WEBVTT
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Every observable that you want to calculate will be an average or an integration over the Bruins zone.
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So in general for any observable a.
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I will have to take an integral if I do it analytically, and integral over the green zone of the evaluation of a at HK point of the green zone, and okay it's normalized by the volume of the end zone cheer.
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Since we have a computer, we are, we want to put these theories into a computer and computers don't like integrals, because they are continuum, and nothing is a continuous continuous in a computer.
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We represent the integral as a sum and now it uses an average over over discrete evaluations of a at HK points, divided by the number of key points. So for example, the total energy I don't know why I'm used to it here will be the some of the total energies
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at each k point with the shifting weight that is the weight that is given by the sampling.
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So what they mean is that if you have symmetries within your unit sell these cemeteries will be translated by the Fourier transform into the end zone as well.
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So if I have a symmetric brilliant zone that is like these square lattice here again.
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Instead of calculating the electronic structure or the condition wave functions at every key point here, I can recognize the same method isn't big, only a little wedge of the, of the breed runs on the disco the irreducible wedge.
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And I can calculate the properties, instead of using four by four is 16 k points in this example, I can calculate all the properties over just three key points, I can average over three key points so three key points represent the whole 16 k points that
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they have there.
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Okay. The thing is, if I have them well into the, into the wedge, their way this food if they have them on the, on the boundary here their weight is hard, so if you calculate the weight.
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How many key points corresponds to each one of these k one k two k three.
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You can just do the game here and pick this one these is if we were into its mirror symmetry on the left hand side, its central symmetry down here. And the other mirror symmetry on the, on the one axis down here so each these points, as a way to four
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out of 16 so the normal nice way to zero point 25.
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This point that is in the, in the rear end zone for the.
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It actually represents. If you do the count yourself. It represents 1234567 and eight k points so its weight is one heart.
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So each gay point there's a different weight when you throw symmetry in.
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Okay, so you'll have to do all these to get the contract way functions and get the total energies and possibly also the forces and that's what the pw coding quantum espresso does.
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Um, so we can list a bunch of advantages of using brainwaves and I tried to convince you that they outweigh the disadvantages, at least for solids.
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So, one good one great advantage that we have over localized business sets, is that we can check convergence, we can tune convergence by tuning, a single number, which is great.
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You don't have to optimize 500 knobs, you're just one.
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Another big advantage is that the busiest said does not depend on the position of the nuclei, we didn't use the nuclei to construct the basis set, if we were taking atomic orbitals, or Gaussian orbitals, they move with the nuclei so every time I move
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the nuclear moving the basis set.
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We can use the Fourier transform in many ways to solve barrios going to choose which chunk of the potential we got it when we calculate the potential when we calculate age sky.
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we can divide it into into different chunks. And we can pick for each chunk whether we want to solve it in real it is convenient to solve it in real space on the reciprocal space, because we have both representation basically for free with the brainwaves.
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An example is the person equation person equation in real space is a pain. This is a passion of the, of the director static potential is equal to the fire or if we throw a Fourier transform in that oppression becomes a multiplicative factor p square and
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person equation becomes easy.
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So, all these chunks of the total end and or the or the total potential of the total energies are calculated quite convenient three density hard three exchange correlation for exchange in correlation and so on are all easily calculated and either big
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advantage of language is that they are also normal, they are naturally an auto normal basis set the normalization is one over the square root of the volume of the sale.
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Okay so anyways these haven't, we're not always.
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You need a lot of them.
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Somebody is writing on the thing, I don't know.
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We have also disadvantages. One is that well this is a small one the business that depends on the shape, and the size of the set when we construct the basis set we are using L.
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So if we have a system where we want to calculate for example I want to do like a custom pressure dynamics with the cell various.
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Then we are a bit in trouble, we have a country but we are changing the business at under the feet though our calculation and do something, you will see in the tutorial when we do the equation of states you will have to be a little careful about that.
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It's not a huge problem, it can be easily circumvented.
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And a big big problem that is not easy to circumvent is that you need a lot of brain waves to represent a function that is not very smooth or not very periodic.
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So, if you think of localized states core states on the atoms. Those are extremely steep, go back to your basic chemistry physical chemistry classes or structure of matter if you do physics.
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Those are very very steep states, deep states steep steep and yeah also deep in energy, and they're not convenient to be treated by by brainwaves because to represent a steep function you need a lot of brain waves.
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So, it really becomes impractical to treat course states. So the way around it, is to us is to get rid of course dates accurately though.
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So that's why we introduce as a part of the brain wave approach to density functional theory we have to introduce them to the potentials.
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And so the basic idea is to them to dump core states into something that is an effective potential.
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And that takes the clue from the fact that after all if we're interested in the in the chemistry of the materials chemistry is carried out by the violence states themselves, and the core states are just doing some nuclear screening.
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Basically, this screen the nuclei, and you have this periodicity in the of the periodic table and it's periodicity exactly because the system of nuclei and core states is Quran, there is a Quran screening, the course states do a quorum screening of the
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nuclei and then you have the repeating properties that come from the, the properties of the valence electrons that come in one shot at a time. Right.
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Okay, so this is the recipe to do that. There is no theory behind it. I mean, there are several different recipes that work more or less well and there are more or less complicated.
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I just give you the general recipe, which is, well, how do we do it. So we want to define app pseudo item.
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That is the nucleus plus the core states that will be the vo via the capital V ovodda that I had in my
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homework and corn equation and confirm equation, right.
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So, the, the thing that happens is that when you construct this cell the atom with some constraints, the violence states will become the lower states, the given Angular cemetery.
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So if I take sodium.
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And I put all the two s and to be states core states into the bottom in the builder potential, then the three state of the, the electron that is the balanced electron or sodium.
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It is my lowest energy state is like a one state for the show go awesome for the screen.
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The good thing with that is that it will not have nodes in the in the core region, because it doesn't have to be orthogonal to anything else. Right.
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So it is move any there's no nodes,
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the constraint that we have to impose is that the chemical behavior of the atom has to be the same as the actual atom and how do we enforce that we, when we construct the show the potential we make sure that for the valence electrons, the orbital energies
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are the same as those that we get in a, in an order that from calculation.
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So we, there is a code there is a code in the quantum espresso package that constructs, so the potentials and one thing that it does is it takes an atom in a large enough box to avoid periodic say finite size effects.
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If calculates all the states for the atom at all electron level. And then it constructs the show the potential in such a way that the valence electrons have the same or better energy as the electrons calculate as they are the same, but instead excellent
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Scott good at that or director level. Okay.
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good at that all that extra level. Okay. The other constraint is that outside the nuclear region we want to have the same we function as in the order ektron case.
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So, we define a parameter that it's called the core radius of the total potential.
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Below that radius.
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It's all these game basically with some constraints as well. But we don't have a sound physical representation of the system. Below that, radius and above that.
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But for for distances for larger than that ranges from the core from the nucleus. We want to have the same decay, as in the, in the old electron calculation.
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So this is pretty much what we do, let's take for example silicon.
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This is the electronic structure of silicon.
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We recognize that we are interested only in the three F's and the three piece dates and everything else below three s is dumped into the soda potential so these are the core states.
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These are the balance states.
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And then these would be the actual potential for the nucleus, and then we will construct the cell the potential that S is is not going to infinity, it is zero, basically, and its corresponding wave function will be a smooth way function without all these
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wiggles and nodes that beyond the surface and cut off behaves exactly as the actual atomic wave function so the observer potential constructed from atoms calculations they're not adapted to materials or whatever.
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Okay.
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So, the observer potential is, is an approximation, but it's not empirical you construct it on the older rectum calculation for one atom, for each item.
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Okay, so this is an accurate calculation. We have the real three s wave function for silicon, that looks like this, we define a courageous record radius is about an axon normally.
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So we define a certain core radius, that these between one and two atomic units. And this is what the upsell or three way function looks like. So, in the longer distance region that is what is interesting for the chemistry, we have this.
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Now, these are the other sailed away functions of silicon we would have, we get rid of the nodes of the other single node of the three p as well. And for the 3d, which is an empty state, but we still have the channel in there so the potential, because
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it could be partially occupied when you start for mixing for bonding, the 3d channel is pretty similar to the otter reference really China.
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Okay, this is my last slide before the tutorial.
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And am I on time, right, good.
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So this is the workflow that we will run when we run the tutorials with in the the self consistent calculation in the, in the tutorials that we come.
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So that's the idea. You have to solve chef consistently these set of equations to each other to confirm equations.
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What do you do you construct the, the first potential with an initial on this is what it is, what the nuclear car. This is what you give any is an input as the geometry of your system, you will have to give the coordinates of each atom, as an input, or
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at least is an educated guess of those.
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And then the code will construct will construct the potential from those which is the sum of them so the potentials basically.
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And it will construct an initial guests of the density here unfortunately score for, I didn't bother redoing the.
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I took it from a tutorial by Shimane pitching.
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And with that, with the, with the initial guests of the density. We'll see what options are out there to start the density, we can compute the hard three and the exchange and correlation potentials.
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We plug it in into these effective potential that goes inside here that comprises the nuclear the hardware and the exchange and correlation.
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We plug it into the single particle
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single particle con Sham equation basically for for each non interacting quantum state.
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And then once we sold these we have the eigenvalues and the eigenvectors of these.
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The eigenvalues will be an approximation of the orbital energies of the bands, and the eigenvectors allow us to compute the new density at the first step.
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If you check the differences in densities and energies you see that the guest, the solution will be very different from the guests so you have to go back.
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re compute hard three and exchange and correlation and loop until you will have a satisfactory solution that is not too different from the previous step right.
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We'll see the details of how we define a tolerance on the energy and an estimate of how accurate it is. And once we are satisfied, we have the energy and the analytical derivative of the, of the potential gives us the forces basic.
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So we have self consistent calculation of the energies and the forces, pretty much at the same time, the force is a minimal overhead.
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Okay.
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So I think we have a break now at three minutes break and then we have a look at the tutorials.
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Oh, I didn't ask if we have questions, which hopes, and perhaps revere if there are questions on the chat as well as on anything rather than you kick them out or No.
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Okay, so if you have questions that have not answered in the chat here.
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You can ask them in the next step in the tutorial. Is there any advantage of doing calculations in reciprocal space. Yes. One is, for example, the one I was mentioning the, the Fourier transform of the, of the persona equation is advantage, I mean you
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calculate the question equation the reciprocal space because it is extremely advantages. Another advantage of doing the key point sampling instead of replicating the cell.
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I mean, you need to do the key point sampling, or you replicate this sale.
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And the advantage is compute is the computational cost because the case sampling is a linear.
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As the linear cost. So if I calculate for one point if I calculate for two key points is twice the cost as 401k point there.
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If you replicate the cell, the cost of the calculation scales like cubicles or n square login.
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So it becomes rapidly computationally expensive whereas the key point the key point calculation is, is the net.
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I should have mentioned it perhaps in the.
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Okay.