WEBVTT 00:00:00.000 --> 00:00:02.640 PROFESSOR 1: --labs based on electronic structure, 00:00:02.640 --> 00:00:05.820 and this will cover the next two computational labs. 00:00:05.820 --> 00:00:07.860 And on Thursday, we'll start looking 00:00:07.860 --> 00:00:09.930 at some of the finite temperature 00:00:09.930 --> 00:00:12.150 ideas that are based on the energy models 00:00:12.150 --> 00:00:14.910 that we have seen. 00:00:14.910 --> 00:00:17.130 This, by the way, is [? Roberto ?] [? Bajo ?] 00:00:17.130 --> 00:00:20.610 missing a penalty shot. 00:00:20.610 --> 00:00:24.540 So a reminder of what we have seen in the last class. 00:00:24.540 --> 00:00:27.360 We have introduced the idea of pseudopotentials. 00:00:27.360 --> 00:00:33.180 In order to remove the cost of carrying out calculation that 00:00:33.180 --> 00:00:37.710 include the core electrons that are very many, especially 00:00:37.710 --> 00:00:41.910 in sort of larger atoms, and that have exceedingly 00:00:41.910 --> 00:00:44.650 high oscillations around the nucleus 00:00:44.650 --> 00:00:48.030 due to the orthogonality constraint. what we have done 00:00:48.030 --> 00:00:53.340 is we have substituted what would be the z over r 00:00:53.340 --> 00:00:56.580 Coulomb potential with-- 00:00:56.580 --> 00:00:59.760 inside of the core region of the nucleus-- 00:00:59.760 --> 00:01:01.110 a pseudopotential. 00:01:01.110 --> 00:01:03.300 That is a potential that reproduces 00:01:03.300 --> 00:01:08.620 the effect of the nucleus, and of the frozen core electrons. 00:01:08.620 --> 00:01:10.890 And as you can see, this pseudopotential 00:01:10.890 --> 00:01:14.910 tends to be repulsive close to the origin, basically again, 00:01:14.910 --> 00:01:17.940 reproducing this sort of angular momentum push 00:01:17.940 --> 00:01:22.140 outwards of the core electrons to the valence electrons. 00:01:22.140 --> 00:01:26.110 And in order to make this pseudopotential very accurate, 00:01:26.110 --> 00:01:28.470 there has been this idea that has been developed 00:01:28.470 --> 00:01:31.320 of norm conserving pseudopotential that 00:01:31.320 --> 00:01:36.210 will act differently on the different components 00:01:36.210 --> 00:01:39.070 of a valence electron wave function. 00:01:39.070 --> 00:01:41.320 So you have an incoming electron, 00:01:41.320 --> 00:01:44.430 so you have a sort of ground state valence wave function. 00:01:44.430 --> 00:01:45.900 You can decompose it. 00:01:45.900 --> 00:01:50.160 You can decide how much of it is SP or the component, 00:01:50.160 --> 00:01:53.460 and you can act differently on the different slices 00:01:53.460 --> 00:01:55.030 of this wave function. 00:01:55.030 --> 00:01:59.940 And so with this, basically, we can solve a new problem 00:01:59.940 --> 00:02:04.350 in an effective external potential in which the lowest 00:02:04.350 --> 00:02:07.260 energy ground state will actually 00:02:07.260 --> 00:02:10.919 be identical in energy to the valence eigenstate 00:02:10.919 --> 00:02:14.730 of the original so-called all electron on the atom with all 00:02:14.730 --> 00:02:15.840 its electrons. 00:02:15.840 --> 00:02:19.230 And the wave function for this pseudo-atom 00:02:19.230 --> 00:02:23.100 will be identical to the wave function of the real atom 00:02:23.100 --> 00:02:25.740 outside of the core radius-- outside, in this case, 00:02:25.740 --> 00:02:28.840 three atomic units in this slide. 00:02:28.840 --> 00:02:32.100 So this was sort of one of the first important technical 00:02:32.100 --> 00:02:34.560 proofs that were introduced in the '70s and '80s 00:02:34.560 --> 00:02:37.740 to make this calculation really feasible. 00:02:37.740 --> 00:02:39.570 The other sort of idea that I want 00:02:39.570 --> 00:02:42.180 to sort of remind you is that whenever 00:02:42.180 --> 00:02:44.400 we deal with extended systems-- 00:02:44.400 --> 00:02:47.280 solids, liquids, and in particular, 00:02:47.280 --> 00:02:49.650 when we have periodic boundary condition-- that 00:02:49.650 --> 00:02:51.750 is we have our unit cell periodically 00:02:51.750 --> 00:02:55.770 repeated in all dimensions-- the eigenstates of our Hamiltonian 00:02:55.770 --> 00:03:00.360 take the form of Bloch Theorem, and get classified according 00:03:00.360 --> 00:03:02.280 to two quantum numbers. 00:03:02.280 --> 00:03:06.090 A discrete number-- the bounding that's number n-- 00:03:06.090 --> 00:03:08.730 and the continuous number k. 00:03:08.730 --> 00:03:11.790 And the overall eigenstates can be written 00:03:11.790 --> 00:03:14.310 as a product of two functions. 00:03:14.310 --> 00:03:17.010 One is just a plane wave that has 00:03:17.010 --> 00:03:21.300 the periodicity of the so-called crystal momentum K, one 00:03:21.300 --> 00:03:23.160 of these quantum numbers. 00:03:23.160 --> 00:03:27.420 And the other is the periodic part of this Bloch orbital 00:03:27.420 --> 00:03:29.070 written here as u. 00:03:29.070 --> 00:03:33.210 And that this is a function with the same periodicity 00:03:33.210 --> 00:03:34.560 of your unit cell. 00:03:34.560 --> 00:03:37.200 So overall, the orbital itself-- 00:03:37.200 --> 00:03:42.630 the [INAUDIBLE] orbital psi can have any periodicity, 00:03:42.630 --> 00:03:45.450 but it can always be decomposed into a part 00:03:45.450 --> 00:03:52.560 of a well-defined wavelength times a periodic part. 00:03:52.560 --> 00:03:54.810 Periodic part is going to be smooth. 00:03:54.810 --> 00:03:58.120 Again, we don't have any more core electrons, 00:03:58.120 --> 00:04:01.020 so close to the nuclei, it will look just 00:04:01.020 --> 00:04:04.560 as 1s, 2s, 2p orbitals. 00:04:04.560 --> 00:04:07.290 And that periodic part can actually 00:04:07.290 --> 00:04:11.910 be expanded in a set of plane waves-- 00:04:11.910 --> 00:04:15.750 plane waves that need to satisfy the periodic boundary 00:04:15.750 --> 00:04:17.050 condition of our system. 00:04:17.050 --> 00:04:20.190 And so we have seen that we can write the functions 00:04:20.190 --> 00:04:25.230 e to the iGr, where the G is a linear combination 00:04:25.230 --> 00:04:29.160 of primitive reciprocal lattice vectors with integer 00:04:29.160 --> 00:04:32.490 coefficients, and all those G vectors 00:04:32.490 --> 00:04:36.990 are such that e to the iGr has the same periodicity 00:04:36.990 --> 00:04:39.150 of your direct lattice. 00:04:39.150 --> 00:04:43.200 And these are the coefficient of the series expansion. 00:04:43.200 --> 00:04:45.750 And we sort of can systematically 00:04:45.750 --> 00:04:49.260 improve the quality of this basic set expansion 00:04:49.260 --> 00:04:52.500 by increasing the number of G vectors 00:04:52.500 --> 00:04:55.740 that we use, and we can do that systematically 00:04:55.740 --> 00:04:59.670 by taking G vectors with longer and longer lengths, 00:04:59.670 --> 00:05:03.630 or as we say, with higher and higher energy cutoff, 00:05:03.630 --> 00:05:06.090 because higher and higher energy [INAUDIBLE] 00:05:06.090 --> 00:05:10.170 longer and longer moduli for the G vectors corresponds 00:05:10.170 --> 00:05:14.200 to plane waves with finer and finer periodicity, 00:05:14.200 --> 00:05:16.630 so with finer and finer resolution. 00:05:16.630 --> 00:05:19.560 So this is a distinct advantage of plane waves. 00:05:19.560 --> 00:05:23.760 And in addition, this basis sets do not 00:05:23.760 --> 00:05:25.750 depend on atomic position. 00:05:25.750 --> 00:05:27.690 So when we calculate forces, say, 00:05:27.690 --> 00:05:30.930 to do molecular dynamics or a structural relaxation 00:05:30.930 --> 00:05:33.090 calculation, we don't need to take into account 00:05:33.090 --> 00:05:35.670 the fact that the basis set changes 00:05:35.670 --> 00:05:38.790 with the position of the atoms. 00:05:38.790 --> 00:05:40.980 Plane waves are not the only choices. 00:05:40.980 --> 00:05:42.630 There are a number of other choices 00:05:42.630 --> 00:05:44.010 that are very successful. 00:05:44.010 --> 00:05:46.020 In particular, a lot of the quantum chemistry 00:05:46.020 --> 00:05:49.110 could use a Gaussian basis sets. 00:05:49.110 --> 00:05:54.420 So atom-centered orbitals that decay as Gaussians. 00:05:54.420 --> 00:05:57.660 This is very convenient to do calculations like the exchange 00:05:57.660 --> 00:05:59.130 term in Hartree-Fock. 00:05:59.130 --> 00:06:01.390 But of course, you could have just a finite difference 00:06:01.390 --> 00:06:02.590 representation. 00:06:02.590 --> 00:06:04.950 So you could represent your orbitals 00:06:04.950 --> 00:06:07.350 on a grid of points in real space. 00:06:07.350 --> 00:06:10.950 That tends to be very efficient to do parallel calculation, 00:06:10.950 --> 00:06:14.100 but it's very difficult to do accurate calculation 00:06:14.100 --> 00:06:15.780 of the second derivative. 00:06:15.780 --> 00:06:18.600 If you remember, when you expand a function in plane waves, 00:06:18.600 --> 00:06:21.510 you have an analytic expression for the second derivative 00:06:21.510 --> 00:06:24.630 just because the first and second derivative of plane 00:06:24.630 --> 00:06:28.170 waves are just ig or minus g squared times the plane wave 00:06:28.170 --> 00:06:29.010 itself. 00:06:29.010 --> 00:06:30.570 And that's very important. 00:06:30.570 --> 00:06:33.630 So that's when sort of real space representation 00:06:33.630 --> 00:06:35.460 becomes a little bit trickier. 00:06:35.460 --> 00:06:36.945 And then there are a number of sort 00:06:36.945 --> 00:06:41.010 of more approximate approaches based 00:06:41.010 --> 00:06:44.730 on a number of sort of atomic-like localized orbitals. 00:06:44.730 --> 00:06:48.360 Or there are a number of sort of accurate approaches 00:06:48.360 --> 00:06:54.080 that are based usually on a combination of a basic set that 00:06:54.080 --> 00:06:58.110 has a plane wave-like characteristic in the regions 00:06:58.110 --> 00:07:00.510 that are far away from the nuclei, 00:07:00.510 --> 00:07:03.810 and then it has atomic like characteristics in the region 00:07:03.810 --> 00:07:05.040 inside the nuclei. 00:07:05.040 --> 00:07:09.030 And this tends to be the most accurate, but also slightly 00:07:09.030 --> 00:07:11.810 more expensive approaches. 00:07:11.810 --> 00:07:14.310 This concludes one of the technical points 00:07:14.310 --> 00:07:18.420 that you need to be careful in a practical calculation. 00:07:18.420 --> 00:07:21.270 The other point that sort of comes over and over 00:07:21.270 --> 00:07:25.260 again has to do with Brillouin Zone integration. 00:07:25.260 --> 00:07:27.257 Let me give you a first example of a molecule. 00:07:27.257 --> 00:07:29.340 So if you are calculating the electronic structure 00:07:29.340 --> 00:07:32.730 of a molecule, in the course of your calculation, 00:07:32.730 --> 00:07:35.430 you'll need to calculate integrated quantity 00:07:35.430 --> 00:07:37.200 like the charge density. 00:07:37.200 --> 00:07:39.420 The charge density of a molecule is 00:07:39.420 --> 00:07:43.500 going to be the sum of the square moduli of all 00:07:43.500 --> 00:07:45.550 the single particle orbitals. 00:07:45.550 --> 00:07:49.410 So if you are sort of studying a molecule like just hydrogen 2, 00:07:49.410 --> 00:07:52.410 well, you just take the sum of the square moduli 00:07:52.410 --> 00:07:53.760 of the first orbital-- 00:07:53.760 --> 00:07:56.160 you have more orbital, more complex molecule. 00:07:56.160 --> 00:07:59.520 You need to sum over all the orbitals up 00:07:59.520 --> 00:08:03.840 to the [INAUDIBLE]---- the highest occupied molecular orbital. 00:08:03.840 --> 00:08:06.120 You do a calculation in a solid, you 00:08:06.120 --> 00:08:08.580 have to do exactly the same thing. 00:08:08.580 --> 00:08:13.920 That is, you need to calculate the charge density by summing 00:08:13.920 --> 00:08:17.730 over the occupied states. 00:08:17.730 --> 00:08:20.310 Now as I said, what are the quantum numbers 00:08:20.310 --> 00:08:24.060 that describe a solid are a [INAUDIBLE] 00:08:24.060 --> 00:08:27.060 and a continuous index scale that we 00:08:27.060 --> 00:08:28.860 call the quasi-momentum. 00:08:28.860 --> 00:08:33.150 And we usually represent, say, the energy of the states 00:08:33.150 --> 00:08:35.370 in a solid with a band diagram. 00:08:35.370 --> 00:08:37.860 And I've plotted here on the left the band 00:08:37.860 --> 00:08:41.190 diagram for silicon. 00:08:41.190 --> 00:08:43.620 In particular, what we are looking here 00:08:43.620 --> 00:08:45.750 is in the Brillouin Zone of silicon 00:08:45.750 --> 00:08:48.540 along certain high symmetry direction. 00:08:48.540 --> 00:08:52.710 I presume these are the high symmetry points, 00:08:52.710 --> 00:08:54.720 like gamma would be 0, 0, 0. 00:08:54.720 --> 00:08:58.530 [? Alpha ?] would be 1/2, 1/2, 1/2. 00:08:58.530 --> 00:08:59.490 Hope this is correct. 00:08:59.490 --> 00:09:02.790 This should be x, and it is 1, 0, 0. 00:09:02.790 --> 00:09:08.760 So we plot the energies of all the occupied bands of silicon 00:09:08.760 --> 00:09:12.960 in different directions along the Brillouin Zone. 00:09:12.960 --> 00:09:15.570 Silicon has two atoms per unit cell. 00:09:15.570 --> 00:09:19.150 Each atom has four valence electrons, 00:09:19.150 --> 00:09:24.030 so we have eight valence electrons per unit cell. 00:09:24.030 --> 00:09:28.450 It's a system in which there is basically spin degeneracy. 00:09:28.450 --> 00:09:32.340 So there is really sort of the same spatial part 00:09:32.340 --> 00:09:34.330 for spin up and spin down. 00:09:34.330 --> 00:09:37.380 So what we usually say is that we can accommodate 00:09:37.380 --> 00:09:40.470 two electrons for each space orbital, 00:09:40.470 --> 00:09:44.290 and they will just have different spin quantum number. 00:09:44.290 --> 00:09:46.170 So with these eight valence electrons, 00:09:46.170 --> 00:09:48.660 we end up with four valence. 00:09:48.660 --> 00:09:51.570 And sometimes, because of symmetry, you have degeneracy, 00:09:51.570 --> 00:09:53.670 but you see that somewhere like here 00:09:53.670 --> 00:09:56.730 at a sort of arbitrary low symmetry point in the Brillouin 00:09:56.730 --> 00:10:01.140 Zone, you can clearly see four bands. 00:10:01.140 --> 00:10:04.340 So if we want to calculate the charge density of the system, 00:10:04.340 --> 00:10:08.840 we need to sum over all the possible occupied states. 00:10:08.840 --> 00:10:12.320 That is, we need to sum over all the four bands, 00:10:12.320 --> 00:10:13.520 and that's trivial. 00:10:13.520 --> 00:10:18.380 But we also need to integrate over all the possible k 00:10:18.380 --> 00:10:18.950 vectors. 00:10:18.950 --> 00:10:22.880 That is, we need to make an integral in the Brillouin Zone. 00:10:22.880 --> 00:10:24.425 That is, we need to sort of really 00:10:24.425 --> 00:10:28.700 sum over all these possible states. 00:10:28.700 --> 00:10:31.460 An integral is obviously an analytical operation. 00:10:31.460 --> 00:10:34.100 In practice, on a computer, what we do is just 00:10:34.100 --> 00:10:37.310 we discretize that integral, and we take a sum. 00:10:37.310 --> 00:10:38.600 So it means that-- 00:10:38.600 --> 00:10:41.840 and again, I'll use two dimension as an example. 00:10:41.840 --> 00:10:46.700 It means that if this were my Brillouin Zone-- 00:10:46.700 --> 00:10:50.120 two-dimensional Brillouin Zone, and my k vector 00:10:50.120 --> 00:10:53.870 can be anywhere in this Brillouin Zone, 00:10:53.870 --> 00:10:57.350 what I would need to do is an integral overall 00:10:57.350 --> 00:10:59.540 the possibilities inside there. 00:10:59.540 --> 00:11:01.730 But in practice, what I'll do, I'll 00:11:01.730 --> 00:11:04.010 just take a discretization. 00:11:04.010 --> 00:11:06.710 And I'll calculate my band structure 00:11:06.710 --> 00:11:11.150 at each one of these points, and say a regular aqueous space 00:11:11.150 --> 00:11:13.430 dimension of k points. 00:11:13.430 --> 00:11:16.220 And that's an expensive operation. 00:11:16.220 --> 00:11:20.390 So each calculation at each k point 00:11:20.390 --> 00:11:22.790 will require a self-consistent diagonalization 00:11:22.790 --> 00:11:24.360 of your problem. 00:11:24.360 --> 00:11:27.410 And so really, the cost of your calculation 00:11:27.410 --> 00:11:31.350 is linearly scaling in the number of k points. 00:11:31.350 --> 00:11:37.280 So in reality, you want to use as few k points as possible. 00:11:37.280 --> 00:11:41.690 For a system like a semiconductor or an insulator, 00:11:41.690 --> 00:11:43.160 in which if you want-- 00:11:43.160 --> 00:11:46.440 the band gap structure is very smooth-- 00:11:46.440 --> 00:11:51.110 it becomes fairly easy to integrate that smooth bands 00:11:51.110 --> 00:11:53.700 with very coarse measures. 00:11:53.700 --> 00:11:58.580 So for something like silicon, you might be already very happy 00:11:58.580 --> 00:12:01.220 when you start to sample the three-dimensional Brillouin 00:12:01.220 --> 00:12:05.570 Zone with a uniform mesh of k points that could have four k 00:12:05.570 --> 00:12:06.830 points in each direction-- 00:12:06.830 --> 00:12:10.580 4 by 4, by 4, 6 by 6, by 6, 8 by 8, by 8. 00:12:10.580 --> 00:12:14.060 And this is sort of the order of magnitude of the k points 00:12:14.060 --> 00:12:16.280 that you need to use. 00:12:16.280 --> 00:12:21.950 If you were to study for a moment unit cell of silicon, 00:12:21.950 --> 00:12:24.320 you might want to study, say, vacancy information 00:12:24.320 --> 00:12:27.260 energy like you were doing in your first computational lab. 00:12:27.260 --> 00:12:30.720 Then you are not going to use two atom units. 00:12:30.720 --> 00:12:34.160 You are going to use a larger unit cell. 00:12:34.160 --> 00:12:36.630 And without sort of dwelling that much on it, 00:12:36.630 --> 00:12:38.330 suppose that, again, in two dimensions, 00:12:38.330 --> 00:12:41.780 you double the size of your unit cell. 00:12:41.780 --> 00:12:45.680 What's really happening is that the reciprocal lattice vectors 00:12:45.680 --> 00:12:47.690 will become one half in length. 00:12:47.690 --> 00:12:49.730 So remember this general concept. 00:12:49.730 --> 00:12:53.180 When you sort of increase the size of the unit cell, 00:12:53.180 --> 00:12:56.210 the reciprocal cell becomes smaller. 00:12:56.210 --> 00:12:58.010 Suppose that we have doubled-- 00:12:58.010 --> 00:13:01.640 we have gone from two atoms in each-- 00:13:01.640 --> 00:13:04.190 two atoms in our unit cell to-- 00:13:04.190 --> 00:13:05.990 well, if we are in two dimensions, 00:13:05.990 --> 00:13:07.430 eight atoms in the unit cell. 00:13:07.430 --> 00:13:10.880 We have doubled in real space-- in reciprocal space. 00:13:10.880 --> 00:13:16.700 Our Brillouin Zone becomes 4 times smaller. 00:13:16.700 --> 00:13:22.040 And so if we want to keep the same quality of integration, 00:13:22.040 --> 00:13:26.030 actually, it means that our k point sampling needs only 00:13:26.030 --> 00:13:29.840 to be one fourth the number of points that we had before. 00:13:29.840 --> 00:13:33.170 That is we still use the blue points 00:13:33.170 --> 00:13:38.450 that are included in this sort of smaller blackish unit cell. 00:13:38.450 --> 00:13:43.460 So that also means that, say, if 64 k points is a good number 00:13:43.460 --> 00:13:47.000 for a regular set of unit cell of two atoms of silicon, 00:13:47.000 --> 00:13:49.310 if now you are going to use a, say, 00:13:49.310 --> 00:13:52.580 unit cell that has maybe 128 atoms-- 00:13:52.580 --> 00:13:55.530 much larger to calculate the vacancy formation energy-- 00:13:55.530 --> 00:13:59.930 you really need only one k point to calculate your total energy 00:13:59.930 --> 00:14:01.200 with the same accuracy. 00:14:01.200 --> 00:14:04.730 So there is this general idea of scaling and folding. 00:14:04.730 --> 00:14:07.730 You make your real space calculation larger. 00:14:07.730 --> 00:14:10.840 You actually need to use fewer k points. 00:14:10.840 --> 00:14:13.340 And if you are familiar with sort of some of the solid state 00:14:13.340 --> 00:14:15.860 ideas, that also just means that when 00:14:15.860 --> 00:14:18.260 you double the size of your unit cell, 00:14:18.260 --> 00:14:21.920 you are really refolding some of this band 00:14:21.920 --> 00:14:24.480 structure in a smaller space. 00:14:24.480 --> 00:14:29.450 So you actually sum over a number of bands that increases. 00:14:29.450 --> 00:14:32.270 The situation is slightly more complex 00:14:32.270 --> 00:14:35.060 for something like a metal. 00:14:35.060 --> 00:14:37.478 The sort of fundamental difference 00:14:37.478 --> 00:14:39.020 when you deal with a metal as opposed 00:14:39.020 --> 00:14:41.300 to a semiconductor or an insulator 00:14:41.300 --> 00:14:44.900 is that there is something called a Fermi energy. 00:14:44.900 --> 00:14:48.680 That is now there isn't any more a gap, 00:14:48.680 --> 00:14:51.050 so the total charge density is again 00:14:51.050 --> 00:14:56.990 given by sum of the states, sum over all the bands, 00:14:56.990 --> 00:15:01.090 and integral over all the k space. 00:15:01.090 --> 00:15:04.810 But really, the total charge density-- 00:15:04.810 --> 00:15:07.300 so the ultimate integral that would 00:15:07.300 --> 00:15:10.180 be the number of electrons depends really 00:15:10.180 --> 00:15:12.130 on where we stop. 00:15:12.130 --> 00:15:14.830 And there isn't any more sort of natural separation 00:15:14.830 --> 00:15:18.080 between empty and occupied states. 00:15:18.080 --> 00:15:22.120 So there is going to be in a metal an energy level 00:15:22.120 --> 00:15:26.290 that determines what is occupied and what is empty, 00:15:26.290 --> 00:15:28.060 and that's the Fermi energy. 00:15:28.060 --> 00:15:33.850 So for copper I think has 11 electrons per unit cell 00:15:33.850 --> 00:15:37.360 sort of FCC metal with one atom per-- 00:15:37.360 --> 00:15:39.460 one atom in each unit cell. 00:15:39.460 --> 00:15:43.420 What we really need to do in our electrons calculation 00:15:43.420 --> 00:15:46.960 is find what is this energy level. 00:15:46.960 --> 00:15:50.380 That is find what is this Fermi energy, such 00:15:50.380 --> 00:15:55.150 that the integral over all the state below that level 00:15:55.150 --> 00:15:57.910 gives us the right charge density-- in particular, 00:15:57.910 --> 00:16:00.520 the right number of electrons. 00:16:00.520 --> 00:16:03.010 And this is sort of one of the difficulties that come out 00:16:03.010 --> 00:16:05.890 in metals, because now what we are trying to do 00:16:05.890 --> 00:16:11.740 is we are trying to integrate the bands below a level-- 00:16:11.740 --> 00:16:13.150 below the black line. 00:16:13.150 --> 00:16:17.270 And so you really introduce a discontinuity in your integral. 00:16:17.270 --> 00:16:19.720 And so to calculate integrals of discontinuous 00:16:19.720 --> 00:16:24.010 function usually requires a much finer accuracy 00:16:24.010 --> 00:16:25.790 in your k point sampling. 00:16:25.790 --> 00:16:30.400 And so the calculation tends to become much more expensive. 00:16:30.400 --> 00:16:32.080 Nothing else. 00:16:32.080 --> 00:16:38.110 So the general solution to this problem besides using a larger 00:16:38.110 --> 00:16:41.800 number of k points, and particular accuracy 00:16:41.800 --> 00:16:45.670 to this sampling issue is that of introducing what is called 00:16:45.670 --> 00:16:48.230 a finite electron temperature. 00:16:48.230 --> 00:16:52.480 So what is actually done in sort of every practical calculation 00:16:52.480 --> 00:16:56.270 is introduce a small amount of temperature 00:16:56.270 --> 00:16:59.840 so that in reality, there's a sharp discontinuity become 00:16:59.840 --> 00:17:00.430 smoother. 00:17:00.430 --> 00:17:02.410 Because when you have an electron temperature, 00:17:02.410 --> 00:17:06.369 states above the Fermi energy can be slightly occupied 00:17:06.369 --> 00:17:09.040 with the Fermi [INAUDIBLE] occupation, and states 00:17:09.040 --> 00:17:13.240 just below the Fermi energy can be slightly empty. 00:17:13.240 --> 00:17:15.339 So to summarize, I mean, we need to be 00:17:15.339 --> 00:17:18.910 careful in sort of sampling. 00:17:18.910 --> 00:17:21.730 And that's the other sort of fundamental parameter 00:17:21.730 --> 00:17:23.109 of your calculation. 00:17:23.109 --> 00:17:24.730 If you are dealing with a metal, you 00:17:24.730 --> 00:17:27.280 need to be particularly careful, and you 00:17:27.280 --> 00:17:30.880 need to use electron temperature techniques. 00:17:30.880 --> 00:17:34.900 If you are using sort of-- if you are studying an insulator 00:17:34.900 --> 00:17:37.390 a semiconductor, what you usually do 00:17:37.390 --> 00:17:41.890 is just use a regular equi-spaced mesh of k points. 00:17:41.890 --> 00:17:44.290 And that sort of known in the community 00:17:44.290 --> 00:17:45.760 with a technical name. 00:17:45.760 --> 00:17:49.370 They are called [INAUDIBLE] for some very good reason. 00:17:49.370 --> 00:17:51.160 Actually, there are some actually 00:17:51.160 --> 00:17:53.230 fairly beautiful symmetry thoughts 00:17:53.230 --> 00:17:58.690 on why equi-spaced coarse mesh can work very well. 00:17:58.690 --> 00:18:00.700 But in practice, it's nothing else 00:18:00.700 --> 00:18:04.690 than choosing the blue points inside of the green Brillouin 00:18:04.690 --> 00:18:05.800 zone. 00:18:05.800 --> 00:18:09.520 If you are studying a really large system, 00:18:09.520 --> 00:18:13.240 you can actually reduce yourself to sampling only one 00:18:13.240 --> 00:18:14.410 point in the Brillouin Zone. 00:18:14.410 --> 00:18:16.570 Brillouin Zone has become so small 00:18:16.570 --> 00:18:20.986 that just taking sort of-- 00:18:20.986 --> 00:18:22.090 how do you call it? 00:18:22.090 --> 00:18:25.840 A mean-- the theorem of the mean. 00:18:25.840 --> 00:18:28.030 That is, you can substitute the integral 00:18:28.030 --> 00:18:31.150 with the value of the function at one point. 00:18:31.150 --> 00:18:34.160 And usually, you have two choices. 00:18:34.160 --> 00:18:36.010 You can just use the gamma point. 00:18:36.010 --> 00:18:38.110 This means that 0, 0, 0-- 00:18:38.110 --> 00:18:39.880 that has a computational advantage. 00:18:39.880 --> 00:18:42.070 When you sample things at the gamma point, 00:18:42.070 --> 00:18:45.820 you can choose your functions to be real instead of complex. 00:18:45.820 --> 00:18:49.360 And so you have right away your computational cost. 00:18:49.360 --> 00:18:53.610 Or you can choose sort of what could be the best single point 00:18:53.610 --> 00:18:55.840 for your given symmetry that's sometimes 00:18:55.840 --> 00:18:57.890 called the Baldereschi point. 00:18:57.890 --> 00:19:00.070 And that can be again sort of useful 00:19:00.070 --> 00:19:03.880 if you need to do an accurate calculation in a large scale 00:19:03.880 --> 00:19:06.040 system. 00:19:06.040 --> 00:19:10.960 Once you have set these two fundamental parameters, 00:19:10.960 --> 00:19:13.120 you are really ready to do actually 00:19:13.120 --> 00:19:14.980 a practical calculation. 00:19:14.980 --> 00:19:18.730 And so as I said, we have a self-consistent Hamiltonian, 00:19:18.730 --> 00:19:20.560 and so what you need to do is you 00:19:20.560 --> 00:19:24.940 need to iterate your problem until the eigenstates 00:19:24.940 --> 00:19:27.670 that you find give you a charge density that 00:19:27.670 --> 00:19:29.770 is identical to the charge density 00:19:29.770 --> 00:19:31.610 that you have done before. 00:19:31.610 --> 00:19:35.240 So in practice, how would your electronic structure code work? 00:19:35.240 --> 00:19:40.690 Well, first you would tell your code where the atoms are. 00:19:40.690 --> 00:19:43.710 So you need to specify the position of the atom. 00:19:43.710 --> 00:19:47.200 Suppose that you are studying silicon, you could sort of-- 00:19:47.200 --> 00:19:49.840 since you know actually what is the structure of silicon, 00:19:49.840 --> 00:19:52.450 you could already put them in the origin, 00:19:52.450 --> 00:19:56.140 and in the position one fourth, one fourth, one fourth. 00:19:56.140 --> 00:19:58.760 And then you need to specify in particular 00:19:58.760 --> 00:20:01.330 which flavor of non-local pseudopotential 00:20:01.330 --> 00:20:02.800 you are going to use. 00:20:02.800 --> 00:20:05.440 That is, there will be a library of pseudopotential 00:20:05.440 --> 00:20:09.760 that basically represents a silicon atom with all the core 00:20:09.760 --> 00:20:11.070 electrons frozen. 00:20:11.070 --> 00:20:13.570 And there are sort of a number of technicalities that you'll 00:20:13.570 --> 00:20:16.180 see in the lab on which one you should choose, 00:20:16.180 --> 00:20:19.420 but they are sort of more or less all the same, at least 00:20:19.420 --> 00:20:21.080 from this point of view. 00:20:21.080 --> 00:20:23.720 Once the code knows where the atoms are-- 00:20:23.720 --> 00:20:27.700 that is, knows the position of the atoms inside the unit cell, 00:20:27.700 --> 00:20:29.950 knows what is the shape of the unit cell, 00:20:29.950 --> 00:20:31.810 and what is the length of the dotted lattice 00:20:31.810 --> 00:20:35.590 vectors, this infinite array of atoms, 00:20:35.590 --> 00:20:38.110 the infinite crystalline, or amorphous, 00:20:38.110 --> 00:20:41.320 or disordered extended system is set. 00:20:41.320 --> 00:20:44.380 And what we really need to do is throw the electron [? seeds ?] 00:20:44.380 --> 00:20:48.110 and let the electron find their own ground state. 00:20:48.110 --> 00:20:51.830 And so we need to make sure that we have the right basis set 00:20:51.830 --> 00:20:52.330 cutoff. 00:20:52.330 --> 00:20:53.705 That is, we are going to describe 00:20:53.705 --> 00:20:56.710 the orbitals [? anchored. ?] We have the right sampling. 00:20:56.710 --> 00:20:59.170 And at this point, we can sort of 00:20:59.170 --> 00:21:02.650 start the self-consistent procedure. 00:21:02.650 --> 00:21:05.110 And in the sort of simplest form, 00:21:05.110 --> 00:21:07.780 well, we first need to figure out what 00:21:07.780 --> 00:21:09.880 is our Hamiltonian operator. 00:21:09.880 --> 00:21:12.970 The Hamiltonian, remember, depends on the charge density 00:21:12.970 --> 00:21:15.730 itself, because some of the terms in the Hamiltonian, 00:21:15.730 --> 00:21:18.310 like the Hartree energy, the Hartree potential 00:21:18.310 --> 00:21:20.140 of the exchange correlation potential 00:21:20.140 --> 00:21:21.620 depends on the density. 00:21:21.620 --> 00:21:25.000 So we need to pick an initial guess for a trial charge 00:21:25.000 --> 00:21:26.080 density. 00:21:26.080 --> 00:21:31.360 It could just be a superposition of atomic charge density. 00:21:31.360 --> 00:21:33.400 Once we have the charge density, we 00:21:33.400 --> 00:21:37.870 can construct the Hamiltonian, the [INAUDIBLE] Hamiltonian. 00:21:37.870 --> 00:21:39.640 The kinetic energy operator is-- 00:21:39.640 --> 00:21:40.540 we always know it. 00:21:40.540 --> 00:21:44.140 But we can construct this Hartree and exchange 00:21:44.140 --> 00:21:47.050 correlation terms that depend on the charge density, 00:21:47.050 --> 00:21:49.870 and then we'll have the external potential 00:21:49.870 --> 00:21:54.640 that is given by this array of non-local pseudopotential. 00:21:54.640 --> 00:21:57.580 At this point, we have the Hamiltonian, 00:21:57.580 --> 00:22:00.800 and we try to find the lowest energy 00:22:00.800 --> 00:22:02.950 eigenstates for Hamiltonian. 00:22:02.950 --> 00:22:07.900 In particular, we just need to calculate a number of states 00:22:07.900 --> 00:22:11.750 that is equal to the number of occupied orbitals 00:22:11.750 --> 00:22:14.080 if we are dealing with a semiconductor, 00:22:14.080 --> 00:22:18.820 or it's equal to the number of sort of electrons 00:22:18.820 --> 00:22:23.080 plus 20%, 30% in a metal to make sure 00:22:23.080 --> 00:22:25.810 that at different points in the Brillouin Zone, 00:22:25.810 --> 00:22:28.960 we calculate all the bends that could 00:22:28.960 --> 00:22:31.930 be below our Fermi energy. 00:22:31.930 --> 00:22:33.350 So we solve this. 00:22:33.350 --> 00:22:36.250 And this is really the expensive step-- 00:22:36.250 --> 00:22:39.940 and I mean very expensive step in any electronic structure 00:22:39.940 --> 00:22:40.900 calculation. 00:22:40.900 --> 00:22:44.470 And there are sort of a number of ways 00:22:44.470 --> 00:22:48.340 of diagonalizing a matrix, of solving 00:22:48.340 --> 00:22:50.200 this eigenstate equation. 00:22:50.200 --> 00:22:53.380 In a basic set, it tends to be very large. 00:22:53.380 --> 00:22:58.390 When you do sort of a realistic calculation, even for silicon, 00:22:58.390 --> 00:23:02.350 you could have hundreds of plane waves, So. 00:23:02.350 --> 00:23:04.750 Hundreds of basic set elements. 00:23:04.750 --> 00:23:08.080 And large scale calculation like you would doing in research 00:23:08.080 --> 00:23:12.400 would contain tens of thousands of plane waves. 00:23:12.400 --> 00:23:15.970 And actually, you don't-- 00:23:15.970 --> 00:23:19.880 you can't really diagonalize on a regular computer 00:23:19.880 --> 00:23:22.180 even a matrix that has [INAUDIBLE].. 00:23:22.180 --> 00:23:26.290 If you think a matrix that has a dimension of 1,000 00:23:26.290 --> 00:23:30.490 requires one million elements. 00:23:30.490 --> 00:23:36.460 And one number-- a complex number requires 16 bytes. 00:23:36.460 --> 00:23:40.780 So just a matrix that has 1,000 sides that 00:23:40.780 --> 00:23:45.790 will require 16 megabytes to be described, and this number 00:23:45.790 --> 00:23:49.070 explodes quadratically very quickly. 00:23:49.070 --> 00:23:51.550 So you can construct the full Hamiltonian, 00:23:51.550 --> 00:23:53.440 and you don't want to calculate-- 00:23:53.440 --> 00:23:56.620 if you have a matrix of dimension 1,000, 00:23:56.620 --> 00:23:59.020 it will have 1,000 eigenstates. 00:23:59.020 --> 00:24:02.290 But you only care, say, if you are studying silicon, 00:24:02.290 --> 00:24:04.930 on the lowest four eigenstates. 00:24:04.930 --> 00:24:07.450 So you want to have numerical techniques that 00:24:07.450 --> 00:24:10.350 calculate for you only the lowest energy eigenstates. 00:24:10.350 --> 00:24:13.840 And there are a number of them that are well-established. 00:24:13.840 --> 00:24:17.230 So now you have obtained with one of these techniques 00:24:17.230 --> 00:24:21.940 your lowest energy eigenstates, and you sum their square moduli 00:24:21.940 --> 00:24:24.160 to obtain the new charge density. 00:24:24.160 --> 00:24:26.650 And with this, you go back to the step. 00:24:26.650 --> 00:24:29.170 You construct the Hamiltonian operator again, 00:24:29.170 --> 00:24:34.420 you diagonalize new density, and iterate. 00:24:34.420 --> 00:24:37.570 Of course, naturally, a recipe like this 00:24:37.570 --> 00:24:41.140 would most likely never converge. 00:24:41.140 --> 00:24:47.380 So one needs to develop a mixing approach-- 00:24:47.380 --> 00:24:51.040 mixing approaches that sort make the change in the charge 00:24:51.040 --> 00:24:55.000 density at every iterative step smoother 00:24:55.000 --> 00:24:56.720 than what I've described. 00:24:56.720 --> 00:24:58.510 So if you calculate a new charge density 00:24:58.510 --> 00:25:02.560 and you diagonalize it again, your second new charge density 00:25:02.560 --> 00:25:04.570 will be even more different from anything 00:25:04.570 --> 00:25:06.470 that you have obtained before. 00:25:06.470 --> 00:25:08.770 So what you really do is you need 00:25:08.770 --> 00:25:13.660 to find some schemes to evolve in a very smooth way 00:25:13.660 --> 00:25:14.930 your charge density. 00:25:14.930 --> 00:25:17.530 So maybe once you have calculated 00:25:17.530 --> 00:25:20.260 the new sum of eigenstates, you don't 00:25:20.260 --> 00:25:22.810 take that as huge as density, but you just 00:25:22.810 --> 00:25:27.310 update your old charge density with 10% 00:25:27.310 --> 00:25:31.210 of what you have calculated now to try to make the iteration 00:25:31.210 --> 00:25:33.730 to self-consistent very smooth. 00:25:33.730 --> 00:25:36.610 And I have to say a lot of the know-how 00:25:36.610 --> 00:25:38.350 in electronic structure calculation 00:25:38.350 --> 00:25:41.590 in the '90s has really gone into trying 00:25:41.590 --> 00:25:46.090 to find sort of mixing approaches that evolve 00:25:46.090 --> 00:25:48.400 our charge density to solve consistency, 00:25:48.400 --> 00:25:52.870 and that converge under a large variety of circumstances 00:25:52.870 --> 00:25:58.030 for large or complex systems, especially for metals. 00:25:58.030 --> 00:26:02.080 So stripped to the bare elements, 00:26:02.080 --> 00:26:05.830 an electronic structure code really needs to do two things. 00:26:05.830 --> 00:26:10.270 It needs to diagonalize inexpensively 00:26:10.270 --> 00:26:14.260 a Hamiltonian that expressed on a plane wave basis 00:26:14.260 --> 00:26:16.270 is a very large order. 00:26:16.270 --> 00:26:18.472 And I just mentioned the names of some 00:26:18.472 --> 00:26:20.680 of the [? algorithm. ?] Things like the [INAUDIBLE],, 00:26:20.680 --> 00:26:23.440 and the [INAUDIBLE],, or some of the conjugate gradient 00:26:23.440 --> 00:26:25.810 algorithms are all algorithms that 00:26:25.810 --> 00:26:30.850 give us reasonable cost the lowest energy eigenstates 00:26:30.850 --> 00:26:32.380 of that Hamiltonian. 00:26:32.380 --> 00:26:34.930 And then once you have that eigenstates, 00:26:34.930 --> 00:26:37.000 you need to calculate charge density, 00:26:37.000 --> 00:26:39.190 and you need to have a mixing strategy. 00:26:39.190 --> 00:26:43.570 You need to have a strategy to evolve your charge density 00:26:43.570 --> 00:26:45.950 towards self-consistency. 00:26:45.950 --> 00:26:50.350 And that is also a very tricky approach. 00:26:50.350 --> 00:26:53.640 There is a sort of completely different approach 00:26:53.640 --> 00:27:01.270 to the problem that sees the ground state solution not 00:27:01.270 --> 00:27:04.950 as a self-consistent iteration, but 00:27:04.950 --> 00:27:09.580 as a nonlinear direct minimization of the functional. 00:27:09.580 --> 00:27:12.590 If you remember, we have the energy function-- 00:27:12.590 --> 00:27:15.220 or I think it's written here in the next slide. 00:27:15.220 --> 00:27:16.150 No, it's not. 00:27:16.150 --> 00:27:19.600 We have sort of written the density functional of theory 00:27:19.600 --> 00:27:24.280 energy functional, and it is a well-defined expression 00:27:24.280 --> 00:27:25.810 of the orbitals only. 00:27:25.810 --> 00:27:27.070 It will be in one of the-- 00:27:27.070 --> 00:27:30.250 so the following slides will pick it up again. 00:27:30.250 --> 00:27:33.250 And so we can also see the problem 00:27:33.250 --> 00:27:40.000 as the problem of minimization of that functional in a space 00:27:40.000 --> 00:27:42.970 that is very large, because the sort of variables that we 00:27:42.970 --> 00:27:47.020 really deal with are the coefficients of our plane wave 00:27:47.020 --> 00:27:48.100 expansion. 00:27:48.100 --> 00:27:50.800 But in principle-- and I'll show you that in a moment-- 00:27:50.800 --> 00:27:54.610 we can actually write out a minimization algorithm. 00:27:54.610 --> 00:27:58.420 The advantage of this approach, if it's done properly, 00:27:58.420 --> 00:28:01.930 it has always a solution. 00:28:01.930 --> 00:28:06.730 If you keep minimizing your energy, at the end, 00:28:06.730 --> 00:28:10.580 you will get to a global or to a local minimum. 00:28:10.580 --> 00:28:14.260 So this approach tends to sort of converge 00:28:14.260 --> 00:28:17.740 under every circumstances if done properly, and done 00:28:17.740 --> 00:28:20.410 properly is not trivial. 00:28:20.410 --> 00:28:23.920 But then sort of the efficiency of the different things 00:28:23.920 --> 00:28:27.590 is really system dependent. 00:28:27.590 --> 00:28:31.175 And I guess without wanting to bore you with sort of math-- 00:28:31.175 --> 00:28:34.130 so just I wanted to remind you again 00:28:34.130 --> 00:28:36.020 what happens in our computer. 00:28:36.020 --> 00:28:39.830 That is what happens when we say we want to solve 00:28:39.830 --> 00:28:42.500 this eigenstate equation. 00:28:42.500 --> 00:28:47.570 Supposing let's say we are in a self-consistent diagonalization 00:28:47.570 --> 00:28:48.410 approach. 00:28:48.410 --> 00:28:52.700 And as always, you have to remember we expand our wave 00:28:52.700 --> 00:28:58.310 function in a well-defined set of orbitals-- 00:28:58.310 --> 00:28:59.420 that is our basis set. 00:28:59.420 --> 00:29:02.180 I represented here them as phi. 00:29:02.180 --> 00:29:05.070 And it could be plane waves, it could be atomic orbitals. 00:29:05.070 --> 00:29:06.950 We use plane waves all the time. 00:29:06.950 --> 00:29:10.340 So really, in our computer, our unknowns 00:29:10.340 --> 00:29:14.720 are these coefficients of this basis set expansion. 00:29:14.720 --> 00:29:20.150 And so our eigenstate equation, once we multiply on the left 00:29:20.150 --> 00:29:26.550 by phi and star, and integrate, is really a matrix problem. 00:29:26.550 --> 00:29:30.530 So I've written it here as just the same eigenfunction 00:29:30.530 --> 00:29:32.520 equation written over there. 00:29:32.520 --> 00:29:37.190 And if we call hmn at the matrix element of the Hamiltonian 00:29:37.190 --> 00:29:40.920 between, say, two plane waves of different wavelengths, 00:29:40.920 --> 00:29:42.380 this is what our problem means. 00:29:42.380 --> 00:29:44.510 It's just a linear algebra problem. 00:29:44.510 --> 00:29:47.990 We need to find the eigenvalues for which there 00:29:47.990 --> 00:29:50.810 is a possible solution, and the possible solution 00:29:50.810 --> 00:29:51.980 will be eigenstates. 00:29:51.980 --> 00:29:54.050 And an eigenstate is nothing else 00:29:54.050 --> 00:29:56.930 than an appropriate set of coefficients 00:29:56.930 --> 00:30:01.190 that satisfy this equation, and those coefficients put back 00:30:01.190 --> 00:30:03.830 in here will give us actually what 00:30:03.830 --> 00:30:06.935 are the full eigenstates of our problem. 00:30:13.780 --> 00:30:16.190 So this was the sort of self-consistent 00:30:16.190 --> 00:30:17.140 diagonalization. 00:30:17.140 --> 00:30:20.110 As I just said a moment ago, we can also 00:30:20.110 --> 00:30:24.220 look at the problem as a nonlinear minimization problem. 00:30:24.220 --> 00:30:28.990 Once we have decided on an approximation for our exchange 00:30:28.990 --> 00:30:32.500 correlation functional, could be a local density approximation, 00:30:32.500 --> 00:30:35.240 could be a generalized gradient approximation. 00:30:35.240 --> 00:30:37.450 This is a well-defined quantity in which, 00:30:37.450 --> 00:30:40.090 again, the external potential is given 00:30:40.090 --> 00:30:43.120 by this array of non-local pseudopotential. 00:30:43.120 --> 00:30:46.510 And the Hartree energy is just the functional of the charge 00:30:46.510 --> 00:30:47.320 density. 00:30:47.320 --> 00:30:49.630 And the charge density itself is thus 00:30:49.630 --> 00:30:52.600 the sum of the square modulus of the orbitals. 00:30:52.600 --> 00:30:57.280 So in reality, this energy is a function of that psi, 00:30:57.280 --> 00:31:02.380 or in other terms, is nothing else than a very complex 00:31:02.380 --> 00:31:07.540 function of those c1 to cn coefficients 00:31:07.540 --> 00:31:09.320 of each eigenvector. 00:31:09.320 --> 00:31:13.780 So this is nothing else than a minimization problem, again, 00:31:13.780 --> 00:31:18.160 on a number of variables that can be 1,000 if you are 00:31:18.160 --> 00:31:20.110 studying two atoms of silicon. 00:31:20.110 --> 00:31:23.060 And it can be in the tens of hundreds of thousands 00:31:23.060 --> 00:31:25.640 if you start to do really serious calculations. 00:31:25.640 --> 00:31:28.300 So again, it's a fairly complex problem. 00:31:28.300 --> 00:31:32.080 A huge number of variables that you need to deal with, 00:31:32.080 --> 00:31:37.450 and a nonlinear expression for the energy. 00:31:37.450 --> 00:31:41.950 But again, in principle, if we have this explicit expression 00:31:41.950 --> 00:31:48.160 for the energy, E of psi, where the psi that we consider 00:31:48.160 --> 00:31:51.850 around only the occupied orbitals, what we can do 00:31:51.850 --> 00:31:56.350 is nothing else than take the functional derivative 00:31:56.350 --> 00:31:57.395 with respect to that psi. 00:31:57.395 --> 00:31:59.020 I'll consider them real here just 00:31:59.020 --> 00:32:02.200 to avoid sort of complex conjugate numbers. 00:32:02.200 --> 00:32:05.350 And at the end, this is nothing else 00:32:05.350 --> 00:32:10.990 than calculates the derivative of the energy with respect 00:32:10.990 --> 00:32:16.720 to all the coefficients, say, for i that goes 1 up 00:32:16.720 --> 00:32:21.740 to the cutoff, all the coefficients of all 00:32:21.740 --> 00:32:23.110 the occupied orbitals. 00:32:26.920 --> 00:32:29.890 So you see, the larger your system becomes, 00:32:29.890 --> 00:32:34.630 the more basis elements you'll need to use. 00:32:34.630 --> 00:32:38.350 I mean, if you double the size of your system, 00:32:38.350 --> 00:32:40.180 if you look at the math, you'll actually 00:32:40.180 --> 00:32:42.460 need the twice the number of plane waves, 00:32:42.460 --> 00:32:45.580 as it makes sense, to describe a charge 00:32:45.580 --> 00:32:48.350 density or a wave function with the same resolution. 00:32:48.350 --> 00:32:50.530 So you double the size of your system. 00:32:50.530 --> 00:32:55.000 The number of plane waves to describe a single particle 00:32:55.000 --> 00:32:58.630 orbital becomes twice as large, but now you 00:32:58.630 --> 00:33:02.440 will also have twice as many occupied orbitals. 00:33:02.440 --> 00:33:04.420 So just the number of this coefficient 00:33:04.420 --> 00:33:07.100 has become four times as large. 00:33:07.100 --> 00:33:09.970 So again, this calculation becomes very expensive 00:33:09.970 --> 00:33:10.910 very quickly. 00:33:10.910 --> 00:33:12.340 But again, this is well-defined. 00:33:12.340 --> 00:33:16.180 I mean, we can just actually write explicitly 00:33:16.180 --> 00:33:18.700 the nonlinear function of the previous slide 00:33:18.700 --> 00:33:22.190 in terms of the coefficients of the plane waves. 00:33:22.190 --> 00:33:26.110 This is actually done in one of the article posted. 00:33:26.110 --> 00:33:28.870 There is a review of modern physics by Mike Payne 00:33:28.870 --> 00:33:33.580 and coworkers-- among others, John [? Gianopulos ?] at MIT. 00:33:33.580 --> 00:33:35.710 And they actually work out the algebra 00:33:35.710 --> 00:33:37.360 of all these derivatives. 00:33:37.360 --> 00:33:39.430 And then once you have the derivatives, 00:33:39.430 --> 00:33:41.350 you have the gradients, and you know 00:33:41.350 --> 00:33:45.340 how to move along and go to the minimum following 00:33:45.340 --> 00:33:46.510 the gradients. 00:33:46.510 --> 00:33:50.530 The only difference with a sort of regular minimization problem 00:33:50.530 --> 00:33:53.560 is that this is a constraint problem. 00:33:53.560 --> 00:33:57.970 That is what we have because these are really electrons, 00:33:57.970 --> 00:34:00.970 and not just sort of arbitrary functions. 00:34:00.970 --> 00:34:06.080 The electrons need to be meaningful quantum states, 00:34:06.080 --> 00:34:08.929 so they need to be orthonormal. 00:34:08.929 --> 00:34:12.880 So this derivatives with respect to the c 00:34:12.880 --> 00:34:17.770 need actually to take place on the hyper surface 00:34:17.770 --> 00:34:21.100 where these conditions are satisfied. 00:34:21.100 --> 00:34:24.310 That is, if you were to evolve the coefficients of the plane 00:34:24.310 --> 00:34:26.889 waves, what you would find is that as soon 00:34:26.889 --> 00:34:29.530 as you have sort of changed them by a little amount, 00:34:29.530 --> 00:34:32.260 your orbitals per se are not going 00:34:32.260 --> 00:34:34.520 to be orthonormal anymore. 00:34:34.520 --> 00:34:38.560 So this constraint is sort of fundamental, 00:34:38.560 --> 00:34:43.389 and this is what ultimately limits sort of-- 00:34:43.389 --> 00:34:47.710 or determines the computational costs of our calculation. 00:34:47.710 --> 00:34:51.580 Because again, if we double the size of the system, 00:34:51.580 --> 00:34:54.909 we'll have twice as many plane waves, 00:34:54.909 --> 00:34:58.610 and we'll have twice as many occupied orbitals. 00:34:58.610 --> 00:35:00.910 So we have already a cost of fours. 00:35:00.910 --> 00:35:03.370 But those occupied orbitals will need 00:35:03.370 --> 00:35:06.410 to be orthogonal to each other. 00:35:06.410 --> 00:35:09.100 And so the number of these matrix elements 00:35:09.100 --> 00:35:12.340 that you need to calculate has become also twice-- 00:35:12.340 --> 00:35:15.000 or the number of orbitals in this matrix 00:35:15.000 --> 00:35:17.260 have become twice, so the number of matrix elements 00:35:17.260 --> 00:35:19.300 has become four times as large. 00:35:19.300 --> 00:35:21.790 You see, we double the size of the system, 00:35:21.790 --> 00:35:23.920 we'll have twice as many orbitals here, 00:35:23.920 --> 00:35:25.780 twice as many orbitals here. 00:35:25.780 --> 00:35:28.840 And this integral is going to take place 00:35:28.840 --> 00:35:32.030 on a region in space that's twice as large. 00:35:32.030 --> 00:35:36.530 So 2 by 2, by 2 gives us a factor of 8, 00:35:36.530 --> 00:35:40.090 and so gives us the, ultimately, cubic scaling 00:35:40.090 --> 00:35:43.180 of cost of density functional calculation. 00:35:43.180 --> 00:35:47.260 You go from two atoms of silicon to four atoms of silicon, 00:35:47.260 --> 00:35:51.700 your calculation has become eight times more expensive. 00:35:51.700 --> 00:35:55.930 Hartree-Fock in its original formulation scales 00:35:55.930 --> 00:35:57.400 as the fourth power. 00:35:57.400 --> 00:36:00.880 Other quantum chemistry approach scales as the fifth, sixth, 00:36:00.880 --> 00:36:02.330 or seventh power. 00:36:02.330 --> 00:36:06.550 So it becomes very easy to sort of reach, really, 00:36:06.550 --> 00:36:08.500 the limit of calculation that you 00:36:08.500 --> 00:36:11.140 can do on a regular computer or even 00:36:11.140 --> 00:36:12.850 on a regular supercomputer. 00:36:12.850 --> 00:36:16.330 And there is a lot of effort to develop what are called 00:36:16.330 --> 00:36:17.980 linear scaling approaches. 00:36:17.980 --> 00:36:21.190 That these electronic structure algorithm that scale linearly 00:36:21.190 --> 00:36:22.960 as the size of the system. 00:36:22.960 --> 00:36:25.360 And somehow, they are all based on the idea 00:36:25.360 --> 00:36:28.840 that sort of physics or quantum mechanics is local. 00:36:28.840 --> 00:36:32.320 So if your orbital at the end is ultimately 00:36:32.320 --> 00:36:35.510 localized in a certain region of space, 00:36:35.510 --> 00:36:38.800 it will be automatically orthogonal to orbitals 00:36:38.800 --> 00:36:40.240 that are very far away. 00:36:40.240 --> 00:36:42.580 Because if this is localized somewhere here, 00:36:42.580 --> 00:36:45.310 and this psi i is localized somewhere there, 00:36:45.310 --> 00:36:48.410 their overlap will be 0 by definition, 00:36:48.410 --> 00:36:51.550 and so we don't need to worry about orthogonality. 00:36:51.550 --> 00:36:54.250 So somehow, locality of physics, locality 00:36:54.250 --> 00:36:56.320 of quantum mechanics in principle 00:36:56.320 --> 00:37:01.870 tells us that there are linear scaling approaches that 00:37:01.870 --> 00:37:04.300 could work, although, none of them 00:37:04.300 --> 00:37:08.140 have really made into sort of production electronic structure 00:37:08.140 --> 00:37:12.010 at this stage, although there is a lot of ongoing effort 00:37:12.010 --> 00:37:14.130 in many groups. 00:37:17.360 --> 00:37:20.720 So with this [INAUDIBLE] we conclude also 00:37:20.720 --> 00:37:22.760 all the technicalities. 00:37:22.760 --> 00:37:24.860 And what we'll do in the rest of the class 00:37:24.860 --> 00:37:27.540 will give you a sort of panorama of what 00:37:27.540 --> 00:37:31.610 typical applications of density function of theory calculation 00:37:31.610 --> 00:37:33.920 are going to do. 00:37:33.920 --> 00:37:42.230 And I'll go very quickly over this. 00:37:42.230 --> 00:37:45.020 I have here sort of cases in which 00:37:45.020 --> 00:37:49.580 we could be interested in structural excitations. 00:37:49.580 --> 00:37:55.370 So when you start warming up a system, a molecule, or a solid, 00:37:55.370 --> 00:37:59.990 you start exciting the different normal modes of your molecule 00:37:59.990 --> 00:38:00.980 or of a solid. 00:38:00.980 --> 00:38:04.190 I've shown you some of the possibilities for something 00:38:04.190 --> 00:38:07.310 like a carbon nanotube, because that band-- 00:38:07.310 --> 00:38:09.110 we have a banding mode up there. 00:38:09.110 --> 00:38:11.830 We have a pinching mode, and we have a breathing mode. 00:38:11.830 --> 00:38:14.030 And so these are all the possible structural 00:38:14.030 --> 00:38:16.010 excitations, and you can actually 00:38:16.010 --> 00:38:20.170 calculate this structural excitations using 00:38:20.170 --> 00:38:21.660 density functional theory. 00:38:21.660 --> 00:38:28.160 And I've given here a comparison between the case of diamond-- 00:38:28.160 --> 00:38:31.790 what is calculated with the sort of state of the art 00:38:31.790 --> 00:38:33.590 [INAUDIBLE] code like you are going 00:38:33.590 --> 00:38:36.800 to see in your laboratory, and what 00:38:36.800 --> 00:38:39.230 is measured actually with the neutron 00:38:39.230 --> 00:38:40.970 scattering, the red dots. 00:38:40.970 --> 00:38:45.140 So you see without really any input parameters. 00:38:45.140 --> 00:38:48.110 And once you have really phonon dispersion, 00:38:48.110 --> 00:38:51.620 you can calculate all the thermodynamics of solids. 00:38:51.620 --> 00:38:55.550 That is really basically based on the statistics 00:38:55.550 --> 00:38:59.250 of excitation of this vibrational degrees of freedom. 00:38:59.250 --> 00:39:02.480 So you could calculate, say, how your elastic constant 00:39:02.480 --> 00:39:05.790 for your bulk models changes with temperature, 00:39:05.790 --> 00:39:08.210 and this is the calculated black line, 00:39:08.210 --> 00:39:11.240 and you could compare it with experiments. 00:39:11.240 --> 00:39:13.370 Or you could take one of your slabs 00:39:13.370 --> 00:39:16.910 like you have seen in the first laboratory in which you were 00:39:16.910 --> 00:39:20.510 calculating the surface energy, and you could actually 00:39:20.510 --> 00:39:21.650 put it in motion. 00:39:21.650 --> 00:39:24.410 You could follow at a given time temperature 00:39:24.410 --> 00:39:26.540 the dynamics of the atoms. 00:39:26.540 --> 00:39:28.970 And you want to have a slab thick enough 00:39:28.970 --> 00:39:33.110 so the atoms in the middle really act as bulk atoms. 00:39:33.110 --> 00:39:35.180 They don't see the presence of the surface, 00:39:35.180 --> 00:39:38.600 and then it becomes very easy to investigate what's 00:39:38.600 --> 00:39:40.710 happening on the surface. 00:39:40.710 --> 00:39:44.990 And so you can have sort of a snapshot of sort 00:39:44.990 --> 00:39:49.820 of how the atoms are moving on the outer layers, what 00:39:49.820 --> 00:39:51.890 are the sort of typical displacements 00:39:51.890 --> 00:39:53.840 of typical mean square displacements. 00:39:53.840 --> 00:39:58.730 Or you can say study how the distance between the layers 00:39:58.730 --> 00:40:00.360 evolved with temperature. 00:40:00.360 --> 00:40:02.900 So you can look at, say, how, as you increase 00:40:02.900 --> 00:40:05.240 the temperature of your slab, the distance 00:40:05.240 --> 00:40:07.370 between the surface layer and the second layer 00:40:07.370 --> 00:40:08.960 changes with temperature. 00:40:08.960 --> 00:40:11.090 And this would be the computation, 00:40:11.090 --> 00:40:14.090 and here we have the experimental value. 00:40:14.090 --> 00:40:17.660 And there are sort of lots of interesting physics 00:40:17.660 --> 00:40:20.840 that takes place, again, if you look 00:40:20.840 --> 00:40:23.540 at the distance between the second and third layer, 00:40:23.540 --> 00:40:26.630 if the red line above and the system expands there. 00:40:26.630 --> 00:40:28.970 And sort of with computation, you can actually 00:40:28.970 --> 00:40:33.920 probe your system deeper and deeper, where experiment starts 00:40:33.920 --> 00:40:36.530 to become very difficult. It's almost impossible 00:40:36.530 --> 00:40:39.590 to look at what the fourth layer in a surface does, 00:40:39.590 --> 00:40:43.190 and what the fifth layer of a surface does. 00:40:43.190 --> 00:40:45.530 I think in order to keep the balance of the lecture, 00:40:45.530 --> 00:40:48.720 I'll actually switch on to Professor [? Sidor's ?] part 00:40:48.720 --> 00:40:51.720 so he can show you some of the other application. 00:40:51.720 --> 00:40:54.710 And if we have time either in one of the next classes, 00:40:54.710 --> 00:40:59.480 I'll show you some of the other applications 00:40:59.480 --> 00:41:00.630 that we have mentioned. 00:41:00.630 --> 00:41:04.370 So I think I'll pass over the lecture to Professor 00:41:04.370 --> 00:41:07.610 [? Sidor. ?] 00:41:07.610 --> 00:41:11.180 PROFESSOR 2: So Professor [? Mazari ?] already gave you 00:41:11.180 --> 00:41:12.788 some generic applications. 00:41:12.788 --> 00:41:14.580 What I want to do for the rest of the class 00:41:14.580 --> 00:41:17.360 is actually give you some numbers, 00:41:17.360 --> 00:41:22.670 and really look seriously at what the typical accuracies are 00:41:22.670 --> 00:41:24.733 that you can expect from what we'll 00:41:24.733 --> 00:41:27.150 call density functional theory, but what we mean with that 00:41:27.150 --> 00:41:30.860 is sort of standard density function theory 00:41:30.860 --> 00:41:33.500 as we kind of explain it to you in the class 00:41:33.500 --> 00:41:36.080 in the local density or generalized gradient 00:41:36.080 --> 00:41:37.710 approximation. 00:41:37.710 --> 00:41:40.370 So these are sort of staples of electronic structure methods 00:41:40.370 --> 00:41:40.870 now. 00:41:40.870 --> 00:41:44.030 But that doesn't mean that there are, in some cases, not 00:41:44.030 --> 00:41:48.020 already better forms out there, but they're often 00:41:48.020 --> 00:41:50.780 very much in the research stage, and there wouldn't be things 00:41:50.780 --> 00:41:53.660 that you would either easily do on real problems. 00:41:53.660 --> 00:41:55.280 You wouldn't get your hands on them, 00:41:55.280 --> 00:41:57.663 and you wouldn't necessarily easily learn from them. 00:41:57.663 --> 00:41:59.330 So again, what we're going to talk about 00:41:59.330 --> 00:42:06.602 is the kind of standard staple of electronic structure. 00:42:06.602 --> 00:42:09.060 Before I did that, I want to say a little bit about a topic 00:42:09.060 --> 00:42:11.350 we haven't touched about, which is 00:42:11.350 --> 00:42:14.730 what's called the spin polarized version of density 00:42:14.730 --> 00:42:16.920 functional theory. 00:42:16.920 --> 00:42:19.990 If you remember, the Hohenberg-Kohn theorem, 00:42:19.990 --> 00:42:22.890 everything is, in essence, expressed 00:42:22.890 --> 00:42:25.710 in terms of the charge density. 00:42:25.710 --> 00:42:27.930 Everything is a function of the charge density. 00:42:27.930 --> 00:42:33.780 And the electron spin never explicitly appears in there. 00:42:33.780 --> 00:42:35.880 But of course, electrons have spin. 00:42:35.880 --> 00:42:38.250 If you don't consider any coupling with the angular 00:42:38.250 --> 00:42:40.650 momentum, it's really just up or down 00:42:40.650 --> 00:42:46.050 spin, so plus or minus a half [INAUDIBLE].. 00:42:46.050 --> 00:42:47.940 And we'll treat here, in the lecture, 00:42:47.940 --> 00:42:50.260 spin as a scalar quantity. 00:42:50.260 --> 00:42:53.070 So really, spin is just up or down-- 00:42:53.070 --> 00:42:55.860 plus 1 or minus 1, or plus a 1/2, or minus a 1/2 00:42:55.860 --> 00:42:57.540 in the appropriate units. 00:42:57.540 --> 00:43:00.390 And in reality, spin, as soon as it 00:43:00.390 --> 00:43:05.010 applies to the angular momentum, becomes a vector quantity. 00:43:05.010 --> 00:43:08.190 Also as soon as it couples to a magnetic field 00:43:08.190 --> 00:43:10.350 from the environment, it becomes a vector quantity. 00:43:10.350 --> 00:43:12.840 And people do that now already, treating 00:43:12.840 --> 00:43:17.880 spin as a vector quantity, but most codes that you will use, 00:43:17.880 --> 00:43:20.760 spin will simply be treated as a scalar, which 00:43:20.760 --> 00:43:24.360 is fine for most purposes. 00:43:24.360 --> 00:43:29.360 We tend to refer to them as just up and down spin. 00:43:29.360 --> 00:43:33.900 I'll often write that with the either up and down arrow. 00:43:33.900 --> 00:43:41.140 Now why do you actually need to treat the electron spin? 00:43:41.140 --> 00:43:44.890 Let me just sort of give you a refresher of why 00:43:44.890 --> 00:43:46.930 you need to deal with the spin. 00:43:46.930 --> 00:43:50.230 Well, the reason is the Pauli Exclusion Principle really. 00:43:50.230 --> 00:43:53.410 Is that the Pauli Exclusion Principle tells you that two 00:43:53.410 --> 00:43:56.290 electrons cannot be in exactly the same quantum state. 00:43:56.290 --> 00:43:57.740 Remember that? 00:43:57.740 --> 00:44:03.690 So that means that if you have two up electrons approaching 00:44:03.690 --> 00:44:10.650 each other, versus say an up and a down electron, 00:44:10.650 --> 00:44:13.050 these will approach each other differently. 00:44:13.050 --> 00:44:18.790 Because these two are in the same spin state, 00:44:18.790 --> 00:44:21.030 so if you bring them very close together, 00:44:21.030 --> 00:44:24.880 they essentially now get the same coordinate as well. 00:44:24.880 --> 00:44:27.930 So they almost have the same quantum numbers now, 00:44:27.930 --> 00:44:31.180 and the Pauli exclusion principle prevents that. 00:44:31.180 --> 00:44:34.410 So the Pauli Exclusion Principle keeps electrons 00:44:34.410 --> 00:44:39.390 with parallel spin essentially away from each other. 00:44:39.390 --> 00:44:43.170 Whereas if you have electrons with empty parallel spin, 00:44:43.170 --> 00:44:44.917 even if these are at the same position, 00:44:44.917 --> 00:44:46.500 they don't have the same spin, so they 00:44:46.500 --> 00:44:48.400 don't have the same set of quantum numbers, 00:44:48.400 --> 00:44:52.020 so the Pauli Exclusion Principle doesn't act on them. 00:44:52.020 --> 00:44:53.730 And the Pauli Exclusion Principle 00:44:53.730 --> 00:44:57.870 is essentially something that keeps the electrons away 00:44:57.870 --> 00:45:01.060 without an explicit term for it in the Hamiltonian. 00:45:01.060 --> 00:45:03.820 It's not like-- the coulombic interaction, of course, 00:45:03.820 --> 00:45:05.362 keeps electrons away from each other. 00:45:05.362 --> 00:45:06.737 But the Pauli Exclusion Principle 00:45:06.737 --> 00:45:08.520 is essentially something on top of that 00:45:08.520 --> 00:45:11.490 that comes from anti-symmeterizing the wave. 00:45:11.490 --> 00:45:14.250 You don't see it directly in the form-- 00:45:14.250 --> 00:45:16.710 as a term in the form of the Hamiltonian. 00:45:16.710 --> 00:45:18.780 And if you remember, consequences of-- 00:45:22.730 --> 00:45:24.380 it's going to take forever-- 00:45:24.380 --> 00:45:26.390 the Pauli Exclusion Principle is Hund's rule. 00:45:26.390 --> 00:45:29.570 That if you remember atomic d level, 00:45:29.570 --> 00:45:35.040 say, for example, if you add electrons to, say, 00:45:35.040 --> 00:45:39.810 five d levels, you're going to add them with parallel spin 00:45:39.810 --> 00:45:43.530 first, because again, then the Pauli Exclusion Principle 00:45:43.530 --> 00:45:45.060 is satisfied. 00:45:45.060 --> 00:45:50.010 And then you start filling them with the anti-parallel levels. 00:45:50.010 --> 00:45:53.580 So this is going to carry over in atoms-- 00:45:53.580 --> 00:45:55.830 in solids-- I'm sorry. 00:45:55.830 --> 00:45:58.440 Basically, if those five, say-- 00:45:58.440 --> 00:46:00.150 let's focus on d levels. 00:46:00.150 --> 00:46:02.730 If they remain degenerate-- 00:46:02.730 --> 00:46:04.620 so they remain roughly at the same level, 00:46:04.620 --> 00:46:07.290 you're going to fill them according to Hund's rule. 00:46:07.290 --> 00:46:08.860 And that's what I've shown here. 00:46:08.860 --> 00:46:11.550 So in solids, these d levels will 00:46:11.550 --> 00:46:14.130 split a little, which is what I've shown here, 00:46:14.130 --> 00:46:17.280 but they don't split a lot. 00:46:17.280 --> 00:46:21.070 Then you're actually going to fill them with parallel spin. 00:46:21.070 --> 00:46:24.600 And so if that's the case, you have a lot 00:46:24.600 --> 00:46:27.420 of magnetic moment on your ion. 00:46:27.420 --> 00:46:30.170 You have five electron spins here. 00:46:30.170 --> 00:46:32.670 You have no down spin, so you have a strong magnetic moment. 00:46:32.670 --> 00:46:35.920 And I'll show in a second where that becomes important. 00:46:35.920 --> 00:46:37.380 On the other hand, let's say you're 00:46:37.380 --> 00:46:41.190 in an environment that splits off two of these, 00:46:41.190 --> 00:46:43.350 but so much higher. 00:46:43.350 --> 00:46:47.340 At some point, you won't satisfy Hund's rule anymore, 00:46:47.340 --> 00:46:49.770 because the energy cost of-- let's say 00:46:49.770 --> 00:46:52.930 after you put in these three green electrons, now 00:46:52.930 --> 00:46:54.310 you have to add two more. 00:46:54.310 --> 00:46:55.930 You have five electrons. 00:46:55.930 --> 00:46:57.430 To fill them with parallel spin, you 00:46:57.430 --> 00:46:59.930 would have to put them here. 00:46:59.930 --> 00:47:02.700 But since those levers are so much higher, 00:47:02.700 --> 00:47:06.620 you basically want to pay the exchange penalty-- 00:47:06.620 --> 00:47:08.810 they call it the Hund's rule penalty-- 00:47:08.810 --> 00:47:10.760 to put them in a lower orbit when you put them 00:47:10.760 --> 00:47:12.300 in with anti-parallel. 00:47:12.300 --> 00:47:16.880 So in some sense, whether you get a lot of magnetic spin-- 00:47:16.880 --> 00:47:18.500 a lot of magnetic moment left over 00:47:18.500 --> 00:47:22.400 depends on how much your orbital will split in the end. 00:47:22.400 --> 00:47:23.900 But I'm going to show you in the end 00:47:23.900 --> 00:47:26.810 that it can actually have significant consequences 00:47:26.810 --> 00:47:29.840 on the physical properties of your material. 00:47:29.840 --> 00:47:35.320 So in your density functional calculation, 00:47:35.320 --> 00:47:38.290 you will carry a magnetic moment locally 00:47:38.290 --> 00:47:41.930 when the up density and the down density are not the same. 00:47:41.930 --> 00:47:45.100 And so I've sort of already given away here 00:47:45.100 --> 00:47:48.220 how this problem is dealt with in density functional theory. 00:47:50.750 --> 00:47:54.290 The interesting thing is that if you think about it very hard, 00:47:54.290 --> 00:47:56.600 you shouldn't have to deal with spin 00:47:56.600 --> 00:47:58.010 in density functional theory. 00:48:07.260 --> 00:48:09.510 Somehow, it put me back at my old picture. 00:48:17.015 --> 00:48:20.000 In principle, remember where we told you that-- 00:48:20.000 --> 00:48:22.160 Professor [? Mazari ?] told you that the energy 00:48:22.160 --> 00:48:27.290 and the potential is a function of the charge density? 00:48:27.290 --> 00:48:30.380 So the charge itself should actually 00:48:30.380 --> 00:48:33.710 also contain the information about electron spin 00:48:33.710 --> 00:48:36.140 and magnetic moment, even though it doesn't explicitly 00:48:36.140 --> 00:48:38.060 contain that. 00:48:38.060 --> 00:48:39.980 But for a given charge density, there 00:48:39.980 --> 00:48:42.590 would probably be a certain amount of spin polarization. 00:48:42.590 --> 00:48:45.980 So it should all be in that functional, but remember, 00:48:45.980 --> 00:48:48.620 that's the functional that we don't know. 00:48:48.620 --> 00:48:52.825 And so in practice, that doesn't work very well. 00:48:52.825 --> 00:48:54.200 So what we do is we really helped 00:48:54.200 --> 00:48:57.387 density functional theory along by treating the up 00:48:57.387 --> 00:48:59.743 and the down density separately. 00:48:59.743 --> 00:49:01.160 But again, you should keep in mind 00:49:01.160 --> 00:49:02.660 that in principle, in the formalism 00:49:02.660 --> 00:49:05.430 of density functional theory, you wouldn't have to do that. 00:49:05.430 --> 00:49:10.370 So if you do what's called, say, the local spin density 00:49:10.370 --> 00:49:13.910 approximation, which is the spin polarized version of the Local 00:49:13.910 --> 00:49:17.750 Density Approximation-- and so goes under the name LSD-- 00:49:17.750 --> 00:49:21.290 Lucy in the Sky with Diamonds, or LSDA. 00:49:21.290 --> 00:49:24.440 And there's a version of that for the GGA, which nobody-- 00:49:24.440 --> 00:49:27.380 often people will just call it LDA or GGA, 00:49:27.380 --> 00:49:30.710 but they will call it sometimes spin polarized LDA or GGA. 00:49:30.710 --> 00:49:35.590 What you do there is that you have a separate density 00:49:35.590 --> 00:49:37.750 for the up electrons and a separate density 00:49:37.750 --> 00:49:42.680 for the down electrons, and the two will interact differently. 00:49:42.680 --> 00:49:44.470 So the up/up will-- 00:49:44.470 --> 00:49:46.950 up will interact differently with up, 00:49:46.950 --> 00:49:49.300 with up, then up with down. 00:49:49.300 --> 00:49:51.670 And that comes from the way the exchange correlation 00:49:51.670 --> 00:49:54.590 potentials are defined. 00:49:54.590 --> 00:49:58.600 So I think Professor [? Mazari ?] mentioned 00:49:58.600 --> 00:50:01.690 restricted and unrestricted to Hartree-Fock before 00:50:01.690 --> 00:50:04.135 very briefly, and this is essentially the same idea. 00:50:11.090 --> 00:50:12.890 Sort of one quick tip I want to give you 00:50:12.890 --> 00:50:17.150 is that if you have spin polarized materials, 00:50:17.150 --> 00:50:20.510 it's often much more useful to look at spin densities 00:50:20.510 --> 00:50:22.700 than at charge densities. 00:50:22.700 --> 00:50:25.220 One of the really cool things about doing quantum mechanics 00:50:25.220 --> 00:50:27.470 is that you can actually look at the charge densities, 00:50:27.470 --> 00:50:30.140 and look at the electrons, which most people get 00:50:30.140 --> 00:50:32.720 very excited about the first time they do quantum mechanics. 00:50:32.720 --> 00:50:35.090 It's kind of cool-- you can look at where the electrons go. 00:50:35.090 --> 00:50:36.507 Well, the first thing you learn is 00:50:36.507 --> 00:50:39.290 that you don't see much when you look at charge densities. 00:50:39.290 --> 00:50:41.720 You typically see big blobs of charge, 00:50:41.720 --> 00:50:44.930 and it's very hard to see any fine structure of bonding 00:50:44.930 --> 00:50:46.700 in blobs of charge density. 00:50:46.700 --> 00:50:48.370 And I wanted to show you an example. 00:50:48.370 --> 00:50:49.760 Here's lithium cobalt oxide. 00:50:49.760 --> 00:50:51.080 It's a transition metal oxide. 00:50:51.080 --> 00:50:54.370 It's a layered material-- layers of oxygen 00:50:54.370 --> 00:50:55.660 here-- the red things. 00:50:55.660 --> 00:50:58.270 And then layers of cobalt, and layers of lithium. 00:50:58.270 --> 00:51:01.180 And if you plot the charges in a plane-- 00:51:01.180 --> 00:51:05.415 this is actually a plane in the plane of the figure. 00:51:05.415 --> 00:51:06.790 If you look at the charges-- this 00:51:06.790 --> 00:51:08.470 is a picture of the charge density. 00:51:08.470 --> 00:51:11.200 You see big blobs, and you see the oxygen layers here. 00:51:11.200 --> 00:51:13.510 And where the oxygens are 2 minus, 00:51:13.510 --> 00:51:16.102 so remember that you're showing only the valence electrons. 00:51:16.102 --> 00:51:17.810 So these have a lot of valence electrons, 00:51:17.810 --> 00:51:20.530 and so you see a lot of intensity. 00:51:20.530 --> 00:51:23.590 Cobalt has somewhat less valence electrons on it, 00:51:23.590 --> 00:51:25.360 so you see less intensity here. 00:51:25.360 --> 00:51:27.770 And lithium is ionized to lithium plus, 00:51:27.770 --> 00:51:29.920 so it has no valence electrons on it, 00:51:29.920 --> 00:51:33.160 so you see almost nothing here. 00:51:33.160 --> 00:51:36.490 But in essence, this doesn't give you a lot of detail. 00:51:36.490 --> 00:51:41.160 If you actually take a material like this, and rather 00:51:41.160 --> 00:51:46.120 than plot the charges, you plot the spin polarization density. 00:51:46.120 --> 00:51:50.100 So let's say up minus down, or down minus up. 00:51:50.100 --> 00:51:52.830 So it's how much magnetic moment there is locally. 00:51:52.830 --> 00:51:54.428 You get much cleaner pictures. 00:51:54.428 --> 00:51:55.470 This is the same picture. 00:51:55.470 --> 00:51:56.790 It's a slightly different material 00:51:56.790 --> 00:51:57.960 with different ions in it. 00:51:57.960 --> 00:52:00.480 It's with nickel and manganese in it. 00:52:00.480 --> 00:52:02.850 But here's the oxygen. By the way, 00:52:02.850 --> 00:52:04.950 I should have told you red is the neutral color 00:52:04.950 --> 00:52:07.740 in this picture, so it's 0. 00:52:07.740 --> 00:52:10.060 So now you don't see the oxygen at all, 00:52:10.060 --> 00:52:11.910 and the reason is oxygen doesn't have spin. 00:52:11.910 --> 00:52:12.888 It's a filled shell. 00:52:12.888 --> 00:52:15.180 Every time you have filled shells, you don't have spin. 00:52:15.180 --> 00:52:18.390 So looking at spin density often allows you to filter out 00:52:18.390 --> 00:52:21.090 certain ions, and you really-- the transition metals 00:52:21.090 --> 00:52:24.390 tend to have spin on them, and you see that very clearly. 00:52:24.390 --> 00:52:27.300 So it's often a trick that I just wanted to share with you. 00:52:31.490 --> 00:52:34.190 So I want to go in sort of the last half hour-- 00:52:34.190 --> 00:52:39.230 is go through the kind of numerical accuracy, 00:52:39.230 --> 00:52:43.340 and then slowly try to connect that to the physical accuracy 00:52:43.340 --> 00:52:44.840 that you get in properties. 00:52:44.840 --> 00:52:47.780 So if you want to use-- 00:52:47.780 --> 00:52:50.120 have initial methods in density functional theory 00:52:50.120 --> 00:52:52.670 to get to engineering properties, 00:52:52.670 --> 00:52:55.280 a lot of steps you have to make, because in the end, 00:52:55.280 --> 00:52:56.408 we calculate simple things. 00:52:56.408 --> 00:52:58.700 We've got to get charge densities, and band structures, 00:52:58.700 --> 00:52:59.720 and energies. 00:52:59.720 --> 00:53:01.320 And you'll talk later to somebody, 00:53:01.320 --> 00:53:04.550 and they want to know what's the corrosion resistance of this. 00:53:04.550 --> 00:53:08.280 And corrosion resistance is not a quantum operator, 00:53:08.280 --> 00:53:10.010 so you need to take a lot of steps 00:53:10.010 --> 00:53:13.730 to go from the simple, what I would call primitive output, 00:53:13.730 --> 00:53:15.260 to engineering properties. 00:53:15.260 --> 00:53:17.060 But before you even take that step, 00:53:17.060 --> 00:53:19.310 you need to understand the kind of accuracy-- 00:53:19.310 --> 00:53:21.920 how reliable your output is. 00:53:21.920 --> 00:53:25.698 And so I collected a lot of results. 00:53:25.698 --> 00:53:27.740 And I was going to start with the simple things-- 00:53:27.740 --> 00:53:30.238 the energies of the atoms. 00:53:30.238 --> 00:53:32.780 So here's a collection, and it looks like a bunch of numbers, 00:53:32.780 --> 00:53:34.980 but there's a very systematic trend in it. 00:53:34.980 --> 00:53:36.627 So what I show for a bunch of atoms-- 00:53:36.627 --> 00:53:38.960 and these all should have a minus sign in front of them, 00:53:38.960 --> 00:53:42.230 because if you sum up all the electronic states in the atom, 00:53:42.230 --> 00:53:46.218 they're obviously binding, so they should be negative. 00:53:46.218 --> 00:53:48.260 So there's always the experimental line, at least 00:53:48.260 --> 00:53:50.670 for most of them. 00:53:50.670 --> 00:53:54.540 The LDA number, and then the GGA with a fairly recent 00:53:54.540 --> 00:53:57.730 implementation of the exchange correlation function. 00:53:57.730 --> 00:54:01.770 And so if you look carefully at the numbers, 00:54:01.770 --> 00:54:05.260 let's take one here-- let's take carbon for example. 00:54:05.260 --> 00:54:09.330 So experiment is 275688. 00:54:09.330 --> 00:54:11.970 Oops, that wasn't good. 00:54:11.970 --> 00:54:17.130 LDA, the binding energy is somewhat weaker-- 00:54:17.130 --> 00:54:19.380 almost is an electron volt weaker. 00:54:19.380 --> 00:54:24.420 And GGA is slightly closer to the experiment. 00:54:24.420 --> 00:54:26.460 That's typically what you'll see. 00:54:26.460 --> 00:54:28.020 If you look at all the other atoms, 00:54:28.020 --> 00:54:31.890 you'll see a very systematic trend. 00:54:31.890 --> 00:54:38.070 In the LDA, in the atoms, the electrons are not bound enough. 00:54:38.070 --> 00:54:43.838 In the GGA, They are somewhat closer to the experiment. 00:54:46.770 --> 00:54:50.040 It gets more interesting when you look at molecules, 00:54:50.040 --> 00:54:52.870 because now you can talk about a physical binding energy. 00:54:52.870 --> 00:54:56.340 And so the one we look at here is for very simple 00:54:56.340 --> 00:54:58.720 diatomic molecules. 00:54:58.720 --> 00:55:00.010 What's their binding energy? 00:55:00.010 --> 00:55:02.690 So what's their energy to pull them apart? 00:55:02.690 --> 00:55:06.900 So if you think about it, you have a molecule-- 00:55:06.900 --> 00:55:10.980 has an ab. 00:55:10.980 --> 00:55:13.830 This is the vector between them. 00:55:13.830 --> 00:55:16.690 You'll have sort of something that looks like this. 00:55:16.690 --> 00:55:20.100 And so we're looking at what's that well depth here-- 00:55:20.100 --> 00:55:21.050 the binding energy. 00:55:23.670 --> 00:55:28.770 If you look at hydrogen, H2, you have the experimental number 00:55:28.770 --> 00:55:31.410 here, the LDA number. 00:55:31.410 --> 00:55:33.450 You see now that the LDA number-- 00:55:33.450 --> 00:55:35.610 that the binding is too strong. 00:55:35.610 --> 00:55:38.310 The H2 molecule is bound too strongly. 00:55:38.310 --> 00:55:39.240 It's not so bad. 00:55:39.240 --> 00:55:43.620 It's only about 5% in hydrogen. And the GGA, the binding 00:55:43.620 --> 00:55:45.720 is too weak. 00:55:45.720 --> 00:55:49.455 For reference, I've also put uncorrected Hartree-Fock here. 00:55:49.455 --> 00:55:51.330 Professor [? Mazari ?] showed you essentially 00:55:51.330 --> 00:55:54.870 what Hartree-Fock is, which is essentially 00:55:54.870 --> 00:55:58.260 having the Hartree term for the self-consistent Coulomb 00:55:58.260 --> 00:56:00.420 interaction from the other electrons, 00:56:00.420 --> 00:56:03.870 and the exact exchange with no correlation effects. 00:56:06.476 --> 00:56:09.810 This is an interesting one that you will actually often use-- 00:56:09.810 --> 00:56:13.920 O2-- any time you look at oxidation reactions, 00:56:13.920 --> 00:56:14.820 for example. 00:56:14.820 --> 00:56:20.160 Experiment-- oxygen is only about minus 5.2 ev binding. 00:56:20.160 --> 00:56:24.590 LDA binds by a whopping 7 and 1/2. 00:56:24.590 --> 00:56:27.390 So you're more than two electron volts off. 00:56:27.390 --> 00:56:30.080 GGA gets you a little closer in this case. 00:56:34.270 --> 00:56:38.815 Uncorrected Hartree-Fock is off the charts. 00:56:38.815 --> 00:56:40.690 And this is something you'll generically see. 00:56:40.690 --> 00:56:43.185 Uncorrected Hartree-Fock, very few people 00:56:43.185 --> 00:56:44.560 would actually use that any more. 00:56:44.560 --> 00:56:46.990 It's sort of for binding energies way off the charts. 00:56:49.966 --> 00:56:51.545 These things are important. 00:56:51.545 --> 00:56:53.170 If you now-- let's say you want to look 00:56:53.170 --> 00:56:55.460 at an oxidation reaction. 00:56:55.460 --> 00:56:57.050 So that means that at some point, 00:56:57.050 --> 00:57:00.320 you're going to calculate the state of an oxide, 00:57:00.320 --> 00:57:03.380 and compare the chemical potential of the oxygen 00:57:03.380 --> 00:57:07.550 there to that of oxygen gas. 00:57:07.550 --> 00:57:11.215 So of course, you have a big error in the oxygen gas. 00:57:11.215 --> 00:57:12.840 The question is, how much of that error 00:57:12.840 --> 00:57:14.590 carries over to the solid? 00:57:14.590 --> 00:57:16.620 If you make exactly the same error in the solid, 00:57:16.620 --> 00:57:19.280 then the reaction energy is perfect, Because. 00:57:19.280 --> 00:57:21.310 You're going to subtract the two. 00:57:21.310 --> 00:57:23.745 And a lot of practical things you 00:57:23.745 --> 00:57:25.120 do with density functional theory 00:57:25.120 --> 00:57:26.680 depend on error cancellation. 00:57:29.410 --> 00:57:32.020 The thing is that you will have more error cancellation 00:57:32.020 --> 00:57:36.603 as the states that you subtract are more physically similar. 00:57:36.603 --> 00:57:38.020 But the problem is, let's say, you 00:57:38.020 --> 00:57:40.810 look at oxidation of a metal-- aluminum plus oxygen going 00:57:40.810 --> 00:57:43.300 to aluminum oxide. 00:57:43.300 --> 00:57:45.190 The oxygen in aluminum oxide is very 00:57:45.190 --> 00:57:47.660 different from the oxygen in the O2 molecule, 00:57:47.660 --> 00:57:50.800 so not all the error will cancel. 00:57:50.800 --> 00:57:56.047 Let's say it was so bad that you kept 2 ev error. 00:57:56.047 --> 00:58:00.160 So 2 ev error in the molecule is one ev per oxygen 00:58:00.160 --> 00:58:01.770 if you want to think of it that way. 00:58:01.770 --> 00:58:06.540 If you're not careful about it, that's an enormous effect. 00:58:06.540 --> 00:58:09.000 Think about it-- the chemical potential which 00:58:09.000 --> 00:58:11.040 relates directly to the energy goes 00:58:11.040 --> 00:58:15.500 like the logarithm of the partial pressure of oxygen. 00:58:15.500 --> 00:58:19.340 Remember that mu is mu 0 plus rt log p? 00:58:19.340 --> 00:58:21.470 So if now you invert that, that means your-- 00:58:21.470 --> 00:58:24.470 if you wanted to calculate, say, a partial pressure of oxygen 00:58:24.470 --> 00:58:28.210 at which something oxidized, your oxygen pressure 00:58:28.210 --> 00:58:32.280 goes exponentially with the energetics. 00:58:32.280 --> 00:58:35.790 So if you have a 1 ev error at room temperature, 00:58:35.790 --> 00:58:40.620 your error in the oxygen pressure 00:58:40.620 --> 00:58:43.200 is the exponential of one ev over kt, 00:58:43.200 --> 00:58:46.440 which is off the charts. 00:58:46.440 --> 00:58:49.670 So you have to be a little careful with these kinds 00:58:49.670 --> 00:58:51.830 of error calibrations. 00:58:51.830 --> 00:58:54.680 Fortunately, we'll see later when 00:58:54.680 --> 00:58:57.140 we look at reaction between solids, most of the error 00:58:57.140 --> 00:59:00.560 tends to cancel, and we get much, much better accuracy. 00:59:00.560 --> 00:59:03.860 If all our reaction energies were wrong by one ev, 00:59:03.860 --> 00:59:05.150 we wouldn't be here. 00:59:05.150 --> 00:59:07.230 We'd be out of business. 00:59:07.230 --> 00:59:09.740 But you have to keep in mind that you get less error 00:59:09.740 --> 00:59:14.517 cancellation as the states you're comparing are different. 00:59:14.517 --> 00:59:16.850 The more they're different, the less error cancellation. 00:59:16.850 --> 00:59:18.380 That's sort of a rule of thumb. 00:59:18.380 --> 00:59:20.240 And so going from a gas to a solid 00:59:20.240 --> 00:59:21.860 is a significant difference. 00:59:21.860 --> 00:59:23.810 Often, what people do is that if you 00:59:23.810 --> 00:59:26.030 want to get practical results, they'll 00:59:26.030 --> 00:59:28.730 add a correction to this which they fit at one point. 00:59:32.470 --> 00:59:34.360 So these are the small molecules. 00:59:34.360 --> 00:59:35.380 Let's go to the solids. 00:59:38.265 --> 00:59:39.890 Here I don't have the binding energies, 00:59:39.890 --> 00:59:41.350 but I have the lattice parameters. 00:59:41.350 --> 00:59:44.200 But you'll see something that's very 00:59:44.200 --> 00:59:46.697 consistent with the molecules. 00:59:46.697 --> 00:59:48.280 If you look at the lattice parameters, 00:59:48.280 --> 00:59:50.620 you compare, say, the experimental ones 00:59:50.620 --> 00:59:52.360 versus the LDA ones. 00:59:52.360 --> 00:59:55.510 What you'll see is that the LDA ones almost always-- 00:59:55.510 --> 00:59:58.707 and I think always in this case are smaller. 00:59:58.707 --> 01:00:00.790 Actually, yeah, because the difference is actually 01:00:00.790 --> 01:00:02.230 negative. 01:00:02.230 --> 01:00:04.450 The GGA results are always bigger. 01:00:07.160 --> 01:00:10.230 This is rather consistent whatever material you do. 01:00:10.230 --> 01:00:15.640 You will find almost always that the LDA gives you 01:00:15.640 --> 01:00:18.480 lattice parameters that are too small by a factor of a percent, 01:00:18.480 --> 01:00:20.720 sometimes, 2%. 01:00:20.720 --> 01:00:24.560 And so people refer to that as the over-binding of LDA. 01:00:24.560 --> 01:00:26.660 LDA binds somewhat too strongly. 01:00:26.660 --> 01:00:29.220 Remember you saw that in the molecules as well. 01:00:29.220 --> 01:00:31.340 Oxygen had a 7 ev binding energy, 01:00:31.340 --> 01:00:34.270 and it should only have a 5 ev binding energy. 01:00:34.270 --> 01:00:36.020 And so in solids, the way that comes out 01:00:36.020 --> 01:00:38.520 is that your equilibrium lattice parameters are slightly too 01:00:38.520 --> 01:00:40.020 small. 01:00:40.020 --> 01:00:43.140 Actually, I'm not sure that I know of a single result 01:00:43.140 --> 01:00:46.800 where LDA gives a lattice parameter that's too big. 01:00:46.800 --> 01:00:49.957 I've seen that on some occasions in papers, 01:00:49.957 --> 01:00:52.290 but it's almost always an indication that the people did 01:00:52.290 --> 01:00:54.843 the calculation wrong. 01:00:54.843 --> 01:00:56.260 Actually, an LDA lattice parameter 01:00:56.260 --> 01:00:58.840 that agrees with experiment is usually wrong-- 01:00:58.840 --> 01:01:00.390 a wrong calculation. 01:01:00.390 --> 01:01:04.032 GGA is much more unpredictable. 01:01:04.032 --> 01:01:06.490 The ones that I've shown here because they're simple metals 01:01:06.490 --> 01:01:08.650 and semiconductors give you a lattice parameter 01:01:08.650 --> 01:01:10.120 that's too large. 01:01:10.120 --> 01:01:12.190 In GGA, it's actually also possible to get 01:01:12.190 --> 01:01:14.630 a lattice parameter that's too small, although it's rare. 01:01:14.630 --> 01:01:17.260 Most of the time, you're on the higher side, 01:01:17.260 --> 01:01:20.200 but it's less predictable, and that's sort of slightly 01:01:20.200 --> 01:01:21.700 problematic with the GGA. 01:01:21.700 --> 01:01:24.315 In LDA, a good guess of the lattice 01:01:24.315 --> 01:01:26.440 parameters-- you calculate your lattice parameters, 01:01:26.440 --> 01:01:28.120 and you know you're on the low side. 01:01:28.120 --> 01:01:29.560 You always know that the real lattice parameter 01:01:29.560 --> 01:01:30.550 is going to be bigger. 01:01:30.550 --> 01:01:34.660 In GGA, it's slightly more difficult to predict 01:01:34.660 --> 01:01:36.370 on which side you are. 01:01:36.370 --> 01:01:38.890 But in metals, you do tend to be on the high side. 01:01:41.410 --> 01:01:44.140 That actually has consequences for other properties, 01:01:44.140 --> 01:01:46.330 like the bulk modulus. 01:01:46.330 --> 01:01:49.420 If you compare, say, the experimental bulk modulus 01:01:49.420 --> 01:01:53.810 to the LDA one, s you'll find is that in almost all cases-- 01:01:53.810 --> 01:01:55.810 and I think in all cases that I've shown-- well, 01:01:55.810 --> 01:01:58.150 all cases except silicon-- 01:01:58.150 --> 01:02:00.650 the LDA bulk modulus is too large, 01:02:00.650 --> 01:02:02.890 so the material is too stiff that means. 01:02:02.890 --> 01:02:06.880 And that kind of goes together with the over-binding. 01:02:06.880 --> 01:02:09.760 Remember, the bonding energy is too high, 01:02:09.760 --> 01:02:11.698 the lattice parameter is too small. 01:02:11.698 --> 01:02:13.990 All that is kind of in agreement with the material also 01:02:13.990 --> 01:02:15.040 being too stiff. 01:02:15.040 --> 01:02:18.220 As you compress the material, it gets stiffer. 01:02:18.220 --> 01:02:20.800 GGA, most of the time, if you see from deviations, 01:02:20.800 --> 01:02:23.110 has to be on the other side. 01:02:23.110 --> 01:02:25.120 It tends to be too soft. 01:02:25.120 --> 01:02:29.350 And bulk modulus effects that will 01:02:29.350 --> 01:02:34.150 transfer, for example, also into vibrational frequencies. 01:02:34.150 --> 01:02:37.360 In material, when you're too hard, too stiff, 01:02:37.360 --> 01:02:39.070 you'll have higher vibrational frequency. 01:02:39.070 --> 01:02:40.870 When you're too soft, you'll have 01:02:40.870 --> 01:02:42.580 lower vibrational frequencies. 01:02:45.650 --> 01:02:46.795 Here's the same for oxides. 01:02:46.795 --> 01:02:48.170 You're not exactly learning a lot 01:02:48.170 --> 01:02:51.050 new by looking at them, except that in oxides, the errors just 01:02:51.050 --> 01:02:52.175 tend to be slightly larger. 01:02:58.090 --> 01:03:02.080 So here's the summary for geometry prediction. 01:03:02.080 --> 01:03:05.170 You almost always-- and I would probably say always-- 01:03:05.170 --> 01:03:07.090 under-predict with LDA. 01:03:07.090 --> 01:03:10.540 Less systematic errors with GGA. 01:03:10.540 --> 01:03:13.450 For normal materials like semiconductors and metals, 01:03:13.450 --> 01:03:17.950 often, your errors are confined to order 1% to 2%. 01:03:17.950 --> 01:03:20.180 In transition metal oxides-- and if I have a chance, 01:03:20.180 --> 01:03:21.722 I'll say a little bit more about that 01:03:21.722 --> 01:03:25.450 later because they have electronic structures where 01:03:25.450 --> 01:03:30.680 the LDA and GGA approximations are particularly harsh on. 01:03:30.680 --> 01:03:32.750 You tend to have somewhat bigger errors. 01:03:32.750 --> 01:03:36.340 But I may say a little more about that if we get to it. 01:03:36.340 --> 01:03:40.570 So I want to say something about predicting structure, 01:03:40.570 --> 01:03:44.300 and about the energy scale that's required. 01:03:44.300 --> 01:03:46.300 So this is often something you want to do. 01:03:46.300 --> 01:03:50.050 You want to know if I put my energy in this arrangement, 01:03:50.050 --> 01:03:52.398 is that lower in energy than some other arrangement, 01:03:52.398 --> 01:03:54.940 so I can kind of predict what the most stable arrangement is. 01:03:54.940 --> 01:03:56.898 So that could be [INAUDIBLE] crystal structure, 01:03:56.898 --> 01:03:59.750 but it's the same for if you look, say, at a surface. 01:03:59.750 --> 01:04:02.140 So I wanted to give you a feeling for the scale 01:04:02.140 --> 01:04:04.340 of energetic differences. 01:04:04.340 --> 01:04:08.260 So for vanadium, I've listed the atomic energy here 01:04:08.260 --> 01:04:09.700 in [INAUDIBLE]. 01:04:09.700 --> 01:04:12.025 This is the energy of all its electrons, 01:04:12.025 --> 01:04:15.820 so not just the valence electrons actually. 01:04:15.820 --> 01:04:19.120 The energy for FCC vanadium-- so remember, 01:04:19.120 --> 01:04:25.160 the first line is the atom, not in a solid. 01:04:25.160 --> 01:04:31.550 The second line is the FCC iron, and the third one is BCC iron. 01:04:31.550 --> 01:04:32.850 So look at the differences. 01:04:32.850 --> 01:04:34.790 First of all, if you go from the atom 01:04:34.790 --> 01:04:39.552 to the solid, your first four digits don't even change. 01:04:39.552 --> 01:04:41.010 And again, that's a reflection of-- 01:04:41.010 --> 01:04:44.140 well, a lot of your deep core states don't change. 01:04:44.140 --> 01:04:45.870 But you would see something similar even 01:04:45.870 --> 01:04:47.995 with a pseudopotential approximation where you only 01:04:47.995 --> 01:04:49.990 deal with the valence electrons. 01:04:49.990 --> 01:04:59.090 So the cohesive energy is only 0.03% of the total energy. 01:04:59.090 --> 01:05:01.610 So if you're calculating-- the reason I'm saying this-- 01:05:01.610 --> 01:05:04.250 if you're calculating the cohesive energy by first 01:05:04.250 --> 01:05:07.160 calculating the total energy of a solid, 01:05:07.160 --> 01:05:08.780 and then calculating atomic energy, 01:05:08.780 --> 01:05:10.977 you'd better do these things damn accurate, 01:05:10.977 --> 01:05:12.560 because you're going to subtract them, 01:05:12.560 --> 01:05:15.170 and most of what you subtract is the same. 01:05:15.170 --> 01:05:18.140 So to get any significance in your result, 01:05:18.140 --> 01:05:22.540 you need to have high numerical accuracy. 01:05:22.540 --> 01:05:24.670 And that's not a big problem with a lot of code, 01:05:24.670 --> 01:05:27.560 but I want you to keep that in mind. 01:05:27.560 --> 01:05:30.040 But few people care about the cohesive energy. 01:05:30.040 --> 01:05:35.770 Let's say you want to know whether vanadium is FCC or BCC. 01:05:35.770 --> 01:05:38.110 So you could calculate it as BCC. 01:05:38.110 --> 01:05:43.750 Now the FCC/BCC difference is only 0.001% 01:05:43.750 --> 01:05:45.640 of the total energy. 01:05:45.640 --> 01:05:48.530 And these are not complicated structures. 01:05:48.530 --> 01:05:51.760 So in many cases, we're going to work with energy differences 01:05:51.760 --> 01:05:53.170 that are fractions of-- 01:05:53.170 --> 01:05:56.590 so that are 10 to the minus 6, 10 to the minus 7 01:05:56.590 --> 01:05:58.318 times the total energy. 01:05:58.318 --> 01:05:59.860 So it's sort of telling you something 01:05:59.860 --> 01:06:04.660 about how much numerical accuracy you need. 01:06:04.660 --> 01:06:06.820 If you want to look at mixing energies-- 01:06:06.820 --> 01:06:09.310 let's say I mix vanadium with something else-- 01:06:09.310 --> 01:06:10.960 platinum-- and you want to know what's 01:06:10.960 --> 01:06:14.620 the mixing enthalpy, because that sets the whole temperature 01:06:14.620 --> 01:06:18.460 scale for mixing, the whole phase diagram topology. 01:06:18.460 --> 01:06:19.810 That tends to be a fraction-- 01:06:19.810 --> 01:06:24.140 10 to minus 6, 10 to the minus 7 of the total energy. 01:06:24.140 --> 01:06:28.030 So my former advisor used to compare this to-- let's say 01:06:28.030 --> 01:06:31.480 you want to know the weight of a captain that 01:06:31.480 --> 01:06:33.490 sails a big supertanker. 01:06:33.490 --> 01:06:35.680 It's like weighing the tanker with the captain 01:06:35.680 --> 01:06:38.210 and without the captain, and looking at the difference, 01:06:38.210 --> 01:06:40.510 and that's the weight of the captain. 01:06:40.510 --> 01:06:44.750 You're almost at a kind of relative scale like that here. 01:06:44.750 --> 01:06:47.710 So the [? cute ?] thing really is 01:06:47.710 --> 01:06:49.990 that all these approximations we make 01:06:49.990 --> 01:06:55.120 to density functional theory are obviously not this accurate. 01:06:55.120 --> 01:06:58.840 The total energy is not accurate up to a fraction of 10 01:06:58.840 --> 01:07:00.370 to the minus 6. 01:07:00.370 --> 01:07:03.400 The only reason we're here and we can get physical behavior 01:07:03.400 --> 01:07:06.430 right is because a lot of the error 01:07:06.430 --> 01:07:08.200 the density functional theory makes in LDA 01:07:08.200 --> 01:07:09.980 and GGA is systematic. 01:07:09.980 --> 01:07:12.070 And so a lot of it cancels away when 01:07:12.070 --> 01:07:14.290 you take energy differences. 01:07:14.290 --> 01:07:18.790 When I do FCC and BCC vanadium, yes, I may have an error of 10 01:07:18.790 --> 01:07:21.640 to minus 4 in the energy, but most of it 01:07:21.640 --> 01:07:24.680 cancels away when I take the energy difference. 01:07:24.680 --> 01:07:26.343 And that's why we're lucky. 01:07:26.343 --> 01:07:27.760 But you have to keep that in mind, 01:07:27.760 --> 01:07:30.070 because again, the less cancellation you have, 01:07:30.070 --> 01:07:40.410 the bigger your error on the result. 01:07:40.410 --> 01:07:44.640 So again, let me show you how well or how badly it does. 01:07:44.640 --> 01:07:46.950 So I did a very simple thing. 01:07:46.950 --> 01:07:50.460 I looked at how well does it predict, say, 01:07:50.460 --> 01:07:53.010 the structure of the elements. 01:07:53.010 --> 01:07:57.430 This is done in GGA, a standard [? Trudel ?] potential method. 01:07:57.430 --> 01:08:00.315 So this comes out-- you may have to look at your hand 01:08:00.315 --> 01:08:04.290 out because this is extremely fuzzy on the screen. 01:08:04.290 --> 01:08:07.860 In red, I did metals that are experimentally 01:08:07.860 --> 01:08:12.330 FCC, and green I did metals that are experimentally BCC. 01:08:12.330 --> 01:08:15.510 Now what I show you is the calculated energy 01:08:15.510 --> 01:08:18.630 difference between FCC and BCC, and it's actually 01:08:18.630 --> 01:08:25.880 the first line below every element kind of like this. 01:08:25.880 --> 01:08:29.920 And so when that's positive, the BCC energy is higher than FCC, 01:08:29.920 --> 01:08:30.670 so it's going to-- 01:08:30.670 --> 01:08:32.859 FCC is preferred over BCC. 01:08:32.859 --> 01:08:40.120 If it's negative like here, then BCC is preferred over FCC. 01:08:40.120 --> 01:08:42.220 And so if you look-- so the color is 01:08:42.220 --> 01:08:45.910 the experimental result, the number is calculated. 01:08:45.910 --> 01:08:49.830 So if you look at them carefully, they're all correct. 01:08:49.830 --> 01:08:51.569 It's negative when we have green, 01:08:51.569 --> 01:08:57.569 it's positive when we have red. 01:08:57.569 --> 01:09:00.420 You can do a more subtle comparison. 01:09:00.420 --> 01:09:02.399 Look at the difference between HCP-- 01:09:02.399 --> 01:09:04.537 Hexagonal Close Pack-- and FCC. 01:09:04.537 --> 01:09:06.120 And the reason that that's more subtle 01:09:06.120 --> 01:09:09.450 is HCP FCC are much more alike. 01:09:09.450 --> 01:09:10.428 They're all close pack. 01:09:10.428 --> 01:09:11.970 It's just the difference in stacking. 01:09:11.970 --> 01:09:15.870 ab, ab, versus abc, abc. 01:09:15.870 --> 01:09:18.030 And so again, you'll see that they're all correct. 01:09:20.640 --> 01:09:22.500 The red ones are the FCC ones. 01:09:22.500 --> 01:09:24.870 They're the ones where that first line is positive. 01:09:24.870 --> 01:09:29.069 By the way, that number is in kilojoules per mole here. 01:09:29.069 --> 01:09:32.790 The yellow ones, that number is negative. 01:09:32.790 --> 01:09:40.840 So we get the structure of the elements essentially correct. 01:09:40.840 --> 01:09:43.720 There are notable exceptions. 01:09:43.720 --> 01:09:45.520 In LDA, iron is wrong. 01:09:45.520 --> 01:09:50.850 Iron is FCC in LDA, not BCC, but in GGA, that's corrected. 01:09:50.850 --> 01:09:53.020 And then there are the weirdos. 01:09:53.020 --> 01:09:56.780 If you go deep down in the periodic table, 01:09:56.780 --> 01:10:01.120 especially f electron metals have-- 01:10:01.120 --> 01:10:04.480 the f states are extremely localized even in metals, 01:10:04.480 --> 01:10:07.540 and so electron correlation becomes very important here. 01:10:07.540 --> 01:10:09.290 And I may say a little more about that. 01:10:09.290 --> 01:10:12.820 And so there you'll start to see failures LDA and GGA. 01:10:12.820 --> 01:10:15.730 An important one is plutonium. 01:10:15.730 --> 01:10:18.650 Plutonium is kind of important for obvious reasons, 01:10:18.650 --> 01:10:22.090 especially if you work at national labs these days. 01:10:22.090 --> 01:10:26.770 And so people are building more sophisticated methods 01:10:26.770 --> 01:10:30.550 to deal with materials such as plutonium. 01:10:30.550 --> 01:10:33.357 Typically, when you work with f electron metals, sometimes, 01:10:33.357 --> 01:10:35.440 you'll get the answer right, sometimes, you won't. 01:10:35.440 --> 01:10:38.005 But you should be a little more careful. 01:10:42.590 --> 01:10:43.910 Let me skip this. 01:10:43.910 --> 01:10:45.650 You can get the-- 01:10:45.650 --> 01:10:52.010 most of the time, you'll get the structure of compounds right. 01:10:52.010 --> 01:10:54.530 If you go to transition metal oxides-- 01:10:54.530 --> 01:10:56.390 so I sort of went from metallic elements 01:10:56.390 --> 01:10:58.460 now to transition metal oxides-- 01:10:58.460 --> 01:11:00.890 most of the time, you also get the structure right, 01:11:00.890 --> 01:11:03.170 but things get more subtle. 01:11:03.170 --> 01:11:05.990 In transition metal oxides, the transition metal 01:11:05.990 --> 01:11:08.120 has local d states. 01:11:08.120 --> 01:11:11.480 And I showed you before they often have significant spin 01:11:11.480 --> 01:11:13.640 polarization, so the first thing you need to do 01:11:13.640 --> 01:11:16.430 is turn spin polarization on, or you really 01:11:16.430 --> 01:11:18.020 get the wrong answer. 01:11:18.020 --> 01:11:19.980 But it often gets worse. 01:11:19.980 --> 01:11:24.090 Remember that your spin is a scalar, so it's up or down. 01:11:24.090 --> 01:11:26.780 So now you have a spatial degree of freedom 01:11:26.780 --> 01:11:29.510 of how to organize that spin on the ions. 01:11:29.510 --> 01:11:31.520 If you have a bunch of ions, you could put them 01:11:31.520 --> 01:11:32.720 all with the same direction. 01:11:32.720 --> 01:11:36.343 That's a ferromagnet, or you could put them 01:11:36.343 --> 01:11:37.760 with sort of alternating direction 01:11:37.760 --> 01:11:39.980 as a kind of a anti-ferromagnet. 01:11:39.980 --> 01:11:43.280 And then there's many ways to make them anti-ferromagnetic. 01:11:43.280 --> 01:11:45.200 And unfortunately, in transition metal oxide, 01:11:45.200 --> 01:11:47.900 it often matters because there's not only 01:11:47.900 --> 01:11:51.590 a strong spin polarization effect on the energy, 01:11:51.590 --> 01:11:53.040 but there's a fairly strong effect 01:11:53.040 --> 01:11:58.940 of the interaction between spins on different ions. 01:11:58.940 --> 01:12:01.260 And I'm showing you a result here. 01:12:01.260 --> 01:12:02.720 This is a simple crystal structure. 01:12:02.720 --> 01:12:04.610 It's a structure of lithium manganese oxide. 01:12:04.610 --> 01:12:08.450 It's an ordered rock salt. These are still very simple. 01:12:08.450 --> 01:12:11.030 But the correct answer-- 01:12:11.030 --> 01:12:14.283 I'm showing the comparison here between two structures 01:12:14.283 --> 01:12:16.200 only labeled by their symmetry, unfortunately. 01:12:16.200 --> 01:12:19.730 One is C2/m, and one is pmmn. 01:12:19.730 --> 01:12:23.060 There are similar structures, but one is orthorhombic, 01:12:23.060 --> 01:12:24.800 and one is monoclinic. 01:12:24.800 --> 01:12:26.900 The correct answer is pmmn-- 01:12:26.900 --> 01:12:28.610 is the real crystal structure. 01:12:28.610 --> 01:12:31.620 If you do a non-spin polarized calculation-- 01:12:31.620 --> 01:12:34.700 so that's not even allowing spin on the ions-- 01:12:34.700 --> 01:12:35.960 you get a whopping error. 01:12:35.960 --> 01:12:41.720 I mean, C2/m is lower in energy by 250 mill-electron volt 01:12:41.720 --> 01:12:43.280 per formal unit. 01:12:43.280 --> 01:12:44.240 That's very large. 01:12:44.240 --> 01:12:49.020 In kilojoules, that'd be 25 kilojoules per mole. 01:12:49.020 --> 01:12:50.420 It's a very large error. 01:12:50.420 --> 01:12:51.890 If you turn on spin polarization, 01:12:51.890 --> 01:12:53.098 will make them ferromagnetic. 01:12:53.098 --> 01:12:53.930 They're degenerate. 01:12:53.930 --> 01:12:56.060 And if you make them anti-ferromagnetic, 01:12:56.060 --> 01:12:59.030 this one is the lowest in energy. 01:12:59.030 --> 01:13:04.100 Now here's a very common mistake that people make. 01:13:04.100 --> 01:13:06.350 If you take this material at room temperature, 01:13:06.350 --> 01:13:08.700 it's paramagnetic. 01:13:08.700 --> 01:13:11.160 Some people say, well, it's paramagnetic, 01:13:11.160 --> 01:13:13.020 so I shouldn't have any spin polarization-- 01:13:13.020 --> 01:13:14.160 there's no net moment. 01:13:14.160 --> 01:13:17.910 That is so wrong, because a paramagnet still 01:13:17.910 --> 01:13:19.200 has a local moment. 01:13:19.200 --> 01:13:21.390 The ions still have a moment on them, 01:13:21.390 --> 01:13:24.120 it's just randomly oriented. 01:13:24.120 --> 01:13:27.710 So you still need to represent that moment. 01:13:27.710 --> 01:13:30.210 Because it turns out that's the biggest effect on the energy 01:13:30.210 --> 01:13:32.460 is the fact that you have that local moment. 01:13:32.460 --> 01:13:34.560 It's not necessarily how they're arranged. 01:13:34.560 --> 01:13:35.970 You can actually see that here. 01:13:35.970 --> 01:13:39.630 How they're arranged makes you go from this difference 01:13:39.630 --> 01:13:41.100 to this difference. 01:13:41.100 --> 01:13:42.810 But turning on the local moment makes 01:13:42.810 --> 01:13:46.750 you go from this difference to that difference. 01:13:46.750 --> 01:13:48.310 So never fall in that trap. 01:13:48.310 --> 01:13:52.090 It's really only non-magnetic materials or diamagnetic 01:13:52.090 --> 01:13:57.770 materials for which you don't really need spin polarization. 01:13:57.770 --> 01:14:01.740 Now why is this effect so important? 01:14:01.740 --> 01:14:06.560 It's really because if you spin polarize an ion, 01:14:06.560 --> 01:14:08.920 you fill different orbitals. 01:14:08.920 --> 01:14:12.470 I mean, I've shown that here with a bunch of d orbitals. 01:14:12.470 --> 01:14:14.840 And this is typically how they split in most oxides. 01:14:14.840 --> 01:14:16.850 Every time an ion is octahedral-- 01:14:16.850 --> 01:14:21.140 five d orbitals tend to split in pairs of 3, 2-- 01:14:21.140 --> 01:14:22.100 3 down, 2 up. 01:14:22.100 --> 01:14:25.520 In some cases, 2 down, 3 up. 01:14:25.520 --> 01:14:27.020 But let's say you have to put four-- 01:14:27.020 --> 01:14:28.490 five electrons in there. 01:14:28.490 --> 01:14:30.830 How many do I have? 01:14:30.830 --> 01:14:33.290 I'm missing-- no, four electrons. 01:14:33.290 --> 01:14:34.280 If you put them-- 01:14:34.280 --> 01:14:35.390 that's called high spin. 01:14:35.390 --> 01:14:38.120 So all parallel spin, you put them like this. 01:14:38.120 --> 01:14:40.980 If you put them low spin, you put them like this. 01:14:40.980 --> 01:14:42.630 So here we have no moment. 01:14:42.630 --> 01:14:47.370 So these two ions have different chemical properties, 01:14:47.370 --> 01:14:50.580 because the electrons occupy different orbitals. 01:14:50.580 --> 01:14:57.120 These are different d orbitals, and so this orbital points 01:14:57.120 --> 01:15:01.540 in a different direction, for example, than this one. 01:15:01.540 --> 01:15:04.720 So by spin polarizing, you create 01:15:04.720 --> 01:15:05.980 essentially a different ion. 01:15:05.980 --> 01:15:08.920 It's not really an issue of magnetism, 01:15:08.920 --> 01:15:11.470 because magnetic effects tend to be small in materials. 01:15:11.470 --> 01:15:14.230 But it's the fact that you create chemically 01:15:14.230 --> 01:15:18.870 a different ion because you fill different levels. 01:15:18.870 --> 01:15:21.680 That's really why these energy differences are so big. 01:15:26.533 --> 01:15:28.700 I sort of want to end with showing you some reaction 01:15:28.700 --> 01:15:30.620 energies very quickly. 01:15:30.620 --> 01:15:33.200 And I'm going to sort make it systematically harder. 01:15:33.200 --> 01:15:34.580 So here's a simple one. 01:15:34.580 --> 01:15:37.460 A metal, lithium, BCC, with another metal, 01:15:37.460 --> 01:15:40.640 aluminum forming a lithium-aluminum compound. 01:15:40.640 --> 01:15:44.480 Here's the experimental reaction energy, here's the LDA one. 01:15:44.480 --> 01:15:46.880 10% off. 01:15:46.880 --> 01:15:48.020 That's classic. 01:15:48.020 --> 01:15:50.840 Metallic reaction energies, you're somewhere 01:15:50.840 --> 01:15:54.020 in the range 5% to 15%. 01:15:54.020 --> 01:15:55.640 The one I show here, copper-gold, 01:15:55.640 --> 01:15:58.740 is a notable exception where you're over 50% off. 01:15:58.740 --> 01:16:02.030 But most of them, it's much simpler. 01:16:06.360 --> 01:16:11.150 So in metals, you tend to get very good reaction energies. 01:16:11.150 --> 01:16:15.770 I want to show you the case where things go wrong. 01:16:15.770 --> 01:16:20.330 Where your errors become bigger is in redox reactions. 01:16:20.330 --> 01:16:22.670 And I've shown sort of here three different ones. 01:16:22.670 --> 01:16:23.630 They're all related. 01:16:23.630 --> 01:16:25.670 They're essentially a reaction between an oxide, 01:16:25.670 --> 01:16:29.180 or in this case, a phosphate with a metal 01:16:29.180 --> 01:16:31.110 to react the two together. 01:16:31.110 --> 01:16:33.530 And if you look at the reaction energies, 01:16:33.530 --> 01:16:34.950 you're not considerably off. 01:16:34.950 --> 01:16:37.760 GGA gives you 2.8 electronvolts for this reaction. 01:16:37.760 --> 01:16:40.320 Experiment is 3.5. 01:16:40.320 --> 01:16:44.150 This one, which is very similar, the error is 30%. 01:16:44.150 --> 01:16:46.190 You get 3.3 electronvolts. 01:16:46.190 --> 01:16:48.260 Experiment is 4.6. 01:16:48.260 --> 01:16:49.880 Why is that? 01:16:49.880 --> 01:16:52.820 Well, it has to do with the lack of error cancellation. 01:16:52.820 --> 01:16:54.380 If you look in detail what happens 01:16:54.380 --> 01:16:56.450 to the electronic structure in these materials, 01:16:56.450 --> 01:16:58.710 these are redox reactions. 01:16:58.710 --> 01:17:02.240 So if you do the math on the valences-- and believe me, 01:17:02.240 --> 01:17:06.220 this ion here is 3 plus. 01:17:06.220 --> 01:17:07.780 Phosphorus, 5 plus. 01:17:07.780 --> 01:17:09.760 Oxygen is 2 minus, so you can do the math. 01:17:09.760 --> 01:17:14.200 Iron here is 2 plus, and lithium is 1 plus. 01:17:14.200 --> 01:17:16.408 So what has happened in this reaction? 01:17:16.408 --> 01:17:18.700 Well, you've taken essentially an electron from lithium 01:17:18.700 --> 01:17:24.910 in it's metallic state, and put it on the iron 3 plus 01:17:24.910 --> 01:17:27.190 to make iron 2 plus. 01:17:27.190 --> 01:17:29.980 So essentially, you've transferred an electron 01:17:29.980 --> 01:17:35.240 from metallic lithium and ionized lithium to the iron 01:17:35.240 --> 01:17:37.940 to reduce it from 3 plus to 2 plus. 01:17:37.940 --> 01:17:39.800 But think about what's that doing? 01:17:39.800 --> 01:17:42.003 That electron in lithium-- 01:17:42.003 --> 01:17:43.670 lithium is there in the alkaline metals. 01:17:43.670 --> 01:17:45.210 That's an s electron. 01:17:45.210 --> 01:17:47.840 So that's a wide delocalized orbital, 01:17:47.840 --> 01:17:49.250 and I think it's metallic-- 01:17:49.250 --> 01:17:54.440 and you're transferring it to a localized d state on the iron. 01:17:54.440 --> 01:17:56.900 So that electron is essentially being transferred 01:17:56.900 --> 01:17:59.750 between extremely different states, 01:17:59.750 --> 01:18:02.750 and this is what's killing you. 01:18:02.750 --> 01:18:05.780 Because you transfer between such different states, 01:18:05.780 --> 01:18:08.900 you start losing a lot of the sort of cancellation of errors 01:18:08.900 --> 01:18:09.920 that you need. 01:18:09.920 --> 01:18:12.650 That's the functional theory. 01:18:12.650 --> 01:18:15.980 And in particular, the error here 01:18:15.980 --> 01:18:17.960 comes from something quite particular. 01:18:23.190 --> 01:18:28.030 It comes from what we call the self-interaction error. 01:18:28.030 --> 01:18:29.880 So I'm trying to sort make you understand 01:18:29.880 --> 01:18:31.590 where these errors come from so that when 01:18:31.590 --> 01:18:35.400 you work on your application, you get a bit of a feeling for. 01:18:35.400 --> 01:18:39.090 If you remember how we solve all these quantum 01:18:39.090 --> 01:18:41.040 mechanical equations, we reduce them 01:18:41.040 --> 01:18:43.575 to one electronic equations, where 01:18:43.575 --> 01:18:45.700 you have the kinetic energy, the nuclear potential. 01:18:45.700 --> 01:18:48.025 And then this effective potential 01:18:48.025 --> 01:18:49.650 which, remember what all goes in there. 01:18:49.650 --> 01:18:51.150 The effective the potential-- that's 01:18:51.150 --> 01:18:53.400 the one that has the exchange correlation in it, 01:18:53.400 --> 01:18:58.350 but it also has the Hartree field, so the coulombic field 01:18:58.350 --> 01:18:59.790 from all the electrons. 01:18:59.790 --> 01:19:04.500 Well, that field includes the electron itself. 01:19:04.500 --> 01:19:06.330 That's the sort of oddity in essence. 01:19:06.330 --> 01:19:08.250 When you calculate the charge density, 01:19:08.250 --> 01:19:10.440 that's the charge density of all the electrons. 01:19:10.440 --> 01:19:12.390 We then calculate the potential coming 01:19:12.390 --> 01:19:16.020 from the charge density, that's the potential coming from all 01:19:16.020 --> 01:19:17.280 the electrons. 01:19:17.280 --> 01:19:20.350 But you operate that now on a single electron. 01:19:20.350 --> 01:19:24.140 So the electron is feeling its own potential. 01:19:24.140 --> 01:19:27.410 Part of the exchange correlation term 01:19:27.410 --> 01:19:31.057 corrects for that, but not all of it. 01:19:31.057 --> 01:19:32.640 And the problem is that the correction 01:19:32.640 --> 01:19:35.670 doesn't operate as well on different forms of the charge 01:19:35.670 --> 01:19:36.520 density. 01:19:36.520 --> 01:19:41.350 In a metal, you have a small self-interaction error. 01:19:41.350 --> 01:19:44.430 And the reason is if you look at a state in a metal, 01:19:44.430 --> 01:19:45.840 sort of very delocalized. 01:19:45.840 --> 01:19:48.220 So very spread out charge density. 01:19:48.220 --> 01:19:51.360 So if you want to think of it, the part of the electron 01:19:51.360 --> 01:19:54.630 here doesn't feel much of the charge density coming-- 01:19:54.630 --> 01:19:56.850 of the potential coming from that piece of the charge 01:19:56.850 --> 01:20:00.820 density, because they're very far away. 01:20:00.820 --> 01:20:05.080 Whereas if you do a very localized state, in some sense, 01:20:05.080 --> 01:20:08.230 then the potential from the electron 01:20:08.230 --> 01:20:11.320 is very high where the electron itself is sitting, 01:20:11.320 --> 01:20:13.090 because it's all very close. 01:20:13.090 --> 01:20:16.840 If you put an electron in a delta function if you-- 01:20:16.840 --> 01:20:19.202 if you didn't have an uncertainty principle, 01:20:19.202 --> 01:20:21.160 and you calculate its potential, it's basically 01:20:21.160 --> 01:20:22.810 sitting on top of itself then. 01:20:22.810 --> 01:20:26.620 You'd have an infinite self-interaction. 01:20:26.620 --> 01:20:30.190 So the more local the state is, the more self-interaction 01:20:30.190 --> 01:20:31.180 you have. 01:20:31.180 --> 01:20:32.980 And the exchange correlation functional 01:20:32.980 --> 01:20:37.000 can't quite correct these two in the same way. 01:20:37.000 --> 01:20:39.760 And remember that the exchange correlation correction comes 01:20:39.760 --> 01:20:43.420 from homogeneous charge densities, 01:20:43.420 --> 01:20:46.540 so it tends to correct the metallic state better 01:20:46.540 --> 01:20:48.400 than the localized state. 01:20:48.400 --> 01:20:52.960 And so this is why that redox reaction I showed you had a big 01:20:52.960 --> 01:20:56.980 error, because we were transferring from a state that 01:20:56.980 --> 01:21:00.040 was metallic-- the electron went from the lithium state-- 01:21:00.040 --> 01:21:02.590 to the transition metal state. 01:21:02.590 --> 01:21:04.510 And somehow, the self-interaction error 01:21:04.510 --> 01:21:07.270 doesn't cancel. 01:21:07.270 --> 01:21:10.030 And you will see things like that 01:21:10.030 --> 01:21:13.460 whenever you transfer electrons between quite different states, 01:21:13.460 --> 01:21:17.056 so that's something to keep in mind. 01:21:17.056 --> 01:21:19.720 I think I'm running out of time here, so yeah, let me 01:21:19.720 --> 01:21:20.230 stop here. 01:21:23.060 --> 01:21:24.830 So I'm summarizing here because this is 01:21:24.830 --> 01:21:26.900 the stuff I went over before. 01:21:26.900 --> 01:21:29.750 In general, you do pretty well. 01:21:29.750 --> 01:21:33.960 I think if I'd given this summary 10 years ago, 01:21:33.960 --> 01:21:37.100 I would have been even more optimistic, because most people 01:21:37.100 --> 01:21:40.460 worked on metals and semiconductors which 01:21:40.460 --> 01:21:42.640 tend to be fairly delocalized state, 01:21:42.640 --> 01:21:45.140 so LDA and GGA do quite well. 01:21:45.140 --> 01:21:49.310 I think as we dig into more complicated materials, 01:21:49.310 --> 01:21:54.140 we have learned more about the errors of LDA and GGA. 01:21:54.140 --> 01:21:55.580 But on sort of classic metals, you 01:21:55.580 --> 01:21:58.130 do pretty well with lattice constants, reaction energies, 01:21:58.130 --> 01:22:00.860 and cohesive energies. 01:22:00.860 --> 01:22:03.500 But now there is a series of methods under development-- 01:22:03.500 --> 01:22:06.230 and if we have some time, we might sort of just broach 01:22:06.230 --> 01:22:07.040 them-- 01:22:07.040 --> 01:22:09.200 to deal better with the correlation energy, 01:22:09.200 --> 01:22:12.020 with the self-interaction energy to solve 01:22:12.020 --> 01:22:13.760 these problems of both energetics 01:22:13.760 --> 01:22:17.090 and also electronic structure-- things such as the band gap. 01:22:17.090 --> 01:22:18.050 So I'll end here. 01:22:18.050 --> 01:22:20.450 And remember, on Tuesday, we have lab. 01:22:20.450 --> 01:22:24.970 So you meet in the lab, and then Thursday, we'll be back here.