1 00:00:00,000 --> 00:00:02,250 NICOLA MARZARI: --for all our applications 2 00:00:02,250 --> 00:00:05,820 and for the lab sessions. 3 00:00:05,820 --> 00:00:07,590 I guess I keep using Albert Einstein. 4 00:00:07,590 --> 00:00:12,580 This is the 100th anniversary of his sort of famous year, 1905, 5 00:00:12,580 --> 00:00:15,180 so just a little celebration. 6 00:00:15,180 --> 00:00:18,300 One slide of a reminder of what we 7 00:00:18,300 --> 00:00:21,790 have seen in the previous lecture, 8 00:00:21,790 --> 00:00:24,060 we have really developed the formalism leading 9 00:00:24,060 --> 00:00:26,730 to the Hartree-Fock equations. 10 00:00:26,730 --> 00:00:29,460 And the Hartree-Fock equation follow 11 00:00:29,460 --> 00:00:33,030 from a set of very simple and very beautiful path. 12 00:00:33,030 --> 00:00:35,010 We have the Schrodinger equation, 13 00:00:35,010 --> 00:00:38,250 and we have reformulated the Schrodinger equation in terms 14 00:00:38,250 --> 00:00:40,060 of the variational principle. 15 00:00:40,060 --> 00:00:41,520 So we have a functional. 16 00:00:41,520 --> 00:00:44,400 And we know that we can throw into that functional 17 00:00:44,400 --> 00:00:47,970 any arbitrary wave function, and it'll give us an expectation 18 00:00:47,970 --> 00:00:49,560 value of the energy. 19 00:00:49,560 --> 00:00:53,580 And sort of the closer we get to the true ground state wave 20 00:00:53,580 --> 00:00:56,760 function, the lower that energy is going to be. 21 00:00:56,760 --> 00:01:00,560 We are never going to go below the ground state energy. 22 00:01:00,560 --> 00:01:03,530 And so it's sort of a very powerful approach 23 00:01:03,530 --> 00:01:07,380 to try out sorts of possibilities and solution. 24 00:01:07,380 --> 00:01:10,440 And in particular, sort of Hartree and Fock 25 00:01:10,440 --> 00:01:11,470 took this approach. 26 00:01:11,470 --> 00:01:16,170 They wrote sort of the most general many-body wave 27 00:01:16,170 --> 00:01:21,180 function that can be written as a product of single particle 28 00:01:21,180 --> 00:01:22,260 orbitals. 29 00:01:22,260 --> 00:01:25,140 That was actually the original Hartree solution. 30 00:01:25,140 --> 00:01:27,570 Wave functions written as data do not 31 00:01:27,570 --> 00:01:31,140 satisfy a fundamental symmetry of interacting fermions. 32 00:01:31,140 --> 00:01:33,000 That is they are not anti-symmetric. 33 00:01:33,000 --> 00:01:34,230 And so what you do? 34 00:01:34,230 --> 00:01:38,160 You take this product of single particle orbitals, 35 00:01:38,160 --> 00:01:41,100 and you sum it with all the possible permutation, 36 00:01:41,100 --> 00:01:43,590 with all the possible signs in front, 37 00:01:43,590 --> 00:01:46,740 so that the overall wave function is anti-symmetric. 38 00:01:46,740 --> 00:01:49,140 And that can be sort of written compactly 39 00:01:49,140 --> 00:01:52,680 as what is called as later determinant here. 40 00:01:52,680 --> 00:02:00,510 And basically, now our unknowns are the n orbitals phi. 41 00:02:00,510 --> 00:02:05,100 And so we need to determine the shape of this n single particle 42 00:02:05,100 --> 00:02:06,180 orbitals. 43 00:02:06,180 --> 00:02:07,980 And we want to determine them such 44 00:02:07,980 --> 00:02:11,070 that they minimize the expectation value 45 00:02:11,070 --> 00:02:13,380 of the variational principle. 46 00:02:13,380 --> 00:02:17,700 And so that leads basically to a set of differential equation 47 00:02:17,700 --> 00:02:19,750 is just functional analysis. 48 00:02:19,750 --> 00:02:22,560 And when you ask yourself what are the conditions 49 00:02:22,560 --> 00:02:24,780 that those single particle orbitals need 50 00:02:24,780 --> 00:02:27,750 to satisfy in order to minimize the variational principle, 51 00:02:27,750 --> 00:02:31,390 well, this is it, the Hartree-Fock equation. 52 00:02:31,390 --> 00:02:35,700 So each single particle orbital phi of lambda 53 00:02:35,700 --> 00:02:39,330 need to satisfy basically a Schrodinger-like equation. 54 00:02:39,330 --> 00:02:42,870 Again, as always, there is a kinetic energy term here. 55 00:02:42,870 --> 00:02:46,440 There is the interaction with the external potential 56 00:02:46,440 --> 00:02:48,810 that is just the potential of the nuclei. 57 00:02:48,810 --> 00:02:51,400 And then come the so-called mean field terms. 58 00:02:51,400 --> 00:02:55,230 So the electron lambda here will interact 59 00:02:55,230 --> 00:02:57,300 with each and every other electron 60 00:02:57,300 --> 00:03:01,390 mu via an electrostatic interaction. 61 00:03:01,390 --> 00:03:05,100 You see phi star times phi is the charge density coming 62 00:03:05,100 --> 00:03:06,510 from the orbital mu. 63 00:03:06,510 --> 00:03:10,350 And the field that the electron lambda fills 64 00:03:10,350 --> 00:03:13,080 is the electrostatic average density. 65 00:03:13,080 --> 00:03:17,280 And in these, we sum over all the electrons 66 00:03:17,280 --> 00:03:19,440 including the electron lambda. 67 00:03:19,440 --> 00:03:24,360 So up to now, we have a system that is self-interacting. 68 00:03:24,360 --> 00:03:28,020 An electron lambda fills the electrostatic interaction 69 00:03:28,020 --> 00:03:28,920 with itself. 70 00:03:28,920 --> 00:03:30,930 That, in principle, is not correct. 71 00:03:30,930 --> 00:03:34,890 But luckily, this next term that is called the exchange term 72 00:03:34,890 --> 00:03:36,840 cancels that exactly. 73 00:03:36,840 --> 00:03:41,160 And the exchange term is the direct consequence 74 00:03:41,160 --> 00:03:44,160 of having written the trial wave function not just 75 00:03:44,160 --> 00:03:46,860 as a product of a single particle orbital, 76 00:03:46,860 --> 00:03:48,780 because up to now we would have sort 77 00:03:48,780 --> 00:03:51,210 of something closer to the Hartree equation, 78 00:03:51,210 --> 00:03:53,600 but written as a proper anti-symmetric wave 79 00:03:53,600 --> 00:03:57,000 function, summing on all the permutation with them 80 00:03:57,000 --> 00:03:58,560 appropriate signs. 81 00:03:58,560 --> 00:04:00,990 And so, basically, we have Schrodinger-like equation. 82 00:04:00,990 --> 00:04:04,230 A great advantage with respect to the Hartree equation 83 00:04:04,230 --> 00:04:09,900 is now the operator doesn't change depending on the index 84 00:04:09,900 --> 00:04:13,560 lambda because this sums if you want to go over all 85 00:04:13,560 --> 00:04:15,480 the electrons including lambda. 86 00:04:15,480 --> 00:04:17,880 So our only constraint here is that we 87 00:04:17,880 --> 00:04:22,440 need to find the n lowest Eigen state 88 00:04:22,440 --> 00:04:24,820 of this single differential equation. 89 00:04:24,820 --> 00:04:27,760 So if we have n electrons, if you want, 90 00:04:27,760 --> 00:04:30,600 it's not that we have n different differential 91 00:04:30,600 --> 00:04:33,840 equation, like it was the case of the Hartree equation. 92 00:04:33,840 --> 00:04:36,780 But we have an identical differential equation 93 00:04:36,780 --> 00:04:37,860 that is written here. 94 00:04:37,860 --> 00:04:42,070 And we need to find the n lowest energy states. 95 00:04:42,070 --> 00:04:45,090 And those will be our single-particle orbitals. 96 00:04:45,090 --> 00:04:48,330 In all of this, we have started from a variational principle. 97 00:04:48,330 --> 00:04:52,320 So it's very easy to go beyond the Hartree-Fock. 98 00:04:52,320 --> 00:04:55,440 We can say, in large r variational cluster, 99 00:04:55,440 --> 00:04:58,560 we can add more Slater determinants 100 00:04:58,560 --> 00:05:00,350 with sort of different coefficients. 101 00:05:00,350 --> 00:05:04,760 We can try to construct a more complex wave function. 102 00:05:04,760 --> 00:05:07,370 And that solution will become better and better. 103 00:05:07,370 --> 00:05:10,460 Or we can sort of use the perturbation theory. 104 00:05:10,460 --> 00:05:13,550 And so quantum chemistry has developed 105 00:05:13,550 --> 00:05:17,750 a number of techniques that are post-Hartree-Fock techniques 106 00:05:17,750 --> 00:05:21,440 that become systematically more and more accurate. 107 00:05:21,440 --> 00:05:24,230 They are also more and more expensive. 108 00:05:24,230 --> 00:05:28,650 And that's if you want, the main limitation of that direction. 109 00:05:28,650 --> 00:05:31,010 What we'll see today is something, 110 00:05:31,010 --> 00:05:34,050 as they say in Monty Python, completely different. 111 00:05:34,050 --> 00:05:36,710 And that will be sort of density functional theory. 112 00:05:36,710 --> 00:05:38,180 That's, if you want a theory that 113 00:05:38,180 --> 00:05:45,080 starts from a very different set of hypotheses, the net result 114 00:05:45,080 --> 00:05:49,280 will be, again, a set of single-particle equations. 115 00:05:49,280 --> 00:05:52,280 The concept are very similar actually, formally, 116 00:05:52,280 --> 00:05:54,500 to the Hartree-Fock equation, but they 117 00:05:54,500 --> 00:05:58,730 have been derived in a completely different spirit. 118 00:05:58,730 --> 00:06:00,470 Density functional theory tends to be 119 00:06:00,470 --> 00:06:03,890 less expensive than Hartree-Fock and, overall, 120 00:06:03,890 --> 00:06:05,910 tends to be more accurate. 121 00:06:05,910 --> 00:06:08,510 Especially for solid, it's much more accurate. 122 00:06:08,510 --> 00:06:11,840 You'll see when we discuss case studies the Hartree-Fock 123 00:06:11,840 --> 00:06:15,500 solution for, say, interacting electron gas 124 00:06:15,500 --> 00:06:18,320 or in general for metals tends to make 125 00:06:18,320 --> 00:06:21,170 them semiconducting or insulating-like. 126 00:06:21,170 --> 00:06:24,320 So Hartree-Fock then works very poorly for solids. 127 00:06:24,320 --> 00:06:26,840 And that's why, if you want density functional theory, 128 00:06:26,840 --> 00:06:28,790 comes from the solid state community, 129 00:06:28,790 --> 00:06:32,330 while Hartree-Fock that tends to work very well for atoms 130 00:06:32,330 --> 00:06:35,390 comes from the quantum chemistry community. 131 00:06:35,390 --> 00:06:40,600 And all the theory was developed by Walter Kohn, and coworkers. 132 00:06:40,600 --> 00:06:42,515 So you'll see the Hohenberg and Kohn 133 00:06:42,515 --> 00:06:46,460 theorem, the Kohn [INAUDIBLE] mapping during the '60s. 134 00:06:46,460 --> 00:06:49,700 But I would say it's only during the '70s 135 00:06:49,700 --> 00:06:52,700 that people started to be able to actually solve 136 00:06:52,700 --> 00:06:55,760 interesting cases using density functional theory. 137 00:06:55,760 --> 00:06:57,620 And it's really the beginning of the '80s-- 138 00:06:57,620 --> 00:06:59,660 you'll see some cases here today-- 139 00:06:59,660 --> 00:07:02,600 in which people started calculating something that had 140 00:07:02,600 --> 00:07:06,090 sort of a direct application. 141 00:07:06,090 --> 00:07:08,900 So we'll see the phrase diagram of silicon 142 00:07:08,900 --> 00:07:11,060 as a function of pressure or volume 143 00:07:11,060 --> 00:07:14,720 and sort of the first principle prediction of properties 144 00:07:14,720 --> 00:07:16,160 of solids. 145 00:07:16,160 --> 00:07:19,700 Walter Kohn, for the development of this eventual theory, 146 00:07:19,700 --> 00:07:23,360 got the Nobel Prize for chemistry in 1998 147 00:07:23,360 --> 00:07:25,040 together with John Pople. 148 00:07:25,040 --> 00:07:30,920 That has been the person that's been fundamental 149 00:07:30,920 --> 00:07:32,900 in the development of Hartree-Fock 150 00:07:32,900 --> 00:07:36,840 and post-Hartree-Fock approaches in quantum chemistry. 151 00:07:36,840 --> 00:07:37,340 OK. 152 00:07:37,340 --> 00:07:41,150 So let's see sort of what is the general idea behind density 153 00:07:41,150 --> 00:07:42,200 functional theory. 154 00:07:42,200 --> 00:07:44,900 And in many ways, we'll sort of start 155 00:07:44,900 --> 00:07:47,240 from ideas that had been developed 156 00:07:47,240 --> 00:07:50,660 at the end of the '20s and the beginning of the '30s, what 157 00:07:50,660 --> 00:07:54,030 is nowadays called the Thomas-Fermi approach. 158 00:07:54,030 --> 00:07:56,210 And again, the basic idea here is 159 00:07:56,210 --> 00:08:00,710 that the wave function of a many-body interacting problem 160 00:08:00,710 --> 00:08:03,500 is an object that is too complex to treat. 161 00:08:03,500 --> 00:08:07,160 And it would be very, very nice if we could instead try 162 00:08:07,160 --> 00:08:09,380 to deal with a simple object. 163 00:08:09,380 --> 00:08:12,610 And sort of one of the choices could be the charge density. 164 00:08:12,610 --> 00:08:15,260 So if you want, Thomas and Fermi independently 165 00:08:15,260 --> 00:08:17,600 were asking themselves, well, could we 166 00:08:17,600 --> 00:08:20,570 try to solve not really a Schrodinger 167 00:08:20,570 --> 00:08:23,180 equation in the many-body wave function, 168 00:08:23,180 --> 00:08:27,500 but solve something else in which our only unknown is 169 00:08:27,500 --> 00:08:28,670 the charge density? 170 00:08:28,670 --> 00:08:30,650 If you think for a moment, the charge density 171 00:08:30,650 --> 00:08:33,350 is one of the sort of fundamental variables 172 00:08:33,350 --> 00:08:37,200 in the description of our interacting electron problem. 173 00:08:37,200 --> 00:08:39,480 And so this was the question. 174 00:08:39,480 --> 00:08:42,830 Can we do something just with the charge density? 175 00:08:42,830 --> 00:08:45,620 And so what they did is writing out 176 00:08:45,620 --> 00:08:47,780 what we would call a heuristic function. 177 00:08:47,780 --> 00:08:54,050 That is trying to devise a set of terms that would give us 178 00:08:54,050 --> 00:08:57,770 the energy of a set of electrons in a potential 179 00:08:57,770 --> 00:09:01,880 just as a functional of their charge density. 180 00:09:01,880 --> 00:09:04,700 And so, by now, you could sort of 181 00:09:04,700 --> 00:09:07,640 think that some of the relevant terms 182 00:09:07,640 --> 00:09:10,370 will be electron-electron interactions, electron 183 00:09:10,370 --> 00:09:11,150 interaction. 184 00:09:11,150 --> 00:09:14,600 And we could write a sort of electrostatic term, 185 00:09:14,600 --> 00:09:17,570 like the Hartree term in the Hartree or Hartree-Fock 186 00:09:17,570 --> 00:09:21,020 equation, that is just a functional of the charge 187 00:09:21,020 --> 00:09:21,640 density. 188 00:09:21,640 --> 00:09:23,850 So this is sort of fairly easy. 189 00:09:23,850 --> 00:09:26,840 It's also very easy to sort of imagine 190 00:09:26,840 --> 00:09:31,010 what could be the interaction of the electrons 191 00:09:31,010 --> 00:09:33,710 with an external potential through the charge density. 192 00:09:33,710 --> 00:09:37,010 It will be just the integral of that external potential 193 00:09:37,010 --> 00:09:38,570 times the charge density. 194 00:09:38,570 --> 00:09:42,950 What becomes really critical is finding 195 00:09:42,950 --> 00:09:47,720 a functional that will give us the quantum kinetic energy. 196 00:09:47,720 --> 00:09:49,610 If you think, in the Schrodinger equation, 197 00:09:49,610 --> 00:09:53,660 the quantum kinetic energy is really the second derivative 198 00:09:53,660 --> 00:09:55,310 of the wave function. 199 00:09:55,310 --> 00:09:59,800 And obtaining from a charge density 200 00:09:59,800 --> 00:10:03,400 only some insight into what could 201 00:10:03,400 --> 00:10:08,330 be the second derivative of the wave function is very complex. 202 00:10:08,330 --> 00:10:12,630 If you think for a moment at the extreme case of a plane wave, 203 00:10:12,630 --> 00:10:17,860 OK, so a sine and cosine sort of in space, if you remember, 204 00:10:17,860 --> 00:10:22,000 the charge density given by a plane wave is a constant. 205 00:10:22,000 --> 00:10:26,590 We just multiply the imaginary exponential 206 00:10:26,590 --> 00:10:28,330 times its complex conjugate. 207 00:10:28,330 --> 00:10:30,230 That gives us a constant. 208 00:10:30,230 --> 00:10:33,250 So all plane waves lead to a constant, 209 00:10:33,250 --> 00:10:36,550 but obviously the quantum kinetic energy of a plane wave 210 00:10:36,550 --> 00:10:39,790 depends on the wavelength of that plane wave 211 00:10:39,790 --> 00:10:43,130 because the second derivative is what counts. 212 00:10:43,130 --> 00:10:45,010 So what I'm trying to say is that, when 213 00:10:45,010 --> 00:10:51,090 we look at this as a possible wave function, a function, 214 00:10:51,090 --> 00:10:56,340 say, of r and the charge density that comes from this 215 00:10:56,340 --> 00:10:58,710 is going to be a constant, this wave function 216 00:10:58,710 --> 00:11:00,630 times this complex conjugate. 217 00:11:00,630 --> 00:11:03,510 But the kinetic energy of this object 218 00:11:03,510 --> 00:11:07,590 is going to be minus 1/2 k square-- 219 00:11:07,590 --> 00:11:09,870 sorry, plus 1/2 k square. 220 00:11:09,870 --> 00:11:13,170 And so there is really not a good way 221 00:11:13,170 --> 00:11:16,770 for this extreme case to correlate its charge 222 00:11:16,770 --> 00:11:18,330 density to its kinetic energy. 223 00:11:18,330 --> 00:11:19,870 It's an ill-defined problem. 224 00:11:19,870 --> 00:11:21,660 And this is really the difficulty. 225 00:11:21,660 --> 00:11:23,790 So there isn't really a good way if you 226 00:11:23,790 --> 00:11:26,010 wanted to extract the information 227 00:11:26,010 --> 00:11:30,150 on the second derivative from just the charge density. 228 00:11:30,150 --> 00:11:32,070 Now, [INAUDIBLE] sort of this objection, 229 00:11:32,070 --> 00:11:37,620 they tried to find a reasonable functional, so 230 00:11:37,620 --> 00:11:41,080 without sort of trying to get the exact solution, 231 00:11:41,080 --> 00:11:45,210 but try to find a reasonable function that would give us 232 00:11:45,210 --> 00:11:49,410 a good estimate to the quantum kinetic energy starting 233 00:11:49,410 --> 00:11:51,750 from the charge density. 234 00:11:51,750 --> 00:11:54,000 And the solution to this problem that 235 00:11:54,000 --> 00:11:56,280 is something very important is what 236 00:11:56,280 --> 00:12:00,550 we could call a local density approximation. 237 00:12:00,550 --> 00:12:05,850 So the problem here is that we have a non-homogeneous charge 238 00:12:05,850 --> 00:12:08,430 density everywhere in space. 239 00:12:08,430 --> 00:12:11,010 And we try to figure out what could 240 00:12:11,010 --> 00:12:16,260 be the quantum kinetic energy of this non-homogeneous problem. 241 00:12:16,260 --> 00:12:20,130 And the approximation that Thomas and Fermi did 242 00:12:20,130 --> 00:12:24,270 was, well, dividing this non-homogeneous problem 243 00:12:24,270 --> 00:12:27,880 in a set of infinitesimal volume in space. 244 00:12:27,880 --> 00:12:30,210 So it's a bit difficult to draw, but suppose 245 00:12:30,210 --> 00:12:33,450 you have the charge density coming 246 00:12:33,450 --> 00:12:36,390 from some atom or some molecule. 247 00:12:36,390 --> 00:12:41,070 This is a non-homogeneous charge density distribution. 248 00:12:41,070 --> 00:12:45,210 Now, what you do is you divide this in space 249 00:12:45,210 --> 00:12:47,970 and set a very small infinitesimal, if you want, 250 00:12:47,970 --> 00:12:48,930 volume. 251 00:12:48,930 --> 00:12:53,400 And inside each volume, the charge density 252 00:12:53,400 --> 00:12:56,040 can be approximated as a constant. 253 00:12:56,040 --> 00:12:57,930 And what Thomas and Fermi said is, 254 00:12:57,930 --> 00:13:04,170 well, the contribution coming from this infinitesimal volume, 255 00:13:04,170 --> 00:13:08,790 say, the first one to the overall quantum kinetic energy 256 00:13:08,790 --> 00:13:14,160 will be given by that volume times the kinetic energy 257 00:13:14,160 --> 00:13:20,430 density of the homogeneous electron gas at that density. 258 00:13:20,430 --> 00:13:23,790 So if, again, we partition all space, 259 00:13:23,790 --> 00:13:27,720 we could have that the density in this little cube is 0.5. 260 00:13:27,720 --> 00:13:28,780 Here is 0.6. 261 00:13:28,780 --> 00:13:30,450 Here is 0.7. 262 00:13:30,450 --> 00:13:34,170 Outside, it goes to 0. 263 00:13:34,170 --> 00:13:38,580 But we can actually calculate in some other way what 264 00:13:38,580 --> 00:13:42,150 would be the quantum kinetic energy 265 00:13:42,150 --> 00:13:44,770 of a homogeneous electron gas. 266 00:13:44,770 --> 00:13:46,430 That's a problem that we can solve 267 00:13:46,430 --> 00:13:49,590 if the homogeneous electron gas is not interacting. 268 00:13:49,590 --> 00:13:53,640 And we can solve it numerically even if it is interacting. 269 00:13:53,640 --> 00:13:57,300 So we can know what is the quantum kinetic energy 270 00:13:57,300 --> 00:14:00,450 of a homogeneous gas with density 0.5, 271 00:14:00,450 --> 00:14:03,060 density 0.6, density 0.7. 272 00:14:03,060 --> 00:14:05,250 And so we can also know what would 273 00:14:05,250 --> 00:14:10,380 be the quantum kinetic energy per unit of volume of data. 274 00:14:10,380 --> 00:14:14,010 And so we'll say that this non-homogeneous system in blue 275 00:14:14,010 --> 00:14:18,180 will have an overall quantum kinetic energy that is given 276 00:14:18,180 --> 00:14:21,810 really by the integral across space-- 277 00:14:21,810 --> 00:14:23,280 and it's written here-- 278 00:14:23,280 --> 00:14:29,250 of the quantum kinetic energy of the homogeneous electron gas 279 00:14:29,250 --> 00:14:31,170 integrated over space. 280 00:14:31,170 --> 00:14:35,040 And, say, for the non-interacting electron gas, 281 00:14:35,040 --> 00:14:36,960 it's actually very easy to do. 282 00:14:36,960 --> 00:14:42,120 So if you have a non-interacting electron gas at a density rho, 283 00:14:42,120 --> 00:14:46,860 its quantum kinetic energy is just the rho to the 2/3 284 00:14:46,860 --> 00:14:49,470 that then integrated times the unit volume 285 00:14:49,470 --> 00:14:52,350 gives us rho to the 5/3. 286 00:14:52,350 --> 00:14:57,180 So by integrating this quantity, we would get an approximation. 287 00:14:57,180 --> 00:14:59,670 This approximation is basically exact 288 00:14:59,670 --> 00:15:03,780 in the limit of our homogeneous system, obviously. 289 00:15:03,780 --> 00:15:07,140 And it will be sort of quite good 290 00:15:07,140 --> 00:15:11,400 in the limit of our non-homogeneous system that 291 00:15:11,400 --> 00:15:15,420 has a very slowly changing charge density. 292 00:15:15,420 --> 00:15:18,300 The more, if you want, inhomogeneous 293 00:15:18,300 --> 00:15:23,700 your system becomes, the less accurate this approximation is. 294 00:15:23,700 --> 00:15:27,060 And of course, something like an atom or a molecule 295 00:15:27,060 --> 00:15:28,950 is a very inhomogeneous system. 296 00:15:28,950 --> 00:15:31,050 You go with a charge density that 297 00:15:31,050 --> 00:15:35,610 goes from 0 to very high volumes close to the core 298 00:15:35,610 --> 00:15:36,270 of the nuclei. 299 00:15:40,600 --> 00:15:43,740 So this is basically the overall answer 300 00:15:43,740 --> 00:15:47,430 for the overall expression that Thomas and Fermi 301 00:15:47,430 --> 00:15:52,080 postulated for the energy of an inhomogeneous system. 302 00:15:52,080 --> 00:15:54,330 They were saying, well, suppose that we 303 00:15:54,330 --> 00:16:00,090 have a system that has a certain distribution of charge rho. 304 00:16:00,090 --> 00:16:04,290 Without trying to solve the Schrodinger equation finding 305 00:16:04,290 --> 00:16:07,860 out the wave function and sort of go through that the very 306 00:16:07,860 --> 00:16:12,510 complex many-body route, we can actually sort of postulate 307 00:16:12,510 --> 00:16:15,300 that the energy could be written, again, 308 00:16:15,300 --> 00:16:17,340 as an electrostatic energy. 309 00:16:17,340 --> 00:16:20,790 You see sort of each infinitesimal volume 310 00:16:20,790 --> 00:16:23,880 interacting with each other infinitesimal volume 311 00:16:23,880 --> 00:16:27,850 times via 1 over R electrostatic interaction. 312 00:16:27,850 --> 00:16:29,730 Then we have got an external potential. 313 00:16:29,730 --> 00:16:32,950 Again, it's usually the Coulombic field of the nuclei. 314 00:16:32,950 --> 00:16:34,890 And so the interaction between the electron 315 00:16:34,890 --> 00:16:38,040 and that external potential is just trivial given by rho 316 00:16:38,040 --> 00:16:38,970 times v. 317 00:16:38,970 --> 00:16:42,270 And the difficult term, the quantum kinetic energy, 318 00:16:42,270 --> 00:16:46,650 has been calculated with a local density approximation. 319 00:16:46,650 --> 00:16:49,920 And this is the term that's not going to be very good, 320 00:16:49,920 --> 00:16:53,130 again, because it's very difficult to figure out 321 00:16:53,130 --> 00:16:56,040 what could be the curvature of our wave function 322 00:16:56,040 --> 00:16:59,670 just from the density that that wave function produces. 323 00:16:59,670 --> 00:17:03,570 But anyhow, this is a very simple expression to deal with. 324 00:17:03,570 --> 00:17:07,020 So for any external potential v, we 325 00:17:07,020 --> 00:17:12,000 can try to find out the rho that minimizes this expression. 326 00:17:12,000 --> 00:17:14,915 And this will be our Thomas-Fermi solution. 327 00:17:23,010 --> 00:17:24,810 There are obviously a number of problems. 328 00:17:24,810 --> 00:17:26,910 I'll show you in a moment an example of what 329 00:17:26,910 --> 00:17:30,780 the Thomas-Fermi solution would give to an atom. 330 00:17:30,780 --> 00:17:32,610 First of all, I mean, there is really 331 00:17:32,610 --> 00:17:35,140 no theoretical basis to this. 332 00:17:35,140 --> 00:17:37,790 It's what we call a heuristic derivation. 333 00:17:37,790 --> 00:17:39,690 Thomas and Fermi just wrote out what 334 00:17:39,690 --> 00:17:42,900 could be a reasonable energy functional, 335 00:17:42,900 --> 00:17:47,220 and then tried to sort of see what results it would give. 336 00:17:47,220 --> 00:17:50,940 But there hasn't been any kind of formal derivation 337 00:17:50,940 --> 00:17:52,050 of that functional. 338 00:17:52,050 --> 00:17:55,740 It's not like the Hartree-Fock equation that sort of derive 339 00:17:55,740 --> 00:17:59,820 just with some analysis from the variational principle. 340 00:17:59,820 --> 00:18:02,850 Another problem is that, again, it doesn't really 341 00:18:02,850 --> 00:18:08,010 sort of introduce the concept of anti-symmetry that fermions 342 00:18:08,010 --> 00:18:11,160 need to have, the fact that the many-body wave function needs 343 00:18:11,160 --> 00:18:14,430 to be anti-symmetric upon exchange. 344 00:18:14,430 --> 00:18:17,910 But you know, there is no conceptual problem 345 00:18:17,910 --> 00:18:23,580 in adding an exchange energy to the previous functional. 346 00:18:23,580 --> 00:18:26,520 Using the same concept, the same idea 347 00:18:26,520 --> 00:18:29,610 of local density approximation, suppose that we 348 00:18:29,610 --> 00:18:32,280 want to add an exchange term. 349 00:18:32,280 --> 00:18:37,380 Well, we could look at what is the exchange energy coming 350 00:18:37,380 --> 00:18:39,840 from the Hartree-Fock equation, say, 351 00:18:39,840 --> 00:18:42,690 for a homogeneous electron gas. 352 00:18:42,690 --> 00:18:47,040 And that gives us rho to the 1/3 term. 353 00:18:47,040 --> 00:18:51,240 And that's basically the exchange energy density. 354 00:18:51,240 --> 00:18:54,090 And so for an inhomogeneous system, 355 00:18:54,090 --> 00:18:57,240 we are going to sort of approximate its overall 356 00:18:57,240 --> 00:19:03,090 exchange energy just by taking the integral of that energy 357 00:19:03,090 --> 00:19:08,250 density that is 1/3 times the sort of local value 358 00:19:08,250 --> 00:19:09,570 of the charge density. 359 00:19:09,570 --> 00:19:11,640 And so we have a rho to the 4/3. 360 00:19:11,640 --> 00:19:17,610 And so, again, it's a local density approximation. 361 00:19:17,610 --> 00:19:22,860 The great consequence of having this energy functional 362 00:19:22,860 --> 00:19:25,080 that depends only on r is that it 363 00:19:25,080 --> 00:19:28,260 is absolutely inexpensive from the computational point 364 00:19:28,260 --> 00:19:28,890 of view. 365 00:19:28,890 --> 00:19:31,830 The only variable that we need to be concerned with 366 00:19:31,830 --> 00:19:36,400 is just the a scalar as a function of three coordinates. 367 00:19:36,400 --> 00:19:39,120 That is the density as a function of rho. 368 00:19:39,120 --> 00:19:42,390 And it's what we call a linear scaling system. 369 00:19:42,390 --> 00:19:45,360 If you double the size of your system, 370 00:19:45,360 --> 00:19:49,380 the computational complexity just becomes double. 371 00:19:49,380 --> 00:19:51,510 So it has a lot of very good things, 372 00:19:51,510 --> 00:19:53,820 but it's got a fundamental defect. 373 00:19:53,820 --> 00:19:57,090 Because of that approximation in the kinetic energy, 374 00:19:57,090 --> 00:20:02,280 it actually does a very poor job in describing 375 00:20:02,280 --> 00:20:04,800 a non-homogeneous system. 376 00:20:04,800 --> 00:20:09,510 So it would work reasonably well for something like a metal. 377 00:20:09,510 --> 00:20:12,570 Suppose that you want to describe sodium, 378 00:20:12,570 --> 00:20:15,360 or suppose you want to describe aluminum. 379 00:20:15,360 --> 00:20:21,030 Those are system in which the valence electron produce 380 00:20:21,030 --> 00:20:23,730 a charge density that is very homogeneous. 381 00:20:23,730 --> 00:20:26,880 So a Thomas-Fermi approach could actually work well. 382 00:20:26,880 --> 00:20:29,820 And it's actually been used even very recently sort 383 00:20:29,820 --> 00:20:33,060 of quite successfully to describe problems 384 00:20:33,060 --> 00:20:36,930 like the surfaces of lithium, the surfaces of aluminum. 385 00:20:36,930 --> 00:20:41,280 What happens, say, when these simple metals melt? 386 00:20:41,280 --> 00:20:43,110 What happens to the sort of formation 387 00:20:43,110 --> 00:20:44,850 of defects in aluminum? 388 00:20:44,850 --> 00:20:46,890 So there are a number of successes. 389 00:20:46,890 --> 00:20:50,070 But sort of clear example of what goes wrong 390 00:20:50,070 --> 00:20:53,130 is, say, if we study an inhomogeneous systems 391 00:20:53,130 --> 00:20:55,090 like the argon atom. 392 00:20:55,090 --> 00:20:59,340 And again, if we think at the charge density of the argon 393 00:20:59,340 --> 00:21:02,640 atom as a function, say, of the radial distance 394 00:21:02,640 --> 00:21:04,650 from the center from the nucleus, 395 00:21:04,650 --> 00:21:06,490 well, it will look something like this. 396 00:21:06,490 --> 00:21:11,280 We have first 1s, and then we have the 2s, and the 2p shells. 397 00:21:11,280 --> 00:21:14,310 This is somewhat a poor depiction 398 00:21:14,310 --> 00:21:15,750 of that charge density. 399 00:21:15,750 --> 00:21:18,870 If we try to solve the argon atom 400 00:21:18,870 --> 00:21:22,110 with a Thomas-Fermi approach, all these sort 401 00:21:22,110 --> 00:21:28,740 of fine structure of the core shells in the atoms 402 00:21:28,740 --> 00:21:30,690 is completely washed out. 403 00:21:30,690 --> 00:21:31,410 OK. 404 00:21:31,410 --> 00:21:34,980 So it gives you a reasonable approximation and sort 405 00:21:34,980 --> 00:21:37,740 of an appropriate decay of the charge density 406 00:21:37,740 --> 00:21:41,130 as we move far away, but a lot of those details 407 00:21:41,130 --> 00:21:43,330 have completely disappeared. 408 00:21:43,330 --> 00:21:47,340 And for this reason really the Thomas-Fermi approach 409 00:21:47,340 --> 00:21:50,310 wasn't developed beyond the '30s apart 410 00:21:50,310 --> 00:21:52,980 from some of these recent applications 411 00:21:52,980 --> 00:21:55,890 for the very specific case of solids 412 00:21:55,890 --> 00:21:59,340 that have a very homogeneous charge density. 413 00:21:59,340 --> 00:22:01,290 The reason why we describe it here 414 00:22:01,290 --> 00:22:03,240 is that because, in many ways, it's 415 00:22:03,240 --> 00:22:06,720 the grandfather of the ideas that 416 00:22:06,720 --> 00:22:10,170 were developed in the '60s in density functional theory 417 00:22:10,170 --> 00:22:13,290 and, in particular, the idea that for a moment 418 00:22:13,290 --> 00:22:16,680 that we should focus not on the wave function, 419 00:22:16,680 --> 00:22:22,830 but on the charge density of the system as the key ingredient. 420 00:22:22,830 --> 00:22:25,860 The great difference between the Thomas-Fermi approach 421 00:22:25,860 --> 00:22:29,310 and density functional theory is that density functional theory 422 00:22:29,310 --> 00:22:30,810 actually is a theory. 423 00:22:30,810 --> 00:22:33,810 It starts with some theorems that are proven. 424 00:22:33,810 --> 00:22:38,700 And then it shows what are the form of the equations 425 00:22:38,700 --> 00:22:42,300 that, say, a charge density need to satisfy in order 426 00:22:42,300 --> 00:22:44,560 to solve exactly the problem. 427 00:22:44,560 --> 00:22:46,890 So in many way, density functional theory 428 00:22:46,890 --> 00:22:50,490 is, in principle at least, an exact theory. 429 00:22:50,490 --> 00:22:54,900 It's sort of writes out what are the equation that the charge 430 00:22:54,900 --> 00:22:56,850 density needs to satisfy. 431 00:22:56,850 --> 00:22:59,670 And those are absolutely equivalent to a Schrodinger 432 00:22:59,670 --> 00:23:01,860 equation for the wave function. 433 00:23:01,860 --> 00:23:03,160 There are some difficulties. 434 00:23:03,160 --> 00:23:06,900 And this is what we are going to sort of go into right now. 435 00:23:06,900 --> 00:23:11,250 But let me first give you the conceptual framework 436 00:23:11,250 --> 00:23:15,300 of density functional theory and how it was derived. 437 00:23:15,300 --> 00:23:18,730 And as usual, we start from the Schrodinger equation. 438 00:23:18,730 --> 00:23:19,440 OK. 439 00:23:19,440 --> 00:23:23,580 So we start from the idea that, in quantum mechanics, 440 00:23:23,580 --> 00:23:26,940 for any given external potential, 441 00:23:26,940 --> 00:23:30,320 you have a well-defined differential equation. 442 00:23:30,320 --> 00:23:30,900 OK. 443 00:23:30,900 --> 00:23:33,120 It's sort of very complex. 444 00:23:33,120 --> 00:23:35,780 It describes a many-body wave function. 445 00:23:35,780 --> 00:23:40,110 So in most practical cases, we might not be able to solve it, 446 00:23:40,110 --> 00:23:42,480 but everything is well-defined. 447 00:23:42,480 --> 00:23:44,370 You have an external potential. 448 00:23:44,370 --> 00:23:46,230 You have the differential equation 449 00:23:46,230 --> 00:23:48,810 that the many-body wave function needs to satisfy. 450 00:23:48,810 --> 00:23:51,910 And so, in principle, you have the solution. 451 00:23:51,910 --> 00:23:54,270 And so in that sense, sort of the first statement 452 00:23:54,270 --> 00:23:55,530 here is summarized. 453 00:23:55,530 --> 00:23:59,970 For a given external potential and knowing how many electrons 454 00:23:59,970 --> 00:24:03,540 are going to fill this potential, 455 00:24:03,540 --> 00:24:07,770 our quantum problem is formally completely defined. 456 00:24:07,770 --> 00:24:11,110 In principle, the solution exists unique. 457 00:24:11,110 --> 00:24:14,610 We might not be able to calculate it, but it exists. 458 00:24:14,610 --> 00:24:18,270 And once we know the many-body wave function, that solution, 459 00:24:18,270 --> 00:24:22,980 we know everything about our quantum system. 460 00:24:22,980 --> 00:24:23,700 OK. 461 00:24:23,700 --> 00:24:27,960 So this is, if you want, the trivial part of the conclusion. 462 00:24:27,960 --> 00:24:31,110 That is, given an external potential, 463 00:24:31,110 --> 00:24:35,400 we find, by the Schrodinger equation, the wave function. 464 00:24:35,400 --> 00:24:39,000 The wave function determine all the properties of our system 465 00:24:39,000 --> 00:24:42,600 and, in particular, determine the ground state charge 466 00:24:42,600 --> 00:24:43,780 density. 467 00:24:43,780 --> 00:24:46,710 So there is a unique pathway that 468 00:24:46,710 --> 00:24:49,560 starts from the external potential 469 00:24:49,560 --> 00:24:52,530 and leads us to the charge density, the ground state 470 00:24:52,530 --> 00:24:53,550 charge density. 471 00:24:53,550 --> 00:24:56,640 Once you have defined the potential, 472 00:24:56,640 --> 00:25:00,090 you, in principle, have uniquely defined 473 00:25:00,090 --> 00:25:04,180 what is the ground state's charge density of your system. 474 00:25:04,180 --> 00:25:07,860 And so in that sense, we say that the ground state charge 475 00:25:07,860 --> 00:25:11,250 density, the ground state energy, and all the properties 476 00:25:11,250 --> 00:25:14,670 of our system are, in some complex way, 477 00:25:14,670 --> 00:25:18,330 a functional of our external potential 478 00:25:18,330 --> 00:25:20,070 and the number of electrons. 479 00:25:20,070 --> 00:25:22,500 Functional, again, can be anything. 480 00:25:22,500 --> 00:25:25,200 And in this case, it goes through the Schrodinger 481 00:25:25,200 --> 00:25:30,180 equation, nothing sort of complex at this point. 482 00:25:30,180 --> 00:25:34,860 The sort of remarkable result that no one had sort of figured 483 00:25:34,860 --> 00:25:41,490 out between 1964 and 1965 is that the opposite is also 484 00:25:41,490 --> 00:25:44,020 true and is not trivial at all. 485 00:25:44,020 --> 00:25:47,740 So what Hohenberg and Kohn stated first, actually, 486 00:25:47,740 --> 00:25:53,310 in 1964, was this, that the ground state charge 487 00:25:53,310 --> 00:25:58,710 density is a fundamental quantity, 488 00:25:58,710 --> 00:26:01,860 as fundamental as the external potential. 489 00:26:01,860 --> 00:26:07,020 And in particular, not only the external potential, 490 00:26:07,020 --> 00:26:11,460 the terms uniquely the ground state's charge density 491 00:26:11,460 --> 00:26:15,510 of your system, but also the vice versa is true. 492 00:26:15,510 --> 00:26:20,250 That is, given a ground state charge density, 493 00:26:20,250 --> 00:26:23,760 in principle, one can prove that there 494 00:26:23,760 --> 00:26:29,730 is a unique external potential for which that ground state's 495 00:26:29,730 --> 00:26:33,270 charge density is the ground state solution 496 00:26:33,270 --> 00:26:35,470 for that external potential. 497 00:26:35,470 --> 00:26:38,160 So if you have the external potential, 498 00:26:38,160 --> 00:26:41,940 conceptually it's trivial to go through the Schrodinger 499 00:26:41,940 --> 00:26:45,120 equation and its solution to the charge density. 500 00:26:45,120 --> 00:26:47,760 What Hohenberg and Kohn are telling us-- 501 00:26:47,760 --> 00:26:51,060 and I'll just show you a sketch of the proof in a moment-- 502 00:26:51,060 --> 00:26:56,250 is that in principle, if someone is giving you a chance density 503 00:26:56,250 --> 00:26:58,770 and is telling you this charge density 504 00:26:58,770 --> 00:27:01,350 is the ground state's charge density 505 00:27:01,350 --> 00:27:04,290 of a number of electrons and electrons 506 00:27:04,290 --> 00:27:07,000 in an external potential, in principle, 507 00:27:07,000 --> 00:27:11,220 what is that external potential is an information that 508 00:27:11,220 --> 00:27:15,510 is completely contained into the charged density. 509 00:27:15,510 --> 00:27:18,300 And it's not contained in a trivial way. 510 00:27:18,300 --> 00:27:21,220 It's not that you can look at the ground state charge density 511 00:27:21,220 --> 00:27:24,220 and guess what the external potentially is. 512 00:27:24,220 --> 00:27:27,910 And that's where all the complexity of practical densely 513 00:27:27,910 --> 00:27:29,680 functional theory comes. 514 00:27:29,680 --> 00:27:33,050 But from the conceptual and mathematical point of view, 515 00:27:33,050 --> 00:27:35,950 these two quantities are absolutely equivalent. 516 00:27:35,950 --> 00:27:39,190 From one, you get the other and vice versa. 517 00:27:39,190 --> 00:27:45,370 And this sort of vice versa was not trivial. 518 00:27:45,370 --> 00:27:47,920 And that is sort of what is contained 519 00:27:47,920 --> 00:27:52,120 in the so-called first Hohenberg and Kohn problem. 520 00:27:52,120 --> 00:27:54,220 I won't go through the derivation. 521 00:27:54,220 --> 00:27:55,540 It's actually very simple. 522 00:27:55,540 --> 00:27:59,420 I've printed it here in case you sort of want to read it. 523 00:27:59,420 --> 00:28:01,870 But it's basically is a derivation ad absurdum. 524 00:28:01,870 --> 00:28:03,730 What they are saying there is that, 525 00:28:03,730 --> 00:28:08,200 if that external potential were not unique, 526 00:28:08,200 --> 00:28:11,440 if there were two external potential that 527 00:28:11,440 --> 00:28:15,640 were different and would give the same ground state energy, 528 00:28:15,640 --> 00:28:18,070 we would get to absurdum. 529 00:28:18,070 --> 00:28:20,750 So typical mathematical demonstration, 530 00:28:20,750 --> 00:28:23,860 we suppose that there are two different external potential 531 00:28:23,860 --> 00:28:26,650 that give the same ground states as density 532 00:28:26,650 --> 00:28:29,410 and we show that we arrive to a conclusion that 533 00:28:29,410 --> 00:28:30,710 doesn't make sense. 534 00:28:30,710 --> 00:28:34,630 So there can be only a single external potential. 535 00:28:34,630 --> 00:28:35,800 And that's the proof. 536 00:28:35,800 --> 00:28:37,330 And again, it wasn't trivial. 537 00:28:37,330 --> 00:28:40,030 I mean, if you want, this is a very basic statement. 538 00:28:40,030 --> 00:28:43,450 But it took 40 years to be formulated. 539 00:28:43,450 --> 00:28:47,500 And it's actually not true in other cases 540 00:28:47,500 --> 00:28:51,370 that to first glance look very similar. 541 00:28:51,370 --> 00:28:56,210 Suppose that for a moment we want to discuss excited states. 542 00:28:56,210 --> 00:28:58,480 You could say, well, if I have a charge density 543 00:28:58,480 --> 00:29:01,840 and I say this is the charge density of an excited 544 00:29:01,840 --> 00:29:04,600 electronic state, maybe I could also 545 00:29:04,600 --> 00:29:07,720 recover the potential that has generated there. 546 00:29:07,720 --> 00:29:09,160 And that's not true, actually. 547 00:29:09,160 --> 00:29:11,080 So there are sort of a number of cases 548 00:29:11,080 --> 00:29:12,640 in which this is not true. 549 00:29:12,640 --> 00:29:16,360 But for this fundamental sort of relation between the charge 550 00:29:16,360 --> 00:29:19,090 density of the ground state and the external potential, 551 00:29:19,090 --> 00:29:20,420 this is true. 552 00:29:20,420 --> 00:29:24,250 So we have sort of moved away now our attention. 553 00:29:24,250 --> 00:29:26,440 It's not any more the many-body wave 554 00:29:26,440 --> 00:29:29,910 function that we want to focus, but is the charge density. 555 00:29:29,910 --> 00:29:34,060 The charge density is as much a fundamental variable 556 00:29:34,060 --> 00:29:34,960 of our problem. 557 00:29:34,960 --> 00:29:36,470 It's not a derived variable. 558 00:29:36,470 --> 00:29:38,740 It's not something that comes from the wave function, 559 00:29:38,740 --> 00:29:40,282 but is something that we can actually 560 00:29:40,282 --> 00:29:44,050 focus all our attention into. 561 00:29:44,050 --> 00:29:49,750 And now, we need to find the equivalent of the Schrodinger 562 00:29:49,750 --> 00:29:51,900 equation for the charged density. 563 00:29:51,900 --> 00:29:55,600 This is what Schrodinger had done in the '20s, in 1925. 564 00:29:55,600 --> 00:29:59,500 He said, this is the equation that quantum objects satisfy. 565 00:29:59,500 --> 00:30:02,110 And I'll call it the Schrodinger equation. 566 00:30:02,110 --> 00:30:05,110 Now, Hohenberg and Kohn has shown 567 00:30:05,110 --> 00:30:08,920 that we don't need to think in terms of the wave function. 568 00:30:08,920 --> 00:30:11,170 We can think in terms of the charge density 569 00:30:11,170 --> 00:30:13,930 as being the fundamental descriptor of our quantum 570 00:30:13,930 --> 00:30:14,710 system. 571 00:30:14,710 --> 00:30:15,640 What is left? 572 00:30:15,640 --> 00:30:18,220 They need to show me that there is an equivalent 573 00:30:18,220 --> 00:30:19,570 of the Schrodinger equation. 574 00:30:19,570 --> 00:30:24,700 That is we can write a density equation that 575 00:30:24,700 --> 00:30:28,540 is sort of what will give me the ground state and sort of all 576 00:30:28,540 --> 00:30:30,230 the properties of the system. 577 00:30:30,230 --> 00:30:35,590 And that's really the second Hohenberg and Kohn theorem. 578 00:30:35,590 --> 00:30:40,960 That is really writing out the equivalent concept 579 00:30:40,960 --> 00:30:44,110 of the Schrodinger equation for the charge density. 580 00:30:44,110 --> 00:30:47,320 And now, sort of, it becomes fairly conceptual. 581 00:30:47,320 --> 00:30:48,100 OK. 582 00:30:48,100 --> 00:30:52,010 So this is the procedure. 583 00:30:52,010 --> 00:30:55,300 And all of this in the next few slides 584 00:30:55,300 --> 00:30:58,130 is still a conceptual procedure. 585 00:30:58,130 --> 00:31:03,280 It will describe objects that are well-defined in principle, 586 00:31:03,280 --> 00:31:05,560 that are conceptually well-defined, 587 00:31:05,560 --> 00:31:09,220 but we still don't have a clue on what 588 00:31:09,220 --> 00:31:11,200 they look like in practice. 589 00:31:11,200 --> 00:31:15,370 And all the sort of density functional application 590 00:31:15,370 --> 00:31:17,200 goes through a procedure that we'll 591 00:31:17,200 --> 00:31:19,870 see later on that is the sort of [INAUDIBLE] mapping 592 00:31:19,870 --> 00:31:23,950 that gives a hint of what these objects look like. 593 00:31:23,950 --> 00:31:26,770 But up to now, we are going to introduce objects 594 00:31:26,770 --> 00:31:29,380 that are well-defined in principle, 595 00:31:29,380 --> 00:31:31,710 but we don't know how they look like. 596 00:31:31,710 --> 00:31:34,240 And so that's why somehow density functional theory 597 00:31:34,240 --> 00:31:37,180 is a much less intuitive theory than something 598 00:31:37,180 --> 00:31:38,750 like Hartree-Fock. 599 00:31:38,750 --> 00:31:39,250 OK. 600 00:31:39,250 --> 00:31:42,830 So this is going towards the second Hohenberg and Kohn 601 00:31:42,830 --> 00:31:46,840 theorem, defining the fundamental equation 602 00:31:46,840 --> 00:31:48,740 for the charge density. 603 00:31:48,740 --> 00:31:52,000 And this is the step. 604 00:31:52,000 --> 00:31:57,190 For any charge density rho, so someone gives you, 605 00:31:57,190 --> 00:32:01,000 someone draws you, an arbitrary charge density. 606 00:32:01,000 --> 00:32:06,160 Well, we know that there is an external potential 607 00:32:06,160 --> 00:32:09,640 of which that charge density is the ground state. 608 00:32:09,640 --> 00:32:11,290 We don't know what it is, honestly, 609 00:32:11,290 --> 00:32:16,450 but we have proven that there is a unique external potential. 610 00:32:16,450 --> 00:32:17,320 OK. 611 00:32:17,320 --> 00:32:21,400 So because there is a unique external potential, 612 00:32:21,400 --> 00:32:25,540 there is a many-body Schrodinger equation 613 00:32:25,540 --> 00:32:27,790 with that potential in there. 614 00:32:27,790 --> 00:32:31,480 And there is a wave function that 615 00:32:31,480 --> 00:32:33,700 is going to be the ground state wave 616 00:32:33,700 --> 00:32:37,220 function of that many-body Schrodinger equation. 617 00:32:37,220 --> 00:32:43,930 So given a certain rho, we know that an external potential 618 00:32:43,930 --> 00:32:45,040 exists. 619 00:32:45,040 --> 00:32:47,230 And it's unique. 620 00:32:47,230 --> 00:32:49,630 It determines a Schrodinger equation. 621 00:32:49,630 --> 00:32:53,575 And that Schrodinger equation determines our ground state 622 00:32:53,575 --> 00:32:57,130 wave function that we call psi. 623 00:32:57,130 --> 00:33:03,520 So what we are saying is that, given a rho, in principle 624 00:33:03,520 --> 00:33:08,590 that psi, the ground state wave function of the Schrodinger 625 00:33:08,590 --> 00:33:10,660 equation in the external potential 626 00:33:10,660 --> 00:33:12,610 that is uniquely defined by the rho, 627 00:33:12,610 --> 00:33:15,010 is also a well-defined object. 628 00:33:15,010 --> 00:33:18,520 Again, we don't know what it is, but it is well-defined. 629 00:33:18,520 --> 00:33:21,640 And because it's a well-defined object, 630 00:33:21,640 --> 00:33:25,510 we can calculate the expectation value 631 00:33:25,510 --> 00:33:31,750 of that well-defined object of the quantum kinetic energy 632 00:33:31,750 --> 00:33:36,670 minus 1/2 sum over all i of the second derivatives 633 00:33:36,670 --> 00:33:40,870 and the electron-electron interaction, just the 1 634 00:33:40,870 --> 00:33:44,080 over ri minus rj term. 635 00:33:44,080 --> 00:33:50,080 So again, this term is, in principle, well-defined. 636 00:33:50,080 --> 00:33:56,050 And we call this term the universal density functional. 637 00:33:56,050 --> 00:34:02,380 That is for any given arbitrary rho, i, in principle, 638 00:34:02,380 --> 00:34:06,970 can define a number that is this number here. 639 00:34:06,970 --> 00:34:08,560 I have the rho. 640 00:34:08,560 --> 00:34:12,429 In principle, from the rho, I have the external potential. 641 00:34:12,429 --> 00:34:15,730 From the external potential, I have the Schrodinger equation. 642 00:34:15,730 --> 00:34:20,139 In principle, I'm able to solve that Schrodinger equation found 643 00:34:20,139 --> 00:34:23,500 in principle the many-body ground state wave function. 644 00:34:23,500 --> 00:34:24,880 That will be psi. 645 00:34:24,880 --> 00:34:28,540 And I can calculate the expectation value 646 00:34:28,540 --> 00:34:32,139 of psi of the quantum kinetic energy 647 00:34:32,139 --> 00:34:35,739 and of the electron-electron interaction term, 648 00:34:35,739 --> 00:34:37,400 all well-defined. 649 00:34:37,400 --> 00:34:40,810 We have really no clue on how to calculate because we can't 650 00:34:40,810 --> 00:34:43,429 really do in practice any of the steps, 651 00:34:43,429 --> 00:34:49,880 but this universal function of the density is well-defined. 652 00:34:49,880 --> 00:34:52,989 So with this universal functional 653 00:34:52,989 --> 00:34:59,350 that is now well-defined, we can write out something. 654 00:34:59,350 --> 00:35:08,580 We can write to an energy for any given external potential 655 00:35:08,580 --> 00:35:11,820 and for any given charge density. 656 00:35:11,820 --> 00:35:14,700 And we write it as this. 657 00:35:14,700 --> 00:35:19,560 So for any given charge density, there 658 00:35:19,560 --> 00:35:25,530 will be a well-defined number that is this universal density 659 00:35:25,530 --> 00:35:28,960 functional of beta rho prime. 660 00:35:28,960 --> 00:35:32,520 And then we add another term that 661 00:35:32,520 --> 00:35:37,020 is just trivially the integral of this v, 662 00:35:37,020 --> 00:35:42,600 this external potential, times the charge density rho prime. 663 00:35:42,600 --> 00:35:46,920 So again, this new expression that we written 664 00:35:46,920 --> 00:35:48,960 is well-defined. 665 00:35:48,960 --> 00:35:54,120 For any rho prime and for any external potential, 666 00:35:54,120 --> 00:35:57,120 we can calculate trivially this term. 667 00:35:57,120 --> 00:36:01,770 And in principle, we know what this number is. 668 00:36:01,770 --> 00:36:08,400 And this is, if you want, 1964, 1965, the reformulation 669 00:36:08,400 --> 00:36:10,110 of quantum mechanics. 670 00:36:10,110 --> 00:36:12,870 Because, now, Hohenberg and Kohn are 671 00:36:12,870 --> 00:36:17,520 able to prove that there is a variational principle. 672 00:36:17,520 --> 00:36:20,700 That is, for this expression written here, 673 00:36:20,700 --> 00:36:24,090 for this functional of rho prime, 674 00:36:24,090 --> 00:36:29,430 we can prove that for any rho prime 675 00:36:29,430 --> 00:36:34,980 that we can throw in the overall numerical value 676 00:36:34,980 --> 00:36:37,650 of this expression is always going 677 00:36:37,650 --> 00:36:46,110 to be either greater or equal to the ground state energy 678 00:36:46,110 --> 00:36:49,300 that we would obtain from the Schrodinger equation 679 00:36:49,300 --> 00:36:51,910 in the presence of this external potential. 680 00:36:51,910 --> 00:36:56,380 So now, we have a well-defined density functional. 681 00:36:56,380 --> 00:36:58,350 So if you have an external potential, 682 00:36:58,350 --> 00:37:04,110 the z over r of your atom, you can try out now not 683 00:37:04,110 --> 00:37:06,480 wave functions that are very difficult, 684 00:37:06,480 --> 00:37:10,140 but you can try out charge density. 685 00:37:10,140 --> 00:37:12,900 And the charged density that gives you 686 00:37:12,900 --> 00:37:16,410 the lowest expectation value, the lowest 687 00:37:16,410 --> 00:37:19,405 value for this functional, will be the ground state, the charge 688 00:37:19,405 --> 00:37:19,905 density. 689 00:37:23,080 --> 00:37:26,530 Small problem, we have no clue what 690 00:37:26,530 --> 00:37:29,700 this looks like as a function of rho prime. 691 00:37:29,700 --> 00:37:35,170 But if we knew, we would have a wonderfully simple approach 692 00:37:35,170 --> 00:37:37,040 to quantum mechanics. 693 00:37:37,040 --> 00:37:40,960 Now, we don't need to deal with the many-body complexity. 694 00:37:40,960 --> 00:37:47,630 We just minimize this expression as a function of rho prime. 695 00:37:47,630 --> 00:37:49,720 And again, it's sort of fairly easy 696 00:37:49,720 --> 00:37:52,808 to prove this variational principle. 697 00:37:52,808 --> 00:37:54,100 But one needs probably to sit-- 698 00:37:54,100 --> 00:37:55,390 I've given you some reading. 699 00:37:55,390 --> 00:37:57,790 So you're welcome, if you are really interested in this, 700 00:37:57,790 --> 00:38:00,610 to go back and read the first Hohenberg and Kohn theorem 701 00:38:00,610 --> 00:38:03,460 and read the second Hohenberg and Kohn theorem. 702 00:38:03,460 --> 00:38:08,065 But in many ways, the proof of this second Hohenberg 703 00:38:08,065 --> 00:38:11,140 and Kohn theorem can be done again 704 00:38:11,140 --> 00:38:13,210 through the variational principle. 705 00:38:13,210 --> 00:38:16,960 That is, if we have a certain rho prime, 706 00:38:16,960 --> 00:38:21,670 well, that, again, uniquely determines the ground state 707 00:38:21,670 --> 00:38:22,690 wave function. 708 00:38:22,690 --> 00:38:26,920 Rho prime will determine an external potential 709 00:38:26,920 --> 00:38:29,650 that, in principle, is different from this. 710 00:38:29,650 --> 00:38:33,160 But rho prime will determine an external potential 711 00:38:33,160 --> 00:38:35,170 and will determine our wave function 712 00:38:35,170 --> 00:38:38,750 that is the solution of the many-body Schrodinger equation. 713 00:38:38,750 --> 00:38:45,010 And if we take the expectation value of our Hamiltonian 714 00:38:45,010 --> 00:38:47,410 with this external potential in this, 715 00:38:47,410 --> 00:38:51,100 but evaluated on the wave function of c prime 716 00:38:51,100 --> 00:38:53,890 that comes from this charge density rho prime, 717 00:38:53,890 --> 00:38:58,150 well, we can show that this expectation value here 718 00:38:58,150 --> 00:39:02,470 is just identical to functional that I have just written. 719 00:39:02,470 --> 00:39:04,630 And for the variational principle, 720 00:39:04,630 --> 00:39:08,380 then it needs to be greater or equal than E0. 721 00:39:08,380 --> 00:39:10,620 I won't sort of dwell into that. 722 00:39:10,620 --> 00:39:13,600 And again, you can look at the sort of detailed description 723 00:39:13,600 --> 00:39:16,480 and in sort of some of the many references that I've given 724 00:39:16,480 --> 00:39:19,850 or that I've also posted on the website. 725 00:39:19,850 --> 00:39:21,970 But what is conceptually important 726 00:39:21,970 --> 00:39:24,530 is that we have a new equation. 727 00:39:24,530 --> 00:39:25,030 OK. 728 00:39:25,030 --> 00:39:29,860 So 1964, '65, quantum mechanics turned around. 729 00:39:29,860 --> 00:39:33,250 We don't have to think at many-body wave functions. 730 00:39:33,250 --> 00:39:36,490 We can think just at charge density. 731 00:39:36,490 --> 00:39:42,820 And all would be well apart from this detail, 732 00:39:42,820 --> 00:39:48,010 that we don't know what that functional f of rho is. 733 00:39:48,010 --> 00:39:50,560 And so we have a conceptual approach, 734 00:39:50,560 --> 00:39:54,360 but we don't have a practical approach 735 00:39:54,360 --> 00:39:57,640 to solve the density functional reformulation of quantum 736 00:39:57,640 --> 00:39:58,700 mechanics. 737 00:39:58,700 --> 00:40:01,540 And this is, if you wanted, true to this day. 738 00:40:01,540 --> 00:40:07,090 We don't know what is the exact form of f of rho. 739 00:40:07,090 --> 00:40:11,620 If we knew it, sort of most of our sort 740 00:40:11,620 --> 00:40:14,170 of quantum mechanical computational problems 741 00:40:14,170 --> 00:40:15,460 would be trivially solved. 742 00:40:15,460 --> 00:40:19,840 Because solving that variational principle in the charge density 743 00:40:19,840 --> 00:40:24,070 would be most likely a trivial thing to do. 744 00:40:24,070 --> 00:40:27,850 The issue is that not only we don't know, 745 00:40:27,850 --> 00:40:31,240 but we have understood a lot of what that exchange 746 00:40:31,240 --> 00:40:34,960 correlation-- of what that universal density functional 747 00:40:34,960 --> 00:40:35,890 is. 748 00:40:35,890 --> 00:40:38,500 And it's very complex. 749 00:40:38,500 --> 00:40:46,150 So it's unlikely that there is a simple analytical expression 750 00:40:46,150 --> 00:40:49,750 of it as a function of the charge density only. 751 00:40:49,750 --> 00:40:53,950 But, you know, the other great piece of, if you want, 752 00:40:53,950 --> 00:40:56,170 quantum engineering by Walter Kohn 753 00:40:56,170 --> 00:40:59,950 was finding out a very good approximation 754 00:40:59,950 --> 00:41:01,930 to that density functional. 755 00:41:01,930 --> 00:41:04,210 We don't know what the exact one is. 756 00:41:04,210 --> 00:41:06,790 But now, what they are doing is, well, 757 00:41:06,790 --> 00:41:12,190 finding out one that is going to be very, very closely 758 00:41:12,190 --> 00:41:14,780 similar to the exact one. 759 00:41:14,780 --> 00:41:18,040 And so they are going to throw in some physical intuition 760 00:41:18,040 --> 00:41:20,530 to this problem that up to now, if you want, 761 00:41:20,530 --> 00:41:23,860 has been a mathematical problem. 762 00:41:23,860 --> 00:41:26,480 It's another layer of complexity in this discussion, 763 00:41:26,480 --> 00:41:28,120 so I hope I'm not losing you. 764 00:41:28,120 --> 00:41:31,240 But sort of what Walter Kohn did-- 765 00:41:31,240 --> 00:41:34,750 I think he had a young postdoc arriving from Cambridge. 766 00:41:34,750 --> 00:41:39,562 Lu Sham had just done his PhD in England and came there. 767 00:41:39,562 --> 00:41:41,020 And sort of, you know, he told him, 768 00:41:41,020 --> 00:41:43,570 I have this new variational principle. 769 00:41:43,570 --> 00:41:48,580 Let's see what we can do to make it into a practical solution. 770 00:41:48,580 --> 00:41:52,270 I think they were in Santa Barbara, in San Diego probably, 771 00:41:52,270 --> 00:41:53,590 at that time. 772 00:41:53,590 --> 00:41:54,400 OK. 773 00:41:54,400 --> 00:41:58,480 So this is what they are going to do. 774 00:41:58,480 --> 00:42:00,520 Remember, sort of, what is the problem. 775 00:42:00,520 --> 00:42:07,510 We need to figure out what is a reasonable approximation 776 00:42:07,510 --> 00:42:10,310 to this functional here. 777 00:42:10,310 --> 00:42:13,270 So what they say is, well, suppose 778 00:42:13,270 --> 00:42:17,060 that someone has given us this charge density. 779 00:42:17,060 --> 00:42:20,320 So we need, in principle, to find out 780 00:42:20,320 --> 00:42:23,260 what would be the many-body wave function that 781 00:42:23,260 --> 00:42:27,100 is solution of this external potential that corresponds 782 00:42:27,100 --> 00:42:29,810 to this charge density. 783 00:42:29,810 --> 00:42:31,670 This is going to be very complex. 784 00:42:31,670 --> 00:42:38,770 Let's invent a problem in which electrons do not 785 00:42:38,770 --> 00:42:41,020 interact between each other. 786 00:42:41,020 --> 00:42:41,860 OK. 787 00:42:41,860 --> 00:42:43,840 So electrons-- so that's the sort 788 00:42:43,840 --> 00:42:46,840 of main problem in the Schrodinger equation, 789 00:42:46,840 --> 00:42:49,720 that electrons interacting with each other 790 00:42:49,720 --> 00:42:53,710 introduce the two-body electrostatic repulsion 791 00:42:53,710 --> 00:42:55,610 in the Schrodinger equation. 792 00:42:55,610 --> 00:42:57,310 And that's what makes it difficult. 793 00:42:57,310 --> 00:42:59,710 Well, what Kohn and Sham say is let's 794 00:42:59,710 --> 00:43:04,030 for a moment suppose that there is a system of electrons 795 00:43:04,030 --> 00:43:05,020 that don't interact. 796 00:43:05,020 --> 00:43:08,410 So the only thing that those so-called Kohn and Sham 797 00:43:08,410 --> 00:43:13,000 electrons fill is the external potential. 798 00:43:13,000 --> 00:43:13,990 OK. 799 00:43:13,990 --> 00:43:17,920 So those Kohn and Sham electrons will solve, 800 00:43:17,920 --> 00:43:23,110 will satisfy, a Schrodinger equation that is much simpler. 801 00:43:23,110 --> 00:43:26,260 Because there is no electron-electron interaction. 802 00:43:26,260 --> 00:43:29,770 Those Kohn and Sham electron, the only thing that they fill 803 00:43:29,770 --> 00:43:33,130 is a new potential. 804 00:43:33,130 --> 00:43:36,620 And they will have their own quantum kinetic energy. 805 00:43:36,620 --> 00:43:45,040 So what they are saying is, for any given charge density rho, 806 00:43:45,040 --> 00:43:49,930 there is going to be a non-interacting set 807 00:43:49,930 --> 00:43:56,440 of electrons who's ground state charge density is 808 00:43:56,440 --> 00:43:58,750 identical to rho. 809 00:43:58,750 --> 00:44:02,260 So we have said, if we have a charge density rho, 810 00:44:02,260 --> 00:44:06,190 you can all go through, find out the external potential that 811 00:44:06,190 --> 00:44:08,260 comes from rho, the Schrodinger equation, 812 00:44:08,260 --> 00:44:12,340 the many-body interacting electrons solution. 813 00:44:12,340 --> 00:44:13,960 But now, what we are going to say 814 00:44:13,960 --> 00:44:20,080 is we can also think at a system of non-interacting electrons. 815 00:44:20,080 --> 00:44:24,490 And we want those non-interacting electrons 816 00:44:24,490 --> 00:44:31,450 to fill a potential that is such that their ground state is 817 00:44:31,450 --> 00:44:34,090 going to give us a charge density that 818 00:44:34,090 --> 00:44:38,050 is identical to the charge density I'm dealing with. 819 00:44:38,050 --> 00:44:38,740 OK. 820 00:44:38,740 --> 00:44:41,710 And we call that external potential 821 00:44:41,710 --> 00:44:43,810 the Kohn-Sham potential. 822 00:44:43,810 --> 00:44:44,590 OK. 823 00:44:44,590 --> 00:44:48,490 So now, for a charge density, you 824 00:44:48,490 --> 00:44:51,010 don't only have to think at all the complexity 825 00:44:51,010 --> 00:44:53,230 that I've discussed up to now. 826 00:44:53,230 --> 00:44:56,650 But you have also to think that, for a charge density, 827 00:44:56,650 --> 00:44:58,750 there is going to be this set of Kohn and Sham 828 00:44:58,750 --> 00:45:00,310 known interacting electrons. 829 00:45:00,310 --> 00:45:03,790 And there is going to be a potential that 830 00:45:03,790 --> 00:45:07,030 is called the Kohn and Sham potential that is such 831 00:45:07,030 --> 00:45:10,480 that the ground state of the Schrodinger 832 00:45:10,480 --> 00:45:12,820 equation for non-interacting electron, that 833 00:45:12,820 --> 00:45:15,497 is without electron-electron interaction, in that Kohn 834 00:45:15,497 --> 00:45:19,240 and Sham potential will give us a wave function and a ground 835 00:45:19,240 --> 00:45:24,010 state that leads to a charge density identical to the charge 836 00:45:24,010 --> 00:45:27,250 density I'm sort of dealing with. 837 00:45:27,250 --> 00:45:27,910 OK. 838 00:45:27,910 --> 00:45:29,750 What do we do with this? 839 00:45:29,750 --> 00:45:35,980 Well, at this stage, there is a sort of great simplification 840 00:45:35,980 --> 00:45:39,040 that, for the Schrodinger equation 841 00:45:39,040 --> 00:45:43,330 of non-interacting electron, we actually know 842 00:45:43,330 --> 00:45:45,500 what is the exact solution. 843 00:45:45,500 --> 00:45:47,950 So it's actually very simple to solve 844 00:45:47,950 --> 00:45:52,630 a Schrodinger equation in which the electrons do not interact. 845 00:45:52,630 --> 00:45:56,320 Because, now, this later determinant 846 00:45:56,320 --> 00:45:59,720 is actually the exact solution. 847 00:45:59,720 --> 00:46:02,530 So if you have a set of non-interacting electrons, 848 00:46:02,530 --> 00:46:05,020 you don't have the electron-electron term 849 00:46:05,020 --> 00:46:08,380 in that Schrodinger equation, this later determinant is not 850 00:46:08,380 --> 00:46:11,410 only a good approximation, but it's actually 851 00:46:11,410 --> 00:46:14,650 the exact solution. 852 00:46:14,650 --> 00:46:18,670 So for this non-interactive set of electrons, 853 00:46:18,670 --> 00:46:21,970 we can solve everything exactly. 854 00:46:21,970 --> 00:46:25,030 And in particular, we can calculate, say, 855 00:46:25,030 --> 00:46:28,090 what is the kinetic energy of this set 856 00:46:28,090 --> 00:46:30,880 of non-interacting electrons. 857 00:46:30,880 --> 00:46:31,810 OK. 858 00:46:31,810 --> 00:46:39,070 So now, we can sort of have a somehow pseudo-physical way 859 00:46:39,070 --> 00:46:43,510 of decompose this mysterious density 860 00:46:43,510 --> 00:46:48,010 functional into different terms. 861 00:46:48,010 --> 00:46:50,800 So what we are actually doing via the Kohn and Sham 862 00:46:50,800 --> 00:46:56,440 mapping is extracting from here terms that are very large 863 00:46:56,440 --> 00:47:00,430 and that we know how to write, we know how to calculate. 864 00:47:00,430 --> 00:47:07,060 And then, hopefully, once we have extracted all these terms 865 00:47:07,060 --> 00:47:10,960 that we know how to define, we remain with something 866 00:47:10,960 --> 00:47:14,200 that is very small and that we'll 867 00:47:14,200 --> 00:47:18,430 find another numerical approximation for it. 868 00:47:18,430 --> 00:47:22,410 So Kohn and Sham say, well, we have this well-defined density 869 00:47:22,410 --> 00:47:23,400 functional. 870 00:47:23,400 --> 00:47:27,240 We extract two terms that are well-defined. 871 00:47:27,240 --> 00:47:31,320 And these two terms that's sort of the great achievement 872 00:47:31,320 --> 00:47:35,160 actually contain most of the physics of our problem. 873 00:47:35,160 --> 00:47:38,710 And the sort of small term that is left over, 874 00:47:38,710 --> 00:47:41,260 we are going to approximate in some simple way. 875 00:47:41,260 --> 00:47:43,500 And actually, the approximation that they found 876 00:47:43,500 --> 00:47:44,775 worked very well. 877 00:47:44,775 --> 00:47:46,650 And that's why some of this functional theory 878 00:47:46,650 --> 00:47:48,540 became a practical theory. 879 00:47:48,540 --> 00:47:52,410 And so in this sort of density functional, 880 00:47:52,410 --> 00:47:57,300 the first physical large term that they extract 881 00:47:57,300 --> 00:48:00,510 is the quantum kinetic energy that we 882 00:48:00,510 --> 00:48:03,720 call Ts not of the real system. 883 00:48:03,720 --> 00:48:06,790 Because, again, even if it's well-defined, 884 00:48:06,790 --> 00:48:09,220 we don't know how to do that. 885 00:48:09,220 --> 00:48:14,130 But what they were able to write is the quantum kinetic energy 886 00:48:14,130 --> 00:48:18,250 of this non-interacting problem. 887 00:48:18,250 --> 00:48:20,490 So for a given charge density, there 888 00:48:20,490 --> 00:48:24,540 is this set of Kohn and Sham non-interacting electrons 889 00:48:24,540 --> 00:48:30,060 that lives in a potential, such that they have the same ground 890 00:48:30,060 --> 00:48:31,650 state charge density. 891 00:48:31,650 --> 00:48:33,900 And their kinetic energy is trivial 892 00:48:33,900 --> 00:48:36,480 because it's going to be just the kinetic energy 893 00:48:36,480 --> 00:48:39,870 of this later determinant, just the sum of a single particle 894 00:48:39,870 --> 00:48:40,650 term. 895 00:48:40,650 --> 00:48:42,870 So for a charged density now, there 896 00:48:42,870 --> 00:48:46,680 is a well-defined quantum kinetic energy 897 00:48:46,680 --> 00:48:50,140 that is not the true quantum kinetic energy of the system, 898 00:48:50,140 --> 00:48:52,080 but is the quantum kinetic energy 899 00:48:52,080 --> 00:48:55,890 of this sort of associated system 900 00:48:55,890 --> 00:48:57,900 of non-interacting electrons. 901 00:48:57,900 --> 00:49:00,720 But this term is now well-defined. 902 00:49:00,720 --> 00:49:04,590 They say, well, let's extract another term that 903 00:49:04,590 --> 00:49:08,880 is well-defined that is just a Hartree electrostatic energy 904 00:49:08,880 --> 00:49:10,920 of a charge density distribution. 905 00:49:10,920 --> 00:49:11,670 OK. 906 00:49:11,670 --> 00:49:14,880 So if we look at the charge density distribution in which 907 00:49:14,880 --> 00:49:16,950 each infinitesimal volume interacts 908 00:49:16,950 --> 00:49:19,320 with each other infinitesimal volume 909 00:49:19,320 --> 00:49:21,180 with an electrostatic interaction, 910 00:49:21,180 --> 00:49:23,430 that's going to be the term. 911 00:49:23,430 --> 00:49:27,960 And you know, what we are left is now something 912 00:49:27,960 --> 00:49:31,560 that they call the exchange correlation term. 913 00:49:31,560 --> 00:49:33,430 That is everything else. 914 00:49:33,430 --> 00:49:34,230 OK. 915 00:49:34,230 --> 00:49:38,010 So F, in principle, is an exact quantity. 916 00:49:38,010 --> 00:49:43,290 We are now able to define our quantum kinetic energy term. 917 00:49:43,290 --> 00:49:47,280 That is an exact quantity, but is not really 918 00:49:47,280 --> 00:49:49,710 the quantum kinetic energy of the true system. 919 00:49:49,710 --> 00:49:52,020 But we sort of say, you know, this 920 00:49:52,020 --> 00:49:55,050 is going to be equal to a well-defined term 921 00:49:55,050 --> 00:49:58,650 plus another well-defined term plus a third term 922 00:49:58,650 --> 00:49:59,730 that we don't know. 923 00:49:59,730 --> 00:50:01,980 So we have sort of decomposed a quantity 924 00:50:01,980 --> 00:50:06,780 that we have no clue what it is into three terms of which two 925 00:50:06,780 --> 00:50:08,430 terms are well-defined. 926 00:50:08,430 --> 00:50:12,840 And all our cluelessness is contained in the third term. 927 00:50:12,840 --> 00:50:16,650 And we call this third term the exchange correlation, 928 00:50:16,650 --> 00:50:19,320 but the sort of physical advantage 929 00:50:19,320 --> 00:50:22,200 of having done this is that it turns out 930 00:50:22,200 --> 00:50:26,850 that these two terms capture a lot of the complexity 931 00:50:26,850 --> 00:50:28,230 of your problem. 932 00:50:28,230 --> 00:50:31,810 And this term tends to be fairly small. 933 00:50:31,810 --> 00:50:32,580 OK. 934 00:50:32,580 --> 00:50:34,680 So that's all, actually. 935 00:50:34,680 --> 00:50:37,770 That's why it works very well because somehow they 936 00:50:37,770 --> 00:50:42,430 manage to capture the complexity of our system. 937 00:50:42,430 --> 00:50:53,470 And so once that exchange correlation term is defined 938 00:50:53,470 --> 00:50:59,370 and it's approximated in some way that we'll see in a moment, 939 00:50:59,370 --> 00:51:04,950 our problem is now well-defined because we really 940 00:51:04,950 --> 00:51:06,900 have a variational principle. 941 00:51:06,900 --> 00:51:09,780 Remember, the universal density functional 942 00:51:09,780 --> 00:51:12,870 plus the external potential plus the charge 943 00:51:12,870 --> 00:51:15,330 density in the field of the external potential 944 00:51:15,330 --> 00:51:18,750 minimizes the sort of new variational principle 945 00:51:18,750 --> 00:51:21,450 that comes from the Hohenberg and Kohn theorem. 946 00:51:21,450 --> 00:51:24,120 And so we can write it, our variational principle. 947 00:51:24,120 --> 00:51:28,320 That is this quantity with the constraint 948 00:51:28,320 --> 00:51:31,020 that the number of electrons should be 949 00:51:31,020 --> 00:51:33,420 equal to n should be minimum. 950 00:51:33,420 --> 00:51:37,350 And as usual, when you sort of write a variational principle, 951 00:51:37,350 --> 00:51:40,680 you are saying that sort of the differential of that quantity 952 00:51:40,680 --> 00:51:42,330 needs to be equal to 0. 953 00:51:42,330 --> 00:51:45,120 Or if you want, I mean, this is a generic term. 954 00:51:45,120 --> 00:51:48,360 You have a set of what are called Euler-Lagrange equation, 955 00:51:48,360 --> 00:51:49,050 basically. 956 00:51:49,050 --> 00:51:51,120 That is nothing else than differential analysis. 957 00:51:51,120 --> 00:51:54,030 That is you are asking yourself, what 958 00:51:54,030 --> 00:51:56,370 are going to be the conditions that 959 00:51:56,370 --> 00:51:59,730 need to be satisfied by the charge density 960 00:51:59,730 --> 00:52:03,270 in order to satisfy the variational principle? 961 00:52:03,270 --> 00:52:06,030 There is always this sort of 1 to 1 correspondence. 962 00:52:06,030 --> 00:52:07,710 You have a variational principle. 963 00:52:07,710 --> 00:52:09,840 It gives you differential equation. 964 00:52:09,840 --> 00:52:11,550 Or you have differential equation. 965 00:52:11,550 --> 00:52:14,010 You can rewrite them in a variational principle. 966 00:52:14,010 --> 00:52:16,230 We have seen that for the Schrodinger equation. 967 00:52:16,230 --> 00:52:21,660 And we see this, in particular, now explicitly for the Kohn 968 00:52:21,660 --> 00:52:23,100 and Sham orbital. 969 00:52:23,100 --> 00:52:28,230 So I'll actually go directly to the explicit expression 970 00:52:28,230 --> 00:52:31,290 of the Kohn and Sham orbitals. 971 00:52:31,290 --> 00:52:33,750 Again, remember that what we have done 972 00:52:33,750 --> 00:52:38,820 is we have defined a variational principle that 973 00:52:38,820 --> 00:52:41,970 acts on a universal density functional 974 00:52:41,970 --> 00:52:46,320 F plus the charge density and the external potential. 975 00:52:46,320 --> 00:52:48,780 And we have decomposed, we have extracted, 976 00:52:48,780 --> 00:52:51,660 from this universal functional sort of terms 977 00:52:51,660 --> 00:52:53,770 that are large and physical. 978 00:52:53,770 --> 00:52:57,720 And we have sort of pushed all the many-body complexity 979 00:52:57,720 --> 00:53:01,050 of the problem in something that we call the exchange 980 00:53:01,050 --> 00:53:02,760 correlation functional. 981 00:53:02,760 --> 00:53:05,680 That is, again, a functional of the charge density. 982 00:53:05,680 --> 00:53:08,580 We don't know yet what that function of the charge density 983 00:53:08,580 --> 00:53:09,120 is. 984 00:53:09,120 --> 00:53:10,940 But luckily, it's going to be small. 985 00:53:10,940 --> 00:53:13,360 So in a moment, we'll approximate it. 986 00:53:13,360 --> 00:53:19,050 And then we ask ourselves, what are the differential 987 00:53:19,050 --> 00:53:22,620 equations that derive from this variational principle? 988 00:53:22,620 --> 00:53:25,540 Well, in principle, I had written them here. 989 00:53:25,540 --> 00:53:26,470 OK. 990 00:53:26,470 --> 00:53:28,500 We just need to take the variation with respect 991 00:53:28,500 --> 00:53:31,380 to the charge density and imposing the Lagrange 992 00:53:31,380 --> 00:53:33,180 multiplication constraint. 993 00:53:33,180 --> 00:53:38,670 And so this would be basically that the charge density needs 994 00:53:38,670 --> 00:53:42,780 to satisfy this set of equation, the sort of functional 995 00:53:42,780 --> 00:53:45,510 derivative of this non-interacting quantum 996 00:53:45,510 --> 00:53:50,100 kinetic energy plus a number of terms that 997 00:53:50,100 --> 00:53:52,590 really contain the external potential, 998 00:53:52,590 --> 00:53:53,940 the Hartree interaction. 999 00:53:53,940 --> 00:53:56,100 And the exchange correlation need 1000 00:53:56,100 --> 00:53:58,770 to be equal to the Lagrange multiplier that 1001 00:53:58,770 --> 00:54:03,870 fixes the number of electrons. 1002 00:54:03,870 --> 00:54:08,940 We are not able to calculate this functional derivative 1003 00:54:08,940 --> 00:54:11,940 because, remember, the quantum kinetic energy 1004 00:54:11,940 --> 00:54:14,340 of the non-interacting system is again 1005 00:54:14,340 --> 00:54:16,823 written as a later determinant. 1006 00:54:16,823 --> 00:54:18,240 And so there is sort of, you know, 1007 00:54:18,240 --> 00:54:23,730 this type of back in which, even if we had written everything 1008 00:54:23,730 --> 00:54:26,970 in terms of a charge density, we are not 1009 00:54:26,970 --> 00:54:32,040 able to explicitly calculate not only 1010 00:54:32,040 --> 00:54:36,450 the derivative of the true interacting electron's 1011 00:54:36,450 --> 00:54:38,190 kinetic energy with respect to rho, 1012 00:54:38,190 --> 00:54:40,680 but we are not even able to calculate the functional 1013 00:54:40,680 --> 00:54:43,590 derivative of the non-interacting kinetic energy 1014 00:54:43,590 --> 00:54:44,890 with respect to rho. 1015 00:54:44,890 --> 00:54:49,530 But what we are able is actually to calculate the derivative 1016 00:54:49,530 --> 00:54:53,640 of that non-interacting kinetic energy with respect 1017 00:54:53,640 --> 00:54:57,940 to the orbitals that describe the Kohn and Sham electrons. 1018 00:54:57,940 --> 00:55:00,660 Remember that these non-independent Kohn and Sham 1019 00:55:00,660 --> 00:55:04,530 electrons have an exact solution that is a later determinant. 1020 00:55:04,530 --> 00:55:07,560 And so we know there are trivial many-body wave function is 1021 00:55:07,560 --> 00:55:12,210 a later determinant composed by a single particle orbitals. 1022 00:55:12,210 --> 00:55:17,490 And the functional derivative of that independent 1023 00:55:17,490 --> 00:55:20,070 non-interacting electron's kinetic energy 1024 00:55:20,070 --> 00:55:23,160 with respect to the single-particle orbital 1025 00:55:23,160 --> 00:55:28,710 is now trivial and is just minus 1/2 del square. 1026 00:55:28,710 --> 00:55:34,050 So at the end of all these sort of complex formulation, what 1027 00:55:34,050 --> 00:55:36,810 we are left with is something very simple 1028 00:55:36,810 --> 00:55:39,720 and probably something you should focus your attention 1029 00:55:39,720 --> 00:55:40,860 from now on. 1030 00:55:40,860 --> 00:55:45,210 We have now a set of Kohn and Sham equation 1031 00:55:45,210 --> 00:55:47,480 that are the differential equation 1032 00:55:47,480 --> 00:55:51,260 that the electrons need to satisfy in order 1033 00:55:51,260 --> 00:55:55,690 to satisfy the variational principle with the caveat 1034 00:55:55,690 --> 00:55:58,430 that, in this Kohn and Sham equation, 1035 00:55:58,430 --> 00:56:02,210 there is at term, an exchange correlation term, 1036 00:56:02,210 --> 00:56:04,460 that we still don't know what it is. 1037 00:56:04,460 --> 00:56:07,430 It's sort the formally defined as the functional derivative 1038 00:56:07,430 --> 00:56:09,170 of the exchange correlation energy 1039 00:56:09,170 --> 00:56:11,000 with respect to the charge density. 1040 00:56:11,000 --> 00:56:14,330 But we'll need to approximate somewhere. 1041 00:56:14,330 --> 00:56:19,610 And what this equation describes is not anymore 1042 00:56:19,610 --> 00:56:21,890 the true electrons in your system, 1043 00:56:21,890 --> 00:56:25,970 but they describe this cousins of the true electrons. 1044 00:56:25,970 --> 00:56:31,190 They describe this Kohn and Sham non-interacting electrons 1045 00:56:31,190 --> 00:56:34,670 that have their own orbital psi i. 1046 00:56:34,670 --> 00:56:38,870 And that will give us a ground state charge 1047 00:56:38,870 --> 00:56:42,470 density that, if the exchange correlation functional was 1048 00:56:42,470 --> 00:56:46,940 exact, would be not only this, as obviously the same ground 1049 00:56:46,940 --> 00:56:50,930 state energy of our interacting electron system, 1050 00:56:50,930 --> 00:56:55,920 but it would be the exact solution of the problem. 1051 00:56:55,920 --> 00:56:56,550 OK. 1052 00:56:56,550 --> 00:57:02,040 So this equation look a lot like a Schrodinger equation. 1053 00:57:02,040 --> 00:57:06,057 They look a lot, if you want, like the Hartree-Fock equation 1054 00:57:06,057 --> 00:57:07,140 that we've written before. 1055 00:57:07,140 --> 00:57:12,370 Because what we are seeing is that the a Kohn and Sham 1056 00:57:12,370 --> 00:57:18,210 electron i fills a quantum kinetic energy operator, 1057 00:57:18,210 --> 00:57:23,370 fills a Hartree operator, fills the external potential, 1058 00:57:23,370 --> 00:57:28,590 and then fills this sort of remaining term 1059 00:57:28,590 --> 00:57:31,830 that is the exchange correlation potential. 1060 00:57:31,830 --> 00:57:36,900 Again, if we knew what were this exact exchange correlation 1061 00:57:36,900 --> 00:57:40,750 potential, we would have an exact solution to the problem. 1062 00:57:40,750 --> 00:57:43,120 But we know a very good approximation. 1063 00:57:43,120 --> 00:57:46,290 And then if you want finding the ground state, 1064 00:57:46,290 --> 00:57:48,690 it's not very different from finding the ground 1065 00:57:48,690 --> 00:57:52,320 state of the Hartree-Fock equation with the caveat 1066 00:57:52,320 --> 00:57:55,080 that actually this term here is going 1067 00:57:55,080 --> 00:57:59,580 to be much simpler than the exchange term 1068 00:57:59,580 --> 00:58:01,350 of the Hartree-Fock equation. 1069 00:58:01,350 --> 00:58:03,330 If you go back to the first slide 1070 00:58:03,330 --> 00:58:06,420 to the Hartree-Fock equation, the last term 1071 00:58:06,420 --> 00:58:10,290 is that numerically very complex expression 1072 00:58:10,290 --> 00:58:13,260 in which we sort of take the orbital, 1073 00:58:13,260 --> 00:58:18,720 and we put it inside an integral differential operator. 1074 00:58:18,720 --> 00:58:21,015 Now, it's become simpler. 1075 00:58:21,015 --> 00:58:22,140 And that's all if you want. 1076 00:58:22,140 --> 00:58:25,560 So the Kohn and Sham equation look very similar. 1077 00:58:25,560 --> 00:58:28,380 In practice, they are simpler to solve. 1078 00:58:28,380 --> 00:58:32,010 They tend to be more accurate in most cases. 1079 00:58:32,010 --> 00:58:34,690 And that's, at the end, what leads to the success. 1080 00:58:34,690 --> 00:58:37,140 But what is critical for all of this 1081 00:58:37,140 --> 00:58:40,470 is having a reasonable approximation to the exchange 1082 00:58:40,470 --> 00:58:41,910 correlation potential. 1083 00:58:41,910 --> 00:58:44,610 If we had the exact exchange correlation potential, 1084 00:58:44,610 --> 00:58:48,420 everything would be exact in this formulation. 1085 00:58:48,420 --> 00:58:53,130 We would find the Kohn and Sham independent electrons 1086 00:58:53,130 --> 00:58:58,920 that were sort of the ground state electrons for that charge 1087 00:58:58,920 --> 00:59:01,740 density that is ultimately equal to the charge 1088 00:59:01,740 --> 00:59:04,500 density of the interacting electrons 1089 00:59:04,500 --> 00:59:06,470 in this external potential. 1090 00:59:10,800 --> 00:59:11,670 OK. 1091 00:59:11,670 --> 00:59:19,080 And we have the Euler-Lagrange or Kohn and Sham differential 1092 00:59:19,080 --> 00:59:20,940 equation in the previous page. 1093 00:59:20,940 --> 00:59:24,480 I written here sort of, you know, just for reference 1094 00:59:24,480 --> 00:59:29,440 also what would be the total energy of the system. 1095 00:59:29,440 --> 00:59:34,410 And usually, if you had an independent electron, 1096 00:59:34,410 --> 00:59:37,710 the total energy of the system is trivially 1097 00:59:37,710 --> 00:59:42,890 the sum of each of the single particle energies. 1098 00:59:42,890 --> 00:59:43,440 OK. 1099 00:59:43,440 --> 00:59:45,690 If you have 10 electrons and they 1100 00:59:45,690 --> 00:59:47,430 don't interact with each other, you 1101 00:59:47,430 --> 00:59:50,970 can calculate what is the energy of each of these 10 electrons. 1102 00:59:50,970 --> 00:59:54,630 Sum all of them, and that's the total energy of the system. 1103 00:59:54,630 --> 00:59:57,960 In this case, it's more complex. 1104 00:59:57,960 --> 01:00:00,270 And the total energy of the system 1105 01:00:00,270 --> 01:00:02,490 can't be really written as that, but it's 1106 01:00:02,490 --> 01:00:05,790 got other terms that depend on the charge density. 1107 01:00:05,790 --> 01:00:08,670 That's sort of this is, in summary, what 1108 01:00:08,670 --> 01:00:10,300 your total energy is. 1109 01:00:10,300 --> 01:00:13,800 And again, there's nothing else than kinetic energy term 1110 01:00:13,800 --> 01:00:17,490 sort of a Hartree term functional charge density, 1111 01:00:17,490 --> 01:00:19,470 this exchange correlation functional, 1112 01:00:19,470 --> 01:00:23,010 and the interaction between the external potential 1113 01:00:23,010 --> 01:00:24,280 and the charge density. 1114 01:00:24,280 --> 01:00:30,560 But this is actually different from the sum 1115 01:00:30,560 --> 01:00:32,600 of the eigenvalues. 1116 01:00:32,600 --> 01:00:40,610 That would be the sum of the expectation values of psi i 1117 01:00:40,610 --> 01:00:48,330 calculated on the single-particle orbital where 1118 01:00:48,330 --> 01:00:53,020 T is, again, just the simple quantum kinetic energy. 1119 01:00:53,020 --> 01:00:56,590 And the VKS is this Kohn and Sham potential. 1120 01:00:56,590 --> 01:01:00,450 So if you want to calculate the total energy of your system, 1121 01:01:00,450 --> 01:01:03,210 even if it's made of independent electron, 1122 01:01:03,210 --> 01:01:07,470 you can't sum just a single particle orbitals. 1123 01:01:07,470 --> 01:01:10,260 But you have to sort of deal with this expression. 1124 01:01:10,260 --> 01:01:13,920 Nothing complex in this, it's just sort of a caveat that 1125 01:01:13,920 --> 01:01:15,795 is relevant when you want to sort of-- you 1126 01:01:15,795 --> 01:01:18,180 know, this is the reason why we can't really 1127 01:01:18,180 --> 01:01:20,280 find out the equivalent of the [INAUDIBLE] 1128 01:01:20,280 --> 01:01:22,060 theorems for Hartree-Fock. 1129 01:01:22,060 --> 01:01:27,690 This is why at the end these single-particle energies 1130 01:01:27,690 --> 01:01:32,010 are ultimately not physically meaningful. 1131 01:01:32,010 --> 01:01:35,850 They sort of don't give us the total energy of the system 1132 01:01:35,850 --> 01:01:40,250 just by taking the sum over all of them. 1133 01:01:40,250 --> 01:01:41,030 OK. 1134 01:01:41,030 --> 01:01:45,470 So in order to make this into a practical algorithm, 1135 01:01:45,470 --> 01:01:51,440 the only part that remains is finding an approximation 1136 01:01:51,440 --> 01:01:55,250 to that exchange correlation term, to that last term. 1137 01:01:55,250 --> 01:01:58,250 Remember, we had sort of defined this density functional. 1138 01:01:58,250 --> 01:02:01,520 We have been able to extract two meaningful terms, 1139 01:02:01,520 --> 01:02:06,290 the Hartree electrostatic energy and the non-interacting Kohn 1140 01:02:06,290 --> 01:02:07,850 and Sham kinetic energy. 1141 01:02:07,850 --> 01:02:11,390 And we have said what is left is a function of the charge 1142 01:02:11,390 --> 01:02:12,860 density that we call the exchange 1143 01:02:12,860 --> 01:02:15,110 correlation functional. 1144 01:02:15,110 --> 01:02:17,540 How we are going to approximate that? 1145 01:02:17,540 --> 01:02:21,270 Well, we go back to the Thomas-Fermi idea. 1146 01:02:21,270 --> 01:02:24,260 We are going to do a local density 1147 01:02:24,260 --> 01:02:29,370 approximation to that exchange correlation functional. 1148 01:02:29,370 --> 01:02:32,150 So again, what we want to calculate 1149 01:02:32,150 --> 01:02:38,360 is the exchange correlation energy for any arbitrary charge 1150 01:02:38,360 --> 01:02:39,560 density. 1151 01:02:39,560 --> 01:02:41,420 Sometimes I call the charge density n. 1152 01:02:41,420 --> 01:02:43,430 Sometimes I call the charge density rho, 1153 01:02:43,430 --> 01:02:45,240 but they are always the same. 1154 01:02:45,240 --> 01:02:47,100 So how do we do this? 1155 01:02:47,100 --> 01:02:51,090 Well, we don't have the full solution. 1156 01:02:51,090 --> 01:02:53,120 But what we can say, again, is that, 1157 01:02:53,120 --> 01:02:58,760 for an inhomogeneous charge density that changes values 1158 01:02:58,760 --> 01:03:02,660 and then drops to 0, I can calculate the exchange 1159 01:03:02,660 --> 01:03:06,860 correlation energy for this charge density distribution 1160 01:03:06,860 --> 01:03:12,260 by sort of decomposing it in infinitesimal volumes. 1161 01:03:12,260 --> 01:03:15,140 Inside each infinitesimal volume, 1162 01:03:15,140 --> 01:03:18,800 I can say the charge density is constant. 1163 01:03:18,800 --> 01:03:22,790 And you see, I make a local density approximation. 1164 01:03:22,790 --> 01:03:28,640 That is I say the contribution to the overall exchange 1165 01:03:28,640 --> 01:03:32,600 correlation energy of this inhomogeneous system 1166 01:03:32,600 --> 01:03:34,130 can be broken down. 1167 01:03:34,130 --> 01:03:39,110 And each infinitesimal volume will give its own contribution 1168 01:03:39,110 --> 01:03:42,352 to the total exchange correlation density. 1169 01:03:42,352 --> 01:03:44,060 You know, in principle, it's not correct. 1170 01:03:44,060 --> 01:03:48,500 I mean, our problem doesn't have to be local in any way. 1171 01:03:48,500 --> 01:03:51,440 Actually, as people say, this exchange correlation 1172 01:03:51,440 --> 01:03:53,300 functional, the true one, although we 1173 01:03:53,300 --> 01:03:57,680 don't know what it is, we know that is ultra non-local. 1174 01:03:57,680 --> 01:04:01,310 So it can't be decomposed into terms 1175 01:04:01,310 --> 01:04:04,040 that independently sum up. 1176 01:04:04,040 --> 01:04:06,380 So in principle, we can't do this. 1177 01:04:06,380 --> 01:04:09,260 But in practice, it tends to be a good approximation 1178 01:04:09,260 --> 01:04:11,110 for a lot of cases. 1179 01:04:11,110 --> 01:04:15,740 And so what is going to be the contribution to the exchange 1180 01:04:15,740 --> 01:04:19,730 correlation energy from this infinitesimal volume? 1181 01:04:19,730 --> 01:04:24,500 Let's say the charge density there is 0.5. 1182 01:04:24,500 --> 01:04:27,590 Well, what we need to do is we need 1183 01:04:27,590 --> 01:04:32,270 to find out what is the exchange correlation 1184 01:04:32,270 --> 01:04:35,750 energy for the homogeneous electron 1185 01:04:35,750 --> 01:04:37,780 gas that is at this density. 1186 01:04:37,780 --> 01:04:41,630 That's something that, with some advanced computational 1187 01:04:41,630 --> 01:04:45,860 techniques, we can actually find out almost exactly. 1188 01:04:45,860 --> 01:04:51,260 So we would know, if we had a homogeneous charge density 1189 01:04:51,260 --> 01:04:54,320 0.5 everywhere, what would be the charge 1190 01:04:54,320 --> 01:04:56,480 density per unit volume. 1191 01:04:56,480 --> 01:05:01,400 And we can find out what is the exchange correlation charge 1192 01:05:01,400 --> 01:05:06,270 density per unit volume not only for 0.5, 0.6, 0.7, anything. 1193 01:05:06,270 --> 01:05:08,990 And what we are saying is that, in this non-homogeneous 1194 01:05:08,990 --> 01:05:12,110 problem, we construct the overall exchange correlation 1195 01:05:12,110 --> 01:05:15,990 energy by summing up these different pieces. 1196 01:05:15,990 --> 01:05:20,360 And so this is what Ceperley and Alder did in 1980. 1197 01:05:20,360 --> 01:05:27,200 They basically found out what was the almost exact sort 1198 01:05:27,200 --> 01:05:30,950 of closely to numerical exact solution 1199 01:05:30,950 --> 01:05:33,560 for the homogeneous electron gas. 1200 01:05:33,560 --> 01:05:38,690 That is for a system in which you have only electrons 1201 01:05:38,690 --> 01:05:42,320 homogeneously, so the charge density is identical 1202 01:05:42,320 --> 01:05:43,230 everywhere. 1203 01:05:43,230 --> 01:05:45,270 And those electrons interact. 1204 01:05:45,270 --> 01:05:49,040 So you can calculate the energy of this interacting electron 1205 01:05:49,040 --> 01:05:52,530 problem exactly as a function of the density. 1206 01:05:52,530 --> 01:05:53,030 OK. 1207 01:05:53,030 --> 01:05:56,810 So you change the density in your sort of volume, 1208 01:05:56,810 --> 01:05:58,850 and you find out what is this energy. 1209 01:05:58,850 --> 01:06:02,060 And then you can calculate, for any 1210 01:06:02,060 --> 01:06:06,140 of this density, what is the Kohn and Sham quantum 1211 01:06:06,140 --> 01:06:06,890 kinetic energy. 1212 01:06:06,890 --> 01:06:11,880 You can find out what is the Hartree electrostatic energy. 1213 01:06:11,880 --> 01:06:15,980 And so you can also find out, for this specific case 1214 01:06:15,980 --> 01:06:19,850 of the homogeneous gas, numerically 1215 01:06:19,850 --> 01:06:23,210 what would be the exchange correlation density. 1216 01:06:23,210 --> 01:06:26,050 And so that's basically a function. 1217 01:06:26,050 --> 01:06:29,240 So for the homogeneous gas, that is for the case in which n 1218 01:06:29,240 --> 01:06:33,710 doesn't depend on r, people found out 1219 01:06:33,710 --> 01:06:39,650 what was basically this exchange correlation energy. 1220 01:06:39,650 --> 01:06:43,200 It was calculated as a function. 1221 01:06:43,200 --> 01:06:49,370 This is a function of what people call rs. 1222 01:06:49,370 --> 01:06:55,080 rs is the radius of a sphere that contains one electron. 1223 01:06:55,080 --> 01:06:57,560 So it's a sort of inverse quantity 1224 01:06:57,560 --> 01:06:59,240 with respect to the density. 1225 01:06:59,240 --> 01:07:02,630 So numerical calculation, what are called quantum Monte Carlo 1226 01:07:02,630 --> 01:07:07,490 calculation, really solved the interacting Schrodinger 1227 01:07:07,490 --> 01:07:08,790 equation problem. 1228 01:07:08,790 --> 01:07:12,830 But for the specific case of an electron gas that 1229 01:07:12,830 --> 01:07:15,200 has a homogeneous density, they were 1230 01:07:15,200 --> 01:07:17,880 able to do that for various density. 1231 01:07:17,880 --> 01:07:22,760 And so, now, we have a function for the homogeneous problem. 1232 01:07:22,760 --> 01:07:27,080 For the non-homogeneous problem, we take a local density 1233 01:07:27,080 --> 01:07:30,530 approximation, and we say that the overall exchange 1234 01:07:30,530 --> 01:07:33,830 correlation energy is given by the integral over all 1235 01:07:33,830 --> 01:07:35,270 the infinitesimal volume. 1236 01:07:35,270 --> 01:07:38,660 And each infinitesimal volume will have a certain density 1237 01:07:38,660 --> 01:07:43,085 and will contribute with its own density. 1238 01:07:43,085 --> 01:07:45,980 If the density is going to be equal to here, 1239 01:07:45,980 --> 01:07:47,990 this will be the value of the contribution 1240 01:07:47,990 --> 01:07:49,730 of that infinitesimal volume. 1241 01:07:49,730 --> 01:07:52,460 If the density somewhere else corresponds to this, 1242 01:07:52,460 --> 01:07:54,030 this will be the corresponding. 1243 01:07:54,030 --> 01:07:59,390 So we really match up this overall exchange correlation 1244 01:07:59,390 --> 01:08:03,050 term from all the little infinitesimal volume 1245 01:08:03,050 --> 01:08:05,600 exactly as Thomas-Fermi had done, 1246 01:08:05,600 --> 01:08:13,490 but now we do it for a term that is a much smaller term 1247 01:08:13,490 --> 01:08:14,900 in our problem. 1248 01:08:14,900 --> 01:08:19,100 Thomas and Fermi had done it for the quantum kinetic energy. 1249 01:08:19,100 --> 01:08:21,560 Instead, what Kohn and Sham do, they 1250 01:08:21,560 --> 01:08:25,550 do it for what is left from the universal functional 1251 01:08:25,550 --> 01:08:28,340 once you have taken out the electrostatic 1252 01:08:28,340 --> 01:08:31,189 and once you have taken out the quantum kinetic energy 1253 01:08:31,189 --> 01:08:34,100 of the non-interacting electrons. 1254 01:08:34,100 --> 01:08:39,590 At this point, if you want 1980 and even 1255 01:08:39,590 --> 01:08:42,380 before without the computation, with some sort 1256 01:08:42,380 --> 01:08:45,140 of analytical approximations to this curve, 1257 01:08:45,140 --> 01:08:48,260 density functional theory becomes not only a theory, 1258 01:08:48,260 --> 01:08:50,870 but also a practical algorithm. 1259 01:08:50,870 --> 01:08:54,380 We have a set of expression for the exchange correlation term. 1260 01:08:54,380 --> 01:08:56,569 And so, now, it's just a matter of trying 1261 01:08:56,569 --> 01:09:00,560 to find out what the solution to these problems are. 1262 01:09:00,560 --> 01:09:03,529 And because somehow conceptually we 1263 01:09:03,529 --> 01:09:07,189 start from the homogeneous electron gas, 1264 01:09:07,189 --> 01:09:12,080 it turns out that this approach worked especially well 1265 01:09:12,080 --> 01:09:13,250 for solids. 1266 01:09:13,250 --> 01:09:17,029 I mean, the valence electrons in a solid 1267 01:09:17,029 --> 01:09:22,790 are much less structured than the electrons in a molecule 1268 01:09:22,790 --> 01:09:24,740 that they need to drop to 0. 1269 01:09:24,740 --> 01:09:27,500 So the charge density in a solid overall 1270 01:09:27,500 --> 01:09:31,130 varies less dramatically as a function of space 1271 01:09:31,130 --> 01:09:34,130 than the electron density in atoms and molecules. 1272 01:09:34,130 --> 01:09:36,200 And these are actually sort of what 1273 01:09:36,200 --> 01:09:41,189 were summarized the numerical result of Ceperley and Alder. 1274 01:09:41,189 --> 01:09:44,720 So they had calculated this exchange correlation energy 1275 01:09:44,720 --> 01:09:46,560 as a function of the density. 1276 01:09:46,560 --> 01:09:48,979 And that was actually a computational curve, 1277 01:09:48,979 --> 01:09:50,359 a set of dots. 1278 01:09:50,359 --> 01:09:54,620 And this is often cited, again, per Perdew and Zunger 1279 01:09:54,620 --> 01:09:58,340 in a sort of paper of theirs, among other things 1280 01:09:58,340 --> 01:10:03,830 sort of suggested the analytical interpolation of all 1281 01:10:03,830 --> 01:10:05,180 the numerical data. 1282 01:10:05,180 --> 01:10:07,970 And so you see it's something somehow exotic. 1283 01:10:07,970 --> 01:10:13,010 But while it's defined, this is just not even a functional. 1284 01:10:13,010 --> 01:10:15,950 It's just a function of the charge density. 1285 01:10:15,950 --> 01:10:17,900 So it's something that is very simple 1286 01:10:17,900 --> 01:10:20,160 to calculate in practice. 1287 01:10:20,160 --> 01:10:23,180 And so at this point, density functional theory, 1288 01:10:23,180 --> 01:10:25,770 is a well-defined theory. 1289 01:10:25,770 --> 01:10:29,420 So you see 1980, Ceperley and Alder do this quantum Monte 1290 01:10:29,420 --> 01:10:33,230 Carlo calculation, find out sort of what is this exchange 1291 01:10:33,230 --> 01:10:34,430 correlation energy. 1292 01:10:34,430 --> 01:10:37,130 Perdew and Zunger write out a simple interpolation. 1293 01:10:37,130 --> 01:10:40,370 1982, sort of the first time that I 1294 01:10:40,370 --> 01:10:44,000 think we see sort of where all of this is going, 1295 01:10:44,000 --> 01:10:48,080 Marvin Cohen in Berkeley sort of has been 1296 01:10:48,080 --> 01:10:49,970 working for two or three years. 1297 01:10:49,970 --> 01:10:51,980 Alex Zunger was there. 1298 01:10:51,980 --> 01:10:54,750 [INAUDIBLE] him, a number of his students, 1299 01:10:54,750 --> 01:10:56,810 they have been able to actually write out 1300 01:10:56,810 --> 01:10:58,940 all the electronic structure codes 1301 01:10:58,940 --> 01:11:03,140 that are able to solve the density functional equation 1302 01:11:03,140 --> 01:11:05,330 for the case of a periodic solid. 1303 01:11:05,330 --> 01:11:07,460 And so they address the case of silicon, 1304 01:11:07,460 --> 01:11:11,510 sort of the most important material in electronics. 1305 01:11:11,510 --> 01:11:13,790 And so what they do is they are able now 1306 01:11:13,790 --> 01:11:17,480 to calculate the energy of that system 1307 01:11:17,480 --> 01:11:21,080 as a function of the atomic position and, in particular, 1308 01:11:21,080 --> 01:11:23,810 as a function of the lattice parameter. 1309 01:11:23,810 --> 01:11:25,760 So you know, first thing that they do 1310 01:11:25,760 --> 01:11:31,190 is they take silicon in its diamond structure, 1311 01:11:31,190 --> 01:11:35,180 so the FCC lattice with two atoms as a basis. 1312 01:11:35,180 --> 01:11:38,150 And they calculate that energy as a function 1313 01:11:38,150 --> 01:11:39,720 of the lattice parameter. 1314 01:11:39,720 --> 01:11:41,610 And it looks something like this. 1315 01:11:41,610 --> 01:11:44,460 And then obviously, as you have learned by now, 1316 01:11:44,460 --> 01:11:46,650 you look at what is the minimum of that energy. 1317 01:11:46,650 --> 01:11:48,510 And that is the theoretical prediction 1318 01:11:48,510 --> 01:11:49,710 of the lattice parameter. 1319 01:11:49,710 --> 01:11:52,440 And there is [INAUDIBLE],, you know, 1% error. 1320 01:11:52,440 --> 01:11:55,890 They look at the second derivative. 1321 01:11:55,890 --> 01:11:58,350 This curvature here is really the bulk models 1322 01:11:58,350 --> 01:12:02,770 of your problem, again, 5% 10% error. 1323 01:12:02,770 --> 01:12:05,340 And then they say, well, let's suppose that we have silicon 1324 01:12:05,340 --> 01:12:07,650 not in the diamond phase, but let's suppose 1325 01:12:07,650 --> 01:12:11,280 that we have silicon in the beta tin phase. 1326 01:12:11,280 --> 01:12:13,590 And so this is also experimentally known. 1327 01:12:13,590 --> 01:12:15,930 And we know in the beta tin what is the lattice 1328 01:12:15,930 --> 01:12:17,100 parameter of silicon. 1329 01:12:17,100 --> 01:12:19,620 And we know from the Maxwell construction 1330 01:12:19,620 --> 01:12:24,390 what is the pressure at which we would 1331 01:12:24,390 --> 01:12:29,310 have a transition from, say, diamond to beta tin. 1332 01:12:29,310 --> 01:12:32,160 And again, you know, I can't remember what was the error, 1333 01:12:32,160 --> 01:12:34,140 but it's substantially correct. 1334 01:12:34,140 --> 01:12:38,310 And they were able to actually sort of calculate 1335 01:12:38,310 --> 01:12:42,450 the sort of complex zoology of all the high pressure 1336 01:12:42,450 --> 01:12:43,650 phases of silicon. 1337 01:12:43,650 --> 01:12:46,530 And it was in remarkable agreement with the experiment. 1338 01:12:46,530 --> 01:12:50,040 So 1982, this is [INAUDIBLE] and Cohen. 1339 01:12:50,040 --> 01:12:57,375 But in particular, Marvin Cohen in Berkeley showed that for a-- 1340 01:12:57,375 --> 01:13:01,620 Marvin Cohen. 1341 01:13:01,620 --> 01:13:05,520 For a realistic case, density functional theory 1342 01:13:05,520 --> 01:13:09,330 is able really to give us quantity of prediction. 1343 01:13:09,330 --> 01:13:11,010 Marvin Cohen has actually become, 1344 01:13:11,010 --> 01:13:15,060 this year, the president of the American Physical Society. 1345 01:13:15,060 --> 01:13:15,750 OK. 1346 01:13:15,750 --> 01:13:19,380 So this is really the beginning of density functional theory 1347 01:13:19,380 --> 01:13:21,750 as a practical approach. 1348 01:13:21,750 --> 01:13:27,420 And in many ways, what has happened between 1982 and today 1349 01:13:27,420 --> 01:13:30,870 is that we have become better and better 1350 01:13:30,870 --> 01:13:35,370 at solving the algorithm for this, overall, still 1351 01:13:35,370 --> 01:13:37,440 complex computational problem. 1352 01:13:37,440 --> 01:13:41,940 And you see a lot of this in the next lecture that follows. 1353 01:13:41,940 --> 01:13:46,740 And we have become somewhat better, not really dramatically 1354 01:13:46,740 --> 01:13:50,970 better, in calculating that exchange correlation energy. 1355 01:13:50,970 --> 01:13:55,110 In a way, sort of the ideas of Kohn and Sham 1356 01:13:55,110 --> 01:13:59,790 from 1965 of having a local density approximation 1357 01:13:59,790 --> 01:14:01,440 is still very good. 1358 01:14:01,440 --> 01:14:06,660 I mean, it's not used nowadays any more that much, 1359 01:14:06,660 --> 01:14:09,690 but it's as close as-- 1360 01:14:09,690 --> 01:14:13,710 what we can do now is not really that much better. 1361 01:14:13,710 --> 01:14:15,630 And as you can imagine, sort of what 1362 01:14:15,630 --> 01:14:18,510 people have done that was a bit better 1363 01:14:18,510 --> 01:14:21,770 was introducing gradients in your problem. 1364 01:14:21,770 --> 01:14:24,840 So you're trying to guess what the energy 1365 01:14:24,840 --> 01:14:28,710 of an inhomogeneous system comes starting 1366 01:14:28,710 --> 01:14:31,620 from what you know about the homogeneous electron gas. 1367 01:14:31,620 --> 01:14:35,640 Well, maybe you should somehow throw in into your problem also 1368 01:14:35,640 --> 01:14:38,530 the first derivative of the gradient of the density. 1369 01:14:38,530 --> 01:14:42,300 And so people did that fairly soon in the early '80s. 1370 01:14:42,300 --> 01:14:47,130 And sort of using the gradients was actually much worse. 1371 01:14:47,130 --> 01:14:51,480 There was a miracle in the local density approximation 1372 01:14:51,480 --> 01:14:55,080 in which the actual expression of the local density 1373 01:14:55,080 --> 01:15:00,390 approximation satisfies a lot of symmetry properties 1374 01:15:00,390 --> 01:15:03,750 and scaling properties of what would be the exact exchange 1375 01:15:03,750 --> 01:15:05,220 correlation functional. 1376 01:15:05,220 --> 01:15:07,860 At the time, people put in gradients. 1377 01:15:07,860 --> 01:15:11,730 All these sort of symmetries and scaling properties 1378 01:15:11,730 --> 01:15:13,830 were sort of thrown to the dogs. 1379 01:15:13,830 --> 01:15:15,990 And actually the GGAs-- 1380 01:15:15,990 --> 01:15:18,840 sorry, the gradient approximation, 1381 01:15:18,840 --> 01:15:21,370 were working much, much worse. 1382 01:15:21,370 --> 01:15:23,610 And so people need to realize, sort of 1383 01:15:23,610 --> 01:15:28,650 in the late '80s, the work of Axel Becke, of John Perdew 1384 01:15:28,650 --> 01:15:31,770 especially, a lot of it, you sort of 1385 01:15:31,770 --> 01:15:35,640 need to introduce gradients in ways 1386 01:15:35,640 --> 01:15:41,100 that still satisfy a lot of these analytical forms. 1387 01:15:41,100 --> 01:15:46,320 And in many ways, by now, there is a sort of generalized 1388 01:15:46,320 --> 01:15:50,100 exchange correlation functional that's been sort of developed 1389 01:15:50,100 --> 01:15:55,200 in the mid-'90s by Perdew, Kieron Burke now at Rutgers, 1390 01:15:55,200 --> 01:15:57,540 and Matthias Ernzerhof. 1391 01:15:57,540 --> 01:15:59,250 That is called the PBE functional. 1392 01:15:59,250 --> 01:16:01,620 That has become sort of the workhorse. 1393 01:16:01,620 --> 01:16:03,750 So a lot of the time, you'll see sort 1394 01:16:03,750 --> 01:16:05,460 of density functional calculation done 1395 01:16:05,460 --> 01:16:09,420 in the PBE, GGA approximation. 1396 01:16:09,420 --> 01:16:13,680 But again, these are important improvements, 1397 01:16:13,680 --> 01:16:16,470 but if you want just sort of very 1398 01:16:16,470 --> 01:16:18,900 little on top of the local density 1399 01:16:18,900 --> 01:16:22,170 approximation of the [INAUDIBLE].. 1400 01:16:22,170 --> 01:16:25,620 The chemistry community has also sort of done 1401 01:16:25,620 --> 01:16:29,940 a number of very intriguing developments. 1402 01:16:29,940 --> 01:16:34,020 In particular, there are things that the Hartree-Fock 1403 01:16:34,020 --> 01:16:35,340 does very well. 1404 01:16:35,340 --> 01:16:37,350 In particular, because you have the sort 1405 01:16:37,350 --> 01:16:41,370 of exchange term in Hartree-Fock, you cancel, 1406 01:16:41,370 --> 01:16:44,610 remember, the self-interaction say, 1407 01:16:44,610 --> 01:16:47,860 in the single-electron problem coming from the Hartree, 1408 01:16:47,860 --> 01:16:50,130 the electrostatic problem. 1409 01:16:50,130 --> 01:16:53,080 Density functional theory, in theory, 1410 01:16:53,080 --> 01:16:56,640 in its exact formulation, would be self-interaction corrected. 1411 01:16:56,640 --> 01:16:58,470 But in practice, it is not. 1412 01:16:58,470 --> 01:17:02,820 If you solve the hydrogen atom with density functional theory, 1413 01:17:02,820 --> 01:17:04,740 you have that the electron interacts 1414 01:17:04,740 --> 01:17:07,950 with the charge density created by the same, 1415 01:17:07,950 --> 01:17:09,970 by the electron itself. 1416 01:17:09,970 --> 01:17:14,880 And so what sort of the quantum chemistry community has done 1417 01:17:14,880 --> 01:17:20,400 is, well, they said let's take LDAs, let's actually take GGAs 1418 01:17:20,400 --> 01:17:22,140 that seem to work very well. 1419 01:17:22,140 --> 01:17:26,520 But let's actually construct an exchange correlation functional 1420 01:17:26,520 --> 01:17:29,535 that has a little bit of this, but it's 1421 01:17:29,535 --> 01:17:32,280 got also a little bit of what we know worthwhile 1422 01:17:32,280 --> 01:17:34,150 in the Hartree-Fock equation. 1423 01:17:34,150 --> 01:17:36,780 So they construct hybrid functional 1424 01:17:36,780 --> 01:17:40,410 which there are sort of pure density financial terms 1425 01:17:40,410 --> 01:17:45,080 and the sort of Hartree-Fock exchange term mixed in. 1426 01:17:45,080 --> 01:17:47,770 It makes the equation much more complex. 1427 01:17:47,770 --> 01:17:52,260 And if you want, it's a sort of less pure formulation 1428 01:17:52,260 --> 01:17:54,780 of density functional theory, but it 1429 01:17:54,780 --> 01:17:58,590 can work reasonably well or very well especially, again, 1430 01:17:58,590 --> 01:18:00,990 for atoms and molecules. 1431 01:18:00,990 --> 01:18:04,950 And this is where we are, basically, with exchange 1432 01:18:04,950 --> 01:18:06,660 correlation functional. 1433 01:18:06,660 --> 01:18:09,150 I think I'll stop here for today because that's actually 1434 01:18:09,150 --> 01:18:10,300 a lot of work. 1435 01:18:10,300 --> 01:18:12,690 What we'll start seeing in the next class 1436 01:18:12,690 --> 01:18:17,010 is how we actually solve this equation in practice. 1437 01:18:17,010 --> 01:18:20,310 On March 8th, you're going to your second lab in which you 1438 01:18:20,310 --> 01:18:23,250 will actually study the energy of a solid using 1439 01:18:23,250 --> 01:18:24,900 density functional theory. 1440 01:18:24,900 --> 01:18:27,180 What I said today is probably the last 1441 01:18:27,180 --> 01:18:28,710 of the conceptual lectures. 1442 01:18:28,710 --> 01:18:30,945 And I understand that some of it is very complex. 1443 01:18:33,540 --> 01:18:36,880 There is reading material posted on the Stella website. 1444 01:18:36,880 --> 01:18:39,030 There is the Kohanoff-Gidopoulos paper 1445 01:18:39,030 --> 01:18:40,590 on density functional theory. 1446 01:18:40,590 --> 01:18:44,280 And some of the readings that I've given are very useful. 1447 01:18:44,280 --> 01:18:46,800 The two best books that are also cited 1448 01:18:46,800 --> 01:18:49,410 at the end of this lecture are probably the one 1449 01:18:49,410 --> 01:18:51,990 by [INAUDIBLE] or the one by [INAUDIBLE].. 1450 01:18:51,990 --> 01:18:54,990 And they are both called Density Functional Theory or Density 1451 01:18:54,990 --> 01:18:56,610 Functional Theory In Practice. 1452 01:18:56,610 --> 01:18:59,520 And they are cited on the last page. 1453 01:18:59,520 --> 01:19:04,880 Otherwise, this is it for today and see you next week.