National Energy Research Scientific Computing Center (NERSC) VASP Workshop at NERSC by Martijn Marsman, U Vienna (November 9-11, 2016), 18 Dec 2016

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From YouTube: VASP Workshop at NERSC: Dielectric Response

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Presented by Martijn Marsman, University of Vienna

Published on December 18, 2016
Slides are available here http://www.nersc.gov/assets/Uploads/VASP-lecture-Dielectric.pdf
Presented at the 3-day VASP workshop at NERSC, November 9-11, 2016

A

Okay, so we'll discuss some aspects of dielectric response as we computed by means of perturbation, Theory, linear response or the use of finite electric fields, so dielectric responds and then we'll speak about frequency, dependent, dielectric properties and static, dielectric response and response from Alecto, electric fields from density, four and a functional perturbation theory and a few other aspects.

A

So why would we want to compute this well in itself? Obviously, the frequency and Static dependent, static and frequency, dependent dielectric functions correspond to two certain things that we measure right. So the option reflectance and things like this and in the long wavelength limit of the frequency, dependent, polarizability and dielectric matrices. We have things that determine the optical properties in the regimes that are accessible to optical and electronic probes.

A

So daresay there's a reason in itself to to compute these properties for the connection to experiment, but they're also needed in in several levels of theory that go beyond well beyond DFT and hybrid functional theory, of which most prominently I'd, like to mention GW, which we'll hear a bit more today and so like like I already alluded to, is that that GW is sort of an screened exchange, interaction and the screening that we apply.

A

There is not not like, in hybrid functional theory, a simple quarter or something, but it's going to be material dependent and it's based on on the computation of the of the frequency dependent microscopic dielectric response right.

A

So we need these quantities to enter them into into these higher-level methods, yeah and then there's as if you other things there well I won't go into that too much one thing I would like to mention is that, for instance, and that's another one of these methods, so GW is one the other one that that I would like to mention is the beta cell peter equation, so where we, where we try to include excitonic effects in in predictions of optical properties, of of materials and their the dialectic screening of the potential is also needed to include these effects and so GW and the beta cell peter equation.

A

There are some some examples on on these calculations in the hands-on sessions.

A

So what kind of quantities we calculate, what frequency dependent quantities will calculate the microscopic dielectric matrix either in the random phase? Approximation or Beyond, including change, including changes in the in the DFT exchange correlation potential. I'll show you later on what that means, and in for the frequency dependent, macroscopic, dielectric tensor, and we compute imaginary real parts of the dielectric function in or excluding local field effects, either in the RPI or including changes in if the exchange correlation potential as well.

A

So anesthetic properties include things like static, dielectric, tensor, borne effective charges so born effective charges are other things out. There often called dynamical charges as well. So if I move an atom, how much of charge moves with my atom that's more an effective charge and as we do in our excluding local field effects and what local field effects are L I'll mention as well, and this is either done from density, perturbation theory or from the self-consistent response to a finite electric field.

A

Right and so there's there's reasons why these two methods that essentially yield the same information problem is the density, functional perturbation, Theory only works for density function, also not for hybrid functionals, and this beat method that allows us to compute the self-consistent response to a finite electric field that works for for hybrid functions as well, but only for isolating systems or not. For metallic systems, so there are some limits to do these different methods. So if you look at dielectric properties, what are we talking about? So in a macroscopic sense?

A

Are we talking about this dielectric tensor here, for instance, that that that connects an externally applied field to the fields in a material, sorry, dielectric, tensor and in a longitudinal field, and which is a field caused by stationary charges? You can formulate this as well an externally applied potential connected to the total potential in a system and where our total potential is is externally applied potential and the induced potential tensor.

A

This is the dielectric tensor some terminology around this and induced potentials generated by an induced charge, density, one that makes sense in a sort of in a way right and in the linear response regime and which means for suffice sufficiently weak fields. We have a relationship by D between the externally applied potential and induced charge density, and that is through the reducible polarizability and and a relationship between the total potential yeah, the total potential and the induced charge through the irreducible, polarizability and then there's a bunch of relationships between them.

A

It's not so important, don't need to memorize these, but we're, obviously what we're going to be using them a lot. So the inverse dielectric tensor is related through through this relationship to the reducible polarizability and this this new here.

A

This is essentially the Coulomb interaction all right, so this polarizability times the Coulomb interaction plus 1, gives us t the inverse dielectric tensor is a relationship between the virtual irreducible, polarizability and the dielectric tensor, and between these both both these polarizabilities there's a relationship of this form, which is a Dyson equation or a Dyson like equation right.

A

So we're going to be toying with with these and quite a bit so well, those not that it's an scription or macroscopic description like we said well, we need a connection to microscopic quantities because we don't in codes like like Eris, we don't deal with macroscopic quantities, right deal with microscopic quantities and essentially well there's.

A

So there are a whole bunch of illnesses come into play. So we have a field and a dependency on on the positions externally applied field and a frequency, and then we have a tensor of this kind. That depends on the distance between two points right, so we write it as yes and if we write it like this so force efficiently- and this is the microscope- the macroscopic tensor depends only on the distance.

A

If we have, if we have only did sorry a difference here and not two coordinates, we can fully I transform this into momentum space using only one coordinate and end up with a relationship like this for macroscopic quantities and a microscopic dielectric function then enters in such a way right and in momentum space. That means I saw in a macroscopic way. It's only the distance between so in a macroscopic descriptions. Only the distance between two points that is of that is of importance.

A

If you then go to a microscopic description of your system, it's the actual points where I am in my system, that are that are of consequence.

A

So in a microscopic electric field and that that is induced by an externally applied field. We have a more complicated a dielectric function where we don't only have the distance between R and R Prime, but it depends actually where I am. With respect to my atomic nucleus or something like this and these microscopic aspects, and if we Fourier transform this, we have no longer only one Fourier component but three positions. So we end up with a function that depends on G and G Prime.

A

So, and that already gives you some inkling of of what is going to be costly, for instance, because this is going to be costly to store, for instance, because it doesn't depend only on one coordinate or on one for VA component, but already on to so, and both of them obviously are linked in the usual manner by some some matter of averaging right, some averaging over a certain volume. All that that's not something that we are going to be using. But yes, that is the link between microscopic and macroscopic worlds.

A

So, assuming that that are our external fields very on the length scale is much larger than the atomic distances.

A

We have a more a simpler connection between the externally applied field and the resulting field, and that's given by the head of this inverse dielectric matrix, and then we have a very, very simple connection between a macroscopic, dielectric, constant and some aspect, some quantity that we that we in fact can calculate a microscopic quantity so that so for materials that are homogeneous on on a microscopic scale. The off diagonal of diagonal parts of this matrix are of no consequence. An air of Dera, zero.

A

Sorry so for G is not G prime they're 0, and then it reduces even to this particular relationship so being able to compute these microscopic quantities. Under the assumptions that we have an external field that varies on a length scale, that's much larger than atomic distances and materials. There are homogeneous on a macroscopic scale.

A

On the recent sanctions, we have a direct access to a quantity to quantity in the macroscopic world, and this is what we call for instance, so the fact that we assume that the materials homogeneous on a microscopic scale- that is what we call a neglected local field effects, I'm just sort of throwing this in, because you will see this in in in literature. It's not I just want to give you a feeling for what what these terms, if you encounter them in literature, what these terms mean?

A

Okay, so these functions that we saw before is these microscopic dielectric functions, I've written out here, more explicitly in terms of well functions and quantities in reciprocal space, and so here we see this, which was called the reducible polarizability, which is the change in density. So the induced density when we change the external potential, so I've changed in the induced density as a response to change in the external potential, well at a certain at a certain Ruby I component at a certain frequency of our field, and so this is spatial variation of the density.

A

This would be with respect to time, but then transfer it into the frequency domain same thing for the external potential that we use. This is what I said. This is Coulomb kernel, so this this represents the Coulomb. Interaction in reciprocal space does look a bit different than what we saw before because it has been symmetrized, so it's symmetric with respect to G and G prime. You are allowed to do this because the the spectrum of such an operator is exactly the same.

A

You're always allowed to either use the Coulomb operator, which would be 1 over G squared, for instance, or 1 over Q plus G squared or it's symmetric counterpart, which is Q plus G times Q plus G Prime. Okay. So- and this is the inverse dielectric function same thing for the dielectric function, as it depends on the irreducible polarizability right, so our irreducible one, the response in the induced charge due to change in the total potential.

A

Well, these are these quantities that we saw before reducible polarizability irreducible one symmetric, Coulomb, kernel and dyson equation that relates the reducible to the irreducible polarizability. So that looks all well. It looks horrible, but this is all sort of clean and and and we can have- we can cast our problem in quantities that that we that we can write down in formulas, but we don't know them right. So how do we compute P and how do we compute the Chi?

A

We don't know these quantities we can't get from our computations, unfortunately, so the only quantity that we can actually easily access in our in our computations in konchem density, functional theory is what is called the irreducible polarizability in the independent particle picture and as commonly labeled a kind note or Chi konchem and kind note is so.

A

This is the change, the induced, the change in the induced charge density due to a change in the effective potential, so not in the total potential, not in the external potential, but in the effective potential, and what's this effective potential actually seeing this Hamiltonian floated a few times here ever we had a kinetic energy and a local potential, or this local potential is often called the effective potential and okay well that we can vary and look at what happens to the density.

A

So this is a quantity that we can actually compute in our in our calculations right. So an other other and wiser derived expressions for this, which are completely well. The only ingredients that go in are things that we that we actually know from our computations. So if you look here, don't don't spend too much on trying to understand this equation per se, but just see that what it contains our block functions, so eigen function solutions of this konchem equations that we calculate and eigenenergies sorts of things that we calculate here.

A

We see these are these occupational numbers of our of our block states. So we see here that only if one of those states is occupied and the other one is unoccupied, it actually contributes sums that we here see r /, / k, +, /, sorry, Alfre, offer and and and prime to offer to banned indices where one is an occupied band and the other one an unoccupied band. So we have two sums of abandoned.

A

It says, but not only after occupied states, so that is that is one aspect of these computations that now comes along for the first time. In all the other things that we have been looking at, it was always occupied States occupied States and give us the density occupied states that that participate in the folk potential.

A

But now we see we have a quantity that depends on on occupied states as well, so our computations will become more expensive, not only because we have here double sum that we and a sum over K, and we need to do this for a number of points, Q and G and G prime and and frequencies Omega.

A

So it's already expensive in in itself, and so this evaluating these scales is n to the power 4 with respect to system size, but not only that one of these indices runs over unoccupied states, which of which they are essentially an infinite number. So obviously we will not be using an infinite number, but yes, the computational effort will will grow.

A

So when we notice this, the irreducible polarizability in the independent particle picture is kind, not the other ones, the ones that we need to add reducible and irreducible polarizability that we need to compute our microscopic dielectric function. They are related that it can be computed out of this out of this irreducible polarizability in the N independent particle picture, so the one that we compute from our conscience- and that is written here. So those are dyes and like equations again, where we have kind note. Well, we can compute, we see our Coulomb kernel here.

A

We see something that depends on the density functional, so this is f. X C have existed change in our exchange correlation potential with respect to the dense, it's the quantity that we can actually compute and are these relationships that hold, and these are the ones that we saw before that give us access to these dielectric functions that were, after so in the random face approximation, and that's one thing that that that we are going to use widely because it's because it's computationally attractive in the random face approximation.

A

We say that the irreducible polarizability is simply equal to the irreducible polarizability in the independent particle picture, which essentially means that we set this FX C to 0 and that's the essence of the of the of the random face approximation so, and that gives us this particular relationship between a dielectric function, a microscopic dielectric function and this polarizability that we compute from our concern system. So we can go beyond and that that was mentioned on one of the first slides.

A

So we can go beyond to random, face approximation and include changes in the exchange correlation potential. So not set this thing to zero and and we get get such an equation, and that is what we commonly call local field effects in in density. Functional theory. So- and this is local field effects in the random face approximation.

B

A

So, as we saw before in the long wavelength limit, so so so now we try to connect to experiment to experiment. So in the long wavelength limits 4q goes to zero of the dielectrics matrix. We, the dielectric matrix, gives us access to two optical properties in regimes that are that are probed by optical measurements and a microscopic dielectric tensor epsilon infinity, for instance, of Omega, is given by this particular relationship right. This quantity we now can can calculate, for instance, in this random phase approximation from from our cone shampoo, Laura's ability. We can calculate this.

A

This inverse dialect matrix in the random face approximation, for instance, and get access to 2 epsilon infinity. So we can do this at various levels of approximation like I mentioned before we can stay within the independent particle picture. That is what we saw before. That was this. This is truly truly trivial relationship, but we can also include local field effects, as I mentioned before, in the random face approximation or beyond, adding them adding changes in the DFT exchange correlation potential as well. So these are tags that are related to this, so you switch on.

A

L optics is true. You get this particular this particular quantity written out. If you want to go beyond you can use, algo is chi. It uses a lot of the machinery that that will be used for 4gw as well, so it's quite a bit more expensive and then you can get a better approximation or a more fancy approximation to your dielectric screening right.

A

A

So how do we do this? How do we do this in the code for for and for L optics is true, for instance well here. This is the thing that we we want to evaluate.

A

So how do we get this? We compute the imaginary part of the office of this tensor directly from from the cell periodic parts of the of the column orbitals and then do a grommet corner gram, grammars clonic transformation to get at at the real part of this particular quantity. So the computational effort in this case lies in computing. These functions, so these ones are, are easily easily are we can get at easily and those are our block states, but actually, in this, in this queue to go to zero going to zero limit.

A

We need the change in the in these orbitals at K, so so we need them at a very small vector, an infinitesimally small vector away from our block factor.

A

So what we actually need to compute is the first order, change in the cell barrier, part of our of our block functions and that we do in in perturbation theory, so that is that is written here so and so an orbital at a cell burial periodic part of a cone. Some function, sorry of a block function slightly away from K, can be written in this way. So it's a Taylor expansion.

A

So it's the function at K and then this is the derivative of that function with respect to K times this small change right and for perturbation theory. We know expressions for this.

A

These are the changes of our Hamiltonian, with respect to K changes of our overlap, matrix with respect to K all kind of things that we can compute and then this is a standard first order, perturbation theoretical expression where we have sums over occupied and unoccupied states,.

A

Well, we can cost it that way, at least that one runs over occupied States and one runs over unoccupied states and yes, all kinds of things that we can evaluate. But again the need to be here, because these such an expansion in perturbation Theory essentially runs over all states all states of your Hamiltonian of the unperturbed Hamiltonian. So for these kind of things again, we have to include many more states than we would need before for this ground state calculation, so ground state calculations are only interested in occupied states.

A

Here we need to convert our result with respect to the number of unoccupied states that we are that we include in this expansion right in this perturbative expression.

A

So does it work? Yes, it does work so there's paper describing the way that we have implemented this, and there are some test cases where we complain where we are where we compared to a PW plus local orbital. So this is the live into K code and, if you're interested so there's examples where we, where we run exactly such a calculation in hands on section and the quantities that that you're after well I've put in the strings that you can search for in the outcome file.

A

And then there's a list of frequencies and and you'll find the components of the dielectric tensor as a function of frequency directly after. And there are some little scripts that allow you to plot this quickly. With with new.

C

A

So yes, well, this I think this is only a repeat of what we already seen before sorry yeah. So this is sort of a recap right, so the quantity that we are, that we use the quantity that we have access to this irreducible polarizability in the independent partial picture- and this is this other and wiser expression that gives us access to this particular quantity and that scales evaluating this scale says n to the power 4. So why is it n to the power 4 so for a large system where we don't have K points?

A

Well, we can neglect K points here we see that we have g and g Prime's on sums of g and g prime, the number of plane waves in our basis scales with system size right then we have n and n prime sort of the bands that scales with system size as well, so we have already four times system size, so this is n to the power 4 right.

A

So what does this screening actually mean right? So we have this in in the are P a so have. Then we saw these these. It's unfortunate that I didn't put them the Dyson equation there cleanly, but what we? What does this RP a give us?

A

Actually you heard this random face approximation that will that will pop up in GW and and another methods later on, so in the random face approximation, the dielectric screening is given by such an expansion, and so our screened potential in the random face approximation is given by Weldon the baculum, the bear, coulumb interaction and then the electronic environment. So the second term is the electronic environment that that reacts to the field generated by a particle.

A

So we have another particle that perturbs us sr as our particle travels to a system through the system, it induces a change in the charge density and that change in the charge density induce changing the potential that works back on our particle.

A

That's the second term, and then- and you can- you can go on and on with this, because if we have an induced change in the potential, then then there is another induced change in the density and that will induce a change in the potential and that so that is this whole chain of of of terms that are written here and so on and so on and so on.

A

And so electrons react to the induced change in the potential and that will give an additional change in the charge density and a corresponding change in the Hartree potential and so on and so on. Advocacy's. These are all changes in the Hartree potential through his school on Cornell in the random face approximation, and this this can be written as a geometrical series, and so this whole collection of terms up to up to infinitely long terms.

A

You can write us into your metrical series given by the naked Coulomb interaction times the inverse of this, and this is exactly the inverse of this dielectric function and right so in the random face approximation. But do we actually compute, we compute a screened, Coulomb interaction where all the older, all the physical processes involve involve the Hartree interaction right, involve changes in the Hartree potential.

A

So that's the essence of the random face approximation right.

B

A

So you can include that, as we saw before in using this elbow is Chi. You can include all these all these processes, this random face approximation in into your computation of the dielectric properties, and you will get them in written into the are out car file right so per default. It's it's all done at the level of our PA, but you can. You can include changes in the dance in the in the exchange correlation potential as well.

A

So yes, so there's a few practical points and that will that will be of importance to a computation of dielectric properties, but it's also of importance to well very much of importance for GW calculation, where I will use this dielectric properties I've been talking about the fact that we are going to be using these these unoccupied States, and the thing is that that we we use iterative matrix diagonalization to refine our states right. We talked about it yesterday and I said okay.

A

This is very nice because we can get at the so many lowest states in our eigen value spectrum of our Hamiltonian matrix, and the thing is that is iterative techniques. They converge most rapidly for the deepest lying States. As you go up into the eigen eigen value spectrum, the states converge less and less rapidly, so in for a ground set calculation, we look to the convergence of the occupied states right because they give us the total energy and our convergence criterion only looks at total energy.

A

It says: okay, if my total energy doesn't change by soul and so much anymore, I say: I'm converged right, but now we will use the virtual States or the empty states in the spectrum as well, and they don't contribute to the total energy. So the quality of those states and whether or not they are converged doesn't express itself in the total energy.

A

So you could, you can't simply do do a calculation that includes simply more bands and then say: okay now I'm converts, because then the virtual states are well less converged than the than the occupied states, so the quality of your virtual states of your unoccupied bands is not guaranteed. So we do a trick to to get around this. We do a ground set calculation, a simple one where we we simply use the the number of bands that that we would use for any normal ground state calculation and then to obtain virtual orbitals of sufficient quality.

A

We do exact, diagnose Asian of the Hamiltonian, so that's the thing that I said yesterday well that that is the thing that we want to avoid it, because why would I do an exact diagonalization and compute 50,000 50,000 eigenstates of my Hamiltonian, when I'm only interested in four well here we'll do one step of this on top of such ground set calculation. So essentially our occupied states have converged. Our Hamiltonian is correct in that sense, and then we build up this mo Tony, so we restart the calculation reading.

A

Those wave functions build up the Hamiltonian that is defined by the occupied States diagonalize at one time, and then we have all unoccupied states correct as well, so the quality of those states will be as good as the ones that we have optimized for occupy part of the spectrum. So that's an additional step that that you would need to do to get to get high-quality virtual orbitals.

B

A

That is, that is something that that that we will have to implement very soon. It would be a better way to do it, but at the moment that's not present yet so one can obviously say I, don't look at the at the, so you wouldn't have to look per se at the total energy to judge whether your converts, you can look at the eigen energies of of any number of states and as they stop changing or are converts with, with a certain degree, you could say: okay, I'm done.

A

That would allow you to keep using these iterative matrix diagonal asian techniques. Yes, that's definitely true, and we should change this, but at the moment that's not possible. Yet it's quite unfortunate, so we are forced to do this to do this three steps and and you're right. This is a very costly step for large systems. It's even so, and that is why we'll definitely will definitely change it pretty soon.

A

It's even so that this step becomes the most costly step, even for some GW calculations, so doing the actual GW afterwards is not as costly as doing this.

A

This exact diagonalization, which is unfortunate on the other hand, and that is why why we why we still will still use this in the future as well, that all the all the convergence criteria that we use and we'll show this later on starters, especially for for total energies in the random face approximation, we do a trick to to extrapolate the result to an infinite basis set size, and for that you would need all states of off your system.

A

So this this will be a valuable thing, even in the future, but for very large systems and for GW. This is something that that might kill you at the moment. That is true. So these are the typical steps that you would do and well in this case it's only a calculation of the dielectric properties per se. So the three steps is a ground set calculation. Then this exact diagonalization step where you, where you ask for asked for a large number of empty states and those aren't so it's done.

A

The Matrix is diagonalized exactly and then a bunch of those empty states are kept and written out to your wave car file and afterwards you can restart again from that file using either of these methods to to compute dielectric properties.

B

D

B

E

B

A

Can repeat the question so so? How do we, so? How do we diagonalize this this matrix exactly so that that is essentially simply calling a skylight back routine, so the matrix is set up and then scale light back is called 2 to diagonalize it.

A

So, yes, so it is distributed and how large are these? Are these matrices so in many cases, they're of the size of 50,000 times 50,000 entries?

A

Yes, so I recently did a calculation on on a cell with 256, so at GW calculation on a cell with 256 silicon atoms and there it was something like 47 thousand fans, so the whole spectrum depends is really yeah. It's quite sizable, yes yeah, so it would pay off to to be able to skip this step. Yes, definitely.

A

Well, that is, that is the thing. So so is there a rule of thumb? Yes, there are many rules of thumb, so I but I mean so for silicon, and that is the well. This is our favorite test system. Let's say, and everything always works for silicon because it has been tested with silicon. I I would use something like sixty four states per silicon atom right so which means that in that particular calculation that I gave for 256 silicon atoms, I kept 16,000 bands. Well, a large search can be quite sizable.

A

I think it depends on how accurate you want your results to be. You could get away, probably 32 bands for for at them as well, go to another system. I, don't know I, don't want to venture I. Don't want to to tell you yes always use 64 bands per atom right, so I I simply don't know, and it's one of the things white white I mean you can get around this step by simply using all bands that the number of bands you get is then simply defined by your cutoff.

A

And then your results again are going to converge with your energy cutoff, because the size of this matrix is directly defined by your basis set cutoff, because it's the size of the f50 grid and that is defined by your cutoff. And so in this way you get around around this uncertainty uncertainty by simply using all bands and then there's clever algorithms, by which we can extrapolate to infinite basis set size, because we know how it should how it should converge with respect to a s. Your cutoff is high enough.

A

We know the convergence behavior with respect to the cutoff, and then we do an extrapolation, but that can only work if you use all bands that there are, if you use all states of your Hamiltonian yeah. So it does pay off to still do that. Actually, especially for a 40 w and a CF DT calculation as we'll see later on.

A

Yes, so here come these GW potentials that maybe people have have already seen and because and already mentioned this yesterday- that we now depend on everything. We need a certain amount of virtual states as well, and so we need potentials that are that are sufficiently accurate, that they describe the scattering properties up to up to higher in the in the energy spectrum.

A

So and if you look at the conventional silicon pows potential, you see that up to two to three Reed Burke's atom the match between the dotted lines and and and the drawn lines is still quite good, so the pseudo atom and all the electron atom have essentially the same scattering properties but higher up in the spectrum.

A

You start to see deviations, so this potential will not yield very high quality virtual states because they depend on scattering properties higher up in the spectrum, and these two W potentials have been constructed to match the scattering properties up to much higher energies, and that is the idea, the essence the essential step which, which is why we why we advise to use these potentials. If you go to any of these methods that include sums of our virtual states.

A

They're good for ground set calculations as well right I mean so it doesn't spoil the the it doesn't spoil the the properties for ground state calculations.

B

A

So, let's turn to static dielectric response and so static. Dielectric response, there's a few things that we have to calculate so that the program can calculate for you. The iron clamp static, macroscopic, dielectric, tensor solecism. This epsilon invent infinity at Omega is 0 at 0 frequency. The board effective charge tensors that I that I spoke about.

A

So those are the the amount of charge that moves with your atoms are sort of changing the polarization of your system when you move an atom electronic contributions to piezo electric tensors now given here, and we can calculate them either using density function, perturbation Theory, using L epsilon is true or from the self consistent response of two orbitals to find out electric fields, and that is using this algorithm.

A

So there's examples in the hands-on session that deal with these aspects right so for so using density, functional perturbation theory so essentially added the thing that we have to compute here is again the derivative of this of of are the cell periodic part of our block functions with respect to the block wave vector in case of density function of perturbation theory. We don't use the sum over empty States to do this. It's not this.

A

This first order perturbation theory where we have a sum over over sorry over empty states, but we we solve a linear, Stern, himer equation. That looks like this I saw this, so this is the number of states for which we have to do. This is of the order of the occupied states right so that that is nice. That's a nice thing from density, functional perturbation theory, so actually so that we solve for this particular quantity, and this particular quantity actually is, is- is needed to express our perturbation. So we solve our perturbation.

A

Is a it's a it's a bit of a strange thing in this in these electro static electric fields, and so this is a static. Electric field at at Q is zero. So that means it's a constant field right over our material, and that poses a problem for for the kind of methods that we use, because the constant electric field is a that is a constantly dropping potential out for a system with periodic boundary conditions. So it's not it's not a perturbation that we can express under periodic boundary conditions. Right is this potential that we add.

A

We cannot cast in periodic boundary conditions, but there's a way to include it, and that is by computing, this quantity from perturbation, Theory and then, essentially, we use this quantity, and we see it here right in a second perturbative expression, and these quantities are the change in our in our functions with respect to K represents our perturbation, so I won't show you here why that is actually true, but it's possible to cast.

A

This is non periodic perturbation into a form that is, that is again self periodic, and that is done by this first by this first.

A

Our equation gives us this gives us actually a representation of our electric field of our perturbation, and then we solve a second second mr. himer equation to get at the response in our orbitals. Due to this due to this perturbation- and this can include all kinds of local field effects so which means that we can include the response in our self consistent hamiltonian due to this perturbation right so which is commonly called these local field effects and they may be included at the RTL level again or or beyond.

A

This, including changes in the DFT exchange correlation potential in our Hamiltonian. But once we know the response in our orbitals due to our perturbation, we can quite easily calculate well microscopic dielectric matrix given by this and where this sum now runs of a virtual / occupied states.

A

Only so there's no sum over unoccupied states anymore, and we can compute a host of other quantities from this like born effective charges and and these beautiful electric tensors by simply looking at changes in the helm environment forces as we change our orbitals with respect to this to the to their response to the perturbation. And we do this in finite differences.

A

And so we add, we add a bit of this of this response to our current wave functions and look at the change in our forces or in our stresses, and that gives us that gives us then, and so the change in the helm environment force with respect to the field is the born effective charge which is equivalent to the change in the polarization of our system.

A

With respect to the atomic displacement same thing here, the change in the in the in the stress tensor, with respect to to our electric field, gives us the electronic content to the Piazza Electric tensors, yeah and.

F

B

D

F

A

Is also supposedly too fast for the irons to react to.

F

A

B

No, no! No at Omega.

A

Equals zero. Yes, yes, yeah.

B

A

Actually, the infinity is really horrible that this that it's common notation, but it really puts you on the on the wrong foot, because.

B

A

The frequency zero for the electrons, but it's like infinite frequency for the ions right I mean they have no time to react. So it's stupid.

B

A

Okay, the output of all this calculation- well, this is this is more relevant for when we turn in turn to the hands-on section. This gives you somewhere where to find all these all this information in the output.

A

Yes and there's a bunch of examples.

A

Well, let's not go through that, because we're approaching long lunch time- let's quickly, go through this through this response to two finite electric fields, so there's another way to compute these quantities, and so did the first we saw was density function of perturbation theory, but there's also a way to include the effects of a finite electric field, even though all we said, okay, that this field is it's the potential that that corresponds to this finite.

A

Electric field is not something that we can represent under periodic boundary conditions, there's a way to actually include the effects for I for insulating systems, and that that is based on this. What is called the modern theory of polymerization? Well, it's always very bad to call anything modern, because it is modern at some point- and this is already some years old, so right so in 100 years. It's definitely not going to be modern anymore, but it's still called this so, and that says we can change.

A

So it's one of the things that there is maybe a bit counter intuitive the fact that we can compute the polarization under periodic boundary conditions is not a trivial statement in itself, and that is maybe I can can sketch again from this nice wall and that is under periodic boundary conditions. We have two following problems, so let's say: if I have, if I have a dipole in a system, it's easy to compute.

A

The other polarization because of this is no there's no essential problem, but if I, if I, not yes, I cannot put repeated dipoles, which we have on the periodic boundary conditions. It becomes a non trivial thing to compute this, because I can choose my cell to be this right and then integrate out for the charge. It will give me one answer, but it can also choose and that's completely equivalent.

A

I can choose my cell to be this, and this will be a completely different dipole right so on the periodic boundary conditions for the longest time, people said there is no way to define the polarization of a truly periodic system and then come along came.

A

The modern theory of polarization is that well, this is possible actually and there's a bunch of equations that that one can evaluate, and they will give us this macroscopic polarization of a periodic system, so I've just sketched the route to do this result here, because in block functions is always very difficult to do this, but we can go to to Avani a function, descriptions or localized charges and for mobilize charges so localize functions.

A

It is easy to define a dipole moment in in a system, and then you can, then you go through a whole bunch of mathematical, manipulations and end up with an equation that tells you that your macroscopic polarization depends on, and that is what we saw before right. I already mentioned it, the derivative of the self periodic part of our block functions with respect to the block wave vector.

A

We determined this before from from these density, functional perturbation equations to include electric fields, and here we see as this essential quantity popping up as well.

A

Okay for the interest of time, we'll we'll simply believe this and- and we say ok, we have a way to compute the polarization of our system, and this opens now a route to include the effects of a finite electric field as well, because, we'll simply add a term. So this is our normal ground state, DFT, energy, right with wave functions and now we'll add a term which is essentially the in product between the finite electric field that we're considering and the macroscopic polarization of our system.

A

Sorry Susan gives us a new total energy that includes the effect of a finite electric field on our system, so an additional term in our total energy. Add a defense of this expression that we saw before that comes out of the modern theory of polarization. An additional term in our total energy will add and corresponding term to the Hamiltonian, and that is well this one and so the difference.

A

The variation of our polarization with respect to an orbital is added to our Hamiltonian, and then we simply do a self-consistent calculation for this particular Hamiltonian and minimize this total energy. So there are some considerations. You cannot make this field too big. Now, because at some point you will, you will start to close your gap, so that is that is sketched on your slide, and that is argued in this particular paper. If you're interested in this, so there's a criterion implemented how strong you can make your field before you before you close your gap.

A

You must have forget, so the system must be insulating, but then you can. Then you can optimize your wave functions with respect to this particular energy, functional and use these use these solutions to compute the quantities that we have seen before right, because we can, we can compute the change in the polarization now with respect to and a finite electric field, and we saw that this change in the polarization with respect to the field gives us our static, macroscopic, dielectric matrix and the born effective charges is associated with this.

A

We can again like before compute from and we apply an electric field. We have a slight change in the wave functions and through this it's changed in the helm, environment forces and the change in these forces with respect to the field give us the born effective charges, so the essentially quantities that we get from density function of perturbations. Here we can get from a response to to this finite fields as well and, of course, all local field effects are included in a very natural manner, because we do a self-consistent calculation.

A

You apply a field and then you do a self-consistent calculation. So all variations. All changes in potentials to to the field are naturally included in this right. You through the self-consistency, so that's very nice, and we can do this for any Hamiltonian, so density, functional perturbation theory. You can do for density functionals, but not for hybrid functions, but this works for hybrid functions. Well, okay, so there's some numbers here that we did we gotten with this and yeah, so I think we'll skip this. These are essentially only details on on this particular terms.

A

You can look at them at your leisure, but but I think you. This is I've already sketched the.

A

The general, the general idea of it I think we'll we'll stop here. Yeah.

B

E

A

Would like to if there's questions here, then I would do the deal.

E

We used the usual calculations instead of having to zetsu. Yes,.

B

E

What about the equivalent response to.

E

A

No it can, it can do so. It can do NMR so that that is what you can do. So you can compute can compute, but you can't compute structural responses to to magnetic fields.

E

A

So the biggest system that I have been able to do is not very big, because we are poor austrians and we don't have computers, but now I know people that do that. Do you see calculations and systems with thousands of atoms completely beyond anything that I have experience with I, don't know how painful it is and how big the machine has to be, but I think what is you have? Maybe an idea of what what a common size at Newark is for.

A

Do you have some feeling.

D

B

A

I got some some benchmarks from from a guy on and he was using a crane machine and he could go up to thousands of course, so he had access to a fair amount, of course and I think he did systems of some thousands of atom or a thousand of atoms. Things like this, so it is possible, it's surely painful, I'm, absolutely sure this. It hurts to do it. Yeah, yes, sometimes.

F

You want to compute like dielectric properties, where you know it's a semiconductor, but GGA gives it it's either a metal or a very, very small guy. We.

B

F

Throughout the country, yes,.

B

F

Your suggestion for those.

A

Probably the first thing I would do is a sister operator.

F

A

If it's, if it's a metal, I'm.

F

Sorry I thought that if your band gap is too small.

F

Where I can just give it? Yes,.

A

But for metallics ISM I, don't know how you would scissor, but then I would probably go for a hybrid functional.

F

A

Although, although- and that is something that will come up, although that then you're already so, if give T gives you such a bad answer for that for the ground state, then I would try with a hybrid functional, but it's something that that I'll discuss tomorrow. What what I mean? There's there's this there's a compensation, there's a compensation of errors involved in using the random face approximation and DFT. So you do get really good screening properties if you use two random face approximation in connection with DFT, orbitals and eigenenergies.

A

So if you go beyond DFT, then the random face approximation will start to show up as it will. Its limits will start to show up more clearly the same thing if you go beyond the random face approximation.

A

So no, yes, it was like I said before. So, if you use anything but DFT with the random face approximation, you do not got get as good screening properties, so especially for GW. If you then there's some aspects of this I will I will I will discuss them. Morrow.

F

Sidra, where, if you had a small band gap, but you wanted to just open it up to the experimental value, what's the procedure for doing that and that's for the verification.

A

You mean in terms of office of sorry in terms of a scissor operator, yeah.

F

A

Have to look it up, but it, but it's I mean it's a scissor operator is, is super easy right. You don't do anything to your orbitals. You just shift. You are an occupied eigenenergies with respect to the occupied ones.

F

A

Do you call it yeah.

F

How do I call it okay, so I have a GJ gives me one right.

F

A

Perfect so yeah, so how do you apply? The sister of yeah I will have to look. Look it up. I said I mean it's in one. It's used in one of the examples: okay, but I will have to make sure that it that it will work in the way that that you, that you need yeah and it's something I mean it. It is so simple that if there's something still missing there, we can easily quickly fix this yeah, yes to make it work in a in a manner that is convenient to you.

A

So would you get the dielectric properties of isolated molecules? I, don't see a reason why why not I, don't know how good DFT or or these methods would or the random phase approximation would be for isolated molecules, though or so so. I have no experience doing this for for isolated molecules, but there's no there's no essential, there's nothing in any approximation. That would would invalidate the procedure for isolated molecules right, yeah,.

F

A

You could put in bunches of molecules, probably in a cell and.

F

A

Guess it should work, I've never tried. So it's interesting interesting question, but I don't see that there's there's not a from from the algorithm algorithmic point of view. There's nothing that speaks against.

C

A

Iiii without a lawyer, I won't answer this question. No I don't know.

F