there you can say imagine that you don't have the dance the wave function you just know the density in this case you know that I mean if you don't have access to the wave function just to the density you need that your standard perturbation is local but since with the density function perturbation theory scheme we have access to the first order derivative of the wave function actually we can treat also non-local perturbations displacements phonon displacements are you can treat the non-local perturbation house sorry I didn't follow that last bit in the sense that when you solve the density function perturbation theory equations the scheme of the three equations that I showed here you have here you actually you have access to the wave function to the first order derivative of this wave function by solving this equation so you you can have whatever right so you can calculate it in this cone sham formalism you don't need to actually write it in this way right okay that makes sense and if I could I've never understood the longitudinal transverse phonon thing because you say okay e is zero it's transverse d is zero it's longitudinal but in my head a system is either e is zero or d is zero so how do you have two phonons at the same time it depends on the what the way I see it depends on the the orientation of the displacement with respect to the to the q vector like here here you see the q vector is on the plane the displacement is in one case is perpendicular to the to the q and the other case parallel I see so it's like you have to consider q and then also this this little capital q whatever you want to I see that makes sense okay thank you so much you're welcome so there is a question by Mukesh Singh if you can unmute yourself hello hello thank you for this thank you for this nice talk so my question is that why do we give so much importance to Helman-Ferman theorem so we know the variational principle and it seems that both are same can you comment on that the importance of Helman-Ferman theorem yes importance of Helman so why do we give so much importance to Helman-Ferman theorem the variational principle right so what is the difference between them the Helman-Ferman theorem allows you to to to calculate the instead of calculate the derivative of the energy with respect to lambda i to the perturbation lambda i allows you to imagine it to apply the Helman-Ferman theorem to the density to functional theorem so we are in this context and we want to calculate the variation of E with respect to lambda i because it's related to the forces and because of Helman-Ferman theorem you don't have to make this derivative but this integral since you have access to the to the density no wait wait I think my question is since we are getting the same let me understand you probably you are saying that there is a variational principle the one you apply to ask to focus on whatever other problem and then as a why are you stressing so much that you just don't use the normal variational principle is it correct yes yes yes so you are applying variation principle to an Hamiltonian problem it's different so you are not calculating the minimum of a trajectory of someone on a skate running on a hill this is a variational problem in terms of a mission object Hamiltonian and a ground state and Helman-Ferman is exactly based on the property that the state you are calculating the average is the ground state of the operator is a specific variational principle and actually makes your life enormously easier did I answer your question yes yes yes yes thank you thank you so if I can so why do we call the perturbation external perturbation as a strength here sorry can you repeat that why do we call external perturbation strength strength yes so let's say we want to displace a better when we do the full on calculation we do some kind of displacement right so in that case displacement is in my head displacement is perturbation so do we call the strength that displacement as a strength the strength yes the displacement the amount of the displacement is the strength actually so whenever I whenever I see strength I start some kind of differences in our potential so I'm not able to relate that but this is calculated at zero strength at zero displacement I think that he has a problem in connecting the displacement to the strength of the perturbation I think Elena partially answered maybe the last piece is that is a Taylor expansion so you have delta R multiplied by a derivative so it's a whole thing that does the perturbation so more you stretch stronger is the perturbation so it's not a potential it's a stretched potential it's written there so okay okay the UR means okay thank you very much I understand okay so we don't have any other question and we thank Elena again thank you